Noise reduction applied to a decanter centrifuge - TU/e

Noise reduction applied to a decantercentrifuge

A.J. van Engelen

DCT2009.069

Research report, research performed at University of Canterbury, New Zealand

New Zealand Supervisor: Dr. J.R. Pearse

University of CanterburyDepartment of Mechanical Engineering

Supervisor: Prof.dr. H. Nijmeijer

Eindhoven University of TechnologyDepartment of Mechanical EngineeringDynamics and Control

Christchurch (New Zealand), April 2009

Summary

Structural vibrations can be problematic for a (part of a) machine. They contribute tothe wear of the machine, but can also produce a high noise level. In this report a noiseand vibration survey has been performed on an existing design of a gearbox cover of adecanter centrifuge. A finite element model of this gearbox cover is developed to pre-dict the structural vibrations, which has been verified by measurements. A boundaryelement model is used to predict the sound power level produced by this vibrating struc-ture. By adapting this model with respect to material properties the noise reduction isforecasted. It can be concluded that if a 2 mm thick steel gearbox cover is replaced bya 4 mm thick ultra high molecular weight polyethylene one, that the sound productionwill increase with 9 dB for frequencies up to 400 Hz and will decrease with 4 dB forfrequencies in the range of 400 to 540 Hz.

iii

iv Summary

Contents

Summary iii

1 Introduction 1

2 Decanter Centrifuge 3

2.1 Main parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Measuring and modeling noise and vibrations 7

3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Running tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Eigenfrequency and mode shape extraction . . . . . . . . . . . . . . . . 10

3.3.1 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . 11

3.4 Noise radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4.1 Boundary Element Method . . . . . . . . . . . . . . . . . . . . . 12

3.4.2 Radiation Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4.3 Radiated Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Implementation, verification and results of a numerical model 15

4.1 Eigenfrequencies and mode shapes extraction . . . . . . . . . . . . . . . 15

4.1.1 Static impact test . . . . . . . . . . . . . . . . . . . . . . . . . . 15


4.1.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

v

vi Contents

4.1.4 Comparison between numerical and experimental results . . . . . 19

4.2 Running mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.1 Measurement of the forces acting on the gearbox cover . . . . . . 22


4.2.3 Numerical results for the response in running mode . . . . . . . . 25

4.3 Noise radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25


4.3.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Design changes 31

5.1 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.1 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.3 Harmonic response . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2.4 Noise radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Conclusion and Recommendations 39

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Bibliography 41

A Decanter Layout and technical data 43

B Matlab Code to import measurement data from PULSE 45

C Ansys input code for the structural dynamic analysis 49

D Structural eigenmodes of the gearbox guard that correspond withstatic impact tests 55

Contents vii

E Results of the forced response of the gearbox guard made out ofUHMWPE without damping 59

viii Contents

Chapter 1

Introduction

G-Tech is a Christchurch based manufacturer of centrifuges. To comply with the de-mands of their costumers their products should produce a minimum amount of noise.So the vibrations that contribute to the radiated noise should be minimized. The sub-ject is to identify the primary modes of vibrations and the contributions of these tothe radiated noise field of one of their products, the G-Tech 1456. The main objectiveis to identify the primary modes of vibration and their contribution to the radiatedsound field. Moreover, using numerical models the influence of design changes can bepredicted. Because of the complex geometry of the machine, in this study only thegearbox cover is examined.

The objective is to collect data for impact excitation of the gearbox cover in boththe operating (with the machine switched on) and non-operating (static) state. In theoperating state the cover will be excited due to vibrations that are generated by therotating parts, in the static case the cover is excited with a rubber hammer, i.e. afteran impact. Measurements in the frequency domain are made with an accelerometer toidentify the most dominant frequencies. The gearbox cover is modeled in SolidWorksso the model can be imported to a finite element modeling package to carry out anumerical study. With a modal analysis the eigenfrequencies and corresponding modeshapes can be calculated. The accuracy of this model can be determined by comparingthe numerical results with the measurements after the impact excitation. Moreover,the numerically calculated vibrations can be used to compute the radiated sound powerusing boundary element method.

The report is organized as follows. In Chapter 2 the working principle of the decantercentrifuge (like the G-tech 1456) is explained. Moreover, the principal componentsare described in detail. In Chapter 3 the response of the main parts after an impactexcitation are investigated. Also the theory used for the experiments and numericalmodel is described. For the gearbox cover in particular in Chapter 4 it is described howa numerical model is implemented and verified by measurements. The most dominantvibrational frequencies using static impact excitations are listed and compared with

1

2 Chapter 1. Introduction

a modal analysis of a finite element model. Next the sound power radiated by thegearbox cover is calculated using boundary element method. In Chapter 5 the model ischanged with respect to material properties to investigate if it is worthwhile to replacethe gearbox cover by a cover made of a different material. Finally in Chapter 6 theconclusions and recommendations will be presented.

Chapter 2

Decanter Centrifuge

In this report the G-Tech 1456 decanter centrifuge is under investigation. Before goinginto detail about the process of noise reduction, the working principle and the mainparts of the centrifuge are described.

2.1 Main parts

In Figure 2.1 a cross-section of the decanter with the main parts is presented (seeAppendix A for a larger one). The bowl, conical and conveyor are supported by twomain bearings that are mounted on the base. Two electric motors (main and back-drive) are coupled to the rotating assembly through v-belt drives. The screw conveyoris connected through a spline coupling to a gearbox, which makes it possible to rotatethe conveyor slightly slower than the bowl. Throughout this work the rotating speed ofthe bowl is 3250 RPM. More technical data can be found in Appendix A.

2.2 Working principle

The decanter centrifuge is a piece of machinery that is used to separate different liquidsor solids from liquids. It uses centrifugal forces that enforce the liquid (or solid) withthe highest density to be near the surface of the bowl while the liquid with the lowerdensity is floating on this layer (at a smaller radial position). Due to the slightlydifferent rotational speed of the conveyor with respect to the bowl (4 to 48 RPM) thehigh density fluid is conveyed upwards into the conical and will finally exit the decanterat the solids discharge. At the conical a beach will be formed, see Figure 2.2. On theopposite side (in axial direction) of this beach the fluid with a lower density will overflowin the liquids discharge.

3

4 Chapter 2. Decanter Centrifuge

Solids EndLiquids End

Base

Clutch Gearbox Main bearing 1 Bowl Conveyor Main Bearing 2 Feed pipeConical

Solids EndLiquids End

Base

Clutch Gearbox Main bearing 1 Bowl Conveyor Main Bearing 2 Feed pipeConical

Figure 2.1: Cross section of the G-tech 1456 decanter centrifuge.

Figure 2.2: Visualization of the separation process.

2.2. Working principle 5

The decanter centrifuge has the advantage that it discharges continuously. In addi-tion it is able to separate fluids with small density difference. Gravity sedimentation,like large-tank clarifiers, have to run an uneconomically long time in this case. Besidesthe decanter can handle a wide range of feed slurry concentrations and produces driersolids than other centrifuges. These are the main reasons that decanter centrifuges arewidely used. Disadvantages are the high power consumption and high wear of the screwconveyor. Examples of applications are found in the chemical industry, waste sludgeprocessing, minerals extracting and processing [Rec01].

At G-Tech the decanter centrifuge is completely modeled with the 3D ComputerAided Design (CAD) software SolidWorks. The focus is set on the production process.The development of new products starts with improving their recent products. Thecompany does not use any model for the vibration analysis for this product.

6 Chapter 2. Decanter Centrifuge

Chapter 3

Measuring and modeling noiseand vibrations

A decanter centrifuge produces a lot of noise while it is in operating mode. To un-derstand the underlying principles of the sound production, first the vibrations of themachine are investigated. These vibrations are measured using a tri-axial accelerome-ter. Next a numerical model is used to identify the shape of the vibrating structure.Finally the model is used to calculate the sound production of the structure. Moreover,the effects of changes in the design are predicted.

3.1 Experimental setup

For the vibrational experiments a tri-axial accelerometer and a Bruel & Kjær PULSEsystem are used. The tri-axial accelerometer is put on the surface of several parts ofthe decanter centrifuge using some wax. The accelerometer is positioned such that thez-axis represents the axial axis of the centrifuge, the y-axis is pointing upwards and thex-axis sidewards. The three generated signals are amplified and processed in the PULSEsystem and visualized on a laptop. From these signals a Fast Fourier Transformation(FFT) is made to investigate which frequencies are dominant in the response. TheFourier transform G(jω) of a time signal g(t) is given by [Bro85] as

G(jω) =∫∞0 g(t)e−jωtdt ≈ ∫ T

0 g(t)e−jωtdt, (3.1)

with T the (large enough) time span of the data and ω the angular frequency. Usingdigital signal processing equipment, like the PULSE system, this time signal will not becontinuous, but consists of N discrete samples. When the frequency range of interestgoes from 0 −W Hz, a sampling frequency of at least 2W should be used to preventaliasing, [Bro85]. This implies a maximal sample time of 1

2W and a total time span of

7

8 Chapter 3. Measuring and modeling noise and vibrations

T = N2W . The discrete Fourier transform is approximated by

Gk ≈ ∑N−1k=0 gke

−j2πk nN . (3.2)

Both static and running measurements have been performed. In case of the staticmeasurements the main parts are excited with a rubber hammer and the response of thepart is measured. The number of samples is set to N = 213 = 8192, with a frequencyspan of 2 Hz. In this way frequencies up to 16384 Hz will be measured. To preventaliasing a sampling frequency of 32765 Hz (two times the highest frequency of interest)is used. A trigger is set to 13.6 m/s2 so that the measurement starts after the impactand a Hanning window with maximum overlap is used to get the best results withoutlosing too much of the original signal. Only one average is taken in the static tests,because the amplitude of the response is decaying fast after the impact.

