Misalignment Effects of the Self-Tracking Laser Doppler ......Misalignment Effects of the...

Misalignment Effects of the Self-Tracking Laser Doppler Vibrometer

by

Andrew D. Zima, Jr.

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

Mechanical Engineering

APPROVED:

_________________________ P.S King, Chairman

_________________________ _________________________ W.F. O’Brien A.L. Wicks

May, 2001

Blacksburg, Virginia

Key Words: Vibration, Rotor, Laser, Vibrometer, Blade, LDV, Rotating, Misalignment

ii

Misalignment Effects of the Self-Tracking Laser Doppler Vibrometer

by

Andrew D. Zima, Jr.

Committee Chairman: Peter S. King

Mechanical Engineering

Abstract

There are many limitations to the current methods used to measure vibration on

rotating structures. These limitations include physical flow blockages, relating the

measurement spot to the structure rotation, data processing issues, and having to

physically alter the engine. This work further describes aspects of a self-tracking laser

vibrometry system that can be used to measure the vibrations of rotating structures. This

method, if setup correctly, has the capability to overcome many of the limitations listed

above. A study of all misalignment effects is presented in this thesis. The study consists

of a parametric sensitivity analysis of misalignment variables, a parametric Monte Carlo

analysis of misalignment variables, and a full interaction Monte Carlo analysis of

misalignment variables. In addition, the results of the misalignment variable analyses

were used to develop a self-tracker test rig for obtaining fan vibration from a Pratt and

Whitney JT15D turbofan engine. A prototype this test rig was designed, built, and tested

on the turbofan. It was found that in order to achieve acceptable amounts of position and

velocity error using the self-tracker LDV system, very strict alignment of the optical

equipment is necessary. Additionally, the alignment criteria can likely be achieved with

the use of digitally controlled high precision linear motion equipment.

iii

Acknowledgments

I would like to thank the members of my advisory committee for the support and

guidance they have given me throughout my undergraduate and postgraduate study. I

would especially like to thank Dr. Peter S. King for first sparking my interest in the field

of turbomachinery. Also, it was with his guidance, as my committee chair and friend,

that I was able to complete my thesis work. Dr. Walter F. O’Brien, despite his busy

schedule, was always there to make sure I was going in the right direction. Dr. A.L.

Wicks always lent a helping hand with both theoretical laser operations and getting

necessary experimental equipment. The golf tips from him were just an added bonus.

There are a few other influential professors I would also like to thank for their

contributions to both my education and life. Dr. Harry Robertshaw taught me how to

keep things fun while working. Dr. Hayden Griffen helped show me the more practical

sides of engineering. Finally, Dr. Michael Alley helped me get through the technical

writing aspects of my thesis in such a way that the knowledge will stay with me.

A special thanks goes to Frank Caldwell for his help running experiments on the

JT15D turbofan engine. He always helped out when needed and shared his expertise.

I do not think I could survived college without my many friends. I would like to

thank John “I’m a fish” Zachary, Dino “Time to wallow” Bednarsky, Monte “M-dog”

Marcum, Adam “Tractorboy” Hanes, and Jeff “That’s my final answer” Glass. These

guys have always managed to keep me in great spirits throughout college.

Another special group of friends deserve thanks for helping me out in obtaining

my masters degree: the “Turbolab” guys. These are the pretty faces that I saw every day

in both classes and working in my lab. Karl Sheldon, Matt Small, Keith Boyer, Scott

Gallimore, Grant Eddy, Wayne Sexton, Mac Chiu, Jon Luedke, and Alexandre Perrig

were always able to make me laugh as well as help me out when I needed it. Thanks

guys!

Lastly, and most importantly, I would like to thank my family. It was with their

unrequited support (and money) that I was able to go after my goals in life. They have

guided me to be who I am today, and their influences on my life will always continue.

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Table of Contents

Abstract ........................................................................................................... ii Acknowledgments ......................................................................................... iii Table of Contents........................................................................................... iv

Table of Figures ............................................................................................. vi Commonly Used Variables..........................................................................viii 1 Introduction.............................................................................................. 1

1.1 Objectives ...................................................................................................................... 4 1.2 Scope of the Work.......................................................................................................... 5

2 Background and Previous Work .............................................................. 7 2.1 Contacting Methods ....................................................................................................... 7

Strain Gage – Telemetry Method .......................................................................................................... 8 Blade Tip Sensors ................................................................................................................................. 9

2.2 Non-Contacting Methods............................................................................................. 11 Blade Tip Sensors ............................................................................................................................... 11 Holographic Methods ......................................................................................................................... 13 Acoustic Doppler Methods.................................................................................................................. 14

2.3 Summary ...................................................................................................................... 21

3 Laser Doppler Vibrometer ..................................................................... 22 3.1 Diffuse Surface Laser Doppler Vibrometry................................................................. 22 3.2 Reflective Surface Laser Doppler Vibrometry ............................................................ 25 3.3 Summary ...................................................................................................................... 27

4 Modeling The Self-Tracker LDV System ............................................. 28 4.1 Scope Of The Model .................................................................................................... 28 4.2 General Procedures ...................................................................................................... 29 4.3 Relevant Variables ....................................................................................................... 30 4.4 Modeling the Position of the Equipment ..................................................................... 31 4.5 Calculating the Path of The Laser Beam...................................................................... 46 4.6 Finding the Position and Position Error of the Laser Spot........................................... 55 4.7 Finding Velocity and Velocity Error of the Measurement System.............................. 56 4.8 Alternate Error Analysis Method................................................................................. 65

5 The Computer Program.......................................................................... 66 5.1 Program Components................................................................................................... 66 5.2 Running the Program ................................................................................................... 69 5.3 Testing the Computer Simulation ................................................................................ 70 5.4 Other Findings From The Computer Simulation ......................................................... 90

6 Input Variable Response Analysis......................................................... 91 6.1 Parametric Studies........................................................................................................ 95 6.2 Interaction Studies...................................................................................................... 119

v

6.3 Summary .................................................................................................................... 138

7 Prototype Self-Tracker LDV System................................................... 140 7.1 Design of Test Stand .................................................................................................. 140 7.2 Results of Prototype Self-Tracker Testing................................................................. 146

8 Conclusions and Future Recommendations......................................... 148 8.1 Results of Misalignment Studies................................................................................ 148 8.2 Results of Prototype Self-Tracker LDV System........................................................ 150 8.3 Future Recommendations .......................................................................................... 150

Appendix A - MATLAB Simulations for Individual Misalignment Parameters................................................................................................... 152

Appendix B - Other Variable Responses of Parametric Monte Carlo Studies..................................................................................................................... 173

Appendix C - Additional Response Plots from Monte Carlo Interaction Studies......................................................................................................... 175

Appendix D – Prototype Self-Tracker Design Drawings (AutoCAD®).... 180

Appendix E – Alignment Procedure For Implementing the Self-tracker On the JT15D.................................................................................................... 195

References................................................................................................... 197

Vita.............................................................................................................. 199

vi

Table of Figures

Figure 1-1 General Schematic of Self-Tracker LDV.......................................................... 3 Figure 2-1 Schematic of Strain Gage/Telemetry Method................................................... 9 Figure 2-2 Schematic of Meander Method ....................................................................... 10 Figure 2-3 Typical Setup for Optical Blade Tip Sensors by Kadambi et al.[2]................ 12 Figure 2-4 Typical Holographic Images at Different Excitation Frequencies by Lesne et

al. [8] ......................................................................................................................... 14 Figure 2-5 Generalized schematic for pulsed beam or stationary beam approach by

Reinheardt et al.[10].................................................................................................. 17 Figure 2-6 Schematic of typical laser Doppler Vibrometer setup using an image de-

rotator by Lesne [8]................................................................................................... 19 Figure 2-7 Schematic of Self-Tracker Laser Vibrometer ................................................. 21 Figure 3-1 Diagram of Laser Beam Intersection the Blade .............................................. 24 Figure 3-2 Schematic of Diffuse Block Tests................................................................... 24 Figure 3-3 Schematic of LDV Perpendicular To Mirror .................................................. 26 Figure 3-4 Incident Laser Non-Orthogonal to Mirror....................................................... 26 Figure 4-1 Design Constraint Variables for Self-Tracker................................................. 31 Figure 4-2 General Schematic of Coordinate Frame Positioning..................................... 32 Figure 4-3 Defining Translational Motion for Eulerian Transformation.......................... 34 Figure 4-4 Defining Rotational Motion for Eulerian Transformations ............................ 34 Figure 4-5 Schematic of Rotor/Vertex Mirror Assembly ................................................. 39 Figure 4-6 Schematic of a Ray extending from point 1 towards point 2.......................... 47 Figure 4-7 How the Laser Beam Spot Moves On Vertex Mirror ..................................... 57 Figure 4-8 The Beam Spot On the Vertex Mirror as the System Rotates......................... 57 Figure 5-1 Ideally Aligned Case ....................................................................................... 72 Figure 5-2 Computer Simulation For Ideally Aligned Case............................................. 74 Figure 5-3 Translational Error in X-Direction.................................................................. 75 Figure 5-4 Computer Simulation Results of 0.001 meter X-direction Translational Error

................................................................................................................................... 77 Figure 5-5 X and Y Translational Misalignments On Blade ............................................ 79 Figure 5-6 Computer Simulation Results of 0.001 meter Y-direction Translational Error

................................................................................................................................... 80 Figure 5-7 Y-Z Plane View of Rotational Misalignment Around X-axis ........................ 82 Figure 5-8 Computer Simulation Results of 0.001 radian (0.057 deg) Rotational Error

About X-axis............................................................................................................. 84 Figure 5-9 X-Z Plane View of Rotational Misalignment Around Y-axis ........................ 86 Figure 5-10 Computer Simulation Results of .001 radian(.057 deg) Rotational Error

About Y-axis............................................................................................................. 87 Figure 5-11 X-direction Translational and Rotational Misalignment Case...................... 89 Figure 6-1 iSIGHT Program Window for Design Integration.......................................... 96 Figure 6-2 iSIGHT Program Window For Parsed Input File ........................................... 97 Figure 6-3 iSIGHT Program Window For Parsed Output File......................................... 98 Figure 6-4 iSIGHT Program Window for Defining Parametric Sensitivity Bounds........ 99 Figure 6-5 Pareto Plot For Position Error Response To Parametric Translational

Misalignments......................................................................................................... 103

vii

Figure 6-6 Pareto Plot for Velocity Error Response To Parametric Translational Misalignments......................................................................................................... 104

Figure 6-7 Blade and Mirror Velocities for 0.001m X-Translation of Laser Frame ...... 105 Figure 6-8 Translational Vertex Mirror Misalignment................................................... 106 Figure 6-9 Pareto Plot For Position Error Response to Rotational Misalignments ........ 110 Figure 6-10 Pareto Plot For Velocity Error Response to Rotational Misalignments ..... 110 Figure 6-11 (a) Rotational Misalignment of Laser (b) Rotational Misalignment of Rotor

................................................................................................................................. 112 Figure 6-12 Resolving Blade Velocity For x(a) and y(b) Rotational Misalignments of The

Rotor ....................................................................................................................... 113 Figure 6-13 iSIGHT Program Window For Monte Carlo Methods ............................... 116 Figure 6-14 Position Error Response to Translational Misalignments........................... 117 Figure 6-15 - Position Error Response to Rotational Misalignments ............................. 118 Figure 6-16 Velocity Response to Rotational Misalignments ........................................ 118 Figure 6-17 Typical iSIGHT Program Window For Interaction Studies ....................... 121 Figure 6-18 Brush Plot For x-direction Interactions of Translational Misalignments.... 123 Figure 6-19 Detailed Brush Plots of Position Error Response to Translational Interaction

Study ....................................................................................................................... 125 Figure 6-20 Brush Plots For Rotational Interaction Monte Carlo Study........................ 127 Figure 6-21 Detailed Brush Plots of Position Error Response to Rotational Interaction

Study ....................................................................................................................... 128 Figure 6-22 Detailed Brush Plots of Velocity Error Response to Rotational Interaction

Study ....................................................................................................................... 129 Figure 6-23 Brush Plot For Total Interaction Monte Carlo Study Using Uniform .001

meter(or radian) Bounds ....................................................................................... 131 Figure 6-24 Detailed Brush Plots of Position Error Response to Influential Interaction

Parameters............................................................................................................... 133 Figure 6-25 Detailed Brush Plots of Velocity Error Response to Influential Interaction

Parameters............................................................................................................... 134 Figure 6-26 Brush Plots For Monte Carlo Full Interaction Study Based On Parametric

Sensitivities ............................................................................................................. 136 Figure 6-27 Detailed Brush Plots of Position Error Response To All Misalignment

Interactions.............................................................................................................. 137 Figure 6-28 Detailed Brush Plots of Velocity Error Response To All Misalignment

Interactions.............................................................................................................. 137 Figure 7-1 3-D Isometric View Of Laser and Fold Mirror Test Stand........................... 141 Figure 7-2 Disassembled Vertex Mirror Assembly........................................................ 143 Figure 7-3 Assembled Vertex Mirror Assembly ............................................................ 144 Figure 7-4 Vertex Mirror Assembly Mounted on JT15D............................................... 144 Figure 7-5 Picture of Prototype Self-Tracker LDV Setup on JT15D Turbofan ............. 146

viii

Commonly Used Variables Misalignment Variable Representation ( )( ) ntMisalignme nalTranslatio - 21e in the (1) in the frame of (2)

( )( ) ntMisalignme Rotational - 21theta about the (1) in the frame of (2)

(1) X – x-direction Y – y-direction Z – z-direction

(2) L – laser R – rotor

V – vertex mirror F – fold mirror B – blade T – target

1

1 Introduction

Historically, one of the key factors that cause the failure of blades in rotating

machinery is cracking due to high cycle fatigue (HCF). Blade failure can be a serious

disaster if it leads to engine failure. The driving force that leads to HCF is excessive

blade vibration. These vibrations are classified under two categories: 1) that which are

related to the rotation of the structure (synchronous) and 2) that which are not related to

the rotation of the structure (asynchronous). If these synchronous and asynchronous blade

vibrations could be better detected and quantified, the life span and likelihood for a blade

to fail could be better predicted. Thus, the possibility of catastrophic engine failure could

also be reduced.

Numerous methods have been applied previously to detect vibrations on rotating

blades. The main methods used are strain gage telemetry, blade tip sensors, holographic

interferometry, and laser Doppler methods. These different methods all present some

problems with their setup. For strain gage telemetry and blade tip sensors, equipment

(strain gages or sensors) must be physically attached to the equipment. For the

holographic method, it is very difficult to exactly correlate interference patterns to

vibration levels due to image movement. For typical previous laser Doppler methods,

image derotators present problems. Also, all of the above methods necessitate a shaft

encoder to relate the rotation of the structure to what is being measured. At high speeds,

this is also a problem. All of these methods, and their related advantages and

disadvantages are discussed in chapter two of this thesis. In an attempt solve some of the

above mentioned problems, a new laser Doppler technique is being studied.

2

Applying laser Doppler vibrometry (LDV) to rotating structures opens a new door

for innovative non-contacting methods for measuring blade vibrations. Typically, the

practice of using a LDV to measure vibrations on rotating structures requires the use of

an electro-mechanical connection between the rotation of the shaft and the actual

measurement point. Additional equipment must be physically mounted on the rotating

structure to detect the shaft rotation. Then, the signal needs to be carefully processed in

order to relate it to the motion of the measurement point. This presents many speed

limitations due to signal processing requirements. Some of these limitations are signal

lag, and synchronization of the rotating shaft to the measurement point. It would be very

desirable to have an in-service method of measuring blade vibrations that would not

require additional equipment to be attached to the structure itself and also not have the

speed limitations mentioned above.

The focus of this study is the analysis and design of a self-tracking LDV method

for measuring blade vibrations on rotating structures. This setup has a mechanical link, a

fixed mirror on the center of rotation, between the shaft rotation and the location of the

measurement point. Thus, the method can theoretically work at any rotational speed as

well as during speed transients. Figure 1-1 shows a generalized schematic of the concept.

The incident laser beam from the LDV is pointed towards the center of rotation where it

intersects an angled vertex mirror that rotates with the structure. This is the only

modification to the fan or engine. The laser is then reflected back towards a stationary

fold mirror, which then reflects the beam to the measurement point on the blade. The

concept was recently presented in a dissertation by Lomenzo [1].

3

Figure 1-1 General Schematic of Self-Tracker LDV

The setup shown above eliminates the problems associated with needing an electro-

mechanical connection (shaft encoder and steering mirror) between the shaft rotation and

measurement point, but introduces alignment issues. These alignment issues are also

present in current laser Doppler methods also. Any amount of misalignment associated

with any of the components of the laser beam and mirror system may result in

measurement point position errors and/or velocity measurement errors. Associated with

each piece of equipment indicated above is some amount of rotational and translational

misalignment. This will cause the laser beam to land on a spot on the blade that is

different from the intended measurement point. The internal optics of the LDV

compares the beam reflected from the blade to the beam incident on the blade. From the

apparent frequency shift between the two beams, the velocity of the blade can be found.

4

If the laser system is misaligned, the frequency shift may change and misrepresent the

actual vibration of the system.

1.1 Objectives

This thesis had two main objectives:

1. Develop a computer simulation to analyze how all the different static

misalignment parameters affect position error and velocity error,

2. Use the simulation to perform misalignment variable analyses to provide

design criteria for an actual physical measurement system, and

The self-tracker simulation calculated the path of the laser beam throughout the

system as well as the intersection of the laser beam with each piece of equipment, and has

the ability to model misalignments in any piece of equipment used in the system. This

was done by designing a computer simulation that referenced each piece of equipment

from its own reference frame. By changing the parameters of these reference frames

from the design specifications, the misalignments were introduced into the system. The

difference in where the beam actually landed on the target blade and where it was

intended to land is referred to as the position error. The velocity that the LDV sensed

was calculated by finding the component of the rigid body velocities in the laser beam

direction where the laser intersected the rotating structure. In the case of the self-tracker,

these were the target blade and vertex mirror velocities. The velocity error due to

misalignments was found by comparing the velocity found in the misaligned case to that

of the perfectly aligned case.

5

Once the simulation was proven to work, it was coupled with a program called

iSIGHT. This program has the capability to run the computer model multiple times while

changing the input alignment parameters. The input parameters can be changed in a

parametric or factorial manner. A parametric study varies one parameter at a time, and a

factorial study looked at variable interactions by changing multiple parameters

simultaneously. The ultimate goal was to obtain a sensitivity of each parameter to

position error and velocity error in order to come up with size and positioning tolerances

for each physical piece of equipment. This sensitivity, along with an assumed allowable

position error and a typical blade velocity for a turbojet fan blade, was used to obtain the

design criteria for a true physical system. Once the sensitivities were found, they were to

be plugged into a simulation that models the full interaction of all misalignment

variables. From this, a better understanding of which variables most affect the position

error and velocity error responses were found. Also, the extent to which all of the

equipment must me aligned to generate acceptable amounts of velocity and position

errors was found.

1.2 Scope of the Work

Prior studies of the self-tracker LDV system by Lomenzo[1] considered the LDV

and fold mirror and a single assembly, and the rotor, vertex mirror and bladed disk as

another single assembly. This study considers each piece of equipment as its own

assembly, and analyzes the misalignment effects of each assembly. In addition, complete

parametric and interaction analyses of the misalignment effects of the self-tracker

measurement system were completed. This thesis also provides complete documentation

6

for mathematics relating the LDV concepts to the self-tracker, and how the computer

simulation modeled the interaction of the LDV and rotating system. The work focused

on static misalignments; although, the computer simulation has the capability to integrate

dynamic misalignments. Additionally, the results provided design criteria for an actual

self-tracker LDV system to be constructed, but the system was not actually constructed

due to funding issues.

The following sections of the report will include a literature review of past work, a

general overview of the scope of the work, methodologies used to model the self-tracking

LDV system, an overview and discussion on the sensitivity analysis, testing and results,

and finally the design tolerances of a self-tracking laser vibrometer system.

7

2 Background and Previous Work

There are several existing are methods used to measure blade vibrations in rotating

structures. These methods can be broken down into two categories, contacting and non-

contacting. In general, the difference between the two categories is that the contacting

methods have a transducer physically placed on a blade while the non-contacting

methods have no physical apparatus attached to the blade itself. These two measurement

techniques have several different classifications, each of which has associated advantages

and disadvantages. The remainder of this chapter discusses the different contacting and

non-contacting measurement techniques, as well as the advantages and disadvantages of

each. Table 2-1 shows a list of the techniques that will be described.

Table 2-1 Different Measurement Techniques

Contacting Methods Non-Contacting Methods 1. Strain Gage Method 1. Blade Tip Sensors

2. Blade Tip Sensors 2. Holographic Methods

3. Acoustic Doppler Methods

4. Laser Doppler Methods

2.1 Contacting Methods

In general, contacting rotor vibration measurement methods have both advantages

and disadvantages. The major advantage is that the measurement point on the blade is

fixed and known at all times. Thus, data is known with high confidence and is much

8

easier to post-process. The major disadvantage arises from the fact that additional

structures must be mounted on the blade surface. When transducers are placed on the

blade surface, the additional eccentric mass creates different mode shapes that change the

dynamics of the blade and system. Additionally, disruption occurs to the flow field due

to the physical obstruction of the transducers.

Strain Gage – Telemetry Method

Kadambi et al. [2] described the strain gage-telemetry method as an established

industry method that consists of placing a series of strain gages on the blade surface and

measuring the alternating strains imposed on the blade. A telemetry system is needed to

transmit the data to a signal-conditioning unit because the wires of the strain gages will

interfere with the rotation of the system. Also, an additional RPM indicator is needed to

obtain frequency data of the rotating structure. Generally, the strain gages are placed

closer to the hub to keep the added eccentric mass from significantly altering the

dynamics of the system. Thus, actual data is known near the hub, but prediction models

have to be used to get data for the tip. Storey [3] discussed the problems associated with

the physical size of the gages and how this limited the number of measurement points on

a blade. Additionally, he mentioned that the gages and wiring created obstructions that

affected the flow path and structural stiffness of the system. These obstructions change

the dynamic response of the system. Another problem mentioned was the limited life of

the strain gages in high speed rotating environments. Figure 2-1 shows a typical

arrangement of a strain gage-telemetry vibration measurement system.

9

Figure 2-1 Schematic of Strain Gage/Telemetry Method

Blade Tip Sensors

The meander method, discussed by Staeheli [4], involves measuring blade tip

vibration amplitudes by installing a permanent magnet at the tip of a blade. Voltage is

induced using a meander-shaped wire winding that is installed in the casing of the engine,

as shown in Figure 2-2. This method is useful for measuring blade tip vibrations, but no

data anywhere else on the blade can be obtained. Many permanent structural

modifications are needed to implement this method and it is not very practical to add

such an eccentric mass to high-speed rotating structures. Adding weight at the tip of a

blade tends to untwist the blade due to centrifugal loading. More recent modifications to

10

this method involved the use of optical sensors; thus, the permanent magnets are not

required. This will be discussed more in the non-contacting section.

Figure 2-2 Schematic of Meander Method

Raby [5], in 1970, described a method similar to the meander method except that

capacitive transmitters were used instead of the magnets as in the meander technique.

This capacitive technique worked for some applications, but in general, the detectors had

to be placed too close to the blade tip passage to obtain accurate results. A larger

clearance was generally needed in the blade tip passage for safety reasons. Thus,

transducer sensitivities were greatly reduced and results were not accurate.

