Noise reduction applied to a decanter centrifuge

Transcription

1 Noise reduction applied to a decanter centrifuge A.J. van Engelen DCT Research report, research performed at University of Canterbury, New ealand New ealand Supervisor: Dr. J.R. Pearse University of Canterbury Department of Mechanical Engineering Supervisor: Prof.dr. H. Nijmeijer Eindhoven University of Technology Department of Mechanical Engineering Dynamics and Control Christchurch (New ealand), April 2009

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3 Summary Structural vibrations can be problematic for a (part of a) machine. They contribute to the wear of the machine, but can also produce a high noise level. In this report a noise and vibration survey has been performed on an existing design of a gearbox cover of a decanter centrifuge. A finite element model of this gearbox cover is developed to predict the structural vibrations, which has been verified by measurements. A boundary element model is used to predict the sound power level produced by this vibrating structure. By adapting this model with respect to material properties the noise reduction is forecasted. It can be concluded that if a 2 mm thick steel gearbox cover is replaced by a 4 mm thick ultra high molecular weight polyethylene one, that the sound production will increase with 9 db for frequencies up to 400 Hz and will decrease with 4 db for frequencies in the range of 400 to 540 Hz. iii

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5 Contents Summary iii Introduction 2 Decanter Centrifuge 3 2. Main parts Working principle Measuring and modeling noise and vibrations 7 3. Experimental setup Running tests Eigenfrequency and mode shape extraction Numerical implementation Noise radiation Boundary Element Method Radiation Efficiency Radiated Power Summary Implementation, verification and results of a numerical model 5 4. Eigenfrequencies and mode shapes extraction Static impact test Numerical implementation Numerical results v

6 vi Contents 4..4 Comparison between numerical and experimental results Running mode Measurement of the forces acting on the gearbox cover Numerical implementation Numerical results for the response in running mode Noise radiation Numerical implementation Numerical results Summary Design changes 3 5. Damping Material Modal analysis Damping Harmonic response Noise radiation Summary Conclusion and Recommendations Conclusion Recommendations Bibliography 4 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 with static impact tests 55

7 Contents vii E Results of the forced response of the gearbox guard made out of UHMWPE without damping 59

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9 Chapter Introduction G-Tech is a Christchurch based manufacturer of centrifuges. To comply with the demands 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 subject is to identify the primary modes of vibrations and the contributions of these to the radiated noise field of one of their products, the G-Tech 456. The main objective is to identify the primary modes of vibration and their contribution to the radiated sound field. Moreover, using numerical models the influence of design changes can be predicted. Because of the complex geometry of the machine, in this study only the gearbox cover is examined. The objective is to collect data for impact excitation of the gearbox cover in both the operating (with the machine switched on) and non-operating (static) state. In the operating state the cover will be excited due to vibrations that are generated by the rotating parts, in the static case the cover is excited with a rubber hammer, i.e. after an impact. Measurements in the frequency domain are made with an accelerometer to identify the most dominant frequencies. The gearbox cover is modeled in SolidWorks so the model can be imported to a finite element modeling package to carry out a numerical study. With a modal analysis the eigenfrequencies and corresponding mode shapes can be calculated. The accuracy of this model can be determined by comparing the numerical results with the measurements after the impact excitation. Moreover, the numerically calculated vibrations can be used to compute the radiated sound power using boundary element method. The report is organized as follows. In Chapter 2 the working principle of the decanter centrifuge (like the G-tech 456) is explained. Moreover, the principal components are described in detail. In Chapter 3 the response of the main parts after an impact excitation are investigated. Also the theory used for the experiments and numerical model is described. For the gearbox cover in particular in Chapter 4 it is described how a numerical model is implemented and verified by measurements. The most dominant vibrational frequencies using static impact excitations are listed and compared with

10 2 Chapter. Introduction a modal analysis of a finite element model. Next the sound power radiated by the gearbox cover is calculated using boundary element method. In Chapter 5 the model is changed with respect to material properties to investigate if it is worthwhile to replace the gearbox cover by a cover made of a different material. Finally in Chapter 6 the conclusions and recommendations will be presented.

11 Chapter 2 Decanter Centrifuge In this report the G-Tech 456 decanter centrifuge is under investigation. Before going into detail about the process of noise reduction, the working principle and the main parts of the centrifuge are described. 2. Main parts In Figure 2. a cross-section of the decanter with the main parts is presented (see Appendix A for a larger one). The bowl, conical and conveyor are supported by two main bearings that are mounted on the base. Two electric motors (main and backdrive) are coupled to the rotating assembly through v-belt drives. The screw conveyor is connected through a spline coupling to a gearbox, which makes it possible to rotate the conveyor slightly slower than the bowl. Throughout this work the rotating speed of the 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 liquids or solids from liquids. It uses centrifugal forces that enforce the liquid (or solid) with the highest density to be near the surface of the bowl while the liquid with the lower density is floating on this layer (at a smaller radial position). Due to the slightly different rotational speed of the conveyor with respect to the bowl (4 to 48 RPM) the high density fluid is conveyed upwards into the conical and will finally exit the decanter at the solids discharge. At the conical a beach will be formed, see Figure 2.2. On the opposite side (in axial direction) of this beach the fluid with a lower density will overflow in the liquids discharge. 3

12 4 Chapter 2. Decanter Centrifuge Clutch Gearbox Main bearing Bowl Conveyor Conical Main Bearing 2 Feed pipe Base Liquids End Solids End Figure 2.: Cross section of the G-tech 456 decanter centrifuge. Figure 2.2: Visualization of the separation process.

13 2.2. Working principle 5 The decanter centrifuge has the advantage that it discharges continuously. In addition 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. Besides the decanter can handle a wide range of feed slurry concentrations and produces drier solids than other centrifuges. These are the main reasons that decanter centrifuges are widely used. Disadvantages are the high power consumption and high wear of the screw conveyor. Examples of applications are found in the chemical industry, waste sludge processing, minerals extracting and processing [Rec0]. At G-Tech the decanter centrifuge is completely modeled with the 3D Computer Aided Design (CAD) software SolidWorks. The focus is set on the production process. The development of new products starts with improving their recent products. The company does not use any model for the vibration analysis for this product.