For the running tests, i.e. with the bowl and conveyor rotating, a FFT of the signalsis made as well. In this case a trigger has not been used, but the measurements aredone in a free run. Because the decanter is vibrating continuously an average of 400samples is taken to obtain reliable results.

3.2 Running tests

To get a first idea of the dominating vibrations that can cause the produced noise arunning test is performed. A hose is attached to the decanter to simulate a processingsituation, see Figure 3.1. The rotational speed of the bowl is set to 3250 RPM (54 Hz).The tri-axial accelerometer is glued to the main parts of the decanter to obtain theacceleration response of these main parts. Several points are used for each part. Theresults of the running tests are saved in a .txt file that is imported into Matlab. Thedata contains the amplitudes of the acceleration at each frequency of the FFT analysis,so 8192 amplitudes for one channel. To investigate which frequencies are dominating theresponse, the data is sorted by descending amplitudes, see Appendix B for the Matlabfile used. For each part the first 3 dominating frequencies are listed in Table 3.1. Ascan be seen most of the dominating frequencies are a multiplication of the rotationalfrequency (54 Hz). This is due to the unbalance of the rotating parts. For example, thegearbox cover is vibrating with the highest amplitude at a frequency of 270 Hz, whichis the 5th harmonic frequency of the rotational frequency of the bowl.

Because of the complex geometry of the decanter, the choice is made to do the noiseand vibration analysis on some parts separately. In this report the gearbox cover isanalyzed. The gearbox cover is chosen, because this cover is vibrating with the highestamplitude (measured with a tri-axial accelerometer) of all measured parts, with thedecanter centrifuge in operating mode. Besides, the experience is that the connectionpoints of the cover to the base fail relatively fast.

The analysis exists of both a measurement of the response after a static impactand a modal analysis of the gearbox cover with the finite element package Ansys. The

3.2. Running tests 9

Part Frequency Harmonic freq. w.r.t. Amplitude perc. of highest[Hz] rotational freq. [-] [m/s2] amplitude [%]

Gearbox Cover 270 5 115.43 100216 4 59.08 51432 8 42.44 37

Grey lid 378 7 15.80 100324 6 14.37 91216 4 10.38 66

Hopper 108 2 10.49 10054 1 9.35 89216 4 8.36 80

Main Bearing 2 54 1 9.15 10010856 6.92 76162 3 6.53 71

Base 54 1 9.14 100378 7 5.74 63270 5 5.51 60

Backdrive guard 732 4.90 100144 4.57 9354 1 2.79 57

Solid end cover 108 2 1.89 1003320 1.88 992978 1.58 84

Base frame 54 1 0.72 1003320 0.42 58108 2 0.37 51

Table 3.1: Most dominant frequencies of some main parts with the decanter running at3250 RPM.


Solids end Liquids end

Water hose Solids end cover Lid Hopper Base Base frame Gearbox cover

Figure 3.1: A water hose was attached to the decanter to perform a running test. Theflow was set to 4 m3/s.

results of the static impact test are used to obtain the eigenfrequencies and the resultsof the modal analysis are used to identify the corresponding mode shapes. Moreover,the results of the computer model of the gearbox cover can be compared with themeasurements.

3.3 Eigenfrequency and mode shape extraction

To extract the eigenfrequencies of the main parts a static impact test is performed.With a rubber hammer the part is excited. After an impact in the normal directionof a surface, the part will start vibrating in the same direction (out of plane). Dueto this impulse the part is excited in all frequencies of interest. By positioning theaccelerometer at different positions on this surface (the same positions as in the runningtest) all eigenmodes within the frequency range of interest can be extracted. When onlyone position would be used, the chance exists that the accelerometer is on a nodal lineof a mode.

3.4. Noise radiation 11

3.3.1 Numerical implementation

To identify the mode shapes that correspond with the eigenfrequencies a finite element(FE) model is used. G-Tech provided Computer Aided Design (CAD) part files that arecreated with SolidWorks software. These files can be exported as IGES files, which canbe imported in the commercial FE software package Ansys. Within Ansys the model ismeshed. Moreover the material properties and boundary conditions are assigned.

As the main parts are made of steel, damping is not taken into account in thenumerical analysis. The structural dynamics of an undamped system are given by theequations of motion [Ans07]:

Mu + Ku = 0, (3.3)

with u the column with degrees of freedom and M and K the mass and stiffness matricesrespectively. The eigenvalue problem corresponding to this free vibrating undampedsystem is given by:

(−ω2M + K)u = 0. (3.4)

The values for ω for which the determinant of the matrix [−ω2M+K] equals zero are theeigenvalues. The modal analysis function in Ansys solves this eigenvalue problem andstores the eigenvalues λ = ω2 in a diagonal matrix λi. The corresponding eigenvectorsφi satisfy:

(K− λiM)φi = 0. (3.5)

Physically the eigenvalues and eigenvectors represent the undamped eigenfrequencies[rad/s] and the corresponding mode shapes respectively.

3.4 Noise radiation

With the results of a vibrational analysis it is possible to compute acoustic pressuresin the surrounded fluid by a structure using the Helmholtz differential equation. Onlythe surface of a vibrating structure in contact with the fluid, also called wetted surface,is able to transfer energy to the fluid. Therefore the Helmholtz differential equationcan be reduced to an integral equation that covers only the boundary surface S. Thecomplete derivation can be found in [Vis04]:

α(~x)p(~x) =∮S

(∂G(r)∂ny

p(~y) + iωρ0G(r)vny(~y))dS + pin(~x). (3.6)

In (3.6) the acoustic pressure and normal velocity are related to the radiated pressurefield in the fluid domain. The term α(~x) is a geometry related coefficient, ~y is a pointon the boundary surface S and ~x is a field point in the fluid domain. The unit normalto the surface at source point ~y, denoted as ~ny, is pointed into the fluid domain. Thedistance r is the length of vector ~r that is directed from the source point ~y to the fieldpoint ~x : r = ||~x − ~y||. The term pin represents the incident acoustic wave in the case


of a scattering analysis and G(r) is the Green’s function, which represents the effectobserved at point ~x created by a unit source located at point ~y.

In order to solve the Helmholtz integral equation (3.6) for a field point ~x the normalvelocity and pressure at the surface S should be known. If only the normal velocitiesare known, first the pressures at the surface have to be calculated by replacing ~x = ~yin (3.6). Secondly the pressures at the field points can be calculated.

3.4.1 Boundary Element Method

The analysis of vibrations and the resulting radiated sound can be done with the useof sophisticated computer software, e.g. for calculating the dynamics of a practicalstructure a Finite Element (FE) Model with appropriate boundary conditions can beused. This demands a discretization of the structure into a number of finite elements.If the structural dynamics are of interest all elements, interior and boundary, are tobe accounted for, as they are a measure of the total mass and stiffness of the system.For the total radiated sound power however, only the elements on the boundary havea contribution. Only these elements are in contact with the fluid to which energyis transferred. For the purpose of determining the radiated sound by a structure aBoundary Element (BE) model will suffice. The advantage of a BE model is that lessequations are to be solved, as in general there are less nodes in a BE model comparedwith a FE model. In this report Ansys is used for the FE analysis and LMS Virtual.Lab,together with SYSNOISE, for the BE analysis.

From the structural vibration data at the nodes the acoustic pressures at the surfacecan be calculated with the Helmholtz integral equation (3.6). This requires a discretiza-tion of this continuous equation, see [Vis04], giving

Ap = Bv, (3.7)

where the matrices A and B are the influence matrices. These matrices are dependenton the geometry of the structure and comply with the Helmholtz integral equation (3.6).So with the velocities from the structural vibration data the sound pressure at each nodecan be calculated.

3.4.2 Radiation Efficiency

A useful measure of the effectiveness of sound radiation by a vibrating surface is thetotal radiated sound power normalized with respect to the specific acoustic impedanceof the fluid medium, the structure area and the velocity of the surface vibration, whichis defined as the radiation efficiency. A commonly used measure of the surface vibrationis the space-average value of the time-averaged squared vibration velocity defined by

v2n = 1

S

∫S

(1T

∫ T0 v2

ny(~y)dt

)dS, (3.8)


where T is a suitable period of time over which to estimate the mean square velocityv2ny

at a point ~y and S extends over the total vibrating surface.

The radiation efficiency is defined by reference to the acoustical power radiated bya uniformly vibrating baffled piston at a frequency for which the piston circumferencegreatly exceeds the acoustic wavelength k: ka À 1. For the radiated power of a baffledpiston the following relation holds:

P = 12ρ0cSv2

n. (3.9)

The definition of the radiation efficiency is thus:

σ = P/ρ0cSv2n. (3.10)

3.4.3 Radiated Power

In the case of a structure modeled within a FE software package the structural vibrationof this structure with R elements can be calculated with FE method. In this way it ispossible to obtain a column vector of complex velocities at each element center causedby a harmonic point force. The velocities are grouped in a column vector, like:

ve =[

ve1 ve2 . . . veR

]T. (3.11)

With the calculated velocities the sound pressure and radiated sound power can becalculated within a BEM package. The obtained sound pressure at each element is alsogrouped in a column vector:

pe =[

pe1 pe2 . . . peR

]T. (3.12)

From a BE model the relation between the elemental velocities and sound pressures canbe found. As a result of (3.7) the sound pressures can be denoted as

pe = A−1Bve (3.13)

Analytical equations for the radiated sound power [Fah87] give the relationship betweenthe velocities and sound pressures as in

P (ω) =R∑

r=1

12AeRe(v∗erper) = S

2RRe(vHe pe), (3.14)

where Ae and S are respectively the areas of each element and of the whole structure.Substituting (3.13) into (3.14) result in

P (ω) = S2RRe(vH

e A−1Bve). (3.15)


3.5 Summary

To compute the radiated sound field by a vibrating structure a computer model can beused. First the dynamics of the structure should be identified. The eigenfrequencies andcorresponding mode shapes extracted by a numerical software program are comparedwith physical measurements of the excited structure after an impact. This comparisonof the eigenfrequencies gives an indication of the accuracy of the model. The vibrationsthat contribute to the radiated sound field can be computed by solving the Helmholtzintegral equation for a certain frequency domain. By this equation the harmonic veloc-ities of the vibrating structure are related to the resulting acoustical pressures at thesurface. The radiated sound power can be computed from this acoustical impedance.