11

2.2 Non-Contacting Methods

Non-contacting methods solved the problems that occurred in contacting methods,

regarding physical apparatus that affects the dynamic response of the structure. The

problems associated with most non-contacting methods arise because of the inability to

measure a known single point on a blade surface. The exception to this is the optical tip

sensor, which is mentioned in the following section. For most methods, such as the laser

methods and the holographic method, it is very difficult to align the measurement system

with the rotating structure, and then predict exactly where the measurement point would

be. This difficulty is especially true with engines running at high speeds that have

asynchronous and synchronous dynamic motions. Additionally, the signal conditioning

of these methods are generally very difficult and tedious. The following sections

describe various non-contacting methods for measuring rotating blade vibrations.

Blade Tip Sensors

Optical tip sensors are the most commonly used type of non-contacting vibration

sensors. Kadambi et al. [2] described optical tip sensors as very similar to the meander

and inductive/capacitive methods except no equipment was mounted to the tip of the

blade, as shown in Figure 2-3. In 1977 Roth [6] found that an optical proximity detector

was suitable to sense when a blade passed the location point of the detector. When the

RPM of the engine was known, an average transit time between blades could be

calculated. The optical probe would sense a certain amount of time between each blade

as it passed the probe. Any deviation of this transit time to the average transit time was

12

considered due to blade vibration. This method worked only for asynchronous vibrations

because average transit time is a function of the RPM of the structure, and any

synchronous vibrations are also a function of the RPM of the structure. Thus, any

information regarding the vibrations associated with the RPM of the structure could not

be determined because the average transit time had the same fluctuations as the blade-to-

blade transit times. This method generally worked well, but had some drawbacks. Nava

et al. [7] discussed that as amplitude and frequency of vibration changed during

measurement time, the accuracy of the data was greatly reduced. Storey [3] described

how it was hard to sample data at high engine speeds due to the extremely high sampling

rate needed. He also discussed that optical probes detected only what occurred at the tip

of the blade; therefore, data was not available for the entire span of the blade. Also, from

a design and modifications standpoint, it is impractical to have a probe opening in the

casing of a jet engine.

Figure 2-3 Typical Setup for Optical Blade Tip Sensors by Kadambi et al.[2]

13

Holographic Methods

Lesne et al. [8] described the basic premise of all holographic methods as the

superposition of a vibrating structure image on top of a stationary structure image. This

produced an interference fringe pattern that correlated to the motion of the structure. For

a rotating structure, such as a bladed rotor, this task is much more difficult than for a

stationary structure. A rotating structure produces an image that is also rotating;

therefore, the image cannot be compared to the image of the stationary object. Storey [2]

described the most common method to counter this problem for aero-engines: an image

derotator that optically compensated for the rotation of the structure. An image derotator

works by light passing through a beam splitter onto a prism. The prism rotates at half the

speed and in the same direction as the structure so that it reflects an image that appears

stationary. The image is then superimposed with the image of the stationary object,

which creates the interference pattern.

Precise alignment of the two images is very important for accurate results. Also,

a shaft encoder coupled with a control system is needed to rotate the prism at exactly half

the RPM of the structure. For the experiments conducted by Lesne et al. [8], small errors

in correlation between the structure RPM and derotator RPM resulted in inaccurate data.

Typically, this method is used on a more qualitative level, but calibration can be done to

generate quantitative data. Shown in Figure 2-4 are two images from a holographic

interferometry test that studied blade vibrations. Image A shows the interference fringe

pattern of a stationary disk with an excitation of 123 Hz. Image B shows the interference

fringe pattern of a rotating disk at 200 Hz. The two images show a definite qualitative

14

difference between the disk interference fringes at different vibratory modes. From this,

it is apparent that different frequencies can be distinguished by different fringe patterns.

Figure 2-4 Typical Holographic Images at Different Excitation Frequencies by Lesne et al. [8]

Acoustic Doppler Methods

Leon and Scheibel [9], in 1986, described that the acoustic Doppler method

worked on the premise that an increase in vibration in associated blades results in

changing sounds due to change in resonant conditions. For their experiment, non-

contacting acoustic sensors were located on the engine casing downstream of the blades

in question. Theoretically, through spectral analysis, the excited blade and its associated

mode and amplitude could be found. This method is only used to provide a time history

of synchronous vibrations of a certain blade, but not instantaneous stress data. This is

because not all blades are exactly the same and resonate slightly differently. Also, there

15

are difficulties in this method due to filtering out background noise and the very high

dependence of resonance on RPM fluctuations. Variations as little as +/- .03% are

considered significant.

Laser Doppler Methods

These methods are based on the concept, as described by Kadambi [2], that when

a surface moves normal to a coherent wave, the motion of the structure adds a Doppler

shift to the frequency of the coherent light wave. From this new wave, the vibration of

the surface is extracted when compared to the original waveform. The major problem

with this method is extracting a frequency shift true to the measured value, due to system

alignment issues.

Another problem experienced using Laser Doppler Vibrometry is a phenomenon

called speckle noise. This topic will not be explored much in much detail as it was

covered by Lomenzo [1] in a previously related paper, but will give a brief overview of

the topic.

Speckle noise occurs when the surface of the vibrating structure has a roughness

larger than the wavelength of the laser from the vibrometer. Essentially, when the light

reflects back from the vibrating surface, it appears to reflect from a number of small

surfaces. This causes the laser vibrometer to sense a number of path length changes not

associated with the actual vibration of the structure. Depending on the actual surface

smoothness, speed of rotation, and vibration signal magnitude, the speckle noise vibration

may or may not be significant. The subsequent sections discuss in more detail some of

16

the different laser Doppler methods for detecting vibrations on moving structures. They

include a stationary method, two laser tracking methods, and a self-tracking method.

Based on experiments done by Reinheardt et al. [10], the stationary/pulsed beam

method works on the same premise that the blade tip sensors worked. A laser from a

laser Doppler Vibrometer was fixed at a certain point in space where it intersected the

rotating blades of a rotor blade structure. As the structure rotates, the blades intersect

with the laser. As in previous methods, a shaft encoder is needed to measure the

rotational speed of the system and get an average transit time between each blade. The

time measured between the intersections of each blade, using the Laser Doppler

Vibrometer, is then compared to the average transit time for the structure. Any

deviations in transit time correlates to an asynchronous structure vibration. Synchronous

vibrations could not be measured using this technique since fluctuations in rotational

speed are accounted for both in the shaft encoder measurements and the laser

measurements.

The same general method has also been incorporated by placing a stationary laser

perpendicular to the blade passage to accumulate small bursts of data from each blade as

it passes through the laser beam. This method was very difficult to perform at high

rotational speeds due to data acquisition limitations. Experiments generating acceptable

data were done only up to about 100 RPM on blades, and about 1800 RPM on a disk.

Figure 2-5 below shows a typical setup, as used by Reinhardt et al. [10], for a stationary

beam measurement technique.

17

Figure 2-5 Generalized schematic for pulsed beam or stationary beam approach by Reinheardt et al.[10]

There are two main problems associated with stationary beam techniques. First,

data is only acquired for the blade at one point in its rotational space. Due to dynamic

excitations, there may be other vibrational modes present at other angular displacements

of the blade. The other problem associated with stationary beam techniques is the

phenomena mentioned previously called speckle noise, where the surface that the beam

reflects off of is rough compared to the wavelength of the laser particles. This is

especially a problem in measuring the small vibration associated with low rotational

speeds. Speckle noise signals are enough of the actual full signal that it is very difficult

to determine the signal content associated with the actual vibration of the blade.

Another laser Doppler measurement technique uses a synchronized mirror to

reflect the laser beam to a point on a blade. The laser beam theoretically tracks a single

point on a blade, thus is able to measure vibrational modes during the full rotational space

18

of the blade. In order to synchronize a mirror to rotate at the same rate as the structure, a

shaft encoder with a control loop is needed. As the speed of the structure increases, so

does the necessary sampling rate of the data acquisition system. Thus, for typical high-

speed jet engine applications, this method seemed limited. Another limitation of this

method is the inability to align the system and actually track a single point. Bucher et al.

[11] used this technique and measured the vibration of rotating disks. He performed

these experiments at relatively low speeds, around 130 RPM. The speed limitations of

this method make it inadequate for high-speed rotational measurements, such as for a jet

engine. The speed limitations were encountered in the mirror positioning equipment.

With a misrepresented signal, significant errors were present at higher speed applications.

Laser tracking using an optical de-rotator is another non-contacting laser method.

Lesne et al. [8] described this method as a system that combined both the technologies

used with holographic techniques and laser vibrometer tracking techniques. For his

experiment, a stationary laser was directed towards a single measurement point on a

blade. The incident beam was first reflected through an optical de-rotator, as for the

holographic method. The de-rotator rotated in the same direction and at one half the

speed of the rotating measured system. Essentially, a de-rotator is a device to

compensate for the rotation of the system and allow for the incident laser beam to be

compared to a stationary measurement point. A diagram of the typical setup for this

system is shown below in Figure 2-6. The main problem with this technique was that the

rotational axes of the rotating system and de-rotator must coincide exactly. Any amount

of misalignment resulted in out of plane amplitudes proportional to the misalignment. In

order to achieve alignment and revolve the de-rotator at exactly one half the speed of the

19

measurement system, a shaft encoder and signal processing equipment was needed. As

mentioned before, the addition of this apparatus can be costly and intrusive to the

structure itself.

Figure 2-6 Schematic of typical laser Doppler Vibrometer setup using an image de-rotator by Lesne [8]

The self-tracking LDV technique, as introduced by Lomenzo et al. [12] in 1998, is

still in a rather young stage of development. In general, it has the unique characteristic of

having a mechanical link between the rotation of the measurement system and the actual

measurement point. This is done by mounting an angled mirror to the rotor, which

rotates with the structure, on the axis of rotation. Due to this feature, no additional

electronic or mechanical equipment, such as a shaft encoder, is needed to relate the

rotational motion to the measurement point. A schematic of the system is shown below

20

in Figure 2-7. The major downfall of the self-tracker system is the errors encountered

due to misalignment of the required components. A study has been done to investigate

the effects on position error and velocity error of static misalignments of the laser and

fold mirror combination to the hub, rotor, and blade system, also viewed as a single

component.

The work assumed that the laser and fold mirror assembly was aligned with each

other, and that the vertex mirror, hub, and bladed disk were aligned to each other. This

simplified the investigation to translational and rotational misalignment between the two

assemblies mentioned. The results showed that the self-tracker system is capable of

tracking a single measurement point without position or velocity errors if there are no

misalignments. Both translational and rotational misalignments generate position and

velocity errors, which can be predicted using models. Also, although position errors can

be minimized to near undetectable levels, velocity errors cannot.

21

Figure 2-7 Schematic of Self-Tracker Laser Vibrometer

2.3 Summary

The previous sections present various contacting and non-contacting methods for

measuring vibrations on rotor blades. The self-tracking LDV technique is not as

developed as the other methods, but shows promise to be a valued stand-alone way to

measure rotor blade vibrations both in the field and in a laboratory setting. For this very

reason, it is important to continue development of the measurement system.

22

3 Laser Doppler Vibrometer

A discussion of many different methods used to measure velocity on rotor blades has

been completed. The method of particular interest to this thesis is the Self-Tracking

Laser Doppler Vibrometer. As shown in the previous chapter, this method implements a

Laser Doppler Vibrometer (LDV) to measure the velocity of rotor blades. These laser

measurement systems are somewhat complicated and require some explanation. In

addition, the use of the LDV with the self-tracker system has some specific relationships

that need to be explained. Drain [13], in 1980, gave a complete description of laser

vibrometry. Additionally, an explanation of laser vibrometry as applied to the self-

tracker system was given in 1998 by Lomenzo [1]. The remainder of this chapter will

briefly explain how the LDV works in conjunction with the self-tracker system, and also

reiterate the main points discussed by Lomenzo.

3.1 Diffuse Surface Laser Doppler Vibrometry

A laser Doppler vibrometer measures the velocity of an object by detecting a

frequency shift of light that the vibrating structure imposes on the incident laser. A

typical LDV is equipped with a helium-neon laser as a light source. These types of lasers

have a wavelength (λL) of 632.8nm and a frequency (ƒL ) of 470 THz. The governing

equation, for a LDV, that relates the frequency shift of a diffuse surface and the velocity

of that surface is shown in the following equation:

)(*2 tvf dsLλ

=∆ , [ 3-1]

23

where ∆ƒ is the frequency shift in Hertz due to surface movement, and νds(t) is the diffuse

surface velocity. For the particular case of the self-tracker, the laser beam is not

perpendicular to the surface of the target blade, as shown in Figure 3-1. Therefore, the

LDV only senses the portion of the velocity component in the direction of the laser. The

following equation shows how the laser component is calculated:

)cos(2 αλL

dsvf =∆ , [ 3-2]

where α is the angle that the laser beam makes with the target blade. Lomenzo [1] ran

some tests to prove this concept by oscillating a diffuse block with a known frequency at

different angles with respect to the laser. Figure 3-2 shows a schematic of the six

different tests performed with this six-sided block. The direction of the velocity of the

block, and the laser beam are shown relative to the block. The results of this test showed

that the LDV measured the velocity of the block in the direction of the laser beam only.

Also, the results showed that a path length change must be accompanied by a velocity

vector in order for the LDV to measure a velocity signal. The inverse is not true though.

When there was no apparent path length change, but a velocity component of the block

was present, the LDV measured a correct velocity signal. This result was important since

it governed how to calculate the velocity of the laser beam on the diffuse blade of the

self-tracker system. A secondary result of the test showed that even when there was no

velocity in the direction of the laser beam, small amounts of velocity were detected by the

LDV. Lomenzo identified this as likely being due to a phenomenon called speckle noise.

This paper does not cover the topic of speckle noise as it has been discussed, as related to

the self-tracker LDV, by Lomenzo.

24

Figure 3-1 Diagram of Laser Beam Intersection the Blade

Figure 3-2 Schematic of Diffuse Block Tests

25

3.2 Reflective Surface Laser Doppler Vibrometry

The concept of laser vibrometry changes slightly, when a mirror is involved, due to

their inherent reflective nature. The self-tracker system consists of two mirrors, one

angled mirror that rotates with the rotor and one fixed mirror that returns the reflected

laser back towards the target blade. Again, Lomenzo [1] has gone through a thorough

description of these concepts as applied to the self-tracker LDV system. The main

theories that apply to the simulation of the self-tracker will be described in this thesis.

When a LDV is shined upon a moving mirror, there is an inherent frequency shift

of incident laser source caused by the mirror motion. If the laser source is perpendicular

to the reflective surface of the mirror, as shown in Figure 3-3, then the frequency shift

caused by the moving mirror is the same as for a diffuse surface (Equation 3-1). Once

the laser source and the surface of the mirror are no longer perpendicular, as shown in

Figure 3-4, the frequency shift must be taken in the direction of the outgoing laser beam.

The following equation represents the resultant frequency shift in the direction of the

outgoing laser due to the velocity of the mirror:

)cos()cos(2 φβλL

mirvf =∆ , [3-3]

where νmir is the velocity of the mirror and the angles β and φ can be seen on the

associated figure. Since, for the case of the self-tracker system, the mirror is generally

rotating and not translating, this case is not particularly useful. However, the rotating

mirror can cause the effect of a translating mirror. This would occur if the beam spot

moved on the surface of the mirror at all.

26

Figure 3-3 Schematic of LDV Perpendicular To Mirror

Figure 3-4 Incident Laser Non-Orthogonal to Mirror

If there is any movement of the beam spot on the vertex mirror surface of the self-

tracker, there will be some induced velocity associated with that movement. Once this

velocity vector is resolved, solving for the velocity components in the outgoing laser

direction is the same as shown in Figure 3-4 and Equation 3-3. If the angular velocity of

27

the mirror and the x, y, and z position of the beam spot on the mirror are known, the

induced velocity can be calculated. This is done using the cross product of the position

and angular velocity, as shown in the following equation:

ω×= rvrotmir , [ 3-4]

where r is the distance from the center of the mirror to the beam spot, and ω is the

angular velocity of the mirror. Now that the velocity of the mirror is known, νrotmir is

substituted into Equation 3-3 for νmir to obtain the following equation.

)cos()cos(2 φβλL

rotmirvf =∆ [ 3-5]

For the self-tracker, the laser beam encounters the vertex mirror both before and after

intersecting with the target blade. Therefore, the actual induced frequency shift is

)cos()cos(4 φβλL

rotmirvf =∆ [ 3-6]

3.3 Summary

The ideas and concepts within the chapter were used as the basis for calculating the

velocities that the LDV would experience due to misalignments. The next chapter will

discuss the overall mathematical and physical concepts behind the computer program,

which includes the concepts discussed in this chapter.

28

4 Modeling The Self-Tracker LDV System

A computer simulation of the self-tracking laser system was developed using

MATLAB® to analyze the different input and output parameters of the setup. The object

of the program was to create a generalized model in which all input variables could be

easily changed and have outputs that truly represent what occurs in a real system. The

model incorporates rules of mathematics, optics, and basic system dynamics to generate

outputs that represent the actual velocity that the LDV would measure. The subsequent

sections of this chapter go through the initial task at hand, followed by the methodologies

and procedures used to create the self-tracker simulation.

4.1 Scope Of The Model

The self-tracker LDV system consists of a laser, fold mirror, vertex mirror, and a

row of blades attached to a rotating hub. All of these components must be modeled in

such a way to allow for misalignments, physical design characteristics, as well as the

dynamics of the system. In this case, the dynamics refers to the rotation of the rotor hub,

vertex mirror, and blades as to simulate a spinning blade. Physical design characteristics

consist of criteria such as the size of the rotor hub, radius of the blades, angle of the

blade, and distance between the different components. The program outputs position

error of the measurement point, velocity error, all of the errors introduced into the

system, as well as the design constraints. In addition, the model has the capability to

simulate dynamic effects for future studies.

29

4.2 General Procedures

There is a basic methodology used to come up with the position and velocity of the

blades. For a case with any given combination of misalignments, the program locates

each piece of equipment to the specified design and misalignment parameters. The

program then rotates, using a time step, the vertex mirror, hub, and bladed disk assembly

around the center of the hub. For each time step, the path of the laser beam is calculated.

Using this approach, the position of the laser spot on the blade, as well as the length of

the beam path can be calculated. The positions of each piece of equipment is set using

both physical design constraints and misalignment characteristics, which essentially sets

the orientation of each component relative to other components in the system.

Once the position of each piece of equipment, and its respective coordinate frame,

was determined, the orientation of each frame was then found in order to come up with a

unit vector in the z-direction of each component. Using the orientation of each piece of

equipment, along with the laws of reflection, the direction of the laser beam path was

found. Some basic mathematical equations were used to find the intersection point

between the laser beam and each piece of equipment. With the intersection points

known, the path of the laser beam, path length of the laser beam, and the velocity that the

LVD would sense, was calculated.

The simulation then calculates the position and velocity of the measurement

system under ideal and misaligned conditions. The program takes the difference between

the ideal and misaligned positions and velocities at each time step to come up with

position and velocity errors. Finally, the program outputs a series of plots and an

information sheet describing the design and misalignment conditions, as well as the

30

results that the program generated. The following sections of this chapter will address, in

detail, the methods used to find coordinate frame positions, coordinate frame orientations,

unit vectors, beam path, beam path length, laser spot position on the target, different

sensed velocities, laser spot position error, and target blade velocity error.

4.3 Relevant Variables

The first step in creating the self-tracker model is to define what the dependant and

independent variables are. The obvious output variables are position error and velocity

error. There are a multitude of input variables that need to be considered. The design

constraint variables will be examined first. Figure 4-1 shows a typical setup of the self-

tracker vibrometer system. The parameters shown represent the physical layout of the

system. The ranges of these parameters are dependent on the size of the structure in

question, the amount of space available, and also range limitations of the laser itself. A

list of frequently used variables and their associated properties is listed prior to the

introduction section. The use of these variables and how they are incorporated into the

program will become more apparent in subsequent sections.

31

Figure 4-1 Design Constraint Variables for Self-Tracker

The next set of variables to be described is those input variables that define the

misalignment of the different components of the self-tracker system. Each physical piece

of equipment has six degrees of freedom. They are x, y, and z translational

misalignments and x, y, and z rotational misalignments. If the piece of equipment has

any of the just mentioned misalignments, the laser spot will not be at the intended place

and the velocity measurement will have errors.

4.4 Modeling the Position of the Equipment

The position of each piece of equipment must include both the physical design

constraints and the misalignments introduced into the system. The basic method used to

define the location and orientation was to locate each piece of equipment by a separate

coordinate frame. Each piece of equipment must first be located using the appropriate

design parameters. Using the concept of superposition, the misalignment affects are then

32

added to the design parameters to obtain the overall position and orientation. By using

the concept of superposition, two assumptions are used: 1) the responses are linear, and

2) the misalignment errors are relatively small. Under the ideal alignment condition, all

misalignment effects are set to zero. Figure 4-2 shows a general representation of the

layout of the coordinate frames.

Figure 4-2 General Schematic of Coordinate Frame Positioning

For the purpose of this model, each piece of equipment should be thought of in

terms of its coordinate frame. The use of coordinate frames makes it very easy to identify

the location of any piece of equipment relative to any other piece of equipment through

the use of coordinate frame transformations. The Eulerian transformation matrix was

33

used to reference one coordinate frame in terms of another. Given a starting position

along with the x, y, and z transformations and rotations referenced from that starting

position, a new coordinate frame can be found in terms of the starting position and

orientation.

In addition to knowing the position of each piece of equipment, it is also important

to know the orientation. Using a method similar to coordinate transformations for

translations, another transformation matrix was used to define the orientation of each

reference frame. The following sections will describe the position and orientation

transformation process.

The Generalized Matrix Transformation Used to Find Position

Before going into the actual transformations used in the simulation, a generalized

case will be described to show the concepts. The two different coordinate frames will be

called frame {A} and frame {B}, with frame {A} being the starting reference frame. The

notation of variables is as follows:

dx = translation of {B} relative to {A} along x-axis of {A},

dy = translation of {B} relative to {A} along y-axis of {A},

dz = translation of {B} relative to {A} along z-axis of {A},

γ = rotation of {B} about x-axis of {A},

β = rotation of {B} about y-axis of {A}, and

α = rotation of {B} about z-axis of {A}.

The translations from {A} to {B}, in terms of {A} should be as shown in Figure 4-3 and

the rotations from {A} to {B} as shown in Figure 4-4.

34

Figure 4-3 Defining Translational Motion for Eulerian Transformation

Figure 4-4 Defining Rotational Motion for Eulerian Transformations

The way to notate the transformation from {B} to {A} in terms of frame {A} is TBA ,

where A is the beginning frame, B is the frame to be transformed from, and T stands for

transformation. The Eulerian transformation matrix, using c = cosine and s = sine, is as

γ

βα

35

shown in Equation 4-1. When TBA is multiplied by a point that is defined in frame

{B}, TB , a representation for {B} in terms of frame {A} is found. Thus, if it were

assumed that the a point in frame {B} was [xB, yB, zB], the position of that point, BT

A, , in

terms of {A} is found as shown in Equation 4-2. Also, note the subscript B is the

position of the point. If there are more than two coordinate frames, the subscript

describes which point is being described in the respective frame. This became important

due to the multiple coordinate frames the model dealt with.

−−++−

=

1000dzscscsdysccssccssscsdxsscsccsssccc

TBA

γβγββγαγβαγαγβαβαγαγβαγαγβαβα

[4-1]

]1[* AAA zyxTBA

TB

TBA

TA

== [4-2]

In addition to using matrix transformations to define the location of a coordinate

frame, transformations are commonly used to describe the relation between many

interrelated coordinate frames. For our example, it is desirable to know the location of

the target blade in terms of the laser starting point. This is done utilizing the rules

associated with multiplying transformation matrices together. Given a set of

transformation matrices that relates the four different coordinate frames {A}, {B}, {C},

and {D}; the representation of any frame in terms of any other frame can be found using

the rules shown in Equation 4-3 and Equation 4-4.