14 6 Chapter 2. Decanter Centrifuge

15 Chapter 3 Measuring and modeling noise and vibrations A decanter centrifuge produces a lot of noise while it is in operating mode. To understand the underlying principles of the sound production, first the vibrations of the machine are investigated. These vibrations are measured using a tri-axial accelerometer. 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. Experimental setup For the vibrational experiments a tri-axial accelerometer and a Brüel & Kjær PULSE system are used. The tri-axial accelerometer is put on the surface of several parts of the decanter centrifuge using some wax. The accelerometer is positioned such that the z-axis represents the axial axis of the centrifuge, the y-axis is pointing upwards and the x-axis sidewards. The three generated signals are amplified and processed in the PULSE system and visualized on a laptop. From these signals a Fast Fourier Transformation (FFT) is made to investigate which frequencies are dominant in the response. The Fourier transform G(jω) of a time signal g(t) is given by [Bro85] as G(jω) = 0 g(t)e jωt dt T 0 g(t)e jωt dt, (3.) with T the (large enough) time span of the data and ω the angular frequency. Using digital signal processing equipment, like the PULSE system, this time signal will not be continuous, but consists of N discrete samples. When the frequency range of interest goes from 0 W Hz, a sampling frequency of at least 2W should be used to prevent aliasing, [Bro85]. This implies a maximal sample time of 2W and a total time span of 7

16 8 Chapter 3. Measuring and modeling noise and vibrations T = N 2W. The discrete Fourier transform is approximated by G k N k=0 g ke j2πk n N. (3.2) Both static and running measurements have been performed. In case of the static measurements the main parts are excited with a rubber hammer and the response of the part is measured. The number of samples is set to N = 2 3 = 892, with a frequency span of 2 Hz. In this way frequencies up to 6384 Hz will be measured. To prevent aliasing a sampling frequency of Hz (two times the highest frequency of interest) is used. A trigger is set to 3.6 m/s 2 so that the measurement starts after the impact and a Hanning window with maximum overlap is used to get the best results without losing 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 signals is made as well. In this case a trigger has not been used, but the measurements are done in a free run. Because the decanter is vibrating continuously an average of 400 samples 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 a running test is performed. A hose is attached to the decanter to simulate a processing situation, see Figure 3.. 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 the acceleration response of these main parts. Several points are used for each part. The results of the running tests are saved in a.txt file that is imported into Matlab. The data contains the amplitudes of the acceleration at each frequency of the FFT analysis, so 892 amplitudes for one channel. To investigate which frequencies are dominating the response, the data is sorted by descending amplitudes, see Appendix B for the Matlab file used. For each part the first 3 dominating frequencies are listed in Table 3.. As can be seen most of the dominating frequencies are a multiplication of the rotational frequency (54 Hz). This is due to the unbalance of the rotating parts. For example, the gearbox cover is vibrating with the highest amplitude at a frequency of 270 Hz, which is 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 noise and vibration analysis on some parts separately. In this report the gearbox cover is analyzed. The gearbox cover is chosen, because this cover is vibrating with the highest amplitude (measured with a tri-axial accelerometer) of all measured parts, with the decanter centrifuge in operating mode. Besides, the experience is that the connection points of the cover to the base fail relatively fast. The analysis exists of both a measurement of the response after a static impact and a modal analysis of the gearbox cover with the finite element package Ansys. The

17 3.2. Running tests 9 Part Frequency Harmonic freq. w.r.t. Amplitude perc. of highest [Hz] rotational freq. [-] [m/s 2 ] amplitude [%] Gearbox Cover Grey lid Hopper Main Bearing Base Backdrive guard Solid end cover Base frame Table 3.: Most dominant frequencies of some main parts with the decanter running at 3250 RPM.

18 0 Chapter 3. Measuring and modeling noise and vibrations Water hose Solids end cover Lid Hopper Base Base frame Gearbox cover Solids end Liquids end Figure 3.: A water hose was attached to the decanter to perform a running test. The flow was set to 4 m 3 /s. results of the static impact test are used to obtain the eigenfrequencies and the results of 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 the measurements. 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 direction of a surface, the part will start vibrating in the same direction (out of plane). Due to this impulse the part is excited in all frequencies of interest. By positioning the accelerometer at different positions on this surface (the same positions as in the running test) all eigenmodes within the frequency range of interest can be extracted. When only one position would be used, the chance exists that the accelerometer is on a nodal line of a mode.

19 3.4. Noise radiation 3.3. 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 are created with SolidWorks software. These files can be exported as IGES files, which can be imported in the commercial FE software package Ansys. Within Ansys the model is meshed. Moreover the material properties and boundary conditions are assigned. As the main parts are made of steel, damping is not taken into account in the numerical analysis. The structural dynamics of an undamped system are given by the equations of motion [Ans07]: Mü + Ku = 0, (3.3) with u the column with degrees of freedom and M and K the mass and stiffness matrices respectively. The eigenvalue problem corresponding to this free vibrating undamped system is given by: ( ω 2 M + K)u = 0. (3.4) The values for ω for which the determinant of the matrix [ ω 2 M+K] equals zero are the eigenvalues. The modal analysis function in Ansys solves this eigenvalue problem and stores the eigenvalues λ = ω 2 in a diagonal matrix λ i. The corresponding eigenvectors φ i satisfy: (K λ i M)φ i = 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 pressures in the surrounded fluid by a structure using the Helmholtz differential equation. Only the 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 equation can be reduced to an integral equation that covers only the boundary surface S. The complete derivation can be found in [Vis04]: α( x)p( x) = ( ) G(r) S p( y) + iωρ 0 G(r)v ny ( y) ds + p in ( x). (3.6) n y In (3.6) the acoustic pressure and normal velocity are related to the radiated pressure field in the fluid domain. The term α( x) is a geometry related coefficient, y is a point on the boundary surface S and x is a field point in the fluid domain. The unit normal to the surface at source point y, denoted as n y, is pointed into the fluid domain. The distance r is the length of vector r that is directed from the source point y to the field point x : r = x y. The term p in represents the incident acoustic wave in the case