Chapter 4

Implementation, verification andresults of a numerical model

A numerical model is used to compute the forced harmonic response of the gearboxcover. Consequently this harmonic response is used to compute the radiated soundfield. Experiments are made to investigate the accuracy of the finite element model.

4.1 Eigenfrequencies and mode shapes extraction

4.1.1 Static impact test

The measured accelerations during the static impact test performed on the gearboxcover can be found in Figure 4.1. Note that the amplitudes are normalized so that thehighest response has an amplitude of 1 m

s2 . Moreover, in Table 4.1 the first 10 frequenciesthat correspond with these accelerations are listed. Some peaks are close to each other,e.g. peaks can be found at 36 and 38 Hz. This is due to the fact that the spectrumis an average of multiple measurements at different positions and that an interval of 2Hz is used during the measurements. These peaks are listed in Table 4.1 as being onenatural frequency, that belongs to the highest peak.


To identify the mode shapes that correspond with the eigenfrequencies a finite element(FE) model is used. The provided CAD file of the gearbox guard existed of an assemblycontaining the back drive belt guard and gearbox cover, see Figure 4.2. Basically thegearbox cover is only connected to the backdrive belt guard with four bolts throughthe small holes in the sides. To simplify the model as much as possible it is decidedto leave the backdrive belt guard out the analysis and to apply constraints to the four

15

16 Chapter 4. Implementation, verification and results of a numerical model

101

102

103

104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Acc

eler

atio

n [m

/s2 ]

XYZ

Figure 4.1: The dominating frequency responses of the gearbox cover during staticimpact test.

Frequency [Hz]X Y Z

122 540 490192 350 1322234 570 442362 114 354350 666 43436 362 728

114 562 114296 1008 1176680 462 542438 442 454

Table 4.1: Most dominant frequencies of the gearbox cover during a static impact test.

4.1. Eigenfrequencies and mode shapes extraction 17

Figure 4.2: CAD file of the assembled gearbox guard, provided by G-Tech.

(a) Gearbox cover, originally. (b) Gearbox cover, adapted with frontplate.

Figure 4.3: The original and adapted gearbox cover, without backdrive belt guard.

connection points. Because the model does not contain the front panel, this plate ismodeled additionally in SolidWorks, see Figure 4.3.

The next step is to save this CAD file as an IGES file that can be imported intoAnsys. Because the gearbox cover is made of 2 mm thick steel, shell elements can beused. This requires Ansys to import only areas instead of a (solid) volume. The choiceis made to keep only the outside areas, that subsequently are meshed with 8 node shellelements. These elements use quadratic shape functions that are more accurate than 4node shell elements, with linear shape functions. Special attention is paid to the backpanel with the ventilation holes in it. Around these holes a finer mesh is used comparedto the other areas. The result is an element size of 20 mm for the large areas and a meshsize of 3 mm around the small holes. In total the model has 9803 elements and 34067nodes with 6 degrees of freedom (DOF). The mesh used is presented in Figure 4.4. Themodel would be less accurate when larger elements would be used as can be seen ina plot of the eigenfrequency versus the mode number in Figure 4.5. In this figure the


1

X

Y

Z

Modalanalysis of Gearbox Guard

APR 3 2009

14:22:15

ELEMENTS

Figure 4.4: The finite element mesh of the gearbox cover.

Property ValueYoungs Modulus [GPa] 207Poisson ratio [-] 0.3Density [kg/m3] 7800

Table 4.2: Material properties of low carbon mild steel.

element sizes correspond with the default element sizes of the larger areas, like the sideand top panels.

The gearbox cover is made of low carbon mild steel, the material properties arelisted in Table 4.2, [Hea97] [Ger99]. In Ansys the linear isotropic elastic material modelis selected. The static impact test is performed with the gearbox cover mounted on thedecanter, so boundary conditions are needed to represent the same situation. Thereforethe translational DOF’s in Y and X direction of the nodes that are attached to themachine are deleted. Moreover all DOF’s, except for the translational degree of freedomin X direction, at the nodes around the small holes in both sides are deleted to representthe situation that the cover is bolted to the backdrive belt guard.

4.1.3 Numerical results

The first 12 mode shapes calculated with Ansys are presented in Figure 4.6. As canbe seen the modes consist of panels (back, front and side) vibrating out of plane. The

4.1. Eigenfrequencies and mode shapes extraction 19

0 50 100 150 2000

200

400

600

800

1000

1200

1400

1600

1800

Mode nr [−]

Fre

quen

cy [H

z]

20 mm40 mm60 mm80 mm

Figure 4.5: Eigenfrequency versus mode number for different default element sizes.

back-panel has the lowest stiffness (due to the holes) and consequently starts resonatingat the lowest eigenfrequency.

4.1.4 Comparison between numerical and experimental results

The results of the static impact measurements contain the dominant frequencies in X, Yand Z directions. The different panels of the gearbox cover will vibrate out of plane. Thisimplies that the dominant frequencies in X direction are generated by the side panels.Even so, the response in the Y-direction is generated by the top panel and the responsein Z direction by the back panel. To identify the corresponding mode shape fromthe modal analysis one needs to check the mode shape that occurs at a correspondingfrequency from the measurements. If the direction (X, Y or Z) corresponds with the outof plane movement of the right panel, this will indicate that the Finite Element Modelcomplies with the measurement.

In Table 4.3 the numerically determined eigenfrequencies that correspond with thedominating frequencies from the static impact test are listed. Sometimes it is impossibleto identify the mode shape that corresponds with a peak in the static impact test. Inthat case a - is denoted for the eigenfrequency calculated with the modal analysis. Thenumerically calculated mode shapes that correspond with the lowest eigenfrequencies inX, Y and Z direction determined during the static impact test can be found in Figure 4.7.Mode shapes for higher frequencies can be found in Appendix D. The relative largeerror between the numerical and experimental result for the first eigenfrequency is dueto the applied boundary conditions at the holes where the cover is bolted to the back


1

MN

MX

X

Y

Z


.226E-05.131505

.263008.394511

.526014.657517

.78902.920523

1.0521.184

APR 5 200914:58:47

PLOT NO. 1

NODAL SOLUTION

STEP=1SUB =1FREQ=42.711USUM (AVG)RSYS=0DMX =1.184SMN =.226E-05SMX =1.184

(a) Mode 1 @ 43 Hz

1

MN

MX

X

YZ


.639E-03.097453

.194266.29108

.387894.484707

.581521.678335

.775148.871962

APR 5 200915:09:29

PLOT NO. 1

NODAL SOLUTION

STEP=1SUB =2FREQ=50.445USUM (AVG)RSYS=0DMX =.871962SMN =.639E-03SMX =.871962

(b) Mode 2 @ 50 Hz

1

MN

MX

X

Y

Z


.547E-04.09996

.199866.299772

.399678.499584

.59949.699395

.799301.899207

APR 5 200915:10:14

PLOT NO. 1

NODAL SOLUTION


(c) Mode 3 @ 54 Hz

1

MN

MX

X

Y

Z


.251E-03.159127

.318002.476878

.635753.794629

.9535041.112

1.2711.43

APR 5 200915:10:53

PLOT NO. 1

NODAL SOLUTION


(d) Mode 4 @ 69 Hz

1

MN

MX

X

YZ


.002337.142421

.282506.42259

.562674.702759

.842843.982928

1.1231.263

APR 5 200915:11:40

PLOT NO. 1

NODAL SOLUTION

STEP=1SUB =5FREQ=74.457USUM (AVG)RSYS=0DMX =1.263SMN =.002337SMX =1.263

(e) Mode 5 @ 74 Hz

1

MN

MX

X

YZ


.179E-03.158863

.317547.476231

.634915.793599

.9522831.111

1.271.428

APR 5 200915:12:00

PLOT NO. 1

NODAL SOLUTION


(f) Mode 6 @ 89 Hz

1

MN

MX

X

YZ


.809E-03.14373

.28665.429571

.572492.715413

.8583331.001

1.1441.287

APR 5 200915:13:30

PLOT NO. 1

NODAL SOLUTION


(g) Mode 7 @ 89 Hz

1

MN

MX

X

YZ


.258E-03.20385

.407442.611034

.8146271.018

1.2221.425

1.6291.833

APR 5 200915:13:54

PLOT NO. 1

NODAL SOLUTION


(h) Mode 8 @ 91 Hz

1

MN

MX

X

YZ


.001895.248321

.494747.741172

.9875981.234

1.481.727

1.9732.22

APR 5 200915:14:10

PLOT NO. 1

NODAL SOLUTION


(i) Mode 9 @ 112 Hz

1

MN

MX

X

Y

Z


.230E-03.160373

.320517.48066

.640803.800947

.961091.121

1.2811.442

APR 5 200915:14:35

PLOT NO. 1

NODAL SOLUTION


(j) Mode 10 @ 115 Hz

1

MNMX

X

Y

Z


.527E-03.128826

.257124.385423

.513722.642021

.77032.898618

1.0271.155

APR 5 200915:15:07

PLOT NO. 1

NODAL SOLUTION


(k) Mode 11 @ 115 Hz

1

MN

MX

X

Y

Z


.860E-04.080178

.16027.240363

.320455.400547

.480639.560732

.640824.720916

APR 5 200915:15:23

PLOT NO. 1

NODAL SOLUTION


(l) Mode 12 @ 129 Hz

Figure 4.6: First 12 structural modes of the Gearbox Guard.