36

TDC

TCB

TBA

TDA

= [4-3]

11 −−== TDC

TDA

TBA

TCD

TDA

TAB

TCB [4-4]

These are generalized equations to give a sense of how the mathematics are implemented.

As shown in the two previous equations, if the transformation matrix is known between

each successive coordinate frame, then transformations can be done between any two

coordinate frames. The usefulness of this will become more apparent as the physical

system and how it was modeled are further explained.

Translational Transformations Applied to the Self-Tracker LDV

Now that a general overview of how transformation matrices are used has been

discussed, the procedure as applied to the specific case of the Self-Tracker LDV will be

described. The Self-Tracker system configuration has been broken down into seven

separate coordinate frames. The different frames are the global frame, laser frame, rotor

frame, vertex mirror frame, fold mirror frame, blade frame, and target frame. Each of

these frames will now be discussed in further detail.

Global Frame

The global coordinate frame is used to represent a world coordinate system as a

point of reference. Its main purpose is to give one reference frame that all other

coordinate frames can be referenced to. The notation for the global frame will be {G}.

37

The initial position of the global frame is specified as [0, 0, 0, 1] and is notated as GT

G, .

The notation for this is described as follows:

2,1

T , [ 4-5]

where the T stands for a transformation, the 1 place identifies the frame which the point is

being defined in, and the 2 place identifies which point is being described in that

respective frame. For example, GT

G, is the global point defined in its own reference

frame. The fourth value in the initial position, 1, is simply a placeholder since

transformation matrices are four rows by four columns. For future work, instead of being

used as a placeholder, it could be used as a time representation for dynamic simulations.

Laser Frame

The laser frame is referenced directly from the global coordinate frame and will

have the notation {L}. For the ideal alignment case, there is no difference between the

origin of {G} and the origin of {L}. Thus, any differences between the two frames are

due to misalignments introduced into the system. The laser frame is placed at the source

of the laser. The z-axis extends in the direction of the laser. The positive y-axis extends

vertically upward and the x-axis in the horizontal direction. The transformation from {L}

to {G} is TLG . The position of the laser in global coordinates,

LTG

, , is found by

multiplying LT

LT

LG

, , where LT

L, is the origin of the laser in its own coordinate frame.

38

Rotor Frame

The rotor frame, {R}, was referenced from the laser frame and was determined by

a number of factors. First off, it was defined as the distance from the laser frame to the

center of rotation of the structure where the vertex mirror mounted to the rotor hub.

Essentially, it specified the distance from the laser to the jet engine. The ideal design

orientation placed the positive z-axis of the rotor frame 180 degrees from the laser frame,

thus, the z-axis faced directly into the positive z-axis of the laser frame. This was done

by entering a design parameter of π radians (180 degrees) for the y-axis rotation. The

transformation to represent frame {R} in global coordinates was TRG . Since {R} was

specified from {L}, TRG was found by multiplying T

RL

TLG . The position of the rotor in

global coordinates, RT

G, , is found by multiplying

RTR

TRG

, , where RT

R, was the origin of

the rotor frame in its own coordinate frame.

Vertex Mirror Frame

The vertex mirror frame, {V}, was referenced from the rotor frame and was used

to define the angle that the laser beam reflected from the center of the rotating structure.

Also, incorporated into the vertex frame was the rotation of the structure. The vertex

mirror attached to the rotor hub and rotated with it. For the parameter αv, the rotation of

the frame about the z-axis, an angular position in time was specified to create a rotation.

The design angle of the vertex mirror affected the radial position of the laser beam, and

was entered into the program as a fixed rotation about the y-axis of the rotor frame. The

angle remained constant about the y-axis as the vertex mirror rotated with the rotor hub

39

as shown in Figure 4-5. The transformation that represented {V} into global coordinates

was TVG . Since {V} was specified in terms of {R}, and we already know T

RG , T

VG was

found by multiplying TVR

TRG . The position of the vertex mirror in global coordinates,

RTG

, , is found by multiplying VT

VT

VG

, , where VT

V, represented the origin of the vertex

mirror in its own coordinate frame.

Figure 4-5 Schematic of Rotor/Vertex Mirror Assembly

Fold Mirror Frame

The fold mirror frame, {F}, was specified in terms of the laser coordinate system

{L}. The design parameter was the translation along the z-axis of the laser frame that

40

represented the distance from the laser to the fold mirror. This distance was solved for

based on the angle of the vertex mirror, the desired location of the beam spot along the

span of the blade, and the specified distance from the laser to the vertex mirror. The

transformation to represent frame {F} in global coordinates was TFG . Since {F} was

referenced from {L}, TFG was found by multiplying T

FL

TLG . The position of the fold

mirror in global coordinates, FT

G, , was found by multiplying

FTF

TFG

, , where FT

F, was

the position of the fold mirror in its own coordinate frame.

Blade Frame

The blade frame, {B}, was referenced from the rotor frame. The main difference

between the two frames was a displacement along the axis of rotation that offset the

blades from the hub, as shown previously in Figure 4-1. Additionally, in this frame,

vibrations could be added to the blade. The vibrations were assumed to be uniform along

the span of the blade and in the direction of the blade frame z-axis. The amplitude and

frequency of the vibration could be easily changed in the simulation. The transformation

that represented {B} in global coordinates was .TBG Since {B} was referenced from the

rotor frame, TBG was found by multiplying T

BR

TRG . The position of the blade in global

coordinates, BT


BTB

TBG

, , where GT

B, was the origin of the

blade frame in its own coordinates.

41

Target Frame

The target frame, {T}, represented the plane where the laser beam spot intersected

the blade. The difference between the blade frame and target frame was that the target

frame could include some twisting of the blades. If the model was to represent a flat

blade, then the blade frame and target frame were identical. {T} was referenced from the

blade frame and any blade twisting was modeled as a rotation about the x-axis of the

blade. The transformation to get from {T} to global coordinates was TTG . Since {T} was

referenced from the blade frame, TTG was found by multiplying T

TB

TBG . The position of

the target in global coordinates, TT


TTT

TTG

, , where TT

T,

represented the origin of the target frame in its own coordinates.

The Generalized Matrix Transformation Used To Find Orientation

In order to find the orientation of each piece of equipment, it is necessary to

perform rotational transformations between the different coordinate frames. The process

is very similar to that used to find the position of the different coordinate frames. The

main difference being that a slightly different transformation matrix is used along with all

the same rotations specified for the translational transformations. The purpose of the

rotational transformation is to find the orientation of each frame of reference as a unit

vector in the laser beam direction, which is specified as the z-direction in each coordinate

frame. Using the generalized frame notations of {A} and {B} again, the rotational

variables are as follows:

42

γ = rotation of {B} about x-axis of {A},

β = rotation of {B} about y-axis of {A}, and

α = rotation of {B} about z-axis of {A}.

The rotations are specified as shown previously in Figure 4-4. The notation to represent

the rotation of {B} in terms of {A} is RBA , and the associated Eulerian rotation

transformation matrix, using c=cosine and s=sine, is shown in Equation 4-6. The same

matrix multiplication rules used with translational matrices apply to the rotation matrices.

The need to find unit vectors for each coordinate frame will become more apparent later

when describing the path of the laser beam throughout the self-tracker LDV system.

−−++−

=γβγββ

γαγβαγαγβαβαγαγβαγαγβαβα

ccscssccssccssscssscsccsssccc

RBA

[4-6]

Rotational Transformations Applied to the Self-Tracker LDV

Now that a generalized overview of rotational transformations is complete, their

application to the specific case of the Self-Tracker LDV system will be examined. As

mentioned before, the Self-Tracker LDV system consisted of seven different coordinate

frame systems. They were the global frame, laser frame, rotor frame, vertex mirror

frame, fold mirror frame, blade frame, and target frame. The following sections describe

each of the different coordinate frames, from an orientation standpoint, and how to

calculate a unit vector in the laser beam direction for each frame.

43

Global Frame

The global frame, {G}, was used as a starting orientation and reference for all of

the other coordinate frames. There was no unit vector calculated for the global frame, but

it was used as a reference for all other coordinate frames.

Laser Frame

The laser frame, {L}, is oriented from the global coordinate frame, {G}. The

laser was designed to have the same orientation as the global frame, therefore, any

rotations experience by the laser frame were due to misalignments introduced into the

system. The rotational transformation to get from {L} to global coordinates is RLG . The

unit vector of the laser frame in global coordinates, LR


LRL

RLG

, . LRL

, was the unit vector in the z-direction of the laser frame, and was set as [ 0

, 0 , 1].

Rotor Frame

The rotor frame, {R}, had an orientation referenced from the laser frame, {L}.

The rotor was designed to have its z-axis rotated π radians (180 deg) about the y-axis,

from the z-axis of the laser frame. The rotational transformation to represent frame {R}

in global coordinates was RRG . Since {R} was specified in terms of the laser coordinate

frame, RRG is found by multiplying R

RL

RLG . The unit vector along the z-axis in the rotor

44

direction, RR


RRR

RRG

, , where RRR

, represented the unit

vector of the rotor frame, in its own coordinates, that described the z-direction. This unit

vector was set as [0, 0, 1].

Vertex Mirror Frame

The vertex mirror frame, {V}, has an orientation referenced from the rotor frame,

{R}. Additionally, a designed rotation around the z-axis of the rotor was modeled by

creating a radial position that changes with each time step of the program. The number

of time steps; hence, positions can be altered easily to achieve different amounts of

accuracy and resolution. The vertex mirror frame was fixed with the rotor frame and

rotated in conjunction with it. Also, the vertex mirror frame was designed to have a

constant angle about the y-axis of the rotor, which represented the vertex mirror angle.

The rotational transformation to represent frame {V} in global coordinates was RVG .

Since {V} was specified in terms of the rotor coordinate frame, RVG was found by

multiplying RVR

RRG . The unit vector along the z-axis of the vertex mirror,

VRG

, , was

found by multiplying VR

VR

VG

, , where VR

V. was the unit vector of the vertex frame, in its

own coordinates, that described the z-direction. This unit vector was set as [0, 0, 1].

.

Fold Mirror Frame

The fold mirror frame, {F}, has an orientation that was referenced from the laser

frame, {L}. The fold mirror frame had the same orientation as the laser frame under

45

ideal alignment conditions. Thus, the only rotations associated with the fold mirror

frame, compared to the laser frame, were due to misalignments introduced into the

system. The rotational transformation to get from frame {F} to global coordinates was

RFG . Since {F} was specified in laser frame coordinates, R

FG was found by multiplying

RFL

RLG . The unit vector in global coordinates along the z-axis for the fold mirror frame,

FRG

, , was found by multiplying FR

FR

FG

, where FR

F., was the unit vector of the fold

mirror frame, in its own coordinates, that described the z-direction. This unit vector was

set as [0, 0, 1].

Blade Frame

The blade frame, {B}, has an orientation that was referenced from the rotor

frame, {R}. The blade frame had the same orientation as the rotor frame under ideal

alignment conditions and rotates along with the rotor and vertex mirror. Thus, any

differences in the orientation between the blade frame, {B}, and the rotor frame, {R}, are

due to misalignments of the blade frame. The rotational transformation to obtain frame

{B} in global coordinates was RBG . Since {B} was specified in rotor frame coordinates,

RBG was found by multiplying R

BR

RRG . The unit vector along the z-axis for the blade

frame, BR


BRB

RBG

, , where BR

B. was the unit vector of the

blade frame, in its own coordinates, that described the z-direction. This unit vector was

set as [0, 0, 1].

46

Target Frame

The target frame, {T}, has an orientation that was referenced from the blade

frame, {B}. The target frame introduced a twist along the x-axis of the blade frame. It

was assumed that the x-axis of the blade and target frames represented the centerline of

the blade along the span. The blade twist was introduced as a rotation about the x-axis.

The rotational transformation to obtain {T} in global coordinates was RTG . Since {T}

was specified in blade frame coordinates, RTG

was found by multiplying RTB

RBG

. The

unit vector along the z-axis for the target frame, TR


TRT

RTG

, , where TRT

. was the unit vector of the target frame, in its own coordinates, that

described the z-direction. This unit vector was set as [0, 0, 1].

4.5 Calculating the Path of The Laser Beam

Each piece of equipment has now been defined as its own coordinate frame.

Additionally, the position and orientation of each coordinate frame has been described in

terms of the reference, or global, coordinate system. The next step is to calculate the path

of the laser beam throughout the rotating system using both the global position and global

orientation of each frame. By calculating the path of the beam, the position of the laser

on the final target frame can be calculated. The following sections will discuss the

general mathematical methods used to find the laser beam path, as well as the methods

applied to each portion of the laser path.

47

The Generalized Mathematical Methods to Calculate the Laser Beam Path

In order to describe the path of the laser between two different pieces of

equipment, or two coordinate frames, a generalized procedure assuming a ray starting at

point 1 extending towards the x-y plane of point 2 was used, as shown in Figure 4-6. It

was desired to find the point where the ray1-2 intersects the x-y plane at point 2. Once

this was found, the starting point was subtracted from the intersections point at 2 to

obtain a representation of the x, y, and z displacements of the path lengths from point 1 to

point 2. The subsequent section(s) describes the steps used to calculate the intersection

of the ray and the x-y plane at point 2.

(x1, y1, z1)

(x2, y2, z2)

Figure 4-6 Schematic of a Ray extending from point 1 towards point 2

48

Using Figure 4-6 as a reference to calculate the intersection of Ray1-2 and the x-y

plane at location 2, the following set of parametric equations that represented the

intersection point in terms of location 1 was defined:

taxx i 1212 +=

tbyy i 1212 += [4-7]

tczz i 1212 += ,

xi2, yi2, and zi2 are the three dimensional intersection point of Ray1-2 with the x-y plane at

2; x1, y1, and z1 is the location of point 1; a12, b12, and c12 defines the direction unit vector

of Ray1-2; and t is the auxiliary variable that describes xi2, yi2, and zi2. From the set of

parametric equations shown in Equation 1-6, we know the starting point and direction of

Ray1-2, but not the intersection point with plane x-y at location 2 or the auxiliary variable

t. Another equation to represent the intersection point was needed since 3 equations were

insufficient to solve for 4 unknowns.

The x-y plane at location 2 can also be defined by its direction normal and any

two points on the plane. In this case, the desired intersection and the origin of location 2

were chosen to give the additional equation necessary to solve the parametric system.

The equation of the x-y plane at location 2, Equation 4-8, was defined as

( ) ( ) ( ) 0222222222 =−+−+− zzcyybxxa iii , [4-8]

where a2, b2, and c2 are known and define the direction normal of the plane in the z-

direction; x2, y2, and z2 are known and define the origin of location 2; and xi2, yi2, and zi2

are the same intersection points referred to in Equation 4-7. The x, y, and z intersection

49

variables from the parametric equations were then plugged into Equation 4-8. The

resulting equation was rearranged to solve for the auxiliary variable, t, as shown in

Equation 4-9. Now that the auxiliary variable was solved for, it can be plugged back

into the original set of parametric equations to solve for the intersection point of Ray1-2

with the x-y plane of location 2.

122122122

212212212 )()()(ccbbaa

zzcyybxxat++

−+−+−−= [4-9]

This procedure was repeated between each coordinate frame to determine the

intersection point, in global coordinates, of the laser beam with the x-y plane of each

coordinate frame. For any two pieces of equipment, the starting laser point was

subtracted from the laser intersection point of the successive piece of equipment, and the

magnitude of this vector was calculated, as shown in Equation 4-10. This represented the

overall beam path length between the two points. Once the beam path length between

each set of components was calculated, they were added together to come up with the

total length of the laser beam as it goes through the system. The subsequent sections

describe the methods used to find the intersection points and beam path length as applied

to the self-tracker model.

( ) ( ) ( )212

212

21221 zzyyxxbeamlength iii −+−+−=− [4-10]

50

Calculating Intersection Points and Path Length Applied to the Self-Tracker LDV Global Frame

All of the coordinate frames were referenced from the global frame. No path

length was defined for the laser beam since the laser source originates from the laser

frame.

Laser Frame to Vertex Frame

The laser frame was where the laser beam originates and extends until it intersects

with the x-y plane of the vertex mirror frame. The starting point of the laser, the unit

vector along the beam path, and the equation of the vertex mirror x-y plane were all

known. The starting point of the laser, LT

G, , was calculated in a previous section of the

simulation. The unit vector along the laser path extending towards the vertex mirror

plane was defined to be the z-axis of the laser frame, LRG

, . Using both the origin and

the unit vector in the z-direction of the vertex mirror, previously been defined as VT

G,

and VR

G, , the x-y plane of the vertex mirror was defined.

The methods discussed previously to find an intersection point were then applied

and the intersection point on the vertex mirror (iVx, iVy, iVz) was found. The origin of

the laser was specified as the origin of the laser frame, LTL

TLG

, , which yielded the point

(Lx, Ly, Lz). The beam path length from the laser to the vertex mirror was calculated

using Equation 4-11.

51

( ) ( ) ( )222 LziVzLyiVyLxiVxbeamlength VL −+−+−=− [4-11]

Vertex Mirror Frame to Fold Mirror Frame

The starting point of the portion of the laser beam path that extended from the

vertex mirror to the fold mirror was previously found as the intersection point of the

vertex mirror. The beam extends until it intersects the x-y plane of the fold mirror. It

was then necessary to define the direction normal of the laser beam as it leaves the

intersection point on the vertex mirror and extends towards the fold mirror. This was

done using rotational transformations and also using laws of reflection. The direction

normal of the vertex mirror was found by first finding the direction normal of the laser

starting point in terms of the vertex mirror frame, then using the laws of reflection to

determine the direction of the beam leaving the vertex mirror. The unit vector, in global

coordinates, of the laser beam as it extends from the vertex mirror towards the laser

starting point, LVR

G−, , was found using the transformation in Equation 4-12.

−

=

−

− LLV RG

RVR

RRG

RV

,,1

[4-12]

Now, the laws of reflection for a mirror were used to find the direction of the laser beam

leaving the vertex mirror and pointing towards the fold mirror. The z component of the

unit vector remained the same as for the laser beam incoming the vertex mirror from the

laser, but the x and y components were negative. The notation of this unit vector was

52

FVRG

−, since the laser beam is now pointing from the vertex mirror towards the fold

mirror. The relation between FVR

G−, and

LVRG

−, is shown in Equation 4-13.

−−

•

= −−

111

,, LVFV RV

RV [4-13]

Now, the parametric set of equations for the laser from the vertex mirror towards

the fold mirror was found. The next step was to create the equation of the fold mirror x-y

plane by using its origin and unit vector along the z-axis. These were previously defined

as FTG

, and FRG

, , respectively.

The methods discussed previously to find an intersection point were then applied

and the intersection point on the fold mirror (iFx, iFy, iFz) was found. Knowing the

origin of the laser beam to be the intersection on the vertex mirror, the beam path length

between the vertex mirror and fold mirror were found using Equation 4-14.

( ) ( ) ( )222 iVziFziVyiFyiVxiFxbeamlength FV −+−+−=− [4-14]

Fold Mirror Frame to Blade Frame

Finding the intersection point on the blade frame and the laser beam path length

between the fold mirror and blade were found using methods similar to that between the

vertex mirror and fold mirror. First, a ray was defined that leaves the fold mirror at the

intersection point. The direction normal to construct the ray was found using the

properties of the laser beam between the fold mirror and vertex mirror, and also the

53

properties of mirror reflectance. The direction of the laser beam pointing from the vertex

mirror towards the fold mirror, FVR

G−, , was already found. The direction of the beam

from the fold mirror back towards the vertex mirror, VFR

G−, , was simply FVR

G−− , . Now,

using the laws of reflectance, the direction of the ray extending from the fold mirror

towards the blade was found, as shown in Equation 4-15.

−−

•= −−

111

,, VFBF RG

RG [4-15]

Note that this is not a cross product, but a vector multiplication. This direction vector,

along with the intersection point on the fold mirror, defined the ray coming from the fold

mirror towards the blade frame. The next step was to define the x-y plane of the blade

frame using its point of origin and direction normal of the z-axis. These were previously

defined as BT

G, and

BRG

, , respectively. The methods discussed previously to find the

intersection point on a plane were then applied to obtain (iBx, iBy, iBz), the intersection

point of the laser beam with the blade frame. Knowing the origin of the laser was the

intersection on the fold mirror, the path length between the fold mirror and blade were

found using Equation 4-16.

( ) ( ) ( )222 iFziBziFyiByiFxiBxbeamlength BF −+−+−=− [4-16]

Blade Frame to Target Frame

The next step was to find the intersection point of the laser beam with the target

frame. Bear in mind that the only difference between the blade frame and target frame

54

was a possible angular offset about the span of the blade to represent angled blades.

Therefore, the direction of the beam incident on the target frame was the same as that

incident on the blade frame. The starting point of the ray is known to be the intersection

point on the blade, (iBx, iBy, iBz). From the intersection point on the blade and its

direction vector, the ray from the blade towards the target was found. The equation for

the x-y plane of the target was found knowing its origin and direction unit vector in the z-

directions. These were previously found as TTG

, and TR

G, , respectively. The methods

for finding and intersection point were applied to obtain the target frame intersection

point, (iTx, iTy, iTz). Knowing the starting point of the ray was the intersection point on

the blade, the beam path length between the blade and the target was found using

Equation 4-16. The intersection point of the laser beam on the target represents the

position of the laser spot on the blade. There will be further discussion regarding the

final position of the laser spot in following sections.

( ) ( ) ( )222 iBziTziByiTyiBxiTxbeamlength TB −+−+−=− [4-17]

Calculating Overall Laser Beam Path Length

The overall beam path length was calculated by simply adding each component of

the beam path. These components were the portions from the laser to the vertex mirror,

the vertex mirror to the fold mirror, the fold mirror to the blade and the blade to the final

target position. The addition of these components is shown in Equation 4-18. The

usefulness of knowing the beam path length is of importance when finding the velocity of

the target blade, which will be discussed in a subsequent section.

55

TBBFFVVLTOT beamlengthbeamlengthbeamlengthbeamlengthbeamlength −−−− +++= [4-18]

4.6 Finding the Position and Position Error of the Laser Spot

The position of the laser spot on the target frame, at any point in time, was

defined as the intersection point of the laser beam with the x-y plane of the target frame.

This point was previously defined as the point (iTx, iTy, iTz). Since the model was

designed to run for one full rotation of the system, a history of the laser location on the

target frame with respect to the angular position of the target, in global coordinates, was

found. For a perfectly aligned system with no simulated blade vibrations, the laser spot

did not move. When the system is misaligned and/or blade vibrations were introduced,

the position of the laser spot on the blade, with respect to angular position of the rotating

system, changed. The laser spot position on the target frame due to misalignments was

compared to the position of the laser spot when the system is perfectly aligned, as shown

in Equation 4-19. This difference represented the position error of the system. To do

this, the program was run two times; once for the ideally aligned case and once for the

misaligned cases. The two run configurations were identical with the exception of

misalignment errors. This allowed separating out the errors specifically due to the

misalignment of equipment. The maximum position error during one full revolution of

the system represents the largest distance that the laser beam spot was compared to where

the beam was intended to be on the target blade. This was the value of most interest.

Therefore, the maximum position error was calculated by finding the largest absolute

position error during one run of the simulation.

56

[ ] [ ] idealmisalignedPOS iTziTyiTxiTziTyiTxError −= [4-19]

4.7 Finding Velocity and Velocity Error of the Measurement System

A laser vibrometer measures velocity in the direction of the laser beam path. The

next step in modeling the system was to calculate the velocity that the laser vibrometer

would encounter as a result of the misalignments and/or blade vibrations. The first step

in showing how the velocity and velocity errors of the measurement system were

calculated is to explain the different types of velocity that the laser will sense. There are

two velocities that the laser will encounter, one is due to motion of the beam spot on the

vertex mirror and the other is due to any rigid body motion of the laser spot on the target

blade.