20 2 Chapter 3. Measuring and modeling noise and vibrations of a scattering analysis and G(r) is the Green s function, which represents the effect observed 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 normal velocity and pressure at the surface S should be known. If only the normal velocities are known, first the pressures at the surface have to be calculated by replacing x = y in (3.6). Secondly the pressures at the field points can be calculated Boundary Element Method The analysis of vibrations and the resulting radiated sound can be done with the use of sophisticated computer software, e.g. for calculating the dynamics of a practical structure a Finite Element (FE) Model with appropriate boundary conditions can be used. 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 to be 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 have a contribution. Only these elements are in contact with the fluid to which energy is transferred. For the purpose of determining the radiated sound by a structure a Boundary Element (BE) model will suffice. The advantage of a BE model is that less equations are to be solved, as in general there are less nodes in a BE model compared with a FE model. In this report Ansys is used for the FE analysis and LMS Virtual.Lab, together with SSNOISE, for the BE analysis. From the structural vibration data at the nodes the acoustic pressures at the surface can be calculated with the Helmholtz integral equation (3.6). This requires a discretization of this continuous equation, see [Vis04], giving Ap = Bv, (3.7) where the matrices A and B are the influence matrices. These matrices are dependent on 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 node can be calculated Radiation Efficiency A useful measure of the effectiveness of sound radiation by a vibrating surface is the total radiated sound power normalized with respect to the specific acoustic impedance of the fluid medium, the structure area and the velocity of the surface vibration, which is defined as the radiation efficiency. A commonly used measure of the surface vibration is the space-average value of the time-averaged squared vibration velocity defined by vn 2 = ) T S S( T 0 v2 n y ( y)dt ds, (3.8)

21 3.4. Noise radiation 3 where T is a suitable period of time over which to estimate the mean square velocity vn 2 y at a point y and S extends over the total vibrating surface. The radiation efficiency is defined by reference to the acoustical power radiated by a uniformly vibrating baffled piston at a frequency for which the piston circumference greatly exceeds the acoustic wavelength k: ka. For the radiated power of a baffled piston the following relation holds: The definition of the radiation efficiency is thus: P = 2 ρ 0cSv 2 n. (3.9) σ = P /ρ 0 csv 2 n. (3.0) Radiated Power In the case of a structure modeled within a FE software package the structural vibration of this structure with R elements can be calculated with FE method. In this way it is possible to obtain a column vector of complex velocities at each element center caused by a harmonic point force. The velocities are grouped in a column vector, like: v e = [ v e v e2... v er ] T. (3.) With the calculated velocities the sound pressure and radiated sound power can be calculated within a BEM package. The obtained sound pressure at each element is also grouped in a column vector: p e = [ p e p e2... p er ] T. (3.2) From a BE model the relation between the elemental velocities and sound pressures can be found. As a result of (3.7) the sound pressures can be denoted as p e = A Bv e (3.3) Analytical equations for the radiated sound power [Fah87] give the relationship between the velocities and sound pressures as in P (ω) = R r= 2 A ere(verp er ) = S 2R Re(vH e p e ), (3.4) where A e and S are respectively the areas of each element and of the whole structure. Substituting (3.3) into (3.4) result in P (ω) = S 2R Re(vH e A Bv e ). (3.5)

22 4 Chapter 3. Measuring and modeling noise and vibrations 3.5 Summary To compute the radiated sound field by a vibrating structure a computer model can be used. First the dynamics of the structure should be identified. The eigenfrequencies and corresponding mode shapes extracted by a numerical software program are compared with physical measurements of the excited structure after an impact. This comparison of the eigenfrequencies gives an indication of the accuracy of the model. The vibrations that contribute to the radiated sound field can be computed by solving the Helmholtz integral equation for a certain frequency domain. By this equation the harmonic velocities of the vibrating structure are related to the resulting acoustical pressures at the surface. The radiated sound power can be computed from this acoustical impedance.

23 Chapter 4 Implementation, verification and results of a numerical model A numerical model is used to compute the forced harmonic response of the gearbox cover. Consequently this harmonic response is used to compute the radiated sound field. Experiments are made to investigate the accuracy of the finite element model. 4. Eigenfrequencies and mode shapes extraction 4.. Static impact test The measured accelerations during the static impact test performed on the gearbox cover can be found in Figure 4.. Note that the amplitudes are normalized so that the highest response has an amplitude of m s 2. Moreover, in Table 4. the first 0 frequencies that 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 spectrum is an average of multiple measurements at different positions and that an interval of 2 Hz is used during the measurements. These peaks are listed in Table 4. as being one natural frequency, that belongs to the highest peak Numerical implementation 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 assembly containing the back drive belt guard and gearbox cover, see Figure 4.2. Basically the gearbox cover is only connected to the backdrive belt guard with four bolts through the small holes in the sides. To simplify the model as much as possible it is decided to leave the backdrive belt guard out the analysis and to apply constraints to the four 5

24 6 Chapter 4. Implementation, verification and results of a numerical model Acceleration [m/s 2 ] Frequency [Hz] Figure 4.: The dominating frequency responses of the gearbox cover during static impact test. Frequency [Hz] Table 4.: Most dominant frequencies of the gearbox cover during a static impact test.

