Mathcad in Teaching Rotor and Structural Dynamics

8/2/2019 Mathcad in Teaching Rotor and Structural Dynamics

1/7

Mathcad in Teaching Rotor andStructural Dynamics*

GO RAN BROMAN and STEFAN O STHOLM

Department of Mechanical Engineering, University of Karlskrona/Ronneby, s-371 79 Karlskrona, Sweden

This paper describes how Mathcad is used to deepen understanding of fundamental aspects in thefield of rotor and structural dynamics in two of the undergraduate courses in the education programof Bachelor of Science in Mechanical Engineering with emphasis on Product Development at theUniversity of Karlskrona/Ronneby, Karlskrona, Sweden. Integrating this mathematical softwarehas clearly given an improved interest for mathematics as well as mechanics among the students.

INTRODUCTION

THE MACHINE Elements Course is given inthe second year of the Bachelor program. Rotordynamics is a main part of this course. Other partsare springs, connections, bearings, lubricationtheories, and brakes. The class-room hours are28 for lectures and 28 for supervised individualpractice on sample problems. When problems ofcommon interest occur the teacher deals with themon the white board in a seminar-like manner.Throughout the course students are also perform-ing a larger compulsory rotor calculation assign-ment. With the theories supplied at lectures theyindividually calculate the critical angular speed of

a given rotor. Individual guidance for this assign-ment is given in the exercise hours, but only asanswers to questions raised, since one of thegeneral ideas with the assignment is to make eachstudent seek the knowledge necessary to solve thetask. The Mathcad[2] features of matrix handling,such as eigenvalue calculations, fits perfectly forthis assignment, making it possible for the studentsto advance further in learning dynamics than theyotherwise would have managed.

The Structural Dynamics Course is given in thethird year of the Bachelor program. The class-room hours are 16 for lectures, 32 for supervisedindividual practice on sample problems and 8hours in preparatory work learning the Mathcadsoftware, I-DEAS Master Series [3] as the dynamicsimulation tool, and the experimental facilities inthe structural mechanics lab. After this intro-duction the students have full access to all thenecessary equipment for solving their problems.The same idea of problem-oriented learning asabove is applied. Throughout the course problemsare given without any answers. Even the textbooklacks answers. Instead the students should presentsolved problems in a weekly seminar where theother students bring their questions. The teacher

has to explain and help if the solution is notcorrect. Normally the students come to theteacher and ask for guidance before the seminar,which was the intention.

In addition to the textbook sample problemssome other limited problems are given to thestudents as real physical models, which have tobe analyzed in three main steps: theoretical model-ing, computer-based simulation and experimen-tal verification. Mathcad is here used to solvedifferential equations, especially to calculate thedynamic response to different excitations with theLaplace transform and to correlate the simulationresults and the experimental verification. Mathcadis also used as the report writing tool.

ROTOR DYNAMICS ASSIGNMENT IN THEMACHINE ELEMENTS COURSE

The assignment is presented for the students inthe first week of the course and is then supposed tobe solved within the seven weeks of course dura-tion. Each student has the total responsibility tohand over a carefully written report including, inall aspects, a correct solution before a stated date.Students who fail to do this are not allowed toparticipate in the ordinary theoretical test at the

end of the course and must perform a new assign-ment, when the course is repeated next year, inorder to achieve their final grades. This rule hasturned out to function very well, not the least inmaking the students active in demanding theteacher to share his knowledge. Only very fewstudents have failed so far. During the worksome intermediate results may be checked fromthe teachers solution. Cooperation between stu-dents is allowed and encouraged, but each studenthas an individual set of data and must produce anindividual report. The students have some Math-cad experience from prior courses in mathematics.

In this course they therefore achieve the necessaryadditional knowledge by consulting the manual* Accepted 5 June 1997.

426

Int. J. Engng Ed. Vol. 13, No. 6, p. 426432, 1997 0949-149X/91 $3.00+0.00Printed in Great Britain. # 1997 TEMPUS Publications.


2/7

and/or the teacher whenever problems occur. Thisis probably one of the most effective ways oflearning this kind of software. A translated versionof the 1995 year assignment sheet follows.

Calculation assignment in the Machine ElementsCourse 1995

Figure 1 shows a rotor from a gear box. Theshaft has diameter d 30 mm, Young's modulusE 2X06 1011 Nam2 and density& 7800 kgam3.The two cog-wheels have masses m1 and m2 andare placed on the shaft according to x1 and x2respectively. The rotor is supported by two bear-ings at a distance of L 0X5 m. The bearingstogether with their surrounding support havelinear stiffnesses c1 and c2 respectively in all direc-tions perpendicular to the shaft. The left bearingalso supports a bending moment proportional tothe alignment of the shaft in this bearing, that isM cM . The right bearing cannot support

bending moment.

Question 1.Calculate the ordinary critical angular speeds

with respect to bending deformation of the shaft.Consider the cog-wheels as point masses. Distri-bute the mass of the shaft to the point masses andthe bearings. After that the shaft is consideredmass-less. The inertia of the mass distributed tothe bearings is neglected.