4.2. Running mode 21

Direction Eigenfreq. static Eigenfreq. modal Rel. errorimpact test [Hz] analysis [Hz] [%]

X 36 50 38.9(X,Y),Z 114 (-,-), 115 (-,-), 0.9

X 122 129 5.7X 192 184 -4.2X 234 239 2.1X 296 302 2.0X 350 356 1.7Y 350 350 0.0Z 354 356 0.6Y 362 379 4.7Z 434 437 0.7X 438 - -Z 442 438 -0.9Y 442 442 0.0Z 454 464 2.2Y 462 - -Z 490 494 0.8Y 540 553 2.4Z 542 551 1.7Y 562 560 -0.4Y 570 - -Y 666 661 -0.8X 680 - -

Table 4.3: Comparison between the eigenfrequencies determined in a static impact testand a modal analysis in Ansys.

drive guard, that is not infinitely stiff in reality. At low frequencies this will result in alarger error than at high frequencies, where the displacements are smaller. NB due tothe high modal density of the numerical model it is not always possible to make a faircomparison, especially for high frequencies.

4.2 Running mode

When the G-Tech decanter is running the whole structure is vibrating, due to theunbalanced rotating mass. To predict the sound radiated by the gearbox cover, firstthe structural vibrations need to be calculated. In Ansys the forces acting on thegearbox cover, transmitted by the connection points with the frame, need to be assigned.Therefore the accelerations at the connection points are measured using the tri-axialaccelerometer.


1

MN

MX

X

YZ


.639E-03.097453

.194266.29108

.387894.484707

.581521.678335

.775148.871962

APR 5 200915:09:29

PLOT NO. 1

NODAL SOLUTION


(a) Mode shape @ 50 Hz, sidepanel is vibrating in X direction

1

MN

MX

X

Y

Z


.567E-03.144555

.288543.432531

.576518.720506

.8644941.008

1.1521.296

APR 8 200910:26:06

PLOT NO. 1

NODAL SOLUTION


(b) Mode shape @ 350 Hz, toppanel is vibrating in Y direction

1

MN

MX

X

Y

Z


.230E-03.160373

.320517.48066

.640803.800947

.961091.121

1.2811.442

APR 8 200910:43:11

PLOT NO. 1

NODAL SOLUTION


(c) Mode shape @ 115 Hz, backpanel is vibrating in Z direction

Figure 4.7: Structural eigenmodes of the gearbox guard that correspond with the staticimpact test measurement.

4.2.1 Measurement of the forces acting on the gearbox cover

The acceleration at the four connection points (two on each side) are measured using atri-axial accelerometer. For each point three measurements (FFT’s) are made which areaveraged. Also the standard deviation between the different measurements is calculatedto see if the measurements are repeatable. In Figure 4.8 and Figure 4.9 the results forthe complete frequency region and a zoomed version up to 1000 Hz are shown. As canbe seen the highest responses occur at harmonic frequencies of 54 Hz (up to the 10th

harmonic of 540 Hz) and in the high frequency region of 5 to 12 kHz. The forces thatare involved with these accelerations can be determined easily by applying Newton’ssecond law

F = ma, (4.1)

with m and a the mass (5.4 grams) and acceleration of the accelerometer respectively.


With the measured forces applied to the numerical model of the gearbox cover it is pos-sible to calculate the structural response for the running mode. Because the amplitudesof the forces are different for each frequency, separate loadcases need to be defined inAnsys. These loadcases contain the average amplitudes of the harmonic forces in X, Yand Z direction as measured during the running test and are applied to the nodes at thevertical edges at the front of the gearbox cover, see Figure 4.10. It is chosen to use onlythe first 10 harmonics of the running frequency of 54 Hz in the simulation. The highfrequency responses have a large standard deviation and therefore they are not takeninto account in the numerical simulation. NB these frequencies could be dominant inthe radiated sound field and should be investigated in more detail. See Table 4.4 forthe input forces used.

The dynamic response to the harmonic forces listed in Table 4.4 is calculated withinAnsys. To reduce the calculation time, the analysis type is set to modal superposition,

4.2. Running mode 23

100

101

102

103

104

105

0

0.5

1

1.5

2

2.5

Frequency [Hz]

Ave

rage

acc

. [m

/s2 ]

XYZ

100

101

102

103

104

105

0

1

2

3

4

Frequency [Hz]

Sta

ndar

d de

viat

ion

[m/s

2 ]

XYZ

Figure 4.8: Average accelerations and standard deviation measured at the connectionpoints of the gearbox cover.

101

102

103

0

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Ave

rage

acc

. [m

/s2 ]

XYZ

101

102

103

0

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Sta

ndar

d de

viat

ion

[m/s

2 ]

XYZ

Figure 4.9: Zoomed, average accelerations and standard deviation measured at theconnection points of the gearbox cover.


Figure 4.10: The harmonic loads and constraints used for the running simulation.

Frequency |Fx|10−3 |Fy|10−3 |Fz|10−3

[Hz] [N] [N] [N]54 5.3 2.1 0.4108 5.0 2.7 1.3162 2.2 2.0 2.7216 1.5 1.3 0.6270 2.6 5.0 5.3324 0.5 0.4 1.7378 1.0 0.7 0.8432 1.0 0.1 0.6486 0.06 0.05 0.02540 0.1 0.1 0.1

Table 4.4: Amplitudes of forces in X, Y and Z direction used in Ansys for the runningmode analysis.


which means that the response is based on the preceding modal analysis. For this modalanalysis the boundary conditions at the connection points as mentioned in Section 4.1.2are deleted. The undamped equations of motion (3.5) become [Ans07]

φjT Mφj yj + φj

T Kφjyj = φjT F, (4.2)

with φj the j-th mode shape and yj a set of modal coordinates, such that u =∑n

j=1 φjyj ,with n the number of modes on which the superposition is based. As a rule of thumbthe number of modes n should contain at least 50% of the eigenfrequencies more thanthe highest frequency of interest in the harmonic response analysis case [Ans07]. Thisimplies that the eigenfrequency of the highest mode in the mode superposition analysisshould be at least 810 Hz for accurate results. 200 Modes are selected in the modesuperposition analysis case, with the highest mode shape and eigenfrequency at 1309Hz. The complete input file used in Ansys, for both the modal and forced responseanalysis, can be found in Appendix C.

4.2.3 Numerical results for the response in running mode

The deformed shape of the gearbox cover due to the harmonic forces, as mentioned inTable 4.4, is presented in Figure 4.11 for each frequency of excitation. The displacementsare the highest for the low frequencies with a maximum of 5.1e−3 mm in X-direction at54 Hz at the lowest connection points, see Figure 4.11(a).

4.3 Noise radiation

The structural (harmonic) displacements calculated within Ansys can be used in aBoundary Element (BE) analysis to predict the radiated sound power.


The results of the harmonic response analysis case in Ansys are stored in a .rst file thatis imported into LMS Virtual.Lab. First a Load Vector Set is created that contains thedisplacement vector for every node of the gearbox cover mesh. Next step is to createa surrogate acoustic mesh on which this structural data is projected. For an acousticalanalysis less elements and nodes can be used than for a structural dynamic analysis case.Therefore a coarse mesh is created in Ansys consisting of larger 4 node quadrilateralshell elements, see Figure 4.12 for both meshes. This coarse mesh consisted of 4902elements and 5360 nodes. The mesh used in the structural analysis consists of 9803elements and 34067 nodes respectively. So the number of equations to be solved isreduced by 84%.

To set up the acoustical analysis in LMS Virtual.Lab first the Harmonic BEMToolbox is activated. Next the model type definition is set to BEM Direct, exterior, as


1

MN

MX

X

YZ


.439E-05.568E-03

.001132.001696

.00226.002824

.003388.003952

.004516.005079

APR 27 200916:35:51

PLOT NO. 1

NODAL SOLUTION

STEP=1SUB =1FREQ=54USUM (AVG)RSYS=0DMX =.005079SMN =.439E-05SMX =.005079

(a) Deformed shape @ 54 Hz

1

MN

MX

X

YZ


.135E-06.175E-03

.349E-03.524E-03

.699E-03.873E-03

.001048.001222

.001397.001571

APR 27 200916:36:07

PLOT NO. 1

NODAL SOLUTION


(b) Deformed shape @ 108 Hz

1

MN

MX

X

Y

Z


.155E-06.579E-04

.116E-03.173E-03

.231E-03.289E-03

.346E-03.404E-03

.462E-03.519E-03

APR 27 200916:36:29

PLOT NO. 1

NODAL SOLUTION

STEP=1SUB =3FREQ=162USUM (AVG)RSYS=0DMX =.519E-03SMN =.155E-06SMX =.519E-03

(c) Deformed shape @ 162 Hz

1

MN

MX

X

Y

Z


.415E-06.157E-04

.310E-04.463E-04

.616E-04.769E-04

.921E-04.107E-03

.123E-03.138E-03

APR 27 200916:36:44

PLOT NO. 1

NODAL SOLUTION


(d) Deformed shape @ 216 Hz

1

MN

MX

X

Y

Z


.709E-06.195E-04

.383E-04.570E-04

.758E-04.946E-04

.113E-03.132E-03

.151E-03.170E-03

APR 27 200916:41:09

PLOT NO. 1

NODAL SOLUTION


(e) Deformed shape @ 270 Hz

1

MN

MXX

Y

Z


.141E-07.253E-05

.504E-05.755E-05

.101E-04.126E-04

.151E-04.176E-04

.201E-04.226E-04

APR 27 200916:41:59

PLOT NO. 1

NODAL SOLUTION


(f) Deformed shape @ 324 Hz

1

MN

MX

X

Y

Z


.936E-07.134E-04

.268E-04.401E-04

.535E-04.668E-04

.801E-04.935E-04

.107E-03.120E-03

APR 27 200916:42:45

PLOT NO. 1

NODAL SOLUTION


(g) Deformed shape @ 378 Hz

1

MN

MXX

Y

Z


.482E-06.180E-04

.356E-04.531E-04

.707E-04.882E-04

.106E-03.123E-03

.141E-03.158E-03

APR 27 200916:43:43

PLOT NO. 1

NODAL SOLUTION


(h) Deformed shape @ 432 Hz

1

MN

MX

X

Y

Z


.290E-07.181E-05

.360E-05.538E-05

.716E-05.895E-05

.107E-04.125E-04

.143E-04.161E-04

APR 27 200916:44:44

PLOT NO. 1

NODAL SOLUTION


(i) Deformed shape @ 486 Hz

1

MN

MX

X

Y

Z


.289E-08.252E-06

.501E-06.750E-06

.998E-06.125E-05

.150E-05.175E-05

.199E-05.224E-05

APR 27 200916:45:38

PLOT NO. 1

NODAL SOLUTION


(j) Deformed shape @ 540 Hz

Figure 4.11: Nodal displacements of the gearbox cover for different frequencies of exci-tation.