Vertex Mirror Velocity

When the laser has any amount of translational misalignment, the beam spot does

intersect with the center of the vertex mirror, or center of rotation. This causes a

movement of the laser spot as the system rotates, as shown in Figure 4-7 and Figure 4-8.

The axes in the figures are that of the vertex mirror frame. The product of the movement

of the laser spot and the rotation of the system causes a velocity in the z-direction of the

vertex mirror, which has components in the laser beam direction. Since the LDV can

only sense the vibrations in the direction of the laser, the velocity in the z-direction of the

vertex frame needed to be resolved in the laser beam direction.

57

Figure 4-7 How the Laser Beam Spot Moves On Vertex Mirror

ωωωω = 0°°°°

ωωωω = 90°°°°

ωωωω = 180°°°°

ωωωω = 270°°°°

Figure 4-8 The Beam Spot On the Vertex Mirror as the System Rotates

58

By using the values already calculated for the intersection points of the laser beam

with the vertex mirror and fold mirror, as well as the starting point of the laser, the

velocity that the LDV sensed on the vertex mirror was found. The first step was to find

the velocity of the beam spot in the z-direction of the vertex mirror. Since the [0,0,0]

coordinate of the vertex mirror frame was known to be the location of the laser beam spot

under ideal alignment conditions, the actual intersection of the laser beam on the vertex

mirror was the position error that contribute to the velocity that the LDV sensed due to

misalignments. Since the intersection of the laser beam with the vertex mirror was

already found in Global coordinates, the same intersection point, in vertex coordinates,

was found using the following transformation equation:

=

−

iVziVyiVx

TVG

iVziVyiVx

V

1 [4-20]

where, [iVx, iVy, iVz]V is the three dimensional coordinate of the intersection in vertex

coordinates. The cross product of the intersection point with the angular rotation (rad/s)

of the system, both in vertex coordinates, was done to find the velocity imparted in the

coordinates of the vertex frame. Equation 4-21 shows how to transform from the rotation

of the system in rotor coordinates to vertex coordinates, and Equation 4-22 shows how to

find the velocity in the coordinates of the vertex frame.

=

−−

z

y

x

Vz

y

x

RVR

RRG

ωωω

ωωω 11

[4-21]

VVz

y

x

Vz

y

x

iVziVyiVx

XVelmirVelmirVelmir

=

ωωω

[4-22]

59

The next step was to resolve the vertex mirror coordinate velocity of the laser

beam spot on the vertex mirror into components in the laser beam direction. This was

done by calculating the x, y, and z direction cosines between the vertex mirror velocity

component and the incoming laser beam. The intersection points of the vertex mirror and

laser starting point , in global coordinates, were previously found to be [iVx, iVy, iVz]

and [iLx, iLy, iLz], respectively. These were put into the coordinates of the vertex mirror

by performing the transformations showed in equation 4-23.

=

−

iVziVyiVx

TVG

iVziVyiVx

V

1

=

−

iLziLyiLx

TVG

iLziLyiLx

V

1

[4-23]

Also, the beam path length between the vertex mirror and fold mirror, VLbeamlength − ,

was previously found. Using the two intersection points for the vertex mirror and fold

mirror, along with the beam length, the direction cosines between the laser beam and

vertex mirror were found. Equation 4-24 shows these calculation.

−=

−

−

VL

vVLV beamlength

iVxiLx1cosγ

−=−

−

VL

vVLV beamlength

iVyiLy1cosβ [4-24]

−=−

−

VL

vVLV beamlength

iVziLz1cosα

γLV, βLV, and αLV are the x, y, and z angles between the laser beam, which extends

between the vertex mirror to the laser starting point, and the vertex mirror.

The velocity that the LDV sensed due to the motion of the laser beam spot on the

vertex mirror was found by multiplying each velocity component of the mirror velocity

by its respective direction cosine. This is shown in Equation 4-25.

60

•

=

LV

LV

LV

Vz

y

x

LBz

y

x

VelmirVelmirVelmir

VelmirVelmirVelmir

αβγ

cos [4-25]

The three components of velocity in the laser beam directions were added together, as

shown in Equation 4-26 to obtain the total velocity that the LDV would sense due to

beam motion on the vertex mirror.

LBLBLB zyxLB VelmirVelmirVelmirVelmirtot ++= [4-26]

Since the laser beam actually encounters the vertex mirror on its forward and return path

to the target blade, the mirror velocity in Equation 4-26 is doubled to obtain the correct

velocity that the LDV sensed. This can be seen in Equation 4-27.

LBLB VelmirtotVelmirtot *2)2( = [4-27]

All of the above calculation were done for both the misaligned and ideally aligned case.

Rigid Body Induced Velocity

The second velocity that the LDV senses is the velocity at the blade due to any

rigid body motion occurring from the movement of the beam spot on the blade. This

rigid body velocity can be broken down into two separate components: 1) rigid body

velocity due to the laser misalignment and 2) rigid body velocity due to blade vibrations.

The total velocity that the LVD senses was the superposition of the velocity due to the

misalignment and the velocity due to blade vibration. As with the vertex mirror velocity,

the rigid body velocities need to be resolved into the laser beam direction.

Since the purpose of this research was to study the misalignment effects of the

system, calculating the actual velocity due to the vibration of the blade was not necessary.

61

The blade vibration was a known input into both the misaligned and aligned cases, and

equally affected the results of both cases. The reasoning behind this becomes more

apparent as this section continues.

The general methodology behind finding the rigid body motion of the misaligned

case was to, first, find the position error of the beam spot on the target blade. This was

shown previously in Equation 1-18, which already has the position in target frame

coordinates. Next, the rotation of the system had to be transformed into the target

coordinates also. This was done as shown in Equation 4-28, where ωx, ωy, and ωz were

originally in global coordinates.

Gz

y

x

Tz

y

x

RTB

RBR

RRG

=

−−−

ωωω

ωωω 111

[4-28]

The cross product of the position error and angular rotation of the system, both in

target coordinates, yielded the rigid body velocity, due to misalignments only, in the

coordinates of the target frame. This is shown in Equation 4-29.

TTz

y

x

Tz

y

x

iTziTyiTx

XVelrbmVelrbmVelrbm

=

ωωω

[4-29]

It was then necessary to resolve the rigid body velocity of the target frame into

components along the laser beam direction. This was done much in the same way as for

the vertex mirror velocity, except the direction of the beam and its intersection point has

changed. Now, the laser beam is pointing from the fold mirror towards the target blade,

and the intersection point is that of the laser beam with the target frame. The process for

62

finding the direction cosines, between the laser beam path from the fold mirror and the

target blade, is discussed.

It was first necessary to have both the fold mirror intersection point and target

blade intersection point in the coordinates of the target frame. This has already been

done for the target frame intersection point. Equation 4-30 shows the transformation for

the fold mirror.

GTiFziFyiFx

TTB

TBR

TRG

iFziFyiFx

=

−−− 111 [4-30]

The beam path length between the fold mirror intersection point and the target frame

intersection point was also needed to find the direction cosines. The length between the

fold mirror and blade frame, BFbeamlength − , and the length between the blade frame and

target frame, TBbeamlength − , have been previously calculated. To obtain the length

between the fold mirror and target frame, the two were simply added, as show in

Equation 4-31.

TBBFTF beamlengthbeamlengthbeamlength −−− += [4-31]

The x, y, and z direction cosines between the laser beam and the target frame are found

using the following set of equations:

−=−

−

TF

TTFT beamlength

iFxiTx1cosγ

−=−

−

TF

TTFT beamlength

iFyiTy1cosβ [4-32]

−=−

−

TF

TTFT beamlength

iFziTz1cosα ,

63

where γFV, βFV, and αFV are x, y, and z direction cosines between the laser beam and

target frame.

The velocity that the LDV sensed due to the rigid body motion was found by

multiplying the x, y, and z values for rigid body velocity values by their respective

direction cosines, as shown in Equation 4-33.

•

=

FT

FT

FT

Tz

y

x

LBz

y

x

VelrbmVelrbmVelrbm

VelrbmVelrbmVelrbm

αβγ

cos [4-33]

These three components of rigid body velocity in the laser beam direction were then

added together, as shown in Equation 4-34, to obtain the total rigid body velocity that the

LDV sensed due to misalignment effects. Note that this calculation was done for both

misaligned and perfectly aligned cases to ensure that the actual velocity induced from the

rigid body motion of the aligned case was actually zero.

LBLBLB zyxLB VelrbmVelrbmVelrbmVelrbmtot ++= [4-34]

The rigid body velocity due to the blade vibration cancelled out of the

mathematics using this procedure. The position of the misaligned laser spot on the target

frame, [iTx, iTy, iTz]misaligned, consisted of both position due to misalignment and position

due to the blade vibration. Likewise, the position of the aligned laser spot on the target

frame, [iTx, iTy, iTz]aligned, consisted of both position due to misalignment and position

due to the blade vibration. Since the influence of the vibration on position is equal for

64

both cases, when they are subtracted, as shown previously in Equation 4-19, the influence

from the vibration of the blade on the position error is cancelled out.

Total LDV Sensed Velocity

The total velocity that the LDV sensed was simply the sum of the total velocity, in

the laser beam direction, induced by the vertex mirror and the total velocity, in the laser

beam direction, induced by the rigid body motion of the misaligned structure. To ensure

that the sensed vibration of the ideally aligned system was actually zero, the total velocity

of the aligned case was calculated also. These two total LDV sensed velocities are

shown in the following two equations:

misalignLBmislignLBmisalignLDV VelrbmtotVelmirtotVeltot ,,, )2( +=

[4-35]

alignLBalignLBalignLDV VelrbmtotVelmirtotVeltot ,,, )2( +=

Calculating Velocity Error

The result of most importance, with respect to the sensed velocity, is the velocity

error between the misaligned and ideally aligned cases. This represents the portion of the

LDV output signal that would be due to the misalignment of equipment, and not the

vibration of the structure, which is the desired quantity. The total velocity error was

found by taking the difference between the total velocity in the path of the laser beam for

the misaligned case and the total velocity in the path of the laser beam for the ideally

aligned case. This is shown in Equation 4-36.

alignedLDVmisalignedLDVLDV VeltotVeltotrortotVelocityEr ,, −= [4-36]

65

4.8 Alternate Error Analysis Method

The thesis work presented here describes methods and studies related to reducing the

misalignment parameters to minimize the position and velocity errors to within

acceptable levels. There is another way to approach the problem of having position

errors. This method suggests that having some small, non-zero position variation of the

beam spot on the blade is acceptable as long as the position is known. If the rotational

position of the rotor can be related to the position error of the beam on the blade, then

that is sufficient to characterize the vibratory mode on the blade. This method, however,

presents another problem related to the speckle noise phenomena mentioned previously.

As the speckle pattern rotates further from a centralized measurement point, the induced

LDV velocity due to the changing of the speckle pattern is increased. See Lomenzo [1]

for a more detailed explanation of the speckle noise influence as related to position and

velocity errors.

66

5 The Computer Program

The computer program utilized the methodologies presented in the previous

chapter to calculate the position error of the beam spot on the blade and the velocity error

that the laser vibrometer will encounter due to misalignments. This chapter discusses the

programming methodology and what the different subroutines accomplished. The reason

for this is so that when future work is done, the programming methods can easily be

understood and tailored to the user’s needs. Additionally, this chapter also presents some

test cases run to prove that the computer simulation models the systems correctly. Note

that in this section, words in italics represent subroutines in the program.

5.1 Program Components

The computer program was broken down into many different subroutines that

accomplished different tasks. The main routine, main, initiated the step size of the

program, called other subroutines, and eventually wrote the position and velocity errors

to output files. The subroutines called from the main routine were as follows:

main_mis – this subroutine was the file that executed the misaligned case of the program.

Within this subroutine were other subroutines: 1) misalign_dim, 2) frames, 3) beampath,

4) anglesmis, and 5) Velmirmis. The misalign_dim subroutine was the file that the user

modified to change the amount of alignment error that each piece of equipment had.

Also, this subroutine initiated the design parameters of the program, such as the distances

between the pieces of equipment and the angle of the fold mirror. All of the translational

inputs were in metric units (meters) and the rotational inputs were in units of radians.

The frames subroutine input all of the design and alignment error parameters into a

function that generated the coordinate frame translational and rotational transformations

67

for each piece of equipment. This routine located the position of each piece of

equipment. The beampath subroutine calculated the path of the laser beam throughout

the system at any time increment. Additionally, the position of the beam spot on the

target blade is calculated in this subroutine. Both the coordinate frame transformations

and beam path calculations were done as shown the previous chapter. The anglesmis

subroutine calculates two different things. The first thing was the angles that the laser

beam makes with the target blade. These angles were used to resolve the rigid body

velocity of the beam on the target blade into the direction of the laser beam path. The

second thing that the subroutine calculated was the angles that the laser beam makes with

the vertex mirror. This allowed the velocity induced by the movement of the laser beam

on the vertex mirror to be resolved into the laser beam direction. The Velmirmis

subroutine calculated the velocity component that the LDV would sense due to the

motion of the beam spot on the vertex mirror. The subroutine incorporated the angles

found in the anglesmis subroutine.

outputmetric/outputenglish – these two subroutines were used to initiate output text files

that contained the system specifications and misalignment parameters. The two

subroutines were virtually identical, with the exception of one outputting the data in

English units, and one in Metric units. The same text files created here were also used

later in the programming sequence to write the velocity and position errors.

main_ideal – this subroutine was used to execute the ideally aligned case of the laser

measurement system. It was very similar to the subroutine main_mis, except all of the

alignment errors were set to zero. All of the system design parameters were held the

same as for the misaligned case. The same subroutines were called and used just as in the

main_mis subroutine. The position of the beam spot on the target blade for the ideal

alignment case was found using the beampath subroutine. The subroutine used to find

the angles for the rigid body motion of the target blade and the vertex mirror was called

anglesideal. The subroutine used to find the velocity that the LDV sensed due to the

movement of the laser beam on the vertex mirror was called Velmirideal.

68

position – this subroutine used both the position of the beam spot of the misaligned case

and the aligned case and calculated the position error generated by the misalignments.

The position error values of particular interest were the minimum and maximum position

errors encountered during one full revolution of the rotating system. The values indicated

if the position error values were concentric about the desired target point. Furthermore,

the values indicated the severity of the misalignment of the measurement system.

rigidbodymotionmis – this subroutine calculated the velocity that the rigid body motion of

the misaligned beam spot on the target blade imparted in the direction of the laser beam.

The subroutine incorporated the angles between the laser beam and target blade, which

were found in the anglesmis subroutine.

rigidbodymotionalign – this subroutine calculated the velocity that the rigid body motion

of the aligned beam spot on the target blade imparted in the direction of the laser beam.

The subroutine incorporated the angles between the laser beam and target blade, which

were found in the anglesideal subroutine.

veldiff – this subroutine calculated the velocity difference between the ideally aligned

case and the misaligned case to obtain a velocity error. This subroutine also calculated

the maximum velocity error, since it is the maximum that the laser vibrometer would

sense during one full revolution of the rotating system. This value is significant because

it represents how the output of the laser would be affected due to the misalignments of

the different equipment.

plotmetric/plotenglish – these two subroutines plotted the results of the program. The

only difference between the two subroutines was that one plotted the results in metric

units, and the other in English units. Four different plots were generated for each

subroutine: the position of the beam spot on the target blade in global coordinates, the

velocity of the aligned and misaligned cases, the velocity error, and the intersection of the

69

beam spot on the blade as seen from the blade coordinate system. Note that all of these

plots were for one full revolution of the rotating system.

5.2 Running the Program

Running the computer simulation was a fairly simple task. Once the time

increment that gave the necessary precision was decided upon, the remaining items were

all modified within the misalign_dim subroutine. 1000 increment points within one full

revolution of the rotating system were used, which yielded one point every 0.0063

radians (0.36 deg). The accuracy of this step size was acceptable compared to the actual

magnitude of the position and velocity calculations. The simulation cases addressed

experimental efforts to be performed on the Pratt and Whitney JT15D turbofan engine.

This engine and test setup was located in the Virginia Tech turbomachinery and

propulsion laboratories. These experimental efforts will be discussed in subsequent

sections of this report. In order to mimic the setup of the experiment, the design

parameters were set to be the same as for the experimental setup. These design

parameters were held constant, for both the misaligned and aligned cases, while the

program executed. All of the translational and rotational misalignment parameters were

also entered in the misalign_dim subroutine.

Once the design and misalignment parameters were input as desired, the main

subroutine was executed to run the program. The program produced two output files as

well as the plots specified previously. The output files, outputmetric and outputenglish,

were text files that displayed the design parameters specified, the misalignment errors

specified, and the resulting maximum position error, minimum position error, and

maximum velocity error encountered during one full revolution of the rotating system.

70

5.3 Testing the Computer Simulation

In order to prove that the computer simulation works as it was intended to, a series

of six different tests were performed. They consisted of:

1. ideally aligned case,

2. 0.001m laser frame x-axis translational misalignment,

3. 0.001m laser frame y-axis translational misalignment,

4. 0.001 rad (0.057 deg) laser frame misalignment about the x-axis,

5. 0.001 rad (0.057 deg) laser frame misalignment about the y-axis, and

6. 0.001m laser frame x-axis translational misalignment and 0.001 rad (0.057

deg) laser frame misalignment about the x-axis.

For each of the above test cases, the position of the laser beam, on the vertex mirror and

target blade, for a single fixed angle of rotation could be found using geometry. Also,

using the geometrically calculated position errors, along with the rotation speed input to

the program, the velocity that the LDV sensed could be found. The reason for testing

both x-direction and y-direction misalignments were to show that symmetry occurs

between the different cases.

For each of the test cases, the computer simulation generated a number of different

outputs. First, there was a set of four plots. The first plot was of the path that the laser

beam traced out in global coordinates. The second plot was of the path of the laser beam

spot on the target blade in target blade coordinates. The third plot had three different

velocity curves on it. They were the rigid body velocity of the misaligned case, the

mirror velocity of the misaligned case, and the total velocity of the misaligned case.

71

Each of these velocities was in the path of the laser beam. The fourth, and final, plot

showed the total velocity of the misaligned case and the aligned case. The difference

between the two was the total velocity error due to misalignments. The computer

simulation also generated an output file that displayed all of the design conditions,

misalignment parameters, and velocity and position errors. Note that all of the test cases

were run with no blade vibrations input into the system. Appendix B shows the plots and

results of each individual X and Y-direction translational and rotational misalignment.

Perfectly Aligned Case

The perfectly aligned case had no misalignments introduced into any of the

measurement equipment. For this case, we would expect the position error and velocity

error that the LDV sensed to be zero. Additionally, the laser beam should intersect the

vertex mirror at its center of rotation. Geometric method used to determine what values

the program should be giving will now be discussed. For the ideally aligned case, Figure

5-1 is used for a reference diagram. The angle φv is the angle of the vertex mirror, dm is

the length of the base of the vertex mirror, dfv is the distance from the fold mirror to the

intersection point on the vertex, and dlv was the length from the laser to the intersection

point on the vertex mirror. The parameter dspot represented the distance, along the span

of the blade, where the beam spot landed. This value was used to compare the results of

the simulation to the geometric method. Thus, for the geometric calculation, the value of

dspot was solved for.

72

Figure 5-1 Ideally Aligned Case

The values for dlv, dfv, dm, and φv were known inputs into the system. The value

for dspot was found by applying the following Equation 5-1:

)2tan()tan(2

)2tan()tan(2

*2 vvvvspotdmdmdfvd φφφφ

+

−= , [5-1]

where dfv = 2.3125 m, dm = 0.0381 m, and φv = π/180 rad (1 deg). This yielded the result

dspot = 0.1615 m.

The computer simulation yielded the same result as the geometric representation.

Figure 1-2 shows the output from the program. The inputs were the same as for the

geometric representation, and no misalignments were entered into the system. The

program used a different method, other than geometry, to find the location of the beam

φφφφv

φφφφv

φφφφv

φφφφv

φφφφv Fold Mirror

Blade Row

φφφφv

73

spot. The path of the laser beam was traced throughout the system, as well as the

intersections it made with the different pieces of equipment. This was done using

parametric equations as shown in the previous chapter. The result was the same as for

the geometric simulation. They both showed the beam spot landing at a radius of 0.1615

meters. The computer simulation result can be seen in the bottom of Figure 5-2, where

the radius of the beam spot is given. Additionally, it should be noted that both methods

assumed a 0 radian (0 deg) angle of rotation for the system.

There are some additional expected outcomes from running the simulation. First,

there should be no position error since the system is perfectly aligned. Also, there should

be velocities due to rigid body motion or mirror velocity, since the beam spot should

never move on the equipment. These results are also shown in Figure 5-2. The plots

show no position error and not velocities occurred. Additionally, the maximum and

minimum position errors were both calculated to be zero, as shown in the text at the

bottom of the results.

74

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1

0.2Intersection of Beam on Blade in Global Coords

Global X-coordinates(m)

Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5

1Misalignment Velocity Errors in Direction of Beam

Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot

The M inimum Position E rror is 0 meters

The Maximum Position Error is 0 meters

The Maximum Veloc ity Error is 0 meters /sec

The radius of the beam spot is 0.1615 meters

-0.1615 -0.1615 -0.1615

-5

0

5

x 10-12Intersection of Beam on Blade in B lade Coords

X-Position(m) on Blade

Y-P

ositi

on(m

) on

Bla

de

100 200 300-1

-0.5

0

0.5

1Veloc ity of A ligned and M isaligned Cases

Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters

Vertex Mirror Angle = 0.017453 rad

Dis tance from Laser to Fold Mirror = 0.40526 m

Blade Angle = 0 rad

Ideal Distance from center to beamspot = 0.1615 m

RPM Of System = 8000

Blade V ibration Amplitude = 0 m

Blade V ibration Frequency = 4188.7902*t(rad/s)

Figure 5-2 Computer Simulation For Ideally Aligned Case

75

Translational X-direction Misalignment

The translational x-direction misalignment case introduced an offset of the laser

frame of .001 meters in the x direction. Figure 5-3 shows the geometric layout of a

translational error in the x-direction. The variable names are the same as for the ideally

aligned case, except the addition of the translation error, dex, was added. For a

translational error only, the angles of the laser beam with the equipment remains the same

as for the ideal case.

φvφv

φv

2φv

2φv

2φv

Figure 5-3 Translational Error in X-Direction

76

The general equation to find the location of the beam spot, dspot, is generally the

same as for the ideal case, but there are additional terms to account for the laser hitting a

spot on the vertex mirror other than the center of rotation. The equation is as follows:

)2tan()tan()tan(2

)2tan()tan()tan(2

*2 vvvvvvspot dexdmdexdmdfvdtxd φφφφφφ

−+

+−+= [5-2]

where all values are the same as for the ideal case, and dex = 0.001 meters. This yielded

a value of 0.1625 meters for the location of the beam spot along the span of the blade.

The next step was to verify this result with the computer simulation. For the

simulation, an x-direction misalignment of 0.001 meters was entered into the

misalign_dim subroutine. There are a few expected outcomes from running the computer

simulation. First, the beam is neither lined up with the center of the vertex mirror or

stays on the same spot on the target blade as the system rotates. This should cause

velocities to be induced due to both rigid body motion of the beam on the blade and

movement of the laser beam spot on the vertex mirror. As the system rotates, the beam

spot on the blade should trace out a path of some sort also.

All of the above mentioned characteristics were found in the results of the

simulation, as seen in Figure 5-4. Additionally, the radial position of the beam spot on

the blade was the same as for the geometric verification, 0.1625 meters. Also, note that

the minimum and maximum position errors are not identical; thus, a perfect circle was

not traced on the blade.