Question 2.For the critical angular speeds (eigenvalues),

calculate the deflections of the shaft at the

masses and bearings in relation to the deflectionat mass m2 (eigenvectors). Draw sketches of theshaft appearance.

Question 3.Make a more accurate analysis by representing

the shaft as a (higher) number of point masses onan otherwise mass-less shaft. Calculate the fourlowest ordinary critical angular speeds withrespect to bending deformation of the shaft, withan estimated error below one percent. Use forexample Mathcad as a calculation tool.

Comments on the rotor dynamics assignmentAfter some work with the laws of dynamics

and solid mechanics, Question 1 gives rise to ahomogenous system of equations, equation (1):

11m1 1

3212 m2

21 m1 22 m2 1

32

PTTR QUUS y1y2 0

0

1

where ij flexibility numbers of the system(inverted stiffness); mi point masses; w angular frequency of the shaft; yi deflections atpoint masses; i 1 F F F 2.

The eigenvalues of this matrix is easily deter-mined by pen, paper and pocket-calculator. Thismethod is however recommended to be comple-mented by Mathcad. Although not necessary itgives a good opportunity to check the results, bothways, and to get familiar with the software for a

relatively simple problem. Mathcad may also be

Fig. 1. Assignment in machine elements course: rotor supported by two bearings.

Mathcad in Teaching Rotor and Structural Dynamics 427


3/7

used for example in calculation of the alpha-values.

Question 2 gives a visualization of the physicalmeaning of eigenvalues and eigenvectors. Toachieve the accuracy required in Question 3 thesize of the matrix widely exceeds what is possible todeal with by pen and paper. To solve this question

some kind of computer aid is necessary. TheMathcad features of matrix handling fits perfectlyfor this task.

s 1

322

A

11 m1 12 m2 1n mn

21 m1 22 m2 2n mnFFF

FFF

FF

FFFF

n1 m1 n2 m2 ann mn

PTTTTR

QUUUUS 3

Y

y1

y2FFF

yn

HffffdIgggge 4

where n is the number point masses, and with I asthe unity matrix, the eigenvalue problem can bedescribed as:

A sIY 0 5

The Mathcad solution then becomes:

s X eigenvalsA 6

and the critical angular frequencies are obtained

from equation (2). The number of point masses isdoubled until the difference between two successiveiterations is negligible.

PROBLEMS IN THE STRUCTURALDYNAMICS COURSE

In parallel with the lectures in the StructuralDynamics Course, the students have to analyze

dynamic characteristics such as natural fre-quencies and mode shapes of some simple struc-tures considering damping and realistic boundaryconditions. The problems are presented as realmechanical models so the answer has to be foundwith experimental testing. The reports must consistof three parts:

theoretical modeling computer based simulation experimental verification.

Two of the problems are described below,followed by a brief example of the solution givenby the students.

Problem 1.A circular solid shaft is connected with a solid

metal sphere and is connected to a plate accord-ing to Fig. 2. Describe the dynamic behavior of

the system if the plate is secured to a rigidsurface. The material in the model is plaincarbon steel.

Solution example.Since the system has one dominant natural

frequency the system is modeled as a dampedSDOF system with lumped parameters. Theapproach with the generalized parameter model,Craig [1], can be used:

m# c # k kg# 0 7

where # is a function of location and time t. The

general assumed-modes form in equation (8) isused to solve the differential equation (7):

#xY t 2x#t 8

As the assumed-mode shape function the staticdeflection under a tip force is used:

2x x2

2L2

x3

6L39

Fig. 2. A solid metal sphere connected to a circular solid shaft.

G. Broman and S. Ostholm428


4/7

The generalized parameters can then be calculatedas follows:

m

L0

&A22x dx ms22L

c ccr

kL

0

EI2 HH2 dx

kg

L0

ms2H2 dx

10

where ccr is the critical damping coefficient. Thenatural frequency is calculated from:

fn 1

2%

k kg

m

r11

The calculation of the integrals and derivatives areperformed using Mathcad.

To calculate the response of the system, thedamping characteristics have to be determinedexperimentally. The viscous damping coefficient is estimated using the half amplitude method. Animpact excitation is performed on the model whilerecording the response from the laser vibrometerinto a PC measuring system, HP 3565. Theresponse curves are imported to Mathcad andanalyzed.

To calculate the response from an impact excita-tion the Laplace transform capabilities in Mathcadare used. Unfortunately Mathcad can't transforma symbolic differential function, but this is theeasier part.

m# c # k kg# pt 3Laplace

ms2V csV

k kgV Ps 12

where pt is the excitation force. The transferfunction of the system becomes:

Hs 1

ms2 cs k kg13

since the excitation can be approximated with aDiraq pulse this is also the response:

Vs HsPs 1

ms2 cs k kg14

The corresponding time response is calculated,using the inverse Laplace transform function, tobe:

#t 2eca2mtsin

4mk kg c2

p2m

t

4 5

4mk kg c

2p 15Diagrams of both the measured and the calculatedresponse are then presented and the correlationbetween them discussed.