(a) Fine structural mesh (b) Coarse acoustical mesh

Figure 4.12: For the acoustical analysis a coarser mesh is used than for the structuralanalysis.

only the exterior problem is to be solved. Now a Mesh Preprocessing Set can be definedfor the coarse acoustical mesh, to ensure that all normal directions to the structureare set consistently. A new Material and Material Property are assigned to the wettedsurface and the properties are set for air (density of 1.225 kg/m3 and speed of sound of340 m/s).

The Load Vector Set is now ready to be transferred to the coarser acoustical mesh.This is done using a Transfer Vector Set. The MaxDistance method is selected and 8influencing nodes and a distance of 12 mm are selected. After the projection is calculateda picture of the deformed acoustical mesh is made to be sure that the projection isproperly done. To compute the acoustic pressures at the surface of the gearbox coverthe velocities at the nodes are needed. These velocities are calculated by differentiatingthe displacements that were calculated in the harmonic loadcase in Ansys.

The Transfer Vector Set can now be used as an acoustical boundary condition. ABoundary Condition and Source Set is defined and the Transfer Vector Set is added asa source.

Finally the Acoustical Response Set can be defined and the Boundary Condition andSource Set is assigned to it. Now the simulation can be started and once it is finishedthe acoustical power can be displayed in a graph. The total setup in LMS Virtual.Labcan be seen in Figure 4.13.


Figure 4.13: Acoustical simulation in LMS Virtual.Lab.

4.3.2 Numerical results

The total radiated sound power (re 10−12 W) and radiation efficiency are plotted ina graph, see Figure 4.14. Note that the lines between the circular data points do notrepresent any prediction on the sound power, as the amplitudes of the forces for thesefrequencies are much lower. The value of the radiated sound power is dependant onboth the amplitude of the velocities and the radiation efficiency. The velocities will behigh when the structure is excited with high forces or near an eigenfrequency. Also, ingeneral, the velocities are higher for low frequencies.

4.4 Summary

A numerical model of the gearbox cover has been developed to compute the structuralvibrations due to a forced harmonic response. The finite element model used to computethese vibrations is verified by measurements. The error between the eigenfrequenciesdetermined with measurements and the numerical model is for most frequencies small(smaller than 6 %), however due to the high modal density of the model a fair comparisoncannot be made for all frequencies.

The structural vibrations that are computed with finite element method are usedin a boundary element model to compute the radiated sound field. For frequencies upto 550 Hz the maximal acoustic power level is 42 dB (re 10−12 W).

4.4. Summary 29

0 100 200 300 400 500 60010

20

30

40

50

Frequency [Hz]

Aco

ustic

Pow

er [d

B]

0 100 200 300 400 500 6000

0.02

0.04

0.06

Frequency [Hz]

Rad

iatio

n ef

ficie

ncy

[−]

Figure 4.14: Radiated sound power (re 10−12 W) and radiation efficiency.

Chapter 5

Design changes

In general there are four methods to control noise and vibrations [Ren]: (1) absorption,(2) use of barriers and enclosures, (3) structural damping and (4) vibration isolation.In case of the gearbox cover, damping is investigated.

5.1 Damping

The gearbox cover is made of low carbon mild steel. The material damping of steel isassumed to be negligible. This results in an infinite high response at the eigenfrequen-cies obtained in the modal analysis. To reduce the amplitudes of the response at theeigenfrequency, (material) damping should be included in the model.

Within Ansys the harmonic response is calculated using a modal superpositionmethod, see (4.2). To include damping in the model, several damping inputs can begiven in Ansys, resulting in a total modal damping factor ξj of [Ans07]:

ξj = α2ωj

+ βωj

2 + ξ + ξmj , (5.1)

with α and β the Rayleigh damping multipliers for mass and stiffness respectively, ξ aconstant damping ratio and ξm

j a modal damping ratio. As modal damping has only aneffect near the eigenfrequencies, the mass and stiffness of the system should be changedto get a better results at other frequencies.

5.2 Material

The model of the gearbox cover can be adapted easily to investigate the effect of achange of material. The main function of the gearbox cover is to protect the rotatingparts from external influences, so mechanical properties such as Young’s Modulus are

31

32 Chapter 5. Design changes

Property ValueYoungs Modulus [GPa] 1.3Poisson ratio [-] 0.4Density [kg/m3] 930

Table 5.1: Material properties of ultra high molecular weight polyethylene.

0 50 100 150 200 250 3000

200

400

600

800

1000

1200

1400

Mode nr [−]

Fre

quen

cy [H

z]

Steel (2 mm)UHMWPE (4 mm)

Figure 5.1: Eigenfrequency versus mode number for both the steel and UHMWPEgearbox cover.

not so important. Therefore it is investigated what the effect is on the response of thegearbox cover, if it would be made out of 4 mm thick polyethylene. This material hasa high (compared with steel) material damping and is cost efficient.

5.2.1 Modal analysis

The material properties in Ansys are replaced by the properties of ultra high molecularweight polyethylene (UHMWPE) [War71], see Table 5.1. Also the thickness of the shellelements is increased to 4 mm.

After changing the material properties a modal analysis is done. Due to the changesin the material properties the eigenfrequencies will be lower compared to the gearboxcover made out of steel. This implies that more modes are needed in a modal super-position harmonic response analysis case, see Figure 5.1. To comply with the rule ofthumb [Ans07] to use at least a number of modes such that the highest eigenfrequencyis 50% higher than the highest frequency of interest, 300 modes are extracted.

5.2.2 Damping

Because of the visco-elastic (time dependant) material behavior of plastics, the materialdamping of UHMWPE will be frequency dependant. The damping for high frequencieswill be higher than for low frequencies. From the options available in Ansys it is chosen

5.2. Material 33

β(e−5) Damping ratio @ 100 Hz3.183 0.016.366 0.029.549 0.03

Table 5.2: Different values for the Rayleigh damping coefficient β that are used in theharmonic simulation.

to model the damping characteristics of the UHMWPE with Rayleigh damping basedon the stiffness matrix, so the modal damping factor becomes

ξj = βωj

2 . (5.2)

The value for β in this linear equation will determine the modal damping for the com-plete frequency domain. However, no literature is found to verify this model of thedamping characteristics. Consequently, it would be good practice to perform mea-surements to get insight in the real damping characteristics of this polymer. For themoment different values for β are used to obtain a first idea of how damping influencesthe response, see Table 5.2. The corresponding damping ratio at 100 Hz is also denoted.

5.2.3 Harmonic response

To predict the harmonic response of the gearbox cover with this new material, theforced response analysis case is repeated. The damping is added with values for β asin Table 5.2. The maximum displacements due to the harmonic forces are presentedin Table 5.3. As can be seen the maximum displacements for the UHMWPE gearboxcover are higher than the displacements for the steel gearbox cover, especially for lowfrequencies. Damping has a high influence at frequencies near the eigenfrequencies ofthe system. This influence can be clearly seen at excitation frequencies of 162, 216 and270 Hz. For the other frequencies the damping has less influence, as the gearbox coveris not excited near the eigenfrequency.

In Figure 5.2 the results of the deformed shape of the gearbox guard made out ofUHMWPE without damping are compared with the results of the steel gearbox guard.This clearly shows that the UHMWPE one has a higher modal density. The results ofthe deformed shape of the gearbox guard made out of UHMWPE without damping forhigher frequencies are presented in Appendix E. The effect of Rayleigh damping, withβ = 9.549e−5, for excitation frequencies of 54 and 486 Hz, is presented in Figure 5.3.As can be seen the amplitudes of the forced harmonic response are lower when dampingis applied.


Maximum displacement [µm]Frequency Steel Plastic Plastic Plastic Plastic

[Hz] β = 0 β = 0 β = 3e−5 β = 6e−5 β = 9e−5

54 5.1 42.5 41.4 38.6 35.1108 1.6 13.4 11.4 8.0 5.5162 0.5 39.2 1.2 1.2 1.2216 0.1 64.9 0.8 0.7 0.6270 0.2 4.7 0.7 0.6 0.5324 0.02 0.5 0.2 0.2 0.1378 0.1 0.6 0.2 0.1 0.1432 0.2 0.6 0.2 0.1 0.1486 0.02 0.06 7.010−3 6.510−3 6.610−3

540 2.210−3 0.08 1.110−3 9.810−3 9.510−3

Table 5.3: Maximum displacements for the running mode analysis for steel andUHMWPE material properties and different values for the Rayleigh damping.

1

MN

MX

X

YZ


.439E-05.568E-03

.001132.001696

.00226.002824

.003388.003952

.004516.005079

APR 27 200916:35:51

PLOT NO. 1

NODAL SOLUTION


(a) Steel: Deformed shape @ 54 Hz.