77

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.02

-0.01

0

0.01

0.02

Misalignment Velocity Errors in Direction of Beam

Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot

The M inimum Position E rror is 0.001 meters

The Maximum Position Error is 0.0010006 meters



-0.163 -0.162 -0.161 -0.16-1

-0.5

0

0.5

1x 10

-3Intersection of Beam on Blade in B lade Coords


Y-P

ositi

on(m

) on

Bla

de

100 200 300-1

-0.5

0

0.5


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





Figure 5-4 Computer Simulation Results of 0.001 meter X-direction Translational Error

78

Translational Y-direction Misalignment The y-direction translational misalignment also was tested with a value of 0.001

meters. Theoretically, it should produce the same results as the x-directional

misalignments, but just be 90 degrees out of phase. Figure 5-5 shows how the x and y

axis translations differently effect the position of the beam spot on the target blade. For

both cases, the angular rotation, ω, of the system started at 0 radians (0 deg). Notice how

as the system rotates the y-direction translational error lags the x-direction translational

error by 90 degrees. This was due to where the beam spot landed on the vertex mirror.

For the geometric verification of the position of the laser spot, when ω = 0 radians (0

deg), the value for the radius along the span of the blade was 0.1615 meters. The same

equation was used as for the ideal alignment case, but at the 0 radian rotation position the

laser will be translated out of the page by 0.001 meters. Refer to Figure 5-1 for this

representation. These results appeared to be correct given the location of the beam spot

on the vertex mirror and the vertex mirror’s orientation.

The computer simulation was executed with a translational misalignment of 0.001

meters. The output of this simulation is shown in Figure 5-6. The results show that the

position of the laser beam spot on the target blade at ω = 0 radians is 0.1615 meters, as

expected. Also, notice how the velocity that the misalignment caused is, in fact, 90

degrees out of phase from the x-direction misalignment results, shown in Figure 5-6.

Additionally, the minimum and maximum position errors due to the misalignments were

equal. This was expected since the effects of the x and y misalignments are equal, but

out of phase.

79

X – Direction Error

0°

90°

180°

270°

ωωωω

X

Y Y – Direction Error

0°

90°

180°

270°

ωωωω

X

Y

Figure 5-5 X and Y Translational Misalignments On Blade

80

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot





-0.163 -0.162 -0.161 -0.16-1

-0.5

0

0.5

1x 10



Y-P

ositi

on(m

) on

Bla

de

100 200 300-1

-0.5

0

0.5


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





Figure 5-6 Computer Simulation Results of 0.001 meter Y-direction Translational Error

81

Rotational X-direction Misalignment

The rotational x-direction test procedure was performed with a positive rotation in

the x-direction of 0.001 radians (0.057 deg). For a rotational misalignment, the laser

beam intersected the vertex mirror at the center of rotation, but was at an angle. For ω =

0 radians (0 deg), the geometry from a top view appeared much like that of the ideal

alignment case, as seen in Figure 5-1. The beam spot does land in the same point in the

x-direction. Due to the rotational misalignment in the x-direction, the beam spot landed

in the negative y-direction. If the system is viewed from the global y-z plane, and ω = 0

radians, it appeared as shown in Figure 5-7. γL is the rotational misalignment angle

around the x-axis of the laser frame. The following equation was used to geometrically

calculate what the y-value for the laser spot on the blade.

)tan()tan(2

)tan()tan(2

2 LVLVYdmdmdfvdspot γφγφ +

−= [5-3]

Solving this equation when ω = 0 radians, the y-position of the beam spot was found to

be 0.00463 meters.

82

φv

γγγγL

2γγγγL

Figure 5-7 Y-Z Plane View of Rotational Misalignment Around X-axis

The computer simulation was executed with a rotational misalignment of 0.001

radians (0.057 deg) about the x-axis. There were a number of expectations from the

results of the rotational misalignment. First, the position of the beam spot on the blade

should trace out an ellipse, centered on the ideal position of the beam spot. For a

rotational misalignment around the x-axis, it is expected that the position error is least at

ω= 0, π radians (0, 180 deg) and highest at ω = 2

3,2

ππ radians (90, 270 degrees). This

was due to where the laser beam intersected the vertex mirror, and the orientation of the

vertex mirror itself. It was also expected that there would be no induced velocity due to

the vertex mirror since the beam intersected with the center of the vertex mirror.

83

Figure 5-8 shows the output from the computer simulation with a 0.001 radian (0.057

deg) rotational misalignment about the x-axis.

The results of the computer simulation were as expected. The position of the

beam spot was 0.1615 meters in the x-direction and 0.00463 meters. Additionally, the

maximum position error was when ω = 2

3,2

ππ radians (90,270 deg) and the minimum

position error was when ω = 0, π radians (0,180 deg). Also, as expected, there was no

velocity error induced due to the vertex mirror.

84

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1

0.2Inters ec tion of B eam on B lade in G lobal Coords

G lobal X-c oordinates (m )

Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.1

-0.05

0

0.05

0.1

M is alignm ent V eloc ity E rrors in Direc t ion of B eam

P os it ion(deg)

Vel

ocity

Err

or(m

/s)

V elm irV elrbmV eltot

The M inim um P os it ion E rror is 0.0046276 m eters

The M ax im um P os ition E rror is 0.0046306 m eters

The M ax im um V eloc ity E rror is -0.1353 m eters /sec

The radius of the beam spot is 0.1615 m eters

-0.17 -0.165 -0.16 -0.155

-4

-2

0

2

4

x 10-3Inters ec tion of B eam on B lade in B lade Coords

X-P os it ion(m ) on B lade

Y-P

ositi

on(m

) on

Bla

de

100 200 300

-0.1

-0.05

0

0.05

0.1

V eloc ity of A ligned and M is aligned Cases

P os it ion(deg)

Vel

ocity

(m/s

)

V elm is V elideal

Des ign P aram eters

V ertex M irror A ngle = 0.017453 rad

Dis tanc e from Las er to Fold M irror = 0.40526 m

B lade A ngle = 0 rad

Ideal Dis tance from c enter to beam s pot = 0.1615 m

RP M Of S ys tem = 8000

B lade V ibrat ion A m plitude = 0 m

B lade V ibrat ion F requenc y = 4188.7902*t(rad/s )

Figure 5-8 Computer Simulation Results of 0.001 radian (0.057 deg) Rotational Error About X-axis

Rotational Y-direction Misalignment

The misalignment about the y-axis was tested using a value of 0.001 radians

(0.057 deg). The y-axis rotational misalignment should behave very similar to the x-axis

misalignment case, but be 90 degrees out of phase. First, the geometric analysis had to

85

be done to verify the position of the laser beam spot on the target blade. Figure 5-9

shows the geometry of the laser beam with the equipment when ω = 0 radians. The laser

beam comes toward the vertex frame with an angle of βL about the y-axis. This angle

was additive to that of the vertex mirror. Therefore, the angle of the laser, both incoming

and leaving the vertex mirror was 2φV +βL. This was used to find the intersection point

of the laser beam with the blade, as shown in Equation 5-4.

)2tan()tan(2

)2tan()tan(2

2 LVVm

LvVm

FRspotdddd βφφβφφ ++

+

−= [ 5-4]

This equation yielded a result of 0.1661 meters in the x-direction of the target blade.

Since the laser beam is parallel to the x-axis when ω = 0 radians, there is no displacement

in the y-directions. These results seemed logical since by adding misalignment we are

adding to the angle of reflection that the vertex mirror causes.

86

φv

φv+βL

φv+βL

φv+βL

φv+βL

φv+βL

βL

Figure 5-9 X-Z Plane View of Rotational Misalignment Around Y-axis

The computer simulation was executed with a rotational misalignment of 0.001

radians (0.057 deg) about the y-axis of the laser frame. There were a number of

expectations from this test case. First, it was expected that the minimum and maximum

position errors were the same as for the x-axis rotational misalignment, but occur 90

degrees out of phase from the x-axis test case. Second, as with the x-axis misalignment

case, there should be no velocity in the laser beam direction due to the motion of the

beam spot on the vertex mirror. Additionally, the velocity and velocity errors should be

identical to that of the x-axis misalignment, but be 90 degrees out of phase.

87

The computer simulation produces results that were as expected. Figure 5-10

shows the output that the simulation generated. The x-position of the laser spot

on the blade was the same as for the geometric verification, 0.1661 meters. Also,

the velocities and velocity error were equal to the x-axis rotational misalignment

case, but 90 degrees out of phase.

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1

0.2Inters ec tion of B eam on B lade in G lobal Coords

G lobal X-coordinates (m )

Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.1

-0.05

0

0.05

0.1

M isalignm ent V eloc ity E rrors in D irec tion of B eam

P os it ion(deg)

Vel

ocity

Err

or(m

/s)

V e lm irV elrbmV eltot

The M inim um P os it ion E rror is 0.0046276 m eters

The M ax im um P os it ion E rror is 0.0046306 m eters

The M ax im um V eloc ity E rror is 0.1353 m eters /s ec

The radius of the beam spot is 0.16613 m eters

-0.17 -0.165 -0.16 -0.155

-4

-2

0

2

4

x 10-3Inters ec tion of B eam on B lade in B lade Coords

X-P os it ion(m ) on B lade

Y-P

ositi

on(m

) on

Bla

de

100 200 300

-0.1

-0.05

0

0.05

0.1

V eloc ity of A ligned and M is aligned Cases

P os it ion(deg)

Vel

ocity

(m/s

)

V e lm is V elideal

Des ign P aram eters

V ertex M irror A ngle = 0.017453 rad

Dis tance from Laser to Fold M irror = 0.40526 m

B lade A ngle = 0 rad

Ideal Dis tanc e from center to beam spot = 0.1615 m

RP M Of S ys tem = 8000

B lade V ibrat ion A m plitude = 0 m

B lade V ibrat ion Frequency = 4188.7902*t(rad/s )

Figure 5-10 Computer Simulation Results of .001 radian(.057 deg) Rotational Error About Y-axis

88

Translational and Rotational X-direction Misalignment

For this test case, the translational misalignment was set to 0.001 meters in the x-

direction, and the rotational misalignment was set to 0.001 radians (0.057 deg) about the

x-axis. This test case was performed to prove that the combination of translational and

rotational misalignment errors was the superposition of the two results, as previously

stated. The geometric verification of the two misalignments was calculated in previous

sections. For the translational misalignment of 0.001 meters, the result was x-direction

position error was 0.001 meters and the y-axis error was 0 meters. For the rotational

misalignment about the x-axis, the x-direction error was 0 meters, and the y-direction

error was -0.00463 meters. Knowing that the intended position of the beam spot was at x

= 0.1615 meters and y = 0 meters, the position of the beam spot on the blade, using

superposition, was found to be x = 0.1625 m and y = -0.00463 meters.

The computer simulation was executed with an x-translational misalignment of

0.001 meters, and a rotational misalignment about the x-axis of 0.001 radians (0.057

deg). The results of the test case are shown in Figure 5-11. As seen in the text in the

lower right of the output, the x and y position of the laser spot, for ω = 0 radians (0 deg),

matched the geometric verification that used the concept of superposition. Additionally,

it was noticed that the velocity output also followed the concept of superposition.

89

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot



The Maximum Veloc ity Error is -0.1353 meters /sec

The radius of the beam spot in x-dir is 0.1625 meters

The radius of the beam spot in y-dir is -0.0046276 meters

-0.17 -0.165 -0.16 -0.155

-4

-2

0

2

4



Y-P

ositi

on(m

) on

Bla

de

100 200 300

-0.1

-0.05

0

0.05

0.1

Veloc ity of A ligned and M isaligned Cases

Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





Figure 5-11 X-direction Translational and Rotational Misalignment Case

90

5.4 Other Findings From The Computer Simulation

There were a number of outcomes noticed as a result of performing the test cases

with the computer simulation. The most important of these results was that a pure

translational error does not cause any velocity error. The velocity induced by the

movement of the laser beam on the vertex mirror was equal and opposite to the velocity

that was induced due to the rigid body motion of the laser beam on the target blade. This

one was evident by looking at Figure 5-4 and Figure 5-6. The misalignment velocities

due to the vertex mirror and target blade cancel out in both cases to give a total

misalignment velocity of zero. This means that the position of the beam on the blade

crossed with the rotation of the system, must be equal and opposite to the position of the

beam spot on the blade crossed with the rotation of the system. This phenomena was also

reported in the dissertation written by Lomenzo[1].

91

6 Input Variable Response Analysis

Now that the MATLAB computer simulation has been proven to correctly model

the self-tracker system, it is desired to mathematically represent the effects of the input

variables on the position error and velocity error. This was done using a computer

optimization program called iSIGHT. The iSIGHT program has the capability to run the

MATLAB simulation and systematically change the input variables while monitoring the

effects they have on the outputs. The input variables can be changed one at a time or

combinations of input variables could be changed simultaneously. By performing a two-

point sensitivity analysis on each input variable, the parametric sensitivities of the

outputs, position error and velocity error, were found. For a two-point sensitivity

analysis, iSIGHT executed the MATLAB program two times for each input misalignment

variable. For these two program runs of MATLAB, all the input variables were set to

zero except one. This variable took on equipment misalignment values that were

specified when setting up the iSIGHT parametric routine. From this, a sensitivity of the

outputs to each input parameter was found. All of the effects were notated as shown in

the following equation:

ValueInput

Output =∆

∆, [6-1]

where Output∆ is the simulation output (velocity error or position error), Input∆ is the

change in misalignment errors, and Value is the output variable sensitivity that iSIGHT

generated. An assumed maximum allowable velocity error and position error could then

be substituted into the equation and the input to achieve that error was solved for.

92

The downfall of a parametric sensitivity study is that since only two data point are

used for each input variable, the effects of the input on the output are assumed to be

linear. This, however, may not be the case. In order to find out whether a given input

affects the output linearly, a parametric Monte Carlo analysis was performed. A Monte

Carlo method utilizes randomly selected inputs that are selected by iSIGHT, based on

user specified bounds for each input variable. The bounds are selected based on the type

of assumed distribution and the standard deviation that yields the desired bounds under

that particular distribution. For this simulation, the bounds were assumed to be ± 0.005

meters for translational misalignments and ± 0.005 radians (0.286 deg) for rotational

misalignments. These are considered the minimum practical precision that can be

achieved by positioning the equipment. The distribution used was a uniform distribution.

Since this particular study was a parametric one, the Monte Carlo method was performed

on each variable, one at a time. For each input misalignment, all other input

misalignment variables were assumed zero while iSIGHT chose 20 random values of the

alignment variable, within the specified bounds, and the outputs were saved. Thus, a

parametric relation between each input and the respective outputs was found. This

portion of the analysis gave a general idea of how the each input affected the outputs over

a range of input values. Whether or not each variable has a linear effect on the outputs,

or if it has any effect at all, was important to know before moving to the next analysis

method.

Since in a real system it is not realistic to have only one isolated misalignment, it

was desired to study velocity and position errors caused by combinations of misalignment

effects. By changing multiple input variables at the same time, the interaction effects that

93

different variables have on velocity error and position error was studied. iSIGHT also

had the capability to run interaction tests of varying complexities. A full factorial test

would test every possible interaction and analyze the effects. Every combination

between every variable would be tested using a full factorial scheme. This would be very

time consuming since we have a large number of different input variables. The Monte

Carlo method again was applicable to study how combinations of input variables affected

the outputs. Once again all of the inputs were bounded the same way as for the

parametric Monte Carlo method. Bounding the variables define what is referred to as a

design space. By setting the design space, it restricted the values that the input variables

could take on. With this method, the input variable that the each output are most

sensitive to were found. Using the specified design space and variable responses, design

criteria for achieving desired levels of position error and velocity error were found.

A total of seven different studies were done to monitor velocity and position

errors caused by the misalignment effects of the self-tracker system. Table 6-1 shows a

list of these. The remainder of this chapter discusses the different methods and the results

obtained.

94

Table 6-1 Parametric and Interaction Studies Completed

Parametric Studies Interaction Studies

Sensitivity Analysis

• Translational Errors

• Rotational Errors

Monte Carlo Analysis



• Combined Translational & Rotational Errors

Monte Carlo Analysis



The Monte Carlo simulations were used as a means to automate the simultaneous

random sampling of the misalignment parameters. For these studies, the misalignment

errors practically relate to the adjustment resolution of the positioning equipment for the

different pieces of self-tracker hardware. No knowledge of the distribution was known

for these adjustment resolutions, nor was it necessary. It was desired to randomly test

occurrences of misalignments within the specified bounds for each parameter. Also, it

was desired to have equal likelihood of selecting any value within the bounds. It was for

this reason that the uniform distribution was selected. By using the uniform distribution,

only the effects of the interactions were explored, which was desired. Typically, errors

bounds are thought of in terms of numbers of standard deviations. For these particular

studies, the bounds were used to simply give the range of misalignment error

possibilities.

95

6.1 Parametric Studies

A parametric study independently varies the input variables and records the output

for each respective input variable change. The iSIGHT program has the capability to do

this if set up properly. iSIGHT works by first specifying the executable by which the

self-tracker simulation code responded. Additionally, input and output files were

specified. Previously, in Chapter 5 where the MATLAB simulation was explained, these

files were identified as misalign_dim and outputmetric. Figure 6-1 shows a typical

iSIGHT program window at this point. The specified program executable filename was

Matlab.exe. The misalign_dim.m input file contained all of the input variables that

iSIGHT needed to manipulate the MATLAB alignment program. The output file,

outputmetric.txt, contained the two main outputs of the simulation, position error and

velocity error.

96

Figure 6-1 iSIGHT Program Window for Design Integration

The next step in setting up iSIGHT for a parametric study was to parse the input

file. In other words, setup iSIGHT to recognize the input variables that it needed to

recognize and modify. The input file, misalign_dim.m, was parsed in two different

manners to accomplish three different parametric analyses. The two methods were for

1. translational misalignments only, and

2. rotational misalignments only.

Figure 6-2 shows an example of a parsed input file for translational misalignments. Once

the input file was parsed, the output file, outputmetric.txt, needed to be parsed. This was

done to configure iSIGHT to recognize the parameters that respond to varying the input

97

variables, or responses. Figure 6-3 shows an example of the iSIGHT program window

for a typical parsed output file. As can be seen, the highlighted output parameters of

velocity error and position error are recognized as the output parameters.

Figure 6-2 iSIGHT Program Window For Parsed Input File

98

Figure 6-3 iSIGHT Program Window For Parsed Output File

Once iSIGHT was programmed to vary the misalignment parameters by parsing

the files, the Design of Experiments (DOE) needed to be setup. The iSIGHT program

varied the input parameters according to the specifications of the DOE section of the

program. Depending on the desired type of analysis, there were many different options to

choose from to setup this part of the experiment. For this case, since a parametric

analysis was desired, the DOE specification was set to “Parameter Analysis.” This

triggered iSIGHT to vary one parameter at a time. Next, iSIGHT had to be told how

much to vary each input variable. Figure 6-4 shows the program screen where the upper

and lower bounds for the misalignment errors were set. For all translation misalignment

errors, the bounds were set as ± 0.001 meters about the zero misalignment case.

99

Figure 6-4 iSIGHT Program Window for Defining Parametric Sensitivity Bounds

The last step before running the iSIGHT studies was to choose the responses, or

outputs, for the iSIGHT program to save. Since both of the outputs chosen in the parsed

output file were of interest, velocity error and position error are both selected as

responses.

Finally, the iSIGHT program used to parametrically study the translational

misalignments of the self-tracker was executed. It should be noted that the filename of

the main MATLAB simulation file should be set as “startup.m.” This is because the

MATLAB program automatically executes any file called “startup.m” when it opens.

Also, the last command line of the main routine should read “exit.” This tells MATLAB

to fully close down after executing the simulation program. By doing these two things,

100

iSIGHT was able to open MATLAB and automatically execute the simulation for a given

misalignment. Then the MATLAB program would close on its own, whereupon iSIGHT

would alter a different input misalignment variable and run the MATLAB program again.

The same methods used to setup iSIGHT for the parametric study was done for

the rotational, and rotational and translational cases. The only difference was that the

bounds for rotational misalignments were specified as ± 0.001 radians (0.057 deg).

Parametric Sensitivity Analyses

The main goal of running the different studies was to find the maximum allowable

amount of translational and rotational misalignment for each piece of equipment. Since

the misalignments can produce both position errors and velocity errors, maximum

allowable misalignments were found using both. The one that resulted in the smaller

(numerically) allowable misalignment was then used as the specified design constraint.

For the approach based on the position error, it was decided that the maximum

allowed position error of the beam spot on the blade was one half the size of the laser

beam spot. Using the focal length of the LDV lens and the distance to the beam spot on

the blade along the path of the laser beam, the size of the beam spot was found to be

approximately 1.0 mm. Therefore, the acceptable amount of position error was set to be

0.50 mm (0.00005m). Now, for each translational misalignment, the parametric

sensitivity was used to find a value for the maximum allowable amount of misalignment.

This was found by plugging in the maximum allowed position error of the beam spot for

the position error term. An example of this process is shown in Equation 6-2.

101

iSightpos SPVar *max=∆ [6-2]

is the equation to find allowable position misalignments using position error as the basis,

where Var∆ is the maximum amount of misalignment in meters or radians , maxposP is the

maximum allowable position error of the beam spot on the blade (0.00050 m), and

iSightS is the variable sensitivity found using iSIGHT. (meters or radians

misalignment/meters position error)

The maximum allowable misalignment based on velocity error is found in much

the same way as for position error. The only difference is that instead of assuming a

maximum amount of allowed position error, a maximum allowable amount of velocity

error is assumed. The value was found by first obtaining typical mid-span velocity

information for a turbojet compressor fan. Since future testing will likely be done in

Virginia Tech laboratories on a Pratt and Whitney JT15D or a Garrett F-109 turbofan,

velocity information of an engine of the same size was desired. This information was

obtained through correspondence with Jeff Lentz[14] of Honeywell Corporation. Typical

velocities at the mid-span of a blade were estimated as approximately 1.35

meters/second. It was then assumed that an allowable amount of error was 5 percent of

this velocity, or 0.0675 meters/second. Equation 6-3 was then used to find the maximum

amount of allowable misalignment due to velocity errors, where maxvelV is the maximum

allowable velocity error in meters/second, and iSightS is the parametric sensitivity of

velocity error. (meters misalignment/(meters/second velocity error))

iSightvel SVVar *max=∆ [6-3]

102

The method that generated the more numerically sensitive alignment criteria was used at

the design specification for that particular piece of equipment.

Translational Parametric Study Results

A number of interesting results were generated by this study. Figure 6-5 shows a

Pareto plot of the position error effects of the translational misalignment errors. Figure

6-6 shows a Pareto plot of the velocity error effects generated by the translational

misalignment errors. A Pareto plot shows how much each input variable affects a

specified output parameter. Thus, the effects are also called influence coefficients, where

each response is represented as a partial derivative of the cumulative output responses.

Variables with a longer bar have a larger effect on the output. The coefficients are

normalized such that the sum of the coefficients always adds up to 100 percent. A Pareto

plot is a good quick way to see which parameters have the most influence on the outputs.

The different colors of the bars denote if the coefficient is positive or negative. For this

study only the magnitude of each error is considered to be important, so the color of the

bar is negligible.

The main effect noted for this case is that translational misalignment errors have

no influence on the velocity errors, but on position errors only. Due to this occurrence,

the positioning of the equipment, in the translational x,y, and z directions, for the self-

tracker system, should be based on the position error alone. Minimizing the position

error of the laser spot on the target blade was important because the expected velocity of

a blade due to vibrations differs based on where on the blade the measurement is taken.