25 4.. Eigenfrequencies and mode shapes extraction 7 Figure 4.2: CAD file of the assembled gearbox guard, provided by G-Tech. (a) Gearbox cover, originally. (b) Gearbox cover, adapted with front plate. 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 is modeled additionally in SolidWorks, see Figure 4.3. The next step is to save this CAD file as an IGES file that can be imported into Ansys. Because the gearbox cover is made of 2 mm thick steel, shell elements can be used. This requires Ansys to import only areas instead of a (solid) volume. The choice is made to keep only the outside areas, that subsequently are meshed with 8 node shell elements. These elements use quadratic shape functions that are more accurate than 4 node shell elements, with linear shape functions. Special attention is paid to the back panel with the ventilation holes in it. Around these holes a finer mesh is used compared to the other areas. The result is an element size of 20 mm for the large areas and a mesh size of 3 mm around the small holes. In total the model has 9803 elements and nodes with 6 degrees of freedom (DOF). The mesh used is presented in Figure 4.4. The model would be less accurate when larger elements would be used as can be seen in a plot of the eigenfrequency versus the mode number in Figure 4.5. In this figure the

26 8 Chapter 4. Implementation, verification and results of a numerical model ELEMENTS APR :22:5 Figure 4.4: The finite element mesh of the gearbox cover. Property Value oungs Modulus [GPa] 207 Poisson ratio [-] 0.3 Density [kg/m 3 ] 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 side and top panels. The gearbox cover is made of low carbon mild steel, the material properties are listed in Table 4.2, [Hea97] [Ger99]. In Ansys the linear isotropic elastic material model is selected. The static impact test is performed with the gearbox cover mounted on the decanter, so boundary conditions are needed to represent the same situation. Therefore the translational DOF s in and direction of the nodes that are attached to the machine are deleted. Moreover all DOF s, except for the translational degree of freedom in direction, at the nodes around the small holes in both sides are deleted to represent the situation that the cover is bolted to the backdrive belt guard Numerical results The first 2 mode shapes calculated with Ansys are presented in Figure 4.6. As can be seen the modes consist of panels (back, front and side) vibrating out of plane. The

27 4.. Eigenfrequencies and mode shapes extraction mm 40 mm 60 mm 80 mm 200 Frequency [Hz] Mode nr [ ] 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 resonating at the lowest eigenfrequency Comparison between numerical and experimental results The results of the static impact measurements contain the dominant frequencies in, and directions. The different panels of the gearbox cover will vibrate out of plane. This implies that the dominant frequencies in direction are generated by the side panels. Even so, the response in the -direction is generated by the top panel and the response in direction by the back panel. To identify the corresponding mode shape from the modal analysis one needs to check the mode shape that occurs at a corresponding frequency from the measurements. If the direction (, or ) corresponds with the out of plane movement of the right panel, this will indicate that the Finite Element Model complies with the measurement. In Table 4.3 the numerically determined eigenfrequencies that correspond with the dominating frequencies from the static impact test are listed. Sometimes it is impossible to identify the mode shape that corresponds with a peak in the static impact test. In that case a - is denoted for the eigenfrequency calculated with the modal analysis. The numerically calculated mode shapes that correspond with the lowest eigenfrequencies in, and 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 large error between the numerical and experimental result for the first eigenfrequency is due to the applied boundary conditions at the holes where the cover is bolted to the back

28 STEP= SUB = FREQ=42.7 RSS=0 DM =.84 S =.226E-05 SM =.84 STEP= SUB =4 FREQ= RSS=0 DM =.43 S =.25E-03 SM =.43 STEP= SUB =7 FREQ= RSS=0 DM =.287 S =.809E-03 SM =.287 STEP= SUB =0 FREQ=4.699 RSS=0 DM =.442 S =.230E-03 SM =.442 M M M M APR :58:47 PLOT NO. APR :0:53 PLOT NO. APR :3:30 PLOT NO. APR :4:35 PLOT NO. STEP= SUB =2 FREQ= RSS=0 DM = S =.639E-03 SM = STEP= SUB =5 FREQ= RSS=0 DM =.263 S = SM =.263 STEP= SUB =8 FREQ=90.9 RSS=0 DM =.833 S =.258E-03 SM =.833 STEP= SUB = FREQ=4.84 RSS=0 DM =.55 S =.527E-03 SM =.55 M M M M APR :09:29 PLOT NO. APR ::40 PLOT NO. APR :3:54 PLOT NO. APR :5:07 PLOT NO. STEP= SUB =3 FREQ= RSS=0 DM = S =.547E-04 SM = STEP= SUB =6 FREQ= RSS=0 DM =.428 S =.79E-03 SM =.428 STEP= SUB =9 FREQ=2.5 RSS=0 DM =2.22 S = SM =2.22 STEP= SUB =2 FREQ=29.0 RSS=0 DM = S =.860E-04 SM = M M M M APR :0:4 PLOT NO. APR :2:00 PLOT NO. APR :4:0 PLOT NO. APR :5:23 PLOT NO. 20 Chapter 4. Implementation, verification and results of a numerical model.226e (a) 43 Hz.639E (b) Mode 50 Hz.547E (c) Mode 54 Hz.25E E (d) Mode 69 Hz (e) Mode 74 Hz (f) Mode 89 Hz.809E E (g) Mode 89 Hz (h) Mode 9 Hz (i) Mode 2 Hz.230E E E (j) Mode 5 Hz (k) 5 Hz (l) Mode 29 Hz Figure 4.6: First 2 structural modes of the Gearbox Guard.

29 4.2. Running mode 2 Direction Eigenfreq. static Eigenfreq. modal Rel. error impact test [Hz] analysis [Hz] [%] (,), 4 (-,-), 5 (-,-), Table 4.3: Comparison between the eigenfrequencies determined in a static impact test and a modal analysis in Ansys. drive guard, that is not infinitely stiff in reality. At low frequencies this will result in a larger error than at high frequencies, where the displacements are smaller. NB due to the high modal density of the numerical model it is not always possible to make a fair comparison, especially for high frequencies. 4.2 Running mode When the G-Tech decanter is running the whole structure is vibrating, due to the unbalanced rotating mass. To predict the sound radiated by the gearbox cover, first the structural vibrations need to be calculated. In Ansys the forces acting on the gearbox 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-axial accelerometer.