1

MN

MX X

Y

Z

Modalanalysis of Plastic Gearbox Guard

.381E-04.004757

.009475.014194

.018912.023631

.028349.033068

.037786.042505

APR 27 200916:55:12

PLOT NO. 1

NODAL SOLUTION


(b) UHMWPE: Deformed shape @54 Hz.

1

MN

MX

X

YZ


.135E-06.175E-03

.349E-03.524E-03

.699E-03.873E-03

.001048.001222

.001397.001571

APR 27 200916:36:07

PLOT NO. 1

NODAL SOLUTION


(c) Steel: Deformed shape @ 108 Hz.

1

MN

MX

X

YZ


.895E-04.001564

.003039.004514

.005989.007464

.008939.010413

.011888.013363

APR 27 200916:57:19

PLOT NO. 1

NODAL SOLUTION


(d) UHMWPE: Deformed shape @108 Hz.

Figure 5.2: Nodal displacements of the forced response of the gearbox cover made outof steel and UHMWPE.

5.2. Material 35

1

MN

MX X

Y

Z


.381E-04.004757

.009475.014194

.018912.023631

.028349.033068

.037786.042505

APR 27 200916:55:12

PLOT NO. 1

NODAL SOLUTION


(a) Deformed shape @ 54 Hz without damp-ing.

1

MN

MX X

Y

Z


.163E-04.003919

.007821.011724

.015627.019529

.023432.027334

.031237.035139

APR 29 200913:30:02

PLOT NO. 1

NODAL SOLUTION


(b) Deformed shape @ 54 Hz with β =9.549e−5

1

MN

MX

X

Y

Z


.860E-07.691E-05

.137E-04.206E-04

.274E-04.342E-04

.410E-04.479E-04

.547E-04.615E-04

APR 27 200917:05:29

PLOT NO. 1

NODAL SOLUTION


(c) Deformed shape @ 486 Hz without damp-ing.

1

MN

MX

X

Y

Z


.141E-07.746E-06

.148E-05.221E-05

.294E-05.367E-05

.441E-05.514E-05

.587E-05.660E-05

APR 29 200913:44:21

PLOT NO. 1

NODAL SOLUTION


(d) Deformed shape @ 486 Hz with β =9.549e−5

Figure 5.3: The influence of Rayleigh damping on the deformed shape of the plasticgearbox cover.


0 100 200 300 400 500 60010

20

30

40

50

60

70

80

Frequency [Hz]

Aco

ustic

Pow

er [d

B]

steel, β = 0plastic, β = 0

plastic, β = 3.183 10−5

plastic, β = 6.366 10−5

plastic, β = 9.549 10−5

Figure 5.4: Radiated sound power for the gearbox cover made out of UHMWPE fordifferent damping values.

5.2.4 Noise radiation

With the harmonic response information for the plastic gearbox cover the radiated soundpower can be predicted in the same way as for the steel gearbox cover. In Figure 5.4the radiated sound powers can be found for the gearbox cover made out of UHMWPEfor different damping values. Also the radiated sound power for the steel gearbox coveris shown for reference.

As can be seen in Figure 5.4 the damping has a high influence on the radiatedsound power at frequencies near the eigenfrequencies of the gearbox cover. For lowfrequencies however, the radiated sound power is higher for the gearbox cover made outof rotational molded plastic compared to the steel one. The shift point is at 378 Hz,for higher frequencies the radiated power is lower in case of a plastic cover. When theaverage difference in acoustical power is calculated for low frequencies (below 378 Hz)and the high frequencies (378 Hz and higher), it is found that the plastic cover radiatessound with a power of 9 dB more in the low frequency region and has a reduction of 4dB in the high frequency region.

To verify the results for the radiated power by the steel cover, it is recommendedto perform a sound intensity scan. However, when the decanter is running, sound willbe transmitted by the rotating parts within the gearbox cover, which will have a con-tribution in the measurement as the gearbox cover is not a closed enclosure. Thereforeit would be good practice to excite the gearbox cover with a shaker with a knownfrequency and amplitude to verify the used model.

5.3. Summary 37

5.3 Summary

The numerical model of the gearbox cover has been changed with respect to the materialproperties of the cover. The material properties are changed to ultra high molecularweight polyethylene. Also the thickness of the cover is increased to 4 mm (was 2 mm forthe steel one). This plastic has a high material damping, so damping is included in themodel. For different choices for this damping model the sound production is computed.Compared with the steel gearbox cover in general the sound production of the plasticgearbox cover will increase with 9 dB in the frequency range to 400 Hz and decreasewith 4 dB for higher frequencies to 550 Hz.

Chapter 6

Conclusion andRecommendations

6.1 Conclusion

This report contains a noise and vibration survey of the G-Tech 1456 gearbox cover. Thevibration survey measurements are done using a tri-axial accelerometer that is mountedon the cover. For both the operational and non-operational state the main vibrationsare measured. In the non-operational state the eigenfrequencies are extracted and usingthe modal analysis toolbox in the finite element package Ansys, the corresponding modeshapes are identified. Moreover, the eigenfrequencies obtained with the modal analysisare compared with the measurements. In the operational mode the forces acting on thegearbox cover are measured and modeled in Ansys. In this way it is possible to simulatethe operational mode. As the decanter centrifuge is rotating at a frequency of 54 Hz,the harmonic input forces are all an integer multiple of this frequency (harmonics up to540 Hz).

In the non-operation mode the results for the numerical model are comparable(except for the first eigenfrequency) with the experimental results. The largest error is5.7%. However, due to the high modal density of the numerical model, it is not alwayspossible to make a fair comparison.

In the frequency range up to 540 Hz the dominant frequency in the operational modeis 270 Hz. However, the responses at 54 and 108 Hz show the biggest displacements.The acoustical response show the biggest response at 432 Hz due to a relative highradiation efficiency for that mode.

When the gearbox cover would be made from 4 mm thick high molecular weightpolyethylene the structural vibrations will have a bigger amplitude compared to thesteel one. The acoustical response for high frequencies (above 378 Hz) however can bereduced by approximately 4 dB. For lower frequencies the radiated power will be on

39

40 Chapter 6. Conclusion and Recommendations

average 9 dB higher. This result is only valid when the damping characteristics thatare used in the model (Rayleigh damping, with β = 9.54910−5) is realistic.

6.2 Recommendations

To verify the acoustic results it is recommended to perform a sound intensity scan of thegearbox cover when it is excited by a shaker. In this way background noise is avoided.

A general remark has to be made about the frequency domain that is used. As can beseen from the measurements the gearbox cover is also excited with higher frequencies (atabout 6000 Hz) than used in this analysis. These frequencies can have a big contributionin the radiated sound field. Therefore these high frequencies should be taken intoaccount in the simulation to be able to compare the steel and plastic gearbox cover inthe complete audible frequency domain. It is expected that the high material dampingof plastic has a larger advantage in this frequency region.

Bibliography

[Ans07] Release 11.0 documentation for ansys. 2007.

[Bro85] R.G. Brown and P.Y.C. Brown. Introduction to random signals and appliedKalman filtering, volume 2. J. Wiley, New York, 1985.

[Fah87] F. Fahy and P. Gardonio. Sound and Structural Vibration, Radiation, Trans-mission and Response, volume 2. Elsevier Acadamic, Amsterdam/London,1987.

[Ger99] J.M. Gere and S.P. Timoshenko. Mechanics of materials 4th SI edition, vol-ume 4. Stanley Thornes, Cheltenham, 1999.

[Hea97] E.J. Hearn. Mechanics of materials 2, volume 3. 1997.

[Rec01] A. Records and K. Sutherland. Decanter Centrifuge Handbook, volume 1. El-sevier Advanced Technologies, Oxford, 2001.

[Ren] J Renninger. Understanding damping techniques for noise and vibration con-trol. www.earsc.com.

[Vis04] R. Vissers. A boundary element approach to acoustic radiation and sourceidentification, volume 1. 2004.

[War71] I.M. Ward and J. Sweeney. The mechanical properties of solid polymers, vol-ume 2. J. Wiley, New York, 1971.

41

42 Bibliography

Appendix A

Decanter Layout and technicaldata

In Figure A.1 a cross-section of the G-Tech 1456 decanter centrifuge is shown. Table A.1is a list of the technical data of this decanter centrifuge.

Technical dataMax. Bowl Speed 4000 RPMCentrifugal Force 3150 G’sDifferential Speed 4-48 RPMRun Up time 2-3 minutesBowl Dimensions 14” (355 mm) Diameter x 56” (1420 mm) LongGross Weight 2150 kgShipping Volume 6 m3

Wetted Parts 316 Stainless SteelBase Assembly Cast IronMain Drive 18-37 kW 230/460 VAC @ 50/60 HzBack Drive 4-7.5 kW 230/460 VAC @ 50/60 Hz

Table A.1: Technical data of the G-Tech 1456 decanter centrifuge.

43

44 Appendix A. Decanter Layout and technical data

So

lid

s E

nd

Liq

uid

s E

nd

Base

Clu

tch

Gearb

ox

Main

beari

ng

1B

ow

lC

on

vey

or

Main

Beari

ng

2F

eed

pip

eC

on

ical

So

lid

s E

nd

Liq

uid

s E

nd

Base

Clu

tch

Gearb

ox

Main

beari

ng

1B

ow

lC

on

vey

or

Main

Beari

ng

2F

eed

pip

eC

on

ical

Figure A.1: Cross section of the G-tech 1456 decanter centrifuge.