Also, it should be noted that all of the alignment parameters have an equal effect, or no

103

effect, on the position error. Looking at Figure 6-5, all of the input variables that

generate a response have the same bar length. Thus, the magnitude of each response is

equal.

eXL

eYR

eXR

eYL

eXB

eYT

eYB

eXT

eYF

eYV

eYV

Tran

slatio

nal M

isalig

nmen

t Err

or

% Total Effect on Position Error5 10

Pareto Plot for Position Error

eXV

eXL

eYR

eXR

eYL

eXB

eYT

eYB

eXT

eYF

eYV

eYV

Tran

slatio

nal M

isalig

nmen

t Err

or



eXL

eYR

eXR

eYL

eXB

eYT

eYB

eXT

eYF

eYV

eYV

Tran

slatio

nal M

isalig

nmen

t Err

or



eXV

Figure 6-5 Pareto Plot For Position Error Response To Parametric Translational Misalignments

104

eXL

eYR

eXR

eYL

eXB

eYT

eYB

eXT

eYF

eYV

eYV

Tran

slatio

nal M

isalig

nmen

t Err

or

% Total Effect on Velocity Error0

Pareto Plot for Velocity Error

eXV

eXL

eYR

eXR

eYL

eXB

eYT

eYB

eXT

eYF

eYV

eYV

Tran

slatio

nal M

isalig

nmen

t Err

or



eXV

Figure 6-6 Pareto Plot for Velocity Error Response To Parametric Translational Misalignments

There are a few main findings from these simulation results. First, a previously

known result discovered by Lomenzo [1] was that, for translation misalignments of the

laser, no velocity errors are generated. This was due to the fact that the rigid body

motion of the blade was equal and opposite to the velocity induced by the vertex mirror.

Figure 6-7 is a plot of the rigid body velocity generated by the blade and the rigid body

velocity generated by the vertex mirror. Both velocities are in the laser beam direction.

Notice how the curves are equal and opposite to cancel have a resultant velocity of zero.

105

50 100 150 200 250 300 350

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbm

Figure 6-7 Blade and Mirror Velocities for 0.001m X-Translation of Laser Frame

The simulation was extended to include all other pieces of equipment of

the self-tracker system. Similar results to the laser misalignment were found

when translational errors were input for the rotor, vertex mirror, fold mirror, and

target blade. No velocity errors were generated due to translational

misalignments of the equipment, but position errors were. It makes sense that that

moving the laser and rotor frames would have similar effects since the same

overall operation is being performed to the system. It is also somewhat easy to

see how moving the vertex mirror in its own x-y plane induced no position error

or velocity error by looking at Figure 6-8. By moving the vertex mirror in either

the x or y-directions, the mirror simply moved parallel to the reflection plane.

106

Thus, there was no difference in where the beam spot landed on the vertex mirror

or the target blade, when compared to the ideal case.

Figure 6-8 Translational Vertex Mirror Misalignment

Table 6-2 also shows that both blade and target frames induce no velocity errors

when translated in the x and y-directions. Since the relative position of the beam spot on

the blade, in global coordinates, never changes if the blade moves in the x or y- directions

of the blade, the beam spot lies in the same spot that it did in the ideal alignment case.

There are; however, position errors associated with the misalignments of the target blade.

Although having a translational misalignment of a blade is not a typical misalignment

that can be controlled, a blade that has moved can still generate a position error along the

span or chord of the blade.

107

Table 6-2 Parametric Translational Misalignment Effects

Variable Misalignment(m)Velocity

Error(m/s)Position Error(m)

Velocity Sensitivity

Position Sensitivity

Allowable Vel Alignment Error

(±m)

Allowable Pos Alignment Error

(±m)Xlaser -0.001 0 -0.001001 Inf 0.999001 Inf 0.000500

0.001 0 0.001001Ylaser -0.001 0 0.001001 Inf -0.999001 Inf 0.000500

0.001 0 -0.001001Xrotor -0.001 0 0.001001 Inf -0.999001 Inf 0.000500

0.001 0 -0.001001Yrotor -0.001 0 -0.001001 Inf 0.999001 Inf 0.000500

0.001 0 0.001001Xvertex -0.001 0 0 Inf Inf Inf Inf

0.001 0 0Yvertex -0.001 0 0 Inf Inf Inf Inf

0.001 0 0Xfold -0.001 0 0 Inf Inf Inf Inf

0.001 0 0Yfold -0.001 0 0 Inf Inf Inf Inf

0.001 0 0Xblade -0.001 0 0.001 Inf -1 Inf 0.000500

0.001 0 -0.001Yblade -0.001 0 0.001 Inf -1 Inf 0.000500

0.001 0 -0.001Xtarget -0.001 0 0.001 Inf -1 Inf 0.000500

0.001 0 -0.001Ytarget -0.001 0 0.001 Inf -1 Inf 0.000500

0.001 0 -0.001

Since there are no velocity errors generated by translational misalignments of the

self-tracker measurement system components, the translational design constraints for

each piece of equipment, as a result of the parametric analysis, were based on position

error only. Table 6-2 shows the maximum allowable misalignment for each piece of

equipment based on the methods described earlier that involved selecting the allowable

position error. Although some of the equipment has no associated velocity or position

errors, the results are correct only if the general arrangement of the apparatus remains

unchanged.

The calculated maximum allowable misalignments for the equipment are very

low. All of them are on the order of 500µm, or 0.5mm. This means that in order to

achieve 0.5mm of position error resolution at the measurement point, each piece of

equipment must be aligned within 0.5mm of its intended design position. This assumed

108

that the misalignment associated with that piece of equipment was the only misalignment

parameter. This did not consider any interactions of misalignment errors caused by

having multiple translations simultaneously in different frames. Therefore, the

parametric study produced results that represented the least constraining criteria for

aligning each piece of equipment. Even for this best-case scenario that does not include

the effects of misalignment interactions, the equipment has to be translationally aligned

with extreme precision.

Rotational Parametric Study Results

A number of different results came from this study. The first outcome to note was

the similarity to the outcomes that Lomenzo [1] found with rotational misalignments.

Again, the previous research analyzed rotational misalignments of the laser/fold mirror

combination to the rest of system. Lomenzo found that rotational misalignments

produced both velocity errors and position errors. The reason for this is due to the fact

that a pure rotational error of the laser still intersects the center of rotation of the vertex

mirror. Therefore, no velocity error is induced due to the laser beam moving on the

vertex mirror. The resultant velocity error was due to the movement of the laser beam on

the blade. The previous chapter shows a more results of a pure rotational misalignment

of the laser frame. (See Figure 5-8) The results are consistent with that of the previous

study by Lomenzo.

Table 6-3 shows the results of the parametric analysis of rotational misalignments

for each piece of equipment for the self-tracker system.

109

Table 6-3 Results of Parametric Rotational Misalignment Study

Variable Misalignment(rad)Velocity

Error(m/s)Position Error(m)

Velocity Sensitivity (rad/m/s)

Position Sensitivity (rad/m)

Allowable Vel Alignment Error (±rad)

Allowable Pos Alignment Error (±rad)

thetaXL -0.001 -0.135298 -0.004631 0.007391 0.215964 0.000493 0.0001080.001 0.135298 0.004630

thetaYL -0.001 0.135298 -0.004631 -0.007391 0.215957 0.000493 0.0001080.001 -0.135298 0.004631

thetaZL -0.001 0.000000 0.000000 Inf Inf Inf Inf0.001 0.000000 0.000000

thetaXR -0.001 0.270523 -0.009252 -0.003697 0.108081 0.000246 0.0000540.001 -0.270523 0.009252

thetaYR -0.001 0.000000 -0.009268 Inf 0.107916 Inf 0.0000540.001 0.000000 0.009265

thetaZR -0.001 0.000000 0.000000 Inf Inf Inf Inf0.001 0.000000 0.000000

thetaXV -0.001 0.270554 -0.009254 -0.003696 0.108064 0.000246 0.0000540.001 -0.270554 0.009254

thetaYV -0.001 0.000000 -0.009261 Inf 0.107982 Inf 0.0000540.001 0.000000 0.009260

thetaZV -0.001 0.004722 0.000162 -0.211782 -6.191950 0.014121 0.0030960.001 -0.004722 -0.000162

thetaXF -0.001 -0.135225 -0.004637 0.007395 0.215663 0.000493 0.0001080.001 0.135225 0.004637

thetaYF -0.001 -0.135225 -0.004637 0.007395 0.215663 0.000493 0.0001080.001 0.135225 0.004637

thetaZF -0.001 0.000000 0.000000 Inf Inf Inf Inf0.001 0.000000 0.000000

thetaXB -0.001 0.000000 0.000000 Inf Inf Inf Inf0.001 0.000000 0.000000

thetaYB -0.001 0.000000 -0.000006 Inf 177.304965 Inf 0.0886520.001 0.000000 0.000006

thetaZB -0.001 -0.004722 0.000162 0.211782 -6.191950 0.014121 0.0030960.001 0.004722 -0.000162

thetaXT -0.001 0.000000 0.000000 Inf Inf Inf Inf0.001 0.000000 0.000000

thetaYT -0.001 0.000000 -0.000006 Inf 177.304965 Inf 0.0886520.001 0.000000 0.000006

thetaZT -0.001 -0.004722 0.000162 0.211782 -6.191950 0.014121 0.0030960.001 0.004722 -0.000162

Figure 6-9 is a Pareto plot of the position error response to rotational misalignments. It

was quickly noticed that, based on position error, the alignment of the rotor and vertex

mirror were the most critical parameters, followed by the alignment of the fold mirror

and laser. Figure 6-10 is a Pareto plot of the velocity error response when rotational

misalignments were parametrically changed. The most critical parameters, based on the

velocity error analysis, were the rotational alignment of the vertex mirror and rotor about

their respective x-axes. The next highest responses were from the x-direction and y-

direction rotations about the laser and fold mirror.

110

thetaYR

thetaYV

thetaXV

thetaXR

thetaYF

thetaXF

thetaXL

thetaYL

thetaZV

thetaZT

thetaYT

Rot

atio

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isal

ignm

ent E

rror

% Total Effect on Position Error5


thetaZB

thetaYB

thetaZL

thetaZF

thetaZR

thetaXT

thetaXB

10 15

thetaYR

thetaYV

thetaXV

thetaXR

thetaYF

thetaXF

thetaXL

thetaYL

thetaZV

thetaZT

thetaYT

Rot

atio

nal M

isal

ignm

ent E

rror

% Total Effect on Position Error5


thetaZB

thetaYB

thetaZL

thetaZF

thetaZR

thetaXT

thetaXB

10 15

Figure 6-9 Pareto Plot For Position Error Response to Rotational Misalignments

thetaYR

thetaYV

thetaXV

thetaXR

thetaYF

thetaXF

thetaXL

thetaYL

thetaZV

thetaZT

thetaYT

Rot

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isalig

nmen

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or



thetaZB

thetaYB

thetaZL

thetaZF

thetaZR

thetaXT

thetaXB

10 15

thetaYR

thetaYV

thetaXV

thetaXR

thetaYF

thetaXF

thetaXL

thetaYL

thetaZV

thetaZT

thetaYT

Rot

atio

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isalig

nmen

t Err

or



thetaZB

thetaYB

thetaZL

thetaZF

thetaZR

thetaXT

thetaXB

10 15

Figure 6-10 Pareto Plot For Velocity Error Response to Rotational Misalignments

111

There are a number of results that need to be explained. Since the case of

rotational misalignments of the laser frame has been previously studied, and verified in

this study, it will not be discussed again. However, all other rotational misalignment

cases need to be explained.

First, it should be noted that all misalignments were entered into each respective

reference frame. Therefore, if the frame rotates, then the misalignment does also. For

example, for an x-direction rotational misalignment of the vertex mirror, the x-axis of the

vertex frame changes with each incremental rotation of the system. If a misalignment

was introduced into the x-axis of the vertex mirror, then the misalignment remains in

relation to the x-axis of the vertex mirror frame. The following sections will describe the

results of the simulation for each piece of equipment and make certain the results are

mathematically logical.

Rotor – Misaligning the rotor frame generated a position error and a velocity error for x-

rotations, but only a position error for y-rotations. The relative magnitude of the position

error was twice that of misaligning the laser frame. The outcome seemed correct

considering how the laser misalignment case was different than the rotor misalignment

case. Figure 6-11 shows general schematics of a rotational misalignment of the laser(a)

and rotational misalignment of the rotor (b). Notice how for the rotor misalignment, the

fold mirror is at an angle with respect to the rotor frame. For the case of the laser

misalignment, the fold mirror is still aligned to the rotor frame. Therefore, case (b)

would have more position error than case (a). By looking at Figure 6-12 shows the two

cases of an x-direction rotational misalignment of the rotor (a) and a y-direction

112

rotational misalignment of the rotor(b). For the x-direction misalignment, there is a

component of velocity that the laser sensed. This was due to the fact that the velocity

vector has a component that projected in the direction of the laser beam. For the y-

direction, there is no rigid body motion of the beam spot moving in the x-direction of the

blade; therefore, the rigid body velocity at that point was zero. This scenario would

change; however, if there was a blade velocity along the span of the blade. Then, the

LDV would sense a velocity due to the movement of the blade in the laser direction. For

vibrations along the chord of the blade, the blade motion would be perpendicular to the

laser beam direction; thus, there would be no velocity sensed by the LDV.

(a) (b)

Figure 6-11 (a) Rotational Misalignment of Laser (b) Rotational Misalignment of Rotor

113

Desired Beam Spot

Center of Rotation

Actual Beam Spot

Velocity Vector

Laser Beam Path

Desired Beam Spot

Center of Rotation

Actual Beam Spot

Velocity Vector

Laser Beam Path

Velocity Component

In Laser Directions (a)

(b)

Figure 6-12 Resolving Blade Velocity For x(a) and y(b) Rotational Misalignments of The Rotor

Vertex Mirror – Rotational misalignments of the vertex mirror had the same results as

misaligning the rotor. Since these two pieces of equipment are attached, this result was

expected. The same methods for verification were used as for the rotor misalignments.

Fold Mirror – Rotational Misalignments of the fold mirror generated both velocity errors

and position errors. The velocity errors and position errors for x-direction misalignments

were the same as for the y-direction misalignments, but they were 90 degrees out of

phase. Also, the position and velocity errors were smaller in magnitude than for

misalignments of the laser, rotor, and vertex mirror. Since the laser did not deviate from

the ideal path until it reached the fold mirror, this result was consistent with what was

expected. Since the velocity and position errors generated by fold mirror rotational

114

misalignments were smaller than the other equipment, its alignment was determined to be

slightly less critical.

Blade/Target – The blade and target are the same for this modeling case since there was

not twist introduced into the blade. Therefore, they will discussed as the same

component. Rotational misalignments of the target blade in the x-direction and y-

direction did not cause any velocity error. Additionally, x-direction rotations did not

cause a position error, but the y-direction misalignment caused a very small position

error. If the target blade has a rotational misalignment in the x-direction or y-direction

the laser beam spot is stationary on the blade, and no velocity errors would be induced.

The beam spot is stationary because the misalignment rotates with the target frame.

Therefore, the measurement point is stationary on the target blade frame. An x-direction

misalignment rotates the target blade about the center of the blade, along the span.

Therefore, since the beam ideally intersects the target blade along this line, there is no

position error. A misalignment about the target blade y-axis changed the distance that the

blade is from the fold mirror. Therefore, the laser intersected the target blade in a

different spot, which created a position error.

As with the translational misalignment study, the rotational parametric study

generated the sensitivities of the position and velocity errors due to the misalignments.

From these sensitivities, the maximum allowable misalignment of each piece of

equipment was found. Since some misalignments cause both position and velocity errors,

the one that results in the more restrictive allowable misalignment was used. The

115

maximum allowable misalignment due to position error was found the same way as for

the parametric translational misalignment study.

Table 6-3 shows the results of the sensitivity analysis, and the resulting allowable

misalignments. The allowable misalignments were calculated based on methods

described earlier. Overall, the position error constraint was more sensitive to the

misalignment of the parameters than the velocity error constraint. This meant that the

maximum allowable misalignment for each piece of equipment was chosen based on the

position error analysis.

Parametric Monte Carlo Studies

It was also desired to know how each input misalignment variable affected the

position error and velocity error within a selected design space. In order to do this, a

Monte Carlo simulation was completed for each variable. While the misalignment of all

the variables was held at zero, twenty random points, selected by iSIGHT within the

chosen design space, were run by the MATLAB program. From this, the way that the

input affected the output was determined. For example, it would tell if the response was

linear, sinusoidal, logarithmic, etc. The design space was determined, as mentioned

before, by the type of distribution, the mean, and standard deviation based on what the

variable bounds were. The desired bounds were -0.005 - 0.005 meters for translational

misalignments and -0.005 – 0.005 radians (-0.286 – 0.286 deg) for rotational

misalignments. Since a uniform distribution with a mean of zero was selected, the

standard deviation, assuming the bounds above, was found using the following equation:

116

5.3*2)005.0(005.0 −−=σ . [6-4]

Figure 6-13 shows a typical iSIGHT program window for entering the type of

distribution, mean value, and standard deviation of the bounds. From this, iSIGHT chose

values within the limits around the mean and the outputs were solved. This was done for

all x and y-direction translational and rotational misalignments for each piece of

equipment. It should be noted that the magnitude of the position error and velocity error

were the desired quantities. Therefore, the position errors and velocity errors are always

shown as positive.

Figure 6-13 iSIGHT Program Window For Monte Carlo Methods

117

Shown in Figure 6-14, Figure 6-15, and Figure 6-16 are the position error

responses to translational misalignments, position error response to rotational

misalignments, and the velocity error response to rotational misalignments, respectively.

No velocity error response to translational misalignments was done because it was

already determined that translational misalignments did not contribute to velocity error.

Also, for graph clarity, only the responses for the x-direction translations and rotations

are displayed. It was already shown that the y-direction misalignments respond the same

as for the x-directions; therefore, the results were the same. It should be noted that the

slope of the response plots of the parametric Monte Carlo simulation coincide with the

influence coefficients from the parametric sensitivity analysis. Appendix B shows the y-

direction misalignment plots.

0

0.01

0.02

0.03

0.04

0.05

-0.006 -0.004 -0.002 0 0.002 0.004 0.006Misalignment(m)

Posi

tion

Erro

r(m

)

eXLeXReXVeXFeXB

Figure 6-14 Position Error Response to Translational Misalignments

118

0

0.01

0.02

0.03

0.04

0.05

-0.006 -0.004 -0.002 0 0.002 0.004 0.006Misalignment(radians)

Posi

tion

Erro

r(m

)thetaXLthetaXRthetaXVthetaXFthetaXB

Figure 6-15 - Position Error Response to Rotational Misalignments

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Misalignment(radians)

Vel

ocity

Err

or(m

/s

thetaXLthetaXRthetaXVthetaXFthetaXB

Figure 6-16 Velocity Response to Rotational Misalignments

By looking at the three previous graphs, it was noticed that all of the inputs,

translational or rotational, have a linear effect on both position error and that rotational

misalignments also have a linear effect on the velocity error. Additionally, you can also

see which variables affect the outputs the most. Although Figure 6-14 is somewhat hard

to read because of the scale, this was done to show the relative magnitude of the

119

rotational misalignment responses compared to the translational misalignment responses.

The rotational misalignments had a much greater influence on the output than the

translational misalignments. Also, the rotational misalignments of the rotor and vertex

mirror were the most influential misalignments of the system, which was expected.

Knowing that each variable had a linear effect on position error and velocity error

gave a better understanding of how each variable affected both position error and velocity

error. For the next section of this study, a more complete Monte Carlo analysis was done

that included all variable interactions was done. Understanding the influence of each

variable helped to determine the design space of each variable. Variables that have more

influence on the output may have required a more restrictive design space. In other

words, since the position error and velocity error are affected more by certain parameters,

they have to be aligned more accurately. Now that the response of velocity error and

position error for each individual misalignment parameter was analyzed, more realistic

analyses that includes variable interactions will be discussed. The next section discusses

the interaction of translational misalignments, rotational misalignments, and the

combination of the two.

6.2 Interaction Studies

In a realistic representation of the self-tracker LDV measurement system, it is not

possible to have absolutely no misalignments in any piece of equipment. Due to

manufacturing tolerances, improper alignments, and anything that may cause a

misalignment, there will be some amount of translation and rotation present. In order to

take into account all of the possible interactions of misalignments present in the

measurement system, Monte Carlo simulations were done that included all input

120

variables. This simulation was setup similar to that of the parametric Monte Carlo

method, except all misalignments had simultaneous bounds set, and all input

misalignments were varied simultaneously within these bounds.

Setting the bounds for each misalignment, or the design space, was an important task

because it was what dictated the range of allowable input misalignments for each piece of

equipment. As with the parametric Monte Carlo method, the range of inputs was

determined by the type of distribution, mean value, and standard deviation. Again, the

assumed distribution was uniform with a mean of zero. For a first attempt, to decide

what the bounds were, it was decided that it was reasonable to construct a test system

with ±0.005 meters of translational misalignment and ±0.005 radians (0.286 deg) of

rotational misalignment for each piece of equipment. As mentioned, this was a starting

point. The level of position error and velocity error generated by the interactions was not

known.

Three different Monte Carlo studies were performed to study the different

interactions between misalignment variables. Since the effects of only translational, only

rotational, and translational and rotational together, were of interest, simulations were

setup to investigate all three. From the parametric Monte Carlo simulation, it was seen

that rotational misalignments had much more influence on position error and velocity

error than translational misalignments. Performing a full interaction study of all effects,

it was difficult to see how the translational misalignment influenced the outputs. Thus,

an interaction study of only translational misalignments was done. Based on the same

premise, an interaction study of only rotational misalignments was completed. Since

having only translational or only rotational misalignments does not represent a real

121

system, simulations were done with all translational and rotational misalignments present.

A typical iSIGHT program window for setting up multiple interactions between variables

is shown in Figure 6-17. The number of points used for each simulation was determined

based on the number of input misalignments.

Figure 6-17 Typical iSIGHT Program Window For Interaction Studies

Translational Misalignment Monte Carlo Study

In order to investigate the interactions of only translational misalignments on the

position error and velocity error, a Monte Carlo interaction study of translations was

performed. There were a total of eight pertinent misalignment variables that were

studied. They were the x and y-direction translational misalignments of the laser, rotor,

122

vertex mirror, and fold mirror. It was decided that, in order to save computational time,

the blade misalignments would not be considered. This was a reasonable assumption

since the blades are attached to the rotor and rotor misalignments were still considered.

Also, since this particular case was done to merely get a sense of how the variables

interact, only x-direction misalignments were used. It was shown earlier that both the x

and y-direction misalignments produce symmetric and equal outcomes. Therefore, the

results from the x-direction findings were the same as the y-direction results would have

been.

Shown in Figure 6-18 is a brush plot of the results from the translational x-direction

misalignment Monte Carlo study. A brush plot shows all of the points that were chosen

for each variable, within the design space. Additionally, the plot shows the position error

and velocity error responses to the variables. Although no quantitative numbers are

visible on these plots, a general sense of the influential variables was obtained. These

plots are particularly useful when there are more variables being studied. By observing

the responses with some pattern, or shape, the responses to focus on were found. The x-

axis extents for each of the plots are the bounds that were selected for that particular

variable. Also, the velocity error responses below were so low that they are essentially

zero.

123

Pos E

rreX

LeX

RV

elEr

reX

F

Vel ErreXReXLeXFeXV

Pos E

rreX

LeX

RV

elEr

reX

F

Vel ErreXReXLeXFeXV

Figure 6-18 Brush Plot For x-direction Interactions of Translational Misalignments

In order to better describe the responses and also show the usefulness of the brush plots, a

more detailed version of the response plots is shown in Figure 6-19. In order to interpret

the brush plots, one must first consider the over design space that was selected. For this

case, each misalignment was bounded between ±0.005 meters. Each plot has as many

dimensions as there are input variables. For this case, there were a total of four variables,

therefore, four dimensions. Looking at any individual plot, assume that all of the other

124

variables are somewhere within their respective design space (bounds). Then, the

possible response, position error, is restricted to the region defined by the brush plot.