30 STEP= SUB =2 FREQ= RSS=0 DM = S =.639E-03 SM = M APR :09:29 PLOT NO. STEP= SUB =37 FREQ= RSS=0 DM =.296 S =.567E-03 SM =.296 M APR :26:06 PLOT NO. STEP= SUB =0 FREQ=4.699 RSS=0 DM =.442 S =.230E-03 SM =.442 M APR :43: PLOT NO. 22 Chapter 4. Implementation, verification and results of a numerical model.639e e e (a) Mode 50 Hz, side panel is vibrating in direction (b) Mode 350 Hz, top panel is vibrating in direction (c) Mode 5 Hz, back panel is vibrating in direction Figure 4.7: Structural eigenmodes of the gearbox guard that correspond with the static impact test measurement Measurement of the forces acting on the gearbox cover The acceleration at the four connection points (two on each side) are measured using a tri-axial accelerometer. For each point three measurements (FFT s) are made which are averaged. Also the standard deviation between the different measurements is calculated to see if the measurements are repeatable. In Figure 4.8 and Figure 4.9 the results for the complete frequency region and a zoomed version up to 000 Hz are shown. As can be seen the highest responses occur at harmonic frequencies of 54 Hz (up to the 0 th harmonic of 540 Hz) and in the high frequency region of 5 to 2 khz. The forces that are involved with these accelerations can be determined easily by applying Newton s second law F = ma, (4.) with m and a the mass (5.4 grams) and acceleration of the accelerometer respectively Numerical implementation With the measured forces applied to the numerical model of the gearbox cover it is possible to calculate the structural response for the running mode. Because the amplitudes of the forces are different for each frequency, separate loadcases need to be defined in Ansys. These loadcases contain the average amplitudes of the harmonic forces in, and direction as measured during the running test and are applied to the nodes at the vertical edges at the front of the gearbox cover, see Figure 4.0. It is chosen to use only the first 0 harmonics of the running frequency of 54 Hz in the simulation. The high frequency responses have a large standard deviation and therefore they are not taken into account in the numerical simulation. NB these frequencies could be dominant in the radiated sound field and should be investigated in more detail. See Table 4.4 for the input forces used. The dynamic response to the harmonic forces listed in Table 4.4 is calculated within Ansys. To reduce the calculation time, the analysis type is set to modal superposition,

31 4.2. Running mode 23 Average acc. [m/s 2 ] Frequency [Hz] Standard deviation [m/s 2 ] Frequency [Hz] Figure 4.8: Average accelerations and standard deviation measured at the connection points of the gearbox cover. Average acc. [m/s 2 ] Frequency [Hz] Standard deviation [m/s 2 ] Frequency [Hz] Figure 4.9: oomed, average accelerations and standard deviation measured at the connection points of the gearbox cover.

32 24 Chapter 4. Implementation, verification and results of a numerical model Figure 4.0: The harmonic loads and constraints used for the running simulation. Frequency F x 0 3 F y 0 3 F z 0 3 [Hz] [N] [N] [N] Table 4.4: Amplitudes of forces in, and direction used in Ansys for the running mode analysis.

33 4.3. Noise radiation 25 which means that the response is based on the preceding modal analysis. For this modal analysis the boundary conditions at the connection points as mentioned in Section 4..2 are deleted. The undamped equations of motion (3.5) become [Ans07] φ j T Mφ j ÿ j + φ j T Kφ j y j = φ j T F, (4.2) with φ j the j-th mode shape and y j a set of modal coordinates, such that u = n j= φ jy j, with n the number of modes on which the superposition is based. As a rule of thumb the number of modes n should contain at least 50% of the eigenfrequencies more than the highest frequency of interest in the harmonic response analysis case [Ans07]. This implies that the eigenfrequency of the highest mode in the mode superposition analysis should be at least 80 Hz for accurate results. 200 Modes are selected in the mode superposition analysis case, with the highest mode shape and eigenfrequency at 309 Hz. The complete input file used in Ansys, for both the modal and forced response analysis, can be found in Appendix C Numerical results for the response in running mode The deformed shape of the gearbox cover due to the harmonic forces, as mentioned in Table 4.4, is presented in Figure 4. for each frequency of excitation. The displacements are the highest for the low frequencies with a maximum of 5.e 3 mm in -direction at 54 Hz at the lowest connection points, see Figure 4.(a). 4.3 Noise radiation The structural (harmonic) displacements calculated within Ansys can be used in a Boundary Element (BE) analysis to predict the radiated sound power Numerical implementation The results of the harmonic response analysis case in Ansys are stored in a.rst file that is imported into LMS Virtual.Lab. First a Load Vector Set is created that contains the displacement vector for every node of the gearbox cover mesh. Next step is to create a surrogate acoustic mesh on which this structural data is projected. For an acoustical analysis 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 quadrilateral shell elements, see Figure 4.2 for both meshes. This coarse mesh consisted of 4902 elements and 5360 nodes. The mesh used in the structural analysis consists of 9803 elements and nodes respectively. So the number of equations to be solved is reduced by 84%. To set up the acoustical analysis in LMS Virtual.Lab first the Harmonic BEM Toolbox is activated. Next the model type definition is set to BEM Direct, exterior, as