Problem 2.A solid beam made of plain carbon steel is to be

analyzed, see Fig. 3. The dimensions and boundaryconditions appears from the experimental modelaccording to the figure. Determine the following:

Natural frequencies and corresponding modeshapes.

What should the dimension of the cross-sectionbe if the first natural frequency in the width-direction should be half of the natural frequencyin the height-direction?

Solution example.The same method as in Problem 1 is often used

to calculate the first natural frequency and thedeformation shape for this mode. Since thissystem doesn't have one clearly dominant naturalfrequency as in Problem 1 the system has to beanalyzed as a MDOF system. The system isanalyzed as free undamped vibration of an uni-

form cantilever beam. The equation of motionbecomes, Craig [1]:

EI#HHHH &A# 0 16

Assuming a harmonic motion given by theequation:

#xY t 2x cos 3t 17

and substitute this into equation (16) gives us theeigenvalue equation:

d42

dx4 !42 0 18

where

!4 &A32

EI19

the general solution is given by:

2x C1 sinh !x C2 cosh !x

C3 sin !x C4 cos !x 20

The boundary conditions are:

20 0Yd2

dx0 0Y

d22

dx2 L 0Yd32

dx3 L 0

21

Fig. 3. A solid beam made of plain carbon steel.



5/7

(a)

Fig. 4. Example of Mathcad calculation.



6/7

After some work this leads to:0 1 0 1

! 0 ! 0

!2 sinh !L !2 cosh !L !2 sin !L !2 cos !L

!3 cosh !L !3 sinh !L !3 cos !L !3 sin !L

PTTTR

QUUUS

C1

C2

C3

C4

PTTTR

QUUUS

0

0

0

0

PTTTR

QUUUS 22

for the non-trivial solution we found the charac-teristic equation:

cos !L cosh !L 1 0 23

whose roots are the eigenvalues !r L. Values of!r L

can be found in the textbook which leads toequation (24) for the natural frequencies:

3r !r L

2

L2EI

&A

1a224

The normalized mode shapes can after some workbe calculated by the equation:

2rx cosh !r x cos !r x

krsinh !rx sin !r x 25

where

kr cosh !r L cos !r L

sinh !r L sin !r L 26

(b)

Fig. 4. (Continued).



7/7

The resulting mode shapes are then plotted inMathcad and compared with the simulated resultsfrom I-DEAS Simulation. As an example, asample calculation is presented in Fig. 4 showinghow the assignment is performed in Mathcad.

CONCLUSIONS

After introducing the mathematical softwareMathcad in our courses we notice an improvedinterest for mathematics as well as mechanicsamong the students. The fear of approachingproblems leading to advanced mathematics hasclearly decreased. This type of mathematical soft-ware gives the opportunity to go further in thestudy of dynamics than otherwise possible. The

problems described gives the students a feeling ofhow to correlate simulated and measured resultswith an analytical model. The awareness of thedifferences between defining the boundary condi-tions theoretically and the possibilities to actuallyfasten structures in practice is very important.Also, presenting problems without written answers

makes the students, after some time, confident intheir way of working and their own results. Theyget a good training in critically analyzing theirresults, which is what reality is about. The poorpossibilities of working with, for example, loopsand conditions as well as the lack of symbolicLaplace transforms in Mathcad has been experi-enced as disadvantageous. We are happy to con-clude that some of these and other drawbacks areeliminated in the latest update of the software.

REFERENCES

1. R. R. Jr. Craig, Structural Dynamics, An Introduction to Computer Methods, The University ofTexas at Austin, J. Wiley & Sons (1981).

2. Mathcad PLUS 5.0, MathSoft, Inc., 101 Main Street, Cambridge, Massachusetts, USA.3. I-DEAS Master Series 2.1, SDRC, 2000 Eastman Drive, Milford, Ohio 45150, USA.

Goran Broman. Dr. Broman received his Ph.D. in Mechanical Engineering 1993 at LundInstitute of Technology, Sweden, on the subject `On The Flat Spiral Groove ThrustBearing'. Dr. Broman has been working as Assistant Professor at the Department ofMechanical Engineering, University of Karlskrona/Ronneby since 1993. He is nowresponsible for the Masters program in Structural Mechanics, research supervisor anddeputy head of the department.

Stefan O stholm. Dr. O stholm received his Ph.D. in Mechanical Engineering 1991 at LundInstitute of Technology, Sweden, on the subject `Simulation and Identification of Mechani-cal Load Distribution in Tool Edges'. Dr. O stholm has been working several years at theDepartment of Production and Materials Engineering, Lund Institute of Technology andsince 1992 as Assistant Professor at the Department of Mechanical Engineering, Universityof Karlskrona/Ronneby. He is the head of the department since 1993 and research leader ofthe research group in mechanical engineering.


Mathcad in Teaching Rotor and Structural Dynamics

Documents

Transcript of Mathcad in Teaching Rotor and Structural Dynamics