For example, all variables other than x-direction laser translation (eXL) are restricted to

their ±0.005 meter bound. If there is absolutely no misalignment of the laser in the x-

direction, then the position error will always fall between 0 - 0.005 meters. For this

particular case, the results show that if only x-direction misalignments were considered,

the misalignments of the laser and rotor are the most critical parameters and that the

relative influence of the fold mirror and vertex mirror misalignments are negligible. This

result proved useful later when running the full interaction simulation. Since the x and y-

direction translations of the vertex mirror and fold mirror did not influence the position

error or velocity error, they could be left out of the study. This saved much

computational time.

125

-0.005 -0.003 -0.001 0.001 0.003 0.005eXL

0.000

0.002

0.004

0.006

0.008

0.010

PosE

rr

-0.005 -0.003 -0.001 0.001 0.003 0.005eXR

0.000

0.002

0.004

0.006

0.008

0.010

PosE

rr

-0.005 -0.003 -0.001 0.001 0.003 0.005eXV

0.000

0.002

0.004

0.006

0.008

0.010

PosE

rr

-0.005 -0.003 -0.001 0.001 0.003 0.005eXF

0.000

0.002

0.004

0.006

0.008

0.010

PosE

rr

Figure 6-19 Detailed Brush Plots of Position Error Response to Translational Interaction Study

Rotational Misalignment Monte Carlo Study

In order to investigate the interactions of only rotational misalignments on the

position error and velocity error, a Monte Carlo interaction study of rotations was

performed. There were a total of thirteen pertinent misalignment variables that were

studied. This number was reduced to five in order to reduce the computational time. The

five variables kept were rotations about x-direction for the laser, rotor, and fold mirror;

rotations about z-direction of vertex mirror; and rotations about the y-direction of the

target. In the parametric sensitivity analysis, it was seen that for the laser, rotor, and fold

mirror that the misalignments about the x-directions and y-directions produce equal

results. Also, rotations about the x and y-directions for the vertex mirror are the same as

for the rotor. Rotations about the x-direction of the target produce no position or velocity

126

error. Thus, since this analysis was for qualitative purposes, reducing the computational

time by only using five variables was acceptable.

The design space for the misalignment variables was setup the same as for the

parametric and translational Monte Carlo methods. The bounds for each variable were

assumed to be ±0.005 radians (0.286 deg). Figure 6-20 shows a brush plot of the position

error and velocity error responses to the different rotational misalignments. For the case

of rotational misalignments, velocity error was not ignored since it was significant in

relative magnitude. Looking at the different plots of Figure 6-21 and Figure 6-22, it was

noticed that the responses to rotation about the rotor showed the most distinct shape.

Since, in the parametric analysis, this was the variable that position error and velocity

error were most sensitive to, this was expected. Also, the responses to rotations about

both the laser and fold mirror show a less apparent shape. This meant that the laser and

fold mirror rotational misalignments have influence on the position error and velocity,

but not to the degree that rotational misalignments about the rotor have. Notice that the

velocity error response seemed to be a scalar multiple of the position error. Since the

velocity error was simply a component of the velocity caused by crossing the position

error with the system rotation, this outcome seemed logical. Also, compared to the

influence of the other variables, the y-rotation of the target blade did not influence the

position error or velocity error much.

127

thetaXR thetaXF thetaZV thetaXL thetaYT Vel Error

thet

aXL

Posi

tion

Err

orth

etaY

TV

eloc

ity E

rror

thet

aZV

thet

aXF

thetaXR thetaXF thetaZV thetaXL thetaYT Vel Error

thet

aXL

Posi

tion

Err

orth

etaY

TV

eloc

ity E

rror

thet

aZV

thet

aXF

Figure 6-20 Brush Plots For Rotational Interaction Monte Carlo Study

The V-shaped patterns for some of the scatter plots were of particular interest.

This shape was most distinct for thetaXR, but was also visible in thetaXF and thetaXL.

What the shape meant, was that if all of the other variables were restricted to within their

bounded design space, the response (position or velocity error) would lie in the region

where the pattern, or V-shape, was present. Additionally, in Figure 6-20, it was noticed

that the velocity error vs. position error line became thicker at higher values of position

error and velocity error. For pure rotational misalignments, velocity error at any given

128

point is either zero or proportional to position error. Depending on the combination of

misalignments present, this caused the trend seen in the position error vs. velocity error

scatter plot. There could have been large amounts of position error in equipment that

generates zero velocity error, or large amounts of position error in equipment that

generates large amounts of velocity error. The Monte Carlo method takes these

considerations into account; thus, giving regions of operation instead of a single point.

-0.005 -0.003 -0.001 0.001 0.003 0.005thetaXL

0.00

0.02

0.04

0.06

0.08

0.10

Posi

tionE

rror

-0.005 -0.003 -0.001 0.001 0.003 0.005thetaXR

0.00

0.02

0.04

0.06

0.08

0.10

Posi

tionE

rror

-0.005 -0.003 -0.001 0.001 0.003 0.005thetaXF

0.00

0.02

0.04

0.06

0.08

0.10

Posi

tionE

rror

-0.005 -0.003 -0.001 0.001 0.003 0.005thetaYT

0.00

0.02

0.04

0.06

0.08

0.10

Posi

tionE

rror

Figure 6-21 Detailed Brush Plots of Position Error Response to Rotational Interaction Study

129

-0.005 -0.003 -0.001 0.001 0.003 0.005thetaXL

0.0

0.5

1.0

1.5

2.0

2.5

Velo

city

Erro

r

-0.005 -0.003 -0.001 0.001 0.003 0.005thetaXR

0.0

0.5

1.0

1.5

2.0

2.5

Velo

city

Erro

r

-0.005 -0.003 -0.001 0.001 0.003 0.005thetaXF

0.0

0.5

1.0

1.5

2.0

2.5

Velo

city

Erro

r

-0.005 -0.003 -0.001 0.001 0.003 0.005thetaYT

0.0

0.5

1.0

1.5

2.0

2.5

Velo

city

Erro

r

Figure 6-22 Detailed Brush Plots of Velocity Error Response to Rotational Interaction Study

Total Misalignment Monte Carlo Study In order to represent how misalignments would truly affect the self-tracker LDV

measurement system, a simulation that simultaneously takes into account all variable

misalignments was completed. In order to do this, a Monte Carlo simulation was done

that took into account twelve different input variables. The different misalignment

variables were chosen based on prior simulation results. It was decided that the x and y-

direction translational misalignments of the vertex mirror and fold mirror did not create

any position or velocity errors; therefore, they were not used in the full interaction study.

Also, since the blade was assumed to be attached to the rotor hub, misalignments would

not be possible to correct. For that reason, no blade misalignments were accounted for.

The main reason for having the ability to add misalignments into the bladed frames was

for the purpose of modeling different blade vibrations.

130

For this simulation, a design space was setup in the same manner as done

previously. The distribution was assumed to be uniform and the mean value was zero for

each misalignment parameter. The bounds for the input variables were changed for this

simulation based on the rotational misalignment Monte Carlo simulation. The position

and velocity errors generated by that study were much higher than the desired position

error of 0.0005 meters and velocity error of 0.0675 meter/second. Since not all

interactions and misalignments were used in that study, the requirements were less

stringent. Thus, having tighter bounds on the input misalignments was necessary for the

full interaction analysis. It was decided that all inputs should be bounded by 0.001

meters for translational misalignments and 0.001 radians (0.057 deg) for rotational

misalignments. Figure 6-23 shows the result of the full interaction Monte Carlo

simulation with new design space.

131

θθθθXR eYL θθθθYL θθθθXF θθθθXV eXR eXL θθθθYR θθθθXL θθθθYF θθθθYV eXR VelErr

θ θθθYV

eXR

PosE

rrV

elE

rrθ θθθ Y

Fθ θθθ X

Lθ θθθY

ReX

LeY

ReY

Lθ θθθX

Vθ θθθ X

Fθ θθθY

L

θθθθXR eYL θθθθYL θθθθXF θθθθXV eXR eXL θθθθYR θθθθXL θθθθYF θθθθYV eXR VelErrθθθθXR eYL θθθθYL θθθθXF θθθθXV eXR eXL θθθθYR θθθθXL θθθθYF θθθθYV eXR VelErr

θ θθθYV

eXR

PosE

rrV

elE

rrθ θθθ Y

Fθ θθθ X

Lθ θθθY

ReX

LeY

ReY

Lθ θθθX

Vθ θθθ X

Fθ θθθY

Lθ θθθY

VeX

RPo

sErr

Vel

Err

θ θθθ YF

θ θθθ XL

θ θθθYR

eXL

eYR

eYL

θ θθθXV

θ θθθ XF

θ θθθYL

Figure 6-23 Brush Plot For Total Interaction Monte Carlo Study Using Uniform .001 meter(or radian) Bounds

A few things are noticed by observing the general brush plots. First, again, the

driving factors for the position error and velocity error responses were the rotational

misalignments about the rotor and vertex mirror. A more important finding was that at

the zero misalignment values for these parameters, the black region on the plots is

noticeably larger than for previous simulations. This meant that as more misalignment

variable were added, the possibility of reaching high levels of position error or velocity

132

error has increased. Also, the correlation between position error and velocity error was

reduced. The bottom right brush plot of Figure 6-23 would be almost perfectly linear for

a high correlation. Compared to the same brush plots on Figure 6-18 and Figure 6-20, the

brush plot is much less linear. This was due to the fact that interactions between

translational and rotational misalignments were present. Translational misalignments did

not influence velocity error, while rotational misalignments did. Additionally, the

translational misalignments, in general, are less critical than the rotational misalignments.

Also, the position error plots are less defined than the velocity error plots. This occurred

because translational misalignments affected the position errors, but not velocity error.

The reason that the position error plots still have the same basic shape as the velocity

error plot, but just less defined, was because translational misalignments did not affect

the position error as much as rotational misalignments did.

In order to better visualize the important misalignment variables, more detailed

brush plots of the more influential misalignment parameters are shown in Figure 6-24 and

Figure 6-25. The figures show only the x-direction responses. Appendix C shows the y-

direction responses, as well as the responses due to translational errors. The position error

range for all of the brush plots was from 0 - 0.03 meters. This meant that if every

misalignment fell within the ±0.001 meter bound for translational misalignments and

±0.001 radian (0.057 deg) bound for rotational misalignments, the position error is

guaranteed to be less than 0.03 meters. This result was very high compared to the 0.0005

meter desired position error. Apparently, the interaction of all the misalignments has a

great effect on the position error. The velocity error range for all of the brush plots was

from 0 – 0.8 meters/second. As for the position error, this meant that if every

133

misalignment fell within the ±0.001 meter bound for translational misalignments and

±0.001 radian (0.057 deg) bound for rotational misalignments, the velocity error would

always be less than 0.8 meters/second. This result was also much greater than the desired

0.0675 meters/second. From these results, it became apparent that in order to achieve

the levels of desired position error and velocity error, extremely precise manufacturing,

construction, and alignment of equipment was necessary.

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaXL

0.00

0.01

0.02

0.03

0.04

PosE

rr

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaXR

0.00

0.01

0.02

0.03

0.04

PosE

rr

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaXV

0.00

0.01

0.02

0.03

0.04

PosE

rr

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaXF

0.00

0.01

0.02

0.03

0.04

PosE

rr

Figure 6-24 Detailed Brush Plots of Position Error Response to Influential Interaction Parameters

134

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaXL

0.0

0.2

0.4

0.6

0.8

VelE

rr

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaXR

0.0

0.2

0.4

0.6

0.8

VelE

rr

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaXV

0.0

0.2

0.4

0.6

0.8

VelE

rr

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaXF

0.0

0.2

0.4

0.6

0.8

VelE

rr

Figure 6-25 Detailed Brush Plots of Velocity Error Response to Influential Interaction Parameters

Since it was obvious that restricting the misalignments to be within ±0.001 meters

for translational misalignments and ±0.001 radians (0.057 deg) for rotational

misalignments was not nearly precise enough, another Monte Carlo simulation was

completed. This time, the bounds for each variable misalignment were set as the

constraints found as the results of the parametric sensitivity. Table 6-4 shows the

selected bounds for each misalignment input. As you can see, the bounds were quite

restrictive and will be hard to achieve in a realistic system without using very high

precision linear positioning equipment with a control device for alignment. Nonetheless,

if the simulation met the specified requirements by using this design space, meaningful

information was found.

135

Table 6-4 Monte Carlo Interaction Study Misalignment Bounds

Misalignment Parameter Bounds(+/-)eXL .0005 meterseYL .0005 meterseXR .0005 meterseYR .0005 meters

thetaXL .0001 radiansthetaYL .0001 radiansthetaXR .00005 radiansthetaYR .00005 radiansthetaXV .00005 radiansthetaYV .00005 radiansthetaXF .0001 radiansthetaYF .0001 radians

Figure 6-26 shows the brush plot for the Monte Carlo simulation with a design

space governed by the results of the parametric sensitivity analysis. The variables that

show the most defined patterns were still the rotations about the rotor and vertex mirror.

One result was that the position error and velocity seemed to be much less correlated that

before. This led me to believe that the translational misalignments may not have been

bounded tight enough. In order to investigate this further, more detailed representations

of the brush plots were generated. Figure 6-27 and Figure 6-28, respectively, show the

position and velocity error responses to the simulation using the parametric sensitivity

analysis. In general, the plots show much more scatter and less defining shapes than

previous simulations. This means that as the alignment criteria became more restrictive,

no single variable contributed to the misalignment than the others. Also, for this

particular study, the bounds chosen between different misalignment parameter were not

equal. Thus, one variable that did was not a major contributor to errors before, now may

because it was allowed to take on a great value compared to what other variables were

136

allowed. This phenomenon was also the reason that the position error vs. velocity error

plot shows more scatters than previously. Additionally, since the simulation combined

both rotational and translational misalignments, some of the position error values could

have been small, but still generate large velocity errors depending on whether the

misalignment was translational or rotational.

Note that the plots showed that, for the specified design space, that the position

error was always less than 0.0025 meters and that the velocity error was always less than

0.006 meters/second. This meant that the position error requirement of 0.0005 meters

was not met, but the velocity error requirement of 0.0675 meters/second was.

θ θθθYV

eXR

PosE

rrV

elEr

rθ θθθY

Fθ θθθX

Lθ θθθY

ReX

LeY

ReY

Lθ θθθX

Vθ θθθX

Fθ θθθY

L

θθθθXR eYL θθθθYL θθθθXF θθθθXV eXR eXL θθθθYR θθθθXL θθθθYF θθθθYV eXR VelErr

θ θθθYV

eXR

PosE

rrV

elEr

rθ θθθY

Fθ θθθX

Lθ θθθY

ReX

LeY

ReY

Lθ θθθX

Vθ θθθX

Fθ θθθY

Lθ θθθY

VeX

RPo

sErr

Vel

Err

θ θθθYF

θ θθθXL

θ θθθYR

eXL

eYR

eYL

θ θθθXV

θ θθθXF

θ θθθYL

θθθθXR eYL θθθθYL θθθθXF θθθθXV eXR eXL θθθθYR θθθθXL θθθθYF θθθθYV eXR VelErrθθθθXR eYL θθθθYL θθθθXF θθθθXV eXR eXL θθθθYR θθθθXL θθθθYF θθθθYV eXR VelErr

Figure 6-26 Brush Plots For Monte Carlo Full Interaction Study Based On Parametric Sensitivities

137

-0.0001 -0.0000 0.0000 0.0000 0.0001thetaXL(rad)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m)

-0.00006 0.00006thetaXR(rad)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m)

-0.00006 0.00006thetaXV(rad)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m)

-0.0001 -0.0000 0.0000 0.0000 0.0001thetaXF(rad)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m)

Figure 6-27 Detailed Brush Plots of Position Error Response To All Misalignment Interactions

-0.0001 -0.0000 0.0000 0.0000 0.0001thetaXL(rad)

0.00

0.02

0.04

0.06

Velo

city

Err

or(m

/s)

-0.00006 0.00006thetaXR(rad)

0.00

0.02

0.04

0.06

Velo

city

Err

or(m

/s)

-0.00006 0.00006thetaXV(rad)

0.00

0.02

0.04

0.06

Velo

city

Err

or(m

/s)

-0.0001 -0.0000 0.0000 0.0000 0.0001thetaXF(rad)

0.00

0.02

0.04

0.06

Velo

city

Err

or(m

/s)

Figure 6-28 Detailed Brush Plots of Velocity Error Response To All Misalignment Interactions

138

The results of this study led to a number of interesting conclusions. The first, rather

obvious, observation was that the position error requirement was more difficult to meet

than the velocity error requirement. For the parametric sensitivity analysis done

previously, the final misalignment design conditions ended up being based on the

position error analysis. This was because the position error analysis resulted in tighter

restrictions on the design criteria. The second result was that designing for each

misalignment without considering the interaction of all misalignments did not achieve the

level of accuracy necessary for position error, but did for velocity error. Lastly, the level

of alignment precision for the equipment of the self-tracker LDV measurement system,

necessary to achieve the desired amount of position error and velocity error, was so small

(numerically) that, based on the assumptions of this study, this system would be nearly

impossible to realistically build and obtain accurate data.

6.3 Summary

It was determined that, although a parametric sensitivity analysis provided a good

representation of how each misalignment affected the position error and velocity error of

the measurement system, it did not give accurate design criteria. This was because of the

interaction effects of the different misalignments. By performing a parametric Monte

Carlo study of individual parameters, it was seen that the individual misalignment

parameters influence the position error and velocity error linearly. By performing a full

interaction study of misalignment effects using a Monte Carlo simulation, it was found

that bounding all translational misalignments to ±0.001 meters and all rotational

misalignments to ±0.001 radians (0.057 deg) was not sufficient. A more restrictive

139

design space was necessary. Also, by bounding the misalignments according to the

results of the parametric sensitivity analysis, the system could be aligned to produce

velocity errors within the desired level. The position error requirement was still not met

though, and was approximately fives times larger than desired. If a 0.0025 meter position

error would be acceptable, and the main quantity of interest was velocity error, then the

criterion from the sensitivity analysis (Table 6-4) was acceptable.

140

7 Prototype Self-Tracker LDV System

A full-scale simulation of the self-tracker LDV measurement system has never been

tested and tried on an actual jet engine. Doing this would be very desirable for many

reasons. First, it would prove that the concept works and can applied to an actual high

speed rotating structure. Second, some real data could be obtained regarding the

vibration of the rotating structure. Since the concept theoretically works during speed

transients, this data could prove to be very useful. Also, in order to continue work on this

project, corporate funding will be necessary to purchase the necessary equipment.

Showing that the concept works will likely interest companies. The remainder of this

chapter describes a preliminary self-tracker LDV test rig that was designed and built to be

used to measure fan blade vibrations on a Pratt & Whitney JT15D turbofan engine. This

engine is maintained and operated at the Virginia Tech airport as a part of the Mechanical

Engineering Turbomachinery research facility.

7.1 Design of Test Stand

Since this project was not funded, the test stand was a first attempt at a prototype,

using the limited funds available. First, the design constraints of the laboratory test

facility and the turbofan dimensions were considered. The available space from the rear

of the test bay to the nose cone of the engine was approximately 124 inches (3.1496

meters), the height of the centerline of the engine was approximately 45 inches (1.143

meters), and the mid-span radius of the fan was 6.5 inches (0.1650 meters). With this

information, a general-purpose laser mounting test stand was designed. Figure 7-1 shows

a 3-D isometric view of the laser and fold mirror mounting test stand. The amount of x

141

and y-direction misalignment between the fold mirror and laser was determined by the

manufacturing tolerances specified, which were 0.001 inches (25.4µm). The tolerance of

the fold mirror did not play a role when determining the amount of misalignment for the

translational case. Previously, it was found that translational misalignments of the fold

mirror do not cause position or velocity errors. Considering that this tolerance was

applied to the laser in two places, the maximum possible translational misalignment was

0.002 inches (50.8µm). This fell within the specifications found as a result of the Monte

Carlo interaction; therefore, it was determined to be adequate. These tolerances were

applied to both the x-direction and y-direction when machining the parts; thus, the

misalignment errors of both were within reason.

Fold Mirror Laser (LDV)

Figure 7-1 3-D Isometric View Of Laser and Fold Mirror Test Stand

142

A method to mount a vertex mirror assembly onto the rotational axis of the turbofan

was developed. The nosecone of the JT15D has a threaded female bolthole in the center

of its rotation axis. This was used as a method to join the vertex mirror to the center of

rotation. The vertex mirror assembly can be seen dissembled in Figure 7-2 and

assembled in Figure 7-3. The assembly was then correctly mounted and torqued onto the

nosecone of the jet using approved aircraft-grade bolts and fasteners. Figure 7-4 shows

the vertex mirror assembly mounted on the nosecone of the JT15D turbofan engine. The

machining tolerances of this apparatus were specified to within 0.0005 inches (12.7µm).

This was because the interaction study found the accuracy of this assembly to be most

critical. The simulation referenced the misalignments of the vertex mirror from rotor;

therefore, the mounting of the mirror onto the rotor was done most carefully. Note that

the bolt pattern is symmetric about the center of rotation as to not cause excess vibrations

and that the mirror used was of high quality reflective standards.

143

Adapter

Mirror Springs

Mirror

Retainer

Adjustment Plate

Nosecone Bolt/Washers

Mirror Screws/Washers

ustment

Mirror

Adjustment

Nosecone

Ad

Adapter

j

Figure 7-2 Disassembled Vertex Mirror Assembly

144

Figure 7-3 Assembled Vertex Mirror Assembly

Nosecone

Vertex Mirror Blades

Figure 7-4 Vertex Mirror Assembly Mounted on JT15D

145

Since the laser and fold mirror are mounted on a single plate, they will now be

considered as a single assembly. The misalignment scheme that best represents this

scenario is that of the rotor misalignment (see Figure 6-11b). The last alignment to

consider was that of the rotor axis (which already has the vertex mirror mounted to it)

relative to the laser beam axis. This alignment was the most difficult task, since there

were two independent assemblies being aligned. A best attempt was made to align the

equipment by visually minimizing the position error of the beam spot on the blade. This

was a rather crude iterative process. First, the laser was set to strike the center of the

vertex mirror. This was done by rotating the fan by hand and making sure the beam spot

did not move any on the mirror. The vertex mirror was then adjusted to cause the laser

beam to land at the 6.5 inch (0.1650 meter) radial location on the span. Next, the rotor

shaft was rotated at angles of 0, 2

3,,2

πππ and radians (0, 90, 180, 270 deg). The position

of the beam on the blade at each location was noted, and the adjustment screws on the

laser/fold mirror frame assembly were adjusted accordingly. This process was repeated

until the laser beam position error was within 16

1 of an inch (0.00158 meters). It should

be noted that the size of the beam spot on the blade was measured to be 16

1 of an inch

(0.00158 meters). This was about 50% larger than the previously calculated spot size of

0.001 meters. Figure 7-5 shows a picture of the self-tracker LDV system in front of the

JT15D turbofan engine. Appendix E contains a more detailed alignment procedure used

for aligning the self-tracker with the JT15D.

146

Laser

Fold Mirror

JT15D Turbofan

Figure 7-5 Picture of Prototype Self-Tracker LDV Setup on JT15D Turbofan

7.2 Results of Prototype Self-Tracker Testing

The self-tracker test setup was aligned to the accuracy described above. The engine

was started and the data acquisition was turned on. There were no associated problems

with the vertex mirror assembly. It rotated with the rotor shaft, as intended, and the jet

did not experience any difficulties with the assembly mounted on the nosecone.