34 M M.55E E-04.6E-03.73E-03.23E E E E E-03.59E E-06.95E E E E E-04.3E-03.32E-03.5E-03.70E-03 M.936E-07.34E E-04.40E E E-04.80E E-04.07E-03.20E-03 M.290E-07.8E E E-05.76E E-05.07E-04.25E-04.43E-04.6E-04 M M.45E-06.57E-04.30E E-04.66E E-04.92E-04.07E-03.23E-03.38E-03.4E E E E-05.0E-04.26E-04.5E-04.76E-04.20E E E-06.80E E-04.53E E E-04.06E-03.23E-03.4E-03.58E E E-06.50E E E-06.25E-05.50E-05.75E-05.99E E-05 M M M M 26 Chapter 4. Implementation, verification and results of a numerical model STEP= SUB = FREQ=54 RSS=0 DM = S =.439E-05 SM = APR :35:5 PLOT NO. STEP= SUB =2 FREQ=08 RSS=0 DM =.0057 S =.35E-06 SM =.0057 APR :36:07 PLOT NO..439E E (a) Deformed 54 Hz.35E-06.75E E E E E (b) Deformed 08 Hz STEP= SUB =3 FREQ=62 RSS=0 DM =.59E-03 S =.55E-06 SM =.59E-03 APR :36:29 PLOT NO. STEP= SUB =4 FREQ=26 RSS=0 DM =.38E-03 S =.45E-06 SM =.38E-03 APR :36:44 PLOT NO. (c) Deformed 62 Hz (d) Deformed 26 Hz STEP= SUB =5 FREQ=270 RSS=0 DM =.70E-03 S =.709E-06 SM =.70E-03 APR :4:09 PLOT NO. STEP= SUB =6 FREQ=324 RSS=0 DM =.226E-04 S =.4E-07 SM =.226E-04 APR :4:59 PLOT NO. (e) Deformed 270 Hz (f) Deformed 324 Hz STEP= SUB =7 FREQ=378 RSS=0 DM =.20E-03 S =.936E-07 SM =.20E-03 APR :42:45 PLOT NO. STEP= SUB =8 FREQ=432 RSS=0 DM =.58E-03 S =.482E-06 SM =.58E-03 APR :43:43 PLOT NO. (g) Deformed 378 Hz (h) Deformed 432 Hz STEP= SUB =9 FREQ=486 RSS=0 DM =.6E-04 S =.290E-07 SM =.6E-04 APR :44:44 PLOT NO. STEP= SUB =0 FREQ=540 RSS=0 DM =.224E-05 S =.289E-08 SM =.224E-05 APR :45:38 PLOT NO. (i) Deformed 486 Hz (j) Deformed 540 Hz Figure 4.: Nodal displacements of the gearbox cover for different frequencies of excitation.

35 4.3. Noise radiation 27 (a) Fine structural mesh (b) Coarse acoustical mesh Figure 4.2: For the acoustical analysis a coarser mesh is used than for the structural analysis. only the exterior problem is to be solved. Now a Mesh Preprocessing Set can be defined for the coarse acoustical mesh, to ensure that all normal directions to the structure are set consistently. A new Material and Material Property are assigned to the wetted surface and the properties are set for air (density of.225 kg/m 3 and speed of sound of 340 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 8 influencing nodes and a distance of 2 mm are selected. After the projection is calculated a picture of the deformed acoustical mesh is made to be sure that the projection is properly done. To compute the acoustic pressures at the surface of the gearbox cover the velocities at the nodes are needed. These velocities are calculated by differentiating the displacements that were calculated in the harmonic loadcase in Ansys. The Transfer Vector Set can now be used as an acoustical boundary condition. A Boundary Condition and Source Set is defined and the Transfer Vector Set is added as a source. Finally the Acoustical Response Set can be defined and the Boundary Condition and Source Set is assigned to it. Now the simulation can be started and once it is finished the acoustical power can be displayed in a graph. The total setup in LMS Virtual.Lab can be seen in Figure 4.3.

36 28 Chapter 4. Implementation, verification and results of a numerical model Figure 4.3: Acoustical simulation in LMS Virtual.Lab Numerical results The total radiated sound power (re 0 2 W) and radiation efficiency are plotted in a graph, see Figure 4.4. Note that the lines between the circular data points do not represent any prediction on the sound power, as the amplitudes of the forces for these frequencies are much lower. The value of the radiated sound power is dependant on both the amplitude of the velocities and the radiation efficiency. The velocities will be high when the structure is excited with high forces or near an eigenfrequency. Also, in general, the velocities are higher for low frequencies. 4.4 Summary A numerical model of the gearbox cover has been developed to compute the structural vibrations due to a forced harmonic response. The finite element model used to compute these vibrations is verified by measurements. The error between the eigenfrequencies determined 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 comparison cannot be made for all frequencies. The structural vibrations that are computed with finite element method are used in a boundary element model to compute the radiated sound field. For frequencies up to 550 Hz the maximal acoustic power level is 42 db (re 0 2 W).

37 4.4. Summary Acoustic Power [db] Frequency [Hz] Radiation efficiency [ ] Frequency [Hz] Figure 4.4: Radiated sound power (re 0 2 W) and radiation efficiency.

38 30 Chapter 4. Implementation, verification and results of a numerical model

39 Chapter 5 Design changes In general there are four methods to control noise and vibrations [Ren]: () absorption, (2) use of barriers and enclosures, (3) structural damping and (4) vibration isolation. In case of the gearbox cover, damping is investigated. 5. Damping The gearbox cover is made of low carbon mild steel. The material damping of steel is assumed to be negligible. This results in an infinite high response at the eigenfrequencies obtained in the modal analysis. To reduce the amplitudes of the response at the eigenfrequency, (material) damping should be included in the model. Within Ansys the harmonic response is calculated using a modal superposition method, see (4.2). To include damping in the model, several damping inputs can be given in Ansys, resulting in a total modal damping factor ξ j of [Ans07]: ξ j = α 2ω j + βω j 2 + ξ + ξ m j, (5.) with α and β the Rayleigh damping multipliers for mass and stiffness respectively, ξ a constant damping ratio and ξj m a modal damping ratio. As modal damping has only an effect near the eigenfrequencies, the mass and stiffness of the system should be changed to 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 a change of material. The main function of the gearbox cover is to protect the rotating parts from external influences, so mechanical properties such as oung s Modulus are 3