Appendix B

Matlab Code to importmeasurement data from PULSE

The Matlab function file that is used to import the measurement response data obtainedwith a tri-axial accelerometer and a Bruel & Kjær PULSE system is listed below.

function [max_freq,max_resp] = read_the_data(filename)

%read the data and save them in cell arrays

fid = fopen(filename, ’r’);

spectrum_Y = textscan(fid, ’%f %f %f’ , 8192, ’headerlines’, 83);


spectrum_X = textscan(fid, ’%f %f %f’ , 8192, ’headerlines’, 83+8192+93);


spectrum_Z = textscan(fid, ’%f %f %f’ , 8192, ’headerlines’, 83+8192+93+8192+93);

%convert the cell arrays in arrays

Freq = spectrum_X{1,2};

spec_X = spectrum_X{1,3};

spec_Y = spectrum_Y{1,3};

spec_Z = spectrum_Z{1,3};

%calculating the first 20 frequencies with the maximum response

[max_X,pos_X] = sort(spec_X,’descend’);

max_X = max_X(1:20);

pos_X = pos_X(1:20);

freq_X = Freq(pos_X);

[max_Y,pos_Y] = sort(spec_Y,’descend’);

max_Y = max_Y(1:20);

pos_Y = pos_Y(1:20);

freq_Y = Freq(pos_Y);

[max_Z,pos_Z] = sort(spec_Z,’descend’);

max_Z = max_Z(1:20);

pos_Z = pos_Z(1:20);

freq_Z = Freq(pos_Z);

%combining the results for the X-Y-Z direction

max_freq = [freq_X freq_Y freq_Z];

45

46 Appendix B. Matlab Code to import measurement data from PULSE

max_resp = [max_X max_Y max_Z];

%max_resp = max_resp./max(max(max_resp)); % scaling wrt the highest response

The m-file to create a frequency response plot of the 20 most dominating responsesis listed below.

clear all

close all

clc

%%%%%% text files with the data

% gearbox cover (liquid end)

filename = [’21_S.txt’; ’22_S.txt’; ’23_S.txt’; ’24_S.txt’; ’25_S.txt’;’26_S.txt’; ’27_S.txt’; ’28_S.txt’; ’29_S.txt’];

% blue base logo

% filename = [’30_S.txt’; ’31_S.txt’; ’32_S.txt’; ’33_S.txt’; ’34_S.txt’; ’35_S.txt’;’36_S.txt’; ’37_S.txt’;’38_S.txt’; ’39_S.txt’];

%%%%%% Extracting the maximum responses with the corresponding frequencies

l = length(filename(:,1)) % number of positions used to measure the response

% defining the 20 frequencies with the max responses

max_freq = zeros(20,3*l);

max_resp = zeros(20,3*l);

% import the measurement data using read_the_data_run.m

for i = 1:l

[max_freq(:,i:l:i+2*l),max_resp(:,i:l:i+2*l)] = read_the_data(filename(i,:));

end

% getting the max and min frequencies in X Y and Z direction

min_freqX = min(min(max_freq(:,1:l)));

max_freqX = max(max(max_freq(:,1:l)));

min_freqY = min(min(max_freq(:,l+1:2*l)));

max_freqY = max(max(max_freq(:,l+1:2*l)));

min_freqZ = min(min(max_freq(:,2*1+1:3*l)));

max_freqZ = max(max(max_freq(:,2*1+1:3*l)));

freqX = [];

respX = [];

freqY = [];

respY = [];

freqZ = [];

respZ = [];

% Sorting the max response and averaging them

q = 1;

for i = min_freqX:2:max_freqX % data captured with freq_span = 2 Hz

[rowX,colX] = find(max_freq(:,1:l) == i); % finding freq between min and max freq (of the 20 frequencies)

s = length(rowX);

if s > 0 % if the frequency is found in the max_freq matrix

freqX(q,1) = i;

respX(q,1) = mean(mean(max_resp(rowX,colX),2)); %average over the number of measurements (@ diff positions)

q = q+1;

end

end

47

[respX,pos] = sort(respX,’descend’); % sort with highest response on top

freqX = freqX(pos);

q = 1;

for i = min_freqY:2:max_freqY

[rowY,colY] = find(max_freq(:,l+1:2*l) == i);

s = length(rowY);

if s > 0

freqY(q,1) = i;

respY(q,1) = mean(mean(max_resp(rowY,l+colY),2));

q = q+1;

end

end

[respY,pos] = sort(respY,’descend’);

freqY = freqY(pos);

q = 1;

for i = min_freqZ:2:max_freqZ

[rowZ,colZ] = find(max_freq(:,2*l+1:3*l) == i);

s = length(rowZ);

if s > 0

freqZ(q,1) = i;

respZ(q,1) = mean(mean(max_resp(rowZ,2*l+colZ),2));

q = q+1;

end

end

[respZ,pos] = sort(respZ,’descend’);

freqZ = freqZ(pos);

%figure with the max (averaged) responses, including bar plots

figure(’name’,’averaged’)

semilogx(freqX,respX,’x’,freqY,respY,’o’,freqZ,respZ,’>’,’MarkerSize’,6,’LineWidth’,2)

hold on

bar(freqX,respX,’k’)

bar(freqY,respY,’k’)

bar(freqZ,respZ,’k’)

xlabel(’Frequency [Hz]’)

ylabel(’Acceleration [m/s^2]’)

legend(’X’,’Y’,’Z’)

grid

48 Appendix B. Matlab Code to import measurement data from PULSE

Appendix C

Ansys input code for thestructural dynamic analysis

The input file for Ansys in the structural dynamic analysis is listed below. In this filethe gearbox cover made out of steel is analyzed. The material properties can be easilyadapted when the simulation with plastic material properties has to be done. The linesstarting with an exclamation mark are comments. The loadcases for 108 up to 486Hz are not listed as they are the same as for 54 Hz, only with an other frequency andamplitude input.

finish

/clear

/CWD,’S:\all scratch\Arjan\Gtech\ANSYS_files’ !change working directory

/title, Gearbox Guard

!***************Input Data********

! Low carbon mild steel used as material

E=207e6 !youngs modulus [kg mm/s/mm2]

v=0.3 !poisson ratio [-]

rho=7800e-9 !Density [kg/mm^3]

!************Importing gearbox guard model from IGES file into ansys**************

/AUX15

IOPTN,MERG,YES !merging of keypoints

IOPTN,SOLID,NO !creating of a solid/volume

IOPTN,GTOLER,DEFA !tolerance of IGES import

IOPTN,SMALL,YES !delete small areas

IGESIN,’GT150A Gearbox Guard wf2’,’IGS’!import the IGES file with geometry

finish

!**************preprocessor**********************

/prep7 !starting preprocessor for defining the material and cleaning up the model for a good mesh

nummrg,all !merge alle coincident lines and points

et,1,shell93 !using 8node shell elements

49

50 Appendix C. Ansys input code for the structural dynamic analysis

MP,EX,1,E !material property, Modulus

MP,PRXY,1,v !material property, poisson ratio

MP,DENS,1,rho !material property, density

!************meshing the geomertry***************

ALLSEL,all

ASEL,S,area,,1287,1290 !Select the areas to keep

ASEL,A,area,,1296

ASEL,A,area,,1300,1304


ASEL,A,area,,1725

ASEL,A,area,,1731

BOPTN, KEEP, YES !boolean operator to delete the inside areas

!LPLOT !Plot the lines

LESIZE,all, 20.0, , , , 1, , , 0!Subdivide different lines

ALLSEL,all

ASEL,S,area,,1722,1725,3 !select the areas on top



ASEL,A,area,,1296

SMRTSIZE,OFF

MSHKEY,2

DESIZE, 3, 1, 1, , ,2 ,20 , ,

mopt,aorder,on

mopt,expnd,1

mopt,trans,2

amesh,all !mesh the areas on top

ALLSEL,all

LSEL,S,line,,4964

LSEL,A,line,,5190

LSEL,A,line,,4976

LSEL,A,line,,5178

LSEL,A,line,,8

LSEL,A,line,,13770

LSEL,A,line,,32

LSEL,A,line,,13730

LSEL,A,line,,5010

LSEL,A,line,,11992

LESIZE,all, 5, , , , 1, , , 0 !Subdivide different lines

ALLSEL,all

ASEL,S,area,,1720,1731,11 !select the areas at sides

SMRTSIZE,OFF

MSHKEY,2

DESIZE, 3, 1, 1, , ,2 ,20 , ,

mopt,aorder,on

mopt,expnd,1

mopt,trans,1.4

amesh,all

ALLSEL,all

51

LSEL,all

LSEL,U,line,,10304

LESIZE,all, 3, , , , 1, , , 0 !Subdivide remaining lines

SMRTSIZE,OFF

MSHKEY,2

DESIZE, 3, 1, 1, , ,2 ,20 , ,

mopt,aorder,on

mopt,expnd,1

mopt,trans,1.4

amesh,1721 !mesh the areas at back

!************BOUNDARY CONDITIONS**********

!Set all dof=0 except UX, at small holes in side

ALLSEL,all

NSEL,S,node,,3856

NSEL,A,node,,3857,3863,2

NSEL,A,node,,3866,3870,2

NSEL,A,node,,3840

NSEL,A,node,,3841,3847,2

NSEL,A,node,,3850,3854,2

NSEL,A,node,,5663

NSEL,A,node,,5664,5670,2

NSEL,A,node,,5673,5677,2

NSEL,A,node,,5647

NSEL,A,node,,5648,5654,2

NSEL,A,node,,5657,5661,2

D,ALL,UY,0

D,ALL,UZ,0

D,ALL,ROTX,0

D,ALL,ROTY,0

D,ALL,ROTZ,0

ALLSEL,ALL

!************OBTAIN SOLUTION MODAL ANALYSIS**************

/solu

antype,2 !set modal analysis

MODOPT,lanb,200 !method used is block lanczos, 200 modes expand

RESVEC,ON !Calculate residual vector

EQSLV,FRONT

MXPAND,200 !200 modes expanded

solve

finish

!************HARMONIC ANALYSIS @ 54 HZ**************

ALLSEL,ALL

/solu

LSCLEAR,all !start with no loadsteps

NSEL,U,node,,all !Select nodes where harmonic input will be given

NSEL,S,node,,5515,5539,2


NSEL,A,node,,5514

NSEL,A,node,,3813,3837,2

NSEL,A,node,,3763

antype,3 !set harmonic analysis

NSUBST,1 !number of subsets in this loadcase

HARFRQ, 54 !Frequency of excitation

HROPT, MSUP,200,1

BETAD,1*3.1831E-5 !Set stiffness damping

F, all, FX, 5.3 !Force in X-direction

F, all, FY, 2.1 !Force in Y-direction

F, all, FZ, 0.4 !Force in Z-direction

KBC,1 !stepped loads

OUTRES,NSOL !write only the nodal dof solution

OUTPR,NSOL !solution printout = dof solu

ALLSEL,all

LSWRITE

.......