Unfortunately, there seemed to be a problem with the signal output of the laser. Upon

viewing the data, it was nothing but noise. Prior to assembling the structure, the laser

was tested and believed to be working correctly. It was at this time that another LDV

was located, but it was in use and could not be removed from its application and setup. A

147

generalized output signal of the second LDV was observed and it was working correctly.

The next step is to use this LDV when it becomes available and again run tests on the

JT15D turbofan. This new LDV has the same dimensions as the previous one; therefore,

will mount to the test rig without additional changes.

Although no meaningful data was obtained from the output of the LDV, a number of

results were found though the testing. First, the position error of the beam spot was

minimized to relatively small levels, despite the low cost, and somewhat crude, test setup.

This led me to believe that with some high precision controlled linear positioning

equipment, the self-tracker system could be aligned to achieve acceptable amounts of

position error. There are commercially available automated systems that can be used to

accurately position equipment. The internal electronics of the LDV allow for the

possibility of using the laser itself as a control mechanism for aligning the self-tracker.

The signals that the LDV senses, before they combined internally in the LDV to give

velocity data, could be extracted and used as the basis for positioning the equipment.

This is due to the fact that the laser reads higher values of velocity as the motion become

more in the direction of the laser. By observing maximum signals from the laser, the

correct alignment could be obtained.

Also, it was noticed that after operation, the vertex mirror had a few scratches on it.

This was likely due to debris that was in the flow field of the jet. To resolve this, a

supply of vertex mirrors should be available to replace scratched ones as necessary.

148

8 Conclusions and Future Recommendations

A model for the self-tracking LDV system has been developed such that it has the

capabilities to model translational and rotational misalignments in any single piece of

equipment in the system. The model also has the capabilities to add dynamic effects if so

desired, although only static effects were evaluated for this study. A parametric

sensitivity analysis was completed that showed the effect that any single misalignment

parameter had on position error and velocity error. For each misalignment variable, a

single variable Monte Carlo simulation was completed. This showed how each

misalignment parameter affected position error and velocity error over the entire range of

misalignment possibilities. Finally, a full interaction Monte Carlo simulation was

completed to investigate the interaction of all input misalignment variables. By

completing these studies, a better understanding of the self-tracker LDV measurement

system and the levels of alignment accuracy necessary to generate feasible results was

found. This understanding was then applied to designing, building, and testing and actual

self-tracker LDV test rig to be used with the Pratt & Whitney JT15D turbofan engine.

8.1 Results of Misalignment Studies

It was found that a parametric sensitivity study was an adequate method to show

which misalignment parameters influenced the position and velocity error the most. It

did not, however, show how each misalignment influences position error and velocity

error over a range of input variables. Also, it did not show the interaction effects of the

different possible misalignments. According to the parametric sensitivity analysis, each

misalignment had to be aligned according to the specifications shown in Table 8-1.

149

Table 8-1 Parametric Sensitivity Based Alignment Criteria

Misalignment Parameter Bounds(+/-)eXL .0005 meterseYL .0005 meterseXR .0005 meterseYR .0005 meters

thetaXL .0001 radiansthetaYL .0001 radiansthetaXR .00005 radiansthetaYR .00005 radiansthetaXV .00005 radiansthetaYV .00005 radiansthetaXF .0001 radiansthetaYF .0001 radians

The Monte Carlo parametric study showed that all misalignment parameters have a linear

effect on both position error and velocity error within the limits specified. These limits

were ±0.005 meters for translational misalignments and ±0.005 radians (0.286 deg) for

rotational misalignments. These bounds were well large enough to include realistic

alignment of the measurement system. Lastly, it was determined that a Monte Carlo

simulation that incorporated all possible misalignment interactions could predict the

maximum amount of position error and velocity error possible given a set of bounds for

each input misalignment. Using the bounds from the parametric sensitivity analysis in

the Monte Carlo interaction study, the position error was within ±0.0025 meters and the

velocity error was within ±0.06 meters/second. The alignment criteria were very

restrictive, in general, to achieve the desired level of measurement accuracy.

150

8.2 Results of Prototype Self-Tracker LDV System

Although no actual blade vibration (velocity) data was obtained by running tests

with the prototype self-tracker LDV setup, much useful information was acquired. The

most useful result was that with the test setup used, the alignment of the system was

rather accurate considering the precision of the test rig. Since this prototype test setup

was somewhat crude, if high precision linear positioning equipment with digitally

controlled devices were used, the necessary position and velocity error requirements can

likely be met. By incorporating machining tolerances as the misalignment errors,

estimates for translational and rotational misalignments were found. Another key result

was that there were no known complications while running the tests due to mounting the

vertex mirror assembly to the nosecone. This was an important result due to the high

rotational speeds, upwards of 8,000 RPM, of the jet.

8.3 Future Recommendations

The scope of this work included the implementation of a computer simulation to

model the misalignment effects of the self-tracker LDV measurement system and to

incorporate these results to design, build, and test a first attempt at a test rig. The next

logical step is to obtain the correctly operating LDV and make trial runs on the JT15D

turbofan engine again. This would add more realistic aspect to the idea by obtaining

actual data from a real jet engine. Along with the design of an actual system, additional

research into the area of linear motion equipment should be done. Linear motion

equipment would eventually be the controlling devices used to align the self-tracker.

Ideally, the system would align itself using a control mechanism, and user intervention

151

would be required. Understanding the limitations and capabilities of this equipment

would be most beneficial in the design of an actual system.

The computer simulation developed for this study has the capabilities to

incorporate dynamic misalignments as well as the static misalignment already studied.

Since a real life system has fluctuating movements driven by the rotation of the system,

as well as other vibrational modes, a theoretical study would prove to be beneficial.

Another possible area that should be explored is the possibility of post-processing

velocity data to extract the misalignment response errors. This thesis describes methods

to minimize the position and velocity response errors. On the other hand, it is possible to

spend less effort minimizing these errors and do data post-processing to remove the

velocities induced by the misalignments. The velocities to extract could be found using a

model such as the one developed for my thesis work. In order to make this a viable

endeavor, system dynamics would need to be incorporated into the model. For the

program written for this thesis work, the dynamics could be incorporated as a time

dependant misalignment that varies according to a dynamic model. This method could

prove to be less costly and more practical than attempting to minimize position and

velocity response error by restricting the adjustment resolution of the equipment to the

extremely small values found in this study.

152

Appendix A - MATLAB Simulations for Individual Misalignment Parameters

153

X-direction Translational Laser Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot


Th M i P iti E i 0 0010006 tThe Maximum Position Error is 0.0010006 meters

The Maximum Veloc ity Error is 4.4537e-006 meters /sec

For Zero Degrees of Rotation


The radius of the beam spot in y-dir is 0 meters

0 100 200 3001

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006x 10


Rotational Position(deg)

Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





154

Y-direction Translational Laser Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot



The Maximum Veloc ity Error is -4.4537e-006 meters /sec



The radius of the beam spot in y-dir is 0.001 meters

0 100 200 3001

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





155

Rotational Misalignment of .001 radians About X-direction of Laser

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot


Th M i P iti E i 0 0046306 tThe Maximum Position Error is -0.0046306 meters





0 100 200 300

4.628

4.6285

4.629

4.6295

4.63

4.6305x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





156

Rotational Misalignment of .001 radians About Y-direction of Laser

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot







0 100 200 300

4.628

4.6285

4.629

4.6295

4.63

4.6305x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





157

X-direction Translational Rotor Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot







0 100 200 3001

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)

Vel

ocity

(m/s

)Velm is Velideal

Design Parameters



Blade Angle = 0 rad





158

Y-direction Translational Rotor Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot







0 100 200 3001

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.02

-0.01

0

0.01

0.02


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





159

Rotational Misalignment of .001 radians About X-direction of Rotor

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.25

-0.2

-0.15

-0.1

-0.05


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot



The Maximum Veloc ity Error is -0.27052 meters/sec




0 100 200 300

9.2524

9.2524

9.2524

9.2524



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.2705

-0.2705

-0.2705

-0.2705

-0.2705

-0.2705


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





160

Rotational Misalignment of .001 radians About Y-direction of Rotor

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot







0 100 200 3009.2651

9.2651

9.2651

9.2651



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10

-3Veloc ity of A ligned and M isaligned Cases

Position(deg)

Vel

ocity

(m/s

)Velm is Velideal

Design Parameters



Blade Angle = 0 rad





161

X-direction Translational Vertex Mirror Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot


Th M i P iti E i 0 tThe Maximum Position Error is 0 meters





0 100 200 3000

0.2

0.4

0.6

0.8

1x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10


Position(deg)

Vel

ocity

(m/s

)Velm is Velideal

Design Parameters



Blade Angle = 0 rad





162

Y-direction Translational Vertex Mirror Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot







0 100 200 3000

0.2

0.4

0.6

0.8

1x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10


Position(deg)

Vel

ocity

(m/s

)Velm is Velideal

Design Parameters



Blade Angle = 0 rad





163

Rotational Misalignment of .001 radians About X-direction of Vertex Mirror

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.25

-0.2

-0.15

-0.1

-0.05


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot







0 100 200 3009.2538

9.2538

9.2538

9.2538



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.2706

-0.2706

-0.2706

-0.2706

-0.2706


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





164

Rotational Misalignment of .001 radians About Y-direction of Vertex Mirror

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot







0 100 200 300

9.2602

9.2602

9.2602

9.2602



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





165

X-direction Translational Fold Mirror Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot







0 100 200 3000

0.2

0.4

0.6

0.8

1x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





166

Y-direction Translational Fold Mirror Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot







0 100 200 3000

0.2

0.4

0.6

0.8

1x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





167

Rotational Misalignment of .001 radians About X-direction of Fold Mirror

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot



The Maximum Veloc ity Error is 0.13523 meters /sec




0 100 200 300

4.626

4.628

4.63

4.632

4.634

4.636



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





168

Rotational Misalignment of .001 radians About Y-direction of Fold Mirror

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot







0 100 200 300

4.626

4.628

4.63

4.632

4.634

4.636



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.1

-0.05

0

0.05

0.1


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





169

X-direction Translational Target Blade Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot







0 100 200 300

0.5

1

1.5

2x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





170

Y-direction Translational Target Blade Misalignment of .001 meters

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300

-0.025

-0.02

-0.015

-0.01

-0.005


Position(deg)V

eloc

ity E

rror

(m/s

)

Velm irVelrbmVeltot







0 100 200 3001

1

1

1

1x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 300

-0.0292

-0.0292

-0.0292

-0.0292

-0.0292

-0.0292Veloc ity of A ligned and M isaligned Cases

Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





171

Rotational Misalignment of .001 radians About X-direction of Target Blade

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot

The M inimum Position E rror is 1.7351e-022 meters

Th M i P iti E i 3 1032 017 tThe Maximum Position Error is 3.1032e-017 meters





0 100 200 300-1

-0.5

0

0.5

1x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





172

Rotational Misalignment of .001 radians About Y-direction of Target Blade

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1



Glo

bal Y

-coo

rdin

ates

(m)

100 200 300-1

-0.5

0

0.5


Position(deg)

Vel

ocity

Err

or(m

/s)

Velm irVelrbmVeltot

The M inimum Position E rror is 5.5588e-006 meters

Th M i P iti E i 5 5588 006 tThe Maximum Position Error is 5.5588e-006 meters





0 100 200 300

5.5588

5.5588

5.5588

5.5588

5.5588

5.5588x 10



Tota

l Pos

ition

Err

or(m

) on

Bla

de

0 100 200 3000

0.2

0.4

0.6

0.8

1x 10


Position(deg)

Vel

ocity

(m/s

)

Velm is Velideal

Design Parameters



Blade Angle = 0 rad





173

Appendix B - Other Variable Responses of Parametric Monte Carlo Studies

174

Position Error Response

0

0.002

0.004

0.006

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Misalignment(m)

Posi

tion

Erro

r(m)

eYLeYReYFeYB

Position Error Response

0

0.02

0.04

0.06

-0.006


Posi

tion

Erro

r(m) thetaYL

thetaYRthetaYVthetaYFthetaYB

Velocity Error Response

0

0.2

0.4

0.6

0.8

-0.006 -0.004 -0.002 0 0.002 0.004 0.006


Velo

city

Err

or(m

/s) thetaYL

thetaYRthetaYVthetaYFthetaYB

175

Appendix C - Additional Response Plots from Monte Carlo Interaction Studies

176

Position Error Responses for Monte Carlo Simulation with ±±±±.001 meter Translational Bounds and ±±±±.001 radian Rotational Bounds

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaYL(rad)

0.00

0.01

0.02

0.03

0.04

Posi

tion

Erro

r(m)

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaYR(rad)

0.00

0.01

0.02

0.03

0.04

Posi

tion

Erro

r(m)

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaYV(rad)

0.00

0.01

0.02

0.03

0.04

Posi

tion

Erro

r(m)

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaYF(rad)

0.00

0.01

0.02

0.03

0.04

Posi

tion

Erro

r(m)

-0.0010 -0.0005 0.0000 0.0005 0.0010eXL(m)

0.00

0.01

0.02

0.03

0.04

Posi

tion

Erro

r(m)

-0.0010 -0.0005 0.0000 0.0005 0.0010eXR(rad)

0.00

0.01

0.02

0.03

0.04

Posi

tion

Erro

r(m)

-0.0010 -0.0005 0.0000 0.0005 0.0010eYL(m)

0.00

0.01

0.02

0.03

0.04

Posi

tion

Erro

r(m)

-0.0010 -0.0005 0.0000 0.0005 0.0010eYR(m)

0.00

0.01

0.02

0.03

0.04

Posi

tion

Erro

r(m)

177

Velocity Error Responses for Monte Carlo Simulation with ±±±±.001 meter Translational Bounds and ±±±±.001 radian Rotational Bounds

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaYL(rad)

0.0

0.2

0.4

0.6

0.8

Velo

city

Err

or(m

/s)

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaYR(rad)

0.0

0.2

0.4

0.6

0.8

Velo

city

Err

or(m

/s)

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaYV(rad)

0.0

0.2

0.4

0.6

0.8

Velo

city

Err

or(m

/s)

-0.0010 -0.0005 0.0000 0.0005 0.0010thetaYF(rad)

0.0

0.2

0.4

0.6

0.8

Velo

city

Err

or(m

/s)

-0.0010 -0.0005 0.0000 0.0005 0.0010eXL(m)

0.0

0.2

0.4

0.6

0.8

Velo

city

Erro

r(m/s

)

-0.0010 -0.0005 0.0000 0.0005 0.0010eXR(rad)

0.0

0.2

0.4

0.6

0.8

Velo

city

Erro

r(m/s

)

-0.0010 -0.0005 0.0000 0.0005 0.0010eYL(m)

0.0

0.2

0.4

0.6

0.8

Velo

city

Erro

r(m/s

)

-0.0010 -0.0005 0.0000 0.0005 0.0010eYR(m)

0.0

0.2

0.4

0.6

0.8

Velo

city

Erro

r(m/s

)

178

Position Error Responses for Monte Carlo Simulation with Bounds Based on Parametric Sensitivity Analysis

-0.0001 -0.0000 0.0000 0.0000 0.0001thetaYL(rad)

0.000

0.001

0.002

0.003Po

sitio

n Er

ror(m

)

-0.00006 0.00006thetaYR(rad)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m)

-0.00006 0.00006thetaYV(rad)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m)

-0.0001 -0.0000 0.0000 0.0000 0.0001thetaYF(rad)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m)

-0.0005 -0.0003 -0.0001 0.0001 0.0003 0.0005eXL(m)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m

)

-0.0005 -0.0003 -0.0001 0.0001 0.0003 0.0005eXR(m)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m

)

-0.0005 -0.0003 -0.0001 0.0001 0.0003 0.0005eYL(m)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m

)

-0.0005 -0.0003 -0.0001 0.0001 0.0003 0.0005eYR(m)

0.000

0.001

0.002

0.003

Posi

tion

Erro

r(m

)

179

Velocity Error Responses for Monte Carlo Simulation with Bounds Based on Parametric Sensitivity Analysis

-0.0001 -0.0000 0.0000 0.0000 0.0001thetaYL(rad)

0.00

0.02

0.04

0.06Ve

loci

ty E

rror(m

/s)

-0.00006 0.00006thetaYR(rad)

0.00

0.02

0.04

0.06

Velo

city

Erro

r(m/s

)

-0.00006 0.00006thetaYV(rad)

0.00

0.02

0.04

0.06

Velo

city

Erro

r(m/s

)

-0.0001 -0.0000 0.0000 0.0000 0.0001thetaYF(rad)

0.00

0.02

0.04

0.06

Velo

city

Erro

r(m/s

)

-0.0005 -0.0003 -0.0001 0.0001 0.0003 0.0005eXL(m)

0.00

0.02

0.04

0.06

Velo

city

Erro

r(m/s

)

-0.0006 0.0006eXR(m)

0.00

0.02

0.04

0.06

Velo

city

Erro

r(m/s

)

-0.0006 0.0006eYL(m)

0.00

0.02

0.04

0.06

Velo

city

Erro

r(m/s

)

-0.0005 -0.0003 -0.0001 0.0001 0.0003 0.0005eYR(m)

0.00

0.02

0.04

0.06

Velo

city

Erro

r(m/s

)

180

Appendix D – Prototype Self-Tracker Design Drawings (AutoCAD®)

195

Appendix E – Alignment Procedure For Implementing the Self-tracker On the

JT15D

1. Bolt LDV and fold mirror to mounting plate.

2. Measure distance between front four corners of LDV perpendicular to the plane of fold mirror. Ensure that they are as equal as possible. Now, the LDV and fold mirror are considered one assembly and aligned to one another.

3. Thread nosecone adaptor to nosecone of JT15D using the provided tab-locking

washer. Torque using a spanner wrench.

4. Using an aircraft grade bolt, install adjustment plate and torque using a torque wrench and sockets.

5. Using hex-head fasteners, lock washers, and adjustment springs, install the vertex

mirror and mirror retainer. Tighten all screws to hex wrench torque limits.

6. Using the four hanging threaded rods on the frame, raise the LDV/fold mirror assembly to the height of the vertex mirror. It is helpful to turn on the laser and use it as a guide.

7. To obtain a starting point, use a level, ensure that the mounting plate for the

LDV/fold mirror are level on the sides, front, and back. Adjust threaded rods accordingly while focusing the laser beam on the center of the vertex mirror.

8. Attach a piece of black construction paper across the fold mirror ensuring to cover

the hole that the laser travels through. Poke a pin size hole where the laser hits the paper. Adjust the vertex mirror so that the reflected beam lands on this pin hole. Then, rotate the rotor to 0°, 90°, 180°, and 270° positions while taking note of the position of the beam spot on the black paper on the fold mirror. Rotate the base of the frame to cause the beam spot to land on the pinhole for all rotations. How the base of the frame should be moved is dependant on where the beam spot landed for the four rotations relative to the pin spot. For example, for clockwise rotations of the rotor of 90° and 270°, if the beam spot lies on the right side (facing the fold mirror) of the pin spot, the laser/fold mirror should be translated a slight amount in the direction of the beam on the paper, then rotated such that the beam spot lands on the center of the vertex mirror. Thus, the beam spot is closer to the pinhole. This is an iterative process that should be done for horizontal and vertical misalignments of the beam spot on the paper. This minimized the rotational misalignment between the LDV/fold mirror and rotor/vertex mirror/blade. Vertical rotational adjustments are done with the threaded feet of the frame. Now, remove the black paper from the fold mirror.

196

9. Slightly unthread the vertex mirror fasteners to cause the beam to move to a desired radial and chord position on a blade. Mark this point by cutting out an arrow using tape and positioning it on the desired blade position. Again, rotate the rotor in increments of 0°, 90°, 180°, and 270° while taking note of the position of the beam relative to the desired measurement point. Adjust the translational aspects of the frame relative to the rotor/vertex mirror according to the position errors on the blade. This is done in the same manner as mentioned in step 8. Check the rotational misalignments again by using the black paper. A new pinhole can be made. Repeat this procedure until the beam spot is within acceptable distance from the desired measurement location.

197

References

[1] Lomenzo, Richard A. 1998, “Static Misalignment Effects in a Self-Tracking Laser Doppler Vibrometry Systm for Rotating Bladed Disks.” Ph.D. Dissertation, Virginia Polytechnic Institute & State University, Blacksburg, Virginia. [2] Kadambi, J.R., Quinn, R.D., and Adams, M.L., 1989, “Turbomachinery Blade Vibration and Dynamic Stress Measurements Using Nonintrusive Techniques,” Transactions of theASME, 111, pp. 468-474. [3] Storey, P., 1982, “Holographic Vibration Measurement of a Rotating Fluttering Fan,” Joint Propulsion Conference, 82, pp. 234-241. [4] Staheli, W. , 1975, “Inductive Method for Measuring Rotor Blade Vibrations on Turbomachines,” Sulzer Tech Review, Vol. 57, No. 3. [5] Raby, H., 1970, “Rotor Blade Vibrations Observed at the Casing,” presented at the Conference on Methods of Transmitting Signals From Rotating Machinery, June. [6] Roth, H., 1980, “Measuring Vibration on Rotor Blades With Optical Probes,” Brown Buveri Review, Vol. 64, No. 1. [7] Nava, P., 1994, “Design and Experimental Characterization of a Nonintrusive Measurement System of Rotating Blade Vibration,” Journal of Engineering for Gas Turbines and Power, Vol. 116, pp. 658-662. [8] Lesne, J., Fevrier, T., Triquigneaux, P., and Le Floc’h C., 1985, “Vibratory Analysis of a Rotating Bladed Disk Using Holographic Interferometry and Laser Vibrometry,” SPIE-Optics in Engineering Measurements, Vol. 599. [9] Leon, R.L., and Scheibel, J.R., 1986, “Current Status of the EPRI Acoustic Doppler Blade Monitor,” presented at EPRI Steam Turbine Blade Reliability Workshop, Los Angeles, CA, March 18-20. [10] Reinheardt, Andrew K., Kadambi, J.R., and Quinn, Roger D., 1994, “Laser Vibrometry Measurements of Rotating Blade Vibrations,” ASME-IGTI Conference, Vol. 9, pp. 453-461. [11] Boucher, I., Schmiechen, P., Robb, D.A., and Ewins, D.J. “A Laser –based Measurement System for Measuring the Vibration on Rotating Discs,” Vibration Measurements, Vol. 2358, pp. 398-408. [12] Lomenzo, R.A., Barker, A.J., Wicks, A.L., and King, P.S., “A Laser Vibrometry System for Measuring Vibrations on Rotating Disks,” presented at the 4th National Turbine Engine High Cycle Fatigue (HCF) Conference, Monterey, CA February 1999.

198

[13] Drain, L.E., 1980, The Laser Doppler Technique, New York: John Wiley & Sons Ltd. [14] Lentze, Jeff, 2001, Fan Vibration Specialist for Honeywell, Corp., Correspondence with author via email.

199

Vita

Andrew D. Zima, Jr. was born on August 14, 1976 in Pittsburgh, Pennsylvania.

After graduating from Monacan High School in Chesterfield, Virginia, he went on to

Virginia Tech to study Mechanical Engineering. Drew graduated with a Bachelor of

Science Degree in May of 1999. His education at Virginia Tech was continued by

pursuing a Master’s Degree working in the Mechanical Engineering Turbomachinery and

Propulsion Laboratory. While working towards this degree, he worked as a Teaching

Assistant for both Mechanical Engineering undergraduate classes and for the Mini-Baja

car team. Upon completion of the Master’s of Science in May of 2001, Drew plans to

move to Richmond, Virginia and work for DuPont Tyvek.

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