40 32 Chapter 5. Design changes Property Value oungs Modulus [GPa].3 Poisson ratio [-] 0.4 Density [kg/m 3 ] 930 Table 5.: Material properties of ultra high molecular weight polyethylene Steel (2 mm) UHMWPE (4 mm) 000 Frequency [Hz] Mode nr [ ] Figure 5.: Eigenfrequency versus mode number for both the steel and UHMWPE gearbox cover. not so important. Therefore it is investigated what the effect is on the response of the gearbox cover, if it would be made out of 4 mm thick polyethylene. This material has a high (compared with steel) material damping and is cost efficient Modal analysis The material properties in Ansys are replaced by the properties of ultra high molecular weight polyethylene (UHMWPE) [War7], see Table 5.. Also the thickness of the shell elements is increased to 4 mm. After changing the material properties a modal analysis is done. Due to the changes in the material properties the eigenfrequencies will be lower compared to the gearbox cover made out of steel. This implies that more modes are needed in a modal superposition harmonic response analysis case, see Figure 5.. To comply with the rule of thumb [Ans07] to use at least a number of modes such that the highest eigenfrequency is 50% higher than the highest frequency of interest, 300 modes are extracted Damping Because of the visco-elastic (time dependant) material behavior of plastics, the material damping of UHMWPE will be frequency dependant. The damping for high frequencies will be higher than for low frequencies. From the options available in Ansys it is chosen

41 5.2. Material 33 β(e 5 ) Damping 00 Hz Table 5.2: Different values for the Rayleigh damping coefficient β that are used in the harmonic simulation. to model the damping characteristics of the UHMWPE with Rayleigh damping based on the stiffness matrix, so the modal damping factor becomes ξ j = βω j 2. (5.2) The value for β in this linear equation will determine the modal damping for the complete frequency domain. However, no literature is found to verify this model of the damping characteristics. Consequently, it would be good practice to perform measurements to get insight in the real damping characteristics of this polymer. For the moment different values for β are used to obtain a first idea of how damping influences the response, see Table 5.2. The corresponding damping ratio at 00 Hz is also denoted Harmonic response To predict the harmonic response of the gearbox cover with this new material, the forced response analysis case is repeated. The damping is added with values for β as in Table 5.2. The maximum displacements due to the harmonic forces are presented in Table 5.3. As can be seen the maximum displacements for the UHMWPE gearbox cover are higher than the displacements for the steel gearbox cover, especially for low frequencies. Damping has a high influence at frequencies near the eigenfrequencies of the system. This influence can be clearly seen at excitation frequencies of 62, 26 and 270 Hz. For the other frequencies the damping has less influence, as the gearbox cover is not excited near the eigenfrequency. In Figure 5.2 the results of the deformed shape of the gearbox guard made out of UHMWPE 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 of the deformed shape of the gearbox guard made out of UHMWPE without damping for higher frequencies are presented in Appendix E. The effect of Rayleigh damping, with β = 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 damping is applied.

42 M M.35E-06.75E E E E E M M.895E Chapter 5. Design changes Maximum displacement [µm] Frequency Steel Plastic Plastic Plastic Plastic [Hz] β = 0 β = 0 β = 3e 5 β = 6e 5 β = 9e Table 5.3: Maximum displacements for the running mode analysis for steel and UHMWPE material properties and different values for the Rayleigh damping. STEP= SUB = FREQ=54 RSS=0 DM = S =.439E-05 SM = APR :35:5 PLOT NO. STEP= SUB = FREQ=54 RSS=0 DM = S =.38E-04 SM = APR :55:2 PLOT NO..439E E E Modalanalysis of Plastic Gearbox Guard (a) Steel: Deformed 54 Hz. (b) UHMWPE: Deformed 54 Hz. STEP= SUB =2 FREQ=08 RSS=0 DM =.0057 S =.35E-06 SM =.0057 APR :36:07 PLOT NO. STEP= SUB =2 FREQ=08 RSS=0 DM = S =.895E-04 SM = APR :57:9 PLOT NO. Modalanalysis of Plastic Gearbox Guard (c) Steel: Deformed 08 Hz. (d) UHMWPE: Deformed 08 Hz. Figure 5.2: Nodal displacements of the forced response of the gearbox cover made out of steel and UHMWPE.

43 M M M M 5.2. Material 35 STEP= SUB = FREQ=54 RSS=0 DM = S =.38E-04 SM = APR :55:2 PLOT NO. STEP= SUB = FREQ=54 RSS=0 DM = S =.63E-04 SM = APR :30:02 PLOT NO..38E Modalanalysis of Plastic Gearbox Guard.63E Modalanalysis of Plastic Gearbox Guard (a) Deformed 54 Hz without damping. (b) Deformed 54 Hz with β = 9.549e 5 STEP= SUB =9 FREQ=486 RSS=0 DM =.65E-04 S =.860E-07 SM =.65E-04 APR :05:29 PLOT NO. STEP= SUB =9 FREQ=486 RSS=0 DM =.660E-05 S =.4E-07 SM =.660E-05 APR :44:2 PLOT NO..860E-07.69E-05.37E E E E-04.40E E E-04.65E-04 Modalanalysis of Plastic Gearbox Guard.4E E-06.48E-05.22E E E-05.44E-05.54E E E-05 Modalanalysis of Plastic Gearbox Guard (c) Deformed 486 Hz without damping. (d) Deformed 486 Hz with β = 9.549e 5 Figure 5.3: The influence of Rayleigh damping on the deformed shape of the plastic gearbox cover.

44 36 Chapter 5. Design changes Acoustic Power [db] steel, β = 0 plastic, β = 0 plastic, β = plastic, β = plastic, β = Frequency [Hz] Figure 5.4: Radiated sound power for the gearbox cover made out of UHMWPE for different damping values Noise radiation With the harmonic response information for the plastic gearbox cover the radiated sound power can be predicted in the same way as for the steel gearbox cover. In Figure 5.4 the radiated sound powers can be found for the gearbox cover made out of UHMWPE for different damping values. Also the radiated sound power for the steel gearbox cover is shown for reference. As can be seen in Figure 5.4 the damping has a high influence on the radiated sound power at frequencies near the eigenfrequencies of the gearbox cover. For low frequencies however, the radiated sound power is higher for the gearbox cover made out of 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 the average 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 radiates sound with a power of 9 db more in the low frequency region and has a reduction of 4 db in the high frequency region. To verify the results for the radiated power by the steel cover, it is recommended to perform a sound intensity scan. However, when the decanter is running, sound will be transmitted by the rotating parts within the gearbox cover, which will have a contribution in the measurement as the gearbox cover is not a closed enclosure. Therefore it would be good practice to excite the gearbox cover with a shaker with a known frequency and amplitude to verify the used model.