!************HARMONIC ANALYSIS @ 540 HZ**************

ALLSEL,ALL

/solu

NSEL,U,node,,all !Select nodes where harmonic input will be given

NSEL,S,node,,5515,5539,2

NSEL,A,node,,5514

NSEL,A,node,,3813,3837,2

NSEL,A,node,,3763

antype,3 !set harmonic analysis

NSUBST, 1

HARFRQ, 540 !Frequency of excitation

HROPT, MSUP,200,1

BETAD,1*3.1831E-5 !Set stiffness damping

F, all, FX, 0.1 !Force in X-direction

F, all, FY, 0.1 !Force in Y-direction

F, all, FZ, 0.0 !Force in Z-direction

KBC,1 !stepped loads

OUTRES,NSOL !write only the nodal dof solution

OUTPR,NSOL !solution printout = dof solu

ALLSEL,all

LSWRITE

finish

/solu

LSSOLVE,1,10,1

finish

/solu

ALLSEL,ALL

EXPASS,on

53

NUMEXP,all,54,540

!BETAD,1*3.1831E-5 !Set stiffness damping

OUTPR,nsol,all

solve

Appendix D

Structural eigenmodes of thegearbox guard that correspondwith static impact tests

A static impact test is performed to identify the eigenfrequencies of the gearbox guard.In Table 4.3 the numerically determined eigenfrequencies that correspond with the dom-inant frequencies from this static impact test are listed. The corresponding mode shapes(first 6) can be found in Figures D.1, D.2 and D.3.

55

56Appendix D. Structural eigenmodes of the gearbox guard that correspond with static impact tests

1

MN

MX

X

YZ


.639E-03.097453

.194266.29108

.387894.484707

.581521.678335

.775148.871962

APR 5 200915:09:29

PLOT NO. 1

NODAL SOLUTION


(a) Mode shape @ 50 Hz

1

MN

MX

X

Y

Z


.860E-04.080178

.16027.240363

.320455.400547

.480639.560732

.640824.720916

APR 5 200915:15:23

PLOT NO. 1

NODAL SOLUTION


(b) Mode shape @ 129 Hz

1

MN

MX

X

Y

Z


.984E-03.172738

.344492.516246

.687999.859753

1.0321.203

1.3751.547

APR 8 200910:08:27

PLOT NO. 1

NODAL SOLUTION


(c) Mode shape @ 184 Hz

1

MN

MX

X

Y

Z


.001207.100182

.199157.298131

.397106.496081

.595056.694031

.793005.89198

APR 8 200910:11:18

PLOT NO. 1

NODAL SOLUTION

STEP=1SUB =24FREQ=239.362USUM (AVG)RSYS=0DMX =.89198SMN =.001207SMX =.89198

(d) Mode shape @ 239 Hz

1

MN

MXX

Y

Z


.139E-03.11835

.236561.354772

.472983.591195

.709406.827617

.9458281.064

APR 8 200910:16:23

PLOT NO. 1

NODAL SOLUTION


(e) Mode shape @ 302 Hz

1

MN

MX

X

Y

Z


.539E-03.095455

.190371.285286

.380202.475118

.570034.66495

.759866.854781

APR 8 200910:14:07

PLOT NO. 1

NODAL SOLUTION


(f) Mode shape @ 356 Hz

Figure D.1: Structural eigenmodes of the Gearbox Guard that correspond with thestatic impact test measurement for X-directions.

1

MN

MX

X

Y

Z


.567E-03.144555

.288543.432531

.576518.720506

.8644941.008

1.1521.296

APR 8 200910:26:06

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

Y

Z


.004804.104606

.204408.304209

.404011.503813

.603615.703417

.803219.903021

APR 8 200910:32:27

PLOT NO. 1

NODAL SOLUTION

STEP=1SUB =41FREQ=379.044USUM (AVG)RSYS=0DMX =.903021SMN =.004804SMX =.903021


1

MN

MX

X

Y

Z


.485E-03.114214

.227944.341673

.455403.569132

.682862.796591

.9103211.024

APR 8 200910:35:18

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

Y

Z


.987E-03.123647

.246307.368967

.491627.614287

.736947.859607

.9822671.105

APR 8 200910:25:37

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

Y

Z


.833E-04.092366

.184648.276931

.369213.461496

.553779.646061

.738344.830626

APR 8 200910:26:51

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

Y

Z


.010398.122709

.23502.347331

.459642.571953

.684264.796575

.9088861.021

APR 8 200910:31:32

PLOT NO. 1

NODAL SOLUTION



Figure D.2: Structural eigenmodes of the Gearbox Guard that correspond with thestatic impact test measurement for Y-directions.

57

1

MN

MX

X

Y

Z


.230E-03.160373

.320517.48066

.640803.800947

.961091.121

1.2811.442

APR 8 200910:43:11

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

Y

Z


.539E-03.095455

.190371.285286

.380202.475118

.570034.66495

.759866.854781

APR 8 200910:41:52

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

Y

Z


.001213.12446

.247707.370953

.4942.617447

.740694.86394

.9871871.11

APR 8 200910:40:37

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

Y

Z


.911E-03.076058

.151204.226351

.301498.376645

.451791.526938

.602085.677231

APR 8 200910:41:00

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

Y

Z


.002212.118767

.235321.351876

.46843.584985

.701539.818093

.9346481.051

APR 8 200910:41:21

PLOT NO. 1

NODAL SOLUTION



1

MN

MX

X

YZ


.008121.271486

.534852.798218

1.0621.325

1.5881.852

2.1152.378

APR 8 200910:37:16

PLOT NO. 1

NODAL SOLUTION



Figure D.3: Structural eigenmodes of the Gearbox Guard that correspond with thestatic impact test measurement for Z-directions.

58Appendix D. Structural eigenmodes of the gearbox guard that correspond with static impact tests

Appendix E

Results of the forced response ofthe gearbox guard made out ofUHMWPE without damping

The results of the deformed shape of the gearbox guard made out of UHMWPE withoutdamping due to applied harmonic forces are presented in Figure E.1.

59

60Appendix E. Results of the forced response of the gearbox guard made out of UHMWPE without damping

1

MN

MX X

Y

Z


.381E-04.004757

.009475.014194

.018912.023631

.028349.033068

.037786.042505

APR 27 200916:55:12

PLOT NO. 1

NODAL SOLUTION


(a) Deformed shape @ 54 Hz.

1

MN

MX

X

YZ


.895E-04.001564

.003039.004514

.005989.007464

.008939.010413

.011888.013363

APR 27 200916:57:19

PLOT NO. 1

NODAL SOLUTION


(b) Deformed shape @ 108 Hz.

1

MN

MX

X

Y

Z


.803E-04.004431

.008781.013131

.017481.021832

.026182.030532

.034883.039233

APR 27 200916:59:22

PLOT NO. 1

NODAL SOLUTION


(c) Deformed shape @ 162 Hz.

1

MN

MX

X

Y

Z


.340E-04.007236

.014438.02164

.028842.036045

.043247.050449

.057651.064853

APR 27 200917:00:33

PLOT NO. 1

NODAL SOLUTION


(d) Deformed shape @ 216 Hz.

1

MN

MXX

Y

Z


.507E-05.526E-03

.001047.001568

.002088.002609

.00313.003651

.004172.004693

APR 27 200917:01:25

PLOT NO. 1

NODAL SOLUTION


(e) Deformed shape @ 270 Hz.

1

MNMX

X

Y

Z


.278E-05.524E-04

.102E-03.152E-03

.201E-03.251E-03

.301E-03.350E-03

.400E-03.450E-03

APR 27 200917:02:18

PLOT NO. 1

NODAL SOLUTION


(f) Deformed shape @ 324 Hz.

1

MN

MX

X

Y

Z


.656E-06.682E-04

.136E-03.203E-03

.271E-03.338E-03

.406E-03.473E-03

.541E-03.609E-03

APR 27 200917:03:25

PLOT NO. 1

NODAL SOLUTION


(g) Deformed shape @ 378 Hz.

1

MN

MX

X

Y

Z


.224E-05.699E-04

.138E-03.205E-03

.273E-03.341E-03

.408E-03.476E-03

.543E-03.611E-03

APR 27 200917:04:36

PLOT NO. 1

NODAL SOLUTION


(h) Deformed shape @ 432 Hz.

1

MN

MX

X

Y

Z


.860E-07.691E-05

.137E-04.206E-04

.274E-04.342E-04

.410E-04.479E-04

.547E-04.615E-04

APR 27 200917:05:29

PLOT NO. 1

NODAL SOLUTION


(i) Deformed shape @ 486 Hz.

1

MN

MXX

Y

Z


.206E-06.945E-05

.187E-04.279E-04

.372E-04.464E-04

.557E-04.649E-04

.742E-04.834E-04

APR 27 200917:07:04

PLOT NO. 1

NODAL SOLUTION


(j) Deformed shape @ 540 Hz.

Figure E.1: Nodal displacements of the gearbox cover for different frequencies of exci-tation.

Noise reduction applied to a decanter centrifuge - TU/e

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