45 5.3. Summary Summary The numerical model of the gearbox cover has been changed with respect to the material properties of the cover. The material properties are changed to ultra high molecular weight polyethylene. Also the thickness of the cover is increased to 4 mm (was 2 mm for the steel one). This plastic has a high material damping, so damping is included in the model. 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 plastic gearbox cover will increase with 9 db in the frequency range to 400 Hz and decrease with 4 db for higher frequencies to 550 Hz.

46 38 Chapter 5. Design changes

47 Chapter 6 Conclusion and Recommendations 6. Conclusion This report contains a noise and vibration survey of the G-Tech 456 gearbox cover. The vibration survey measurements are done using a tri-axial accelerometer that is mounted on the cover. For both the operational and non-operational state the main vibrations are measured. In the non-operational state the eigenfrequencies are extracted and using the modal analysis toolbox in the finite element package Ansys, the corresponding mode shapes are identified. Moreover, the eigenfrequencies obtained with the modal analysis are compared with the measurements. In the operational mode the forces acting on the gearbox cover are measured and modeled in Ansys. In this way it is possible to simulate the 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 to 540 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 is 5.7%. However, due to the high modal density of the numerical model, it is not always possible to make a fair comparison. In the frequency range up to 540 Hz the dominant frequency in the operational mode is 270 Hz. However, the responses at 54 and 08 Hz show the biggest displacements. The acoustical response show the biggest response at 432 Hz due to a relative high radiation efficiency for that mode. When the gearbox cover would be made from 4 mm thick high molecular weight polyethylene the structural vibrations will have a bigger amplitude compared to the steel one. The acoustical response for high frequencies (above 378 Hz) however can be reduced by approximately 4 db. For lower frequencies the radiated power will be on 39

48 40 Chapter 6. Conclusion and Recommendations average 9 db higher. This result is only valid when the damping characteristics that are used in the model (Rayleigh damping, with β = ) is realistic. 6.2 Recommendations To verify the acoustic results it is recommended to perform a sound intensity scan of the gearbox 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 be seen from the measurements the gearbox cover is also excited with higher frequencies (at about 6000 Hz) than used in this analysis. These frequencies can have a big contribution in the radiated sound field. Therefore these high frequencies should be taken into account in the simulation to be able to compare the steel and plastic gearbox cover in the complete audible frequency domain. It is expected that the high material damping of plastic has a larger advantage in this frequency region.

49 Bibliography [Ans07] Release.0 documentation for ansys [Bro85] R.G. Brown and P..C. Brown. Introduction to random signals and applied Kalman filtering, volume 2. J. Wiley, New ork, 985. [Fah87] F. Fahy and P. Gardonio. Sound and Structural Vibration, Radiation, Transmission and Response, volume 2. Elsevier Acadamic, Amsterdam/London, 987. [Ger99] J.M. Gere and S.P. Timoshenko. Mechanics of materials 4th SI edition, volume 4. Stanley Thornes, Cheltenham, 999. [Hea97] E.J. Hearn. Mechanics of materials 2, volume [Rec0] A. Records and K. Sutherland. Decanter Centrifuge Handbook, volume. Elsevier Advanced Technologies, Oxford, 200. [Ren] J Renninger. Understanding damping techniques for noise and vibration control. [Vis04] R. Vissers. A boundary element approach to acoustic radiation and source identification, volume [War7] I.M. Ward and J. Sweeney. The mechanical properties of solid polymers, volume 2. J. Wiley, New ork, 97. 4

50 42 Bibliography

51 Appendix A Decanter Layout and technical data In Figure A. a cross-section of the G-Tech 456 decanter centrifuge is shown. Table A. is a list of the technical data of this decanter centrifuge. Technical data Max. Bowl Speed 4000 RPM Centrifugal Force 350 G s Differential Speed 4-48 RPM Run Up time 2-3 minutes Bowl Dimensions 4 (355 mm) Diameter x 56 (420 mm) Long Gross Weight 250 kg Shipping Volume 6 m 3 Wetted Parts 36 Stainless Steel Base Assembly Cast Iron Main Drive 8-37 kw 230/460 50/60 Hz Back Drive kw 230/460 50/60 Hz Table A.: Technical data of the G-Tech 456 decanter centrifuge. 43

52 44 Appendix A. Decanter Layout and technical data Clutch Gearbox Main bearing Bowl Conveyor Conical Main Bearing 2 Feed pipe Base Liquids End Solids End Figure A.: Cross section of the G-tech 456 decanter centrifuge.

53 Appendix B Matlab Code to import measurement data from PULSE The Matlab function file that is used to import the measurement response data obtained with a tri-axial accelerometer and a Brüel & 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_ = textscan(fid, %f %f %f, 892, headerlines, 83); fid = fopen(filename, r ); spectrum_ = textscan(fid, %f %f %f, 892, headerlines, ); fid = fopen(filename, r ); spectrum_ = textscan(fid, %f %f %f, 892, headerlines, ); %convert the cell arrays in arrays Freq = spectrum_{,2}; spec_ = spectrum_{,3}; spec_ = spectrum_{,3}; spec_ = spectrum_{,3}; %calculating the first 20 frequencies with the maximum response [max_,pos_] = sort(spec_, descend ); max_ = max_(:20); pos_ = pos_(:20); freq_ = Freq(pos_); [max_,pos_] = sort(spec_, descend ); max_ = max_(:20); pos_ = pos_(:20); freq_ = Freq(pos_); [max_,pos_] = sort(spec_, descend ); max_ = max_(:20); pos_ = pos_(:20); freq_ = Freq(pos_); %combining the results for the -- direction max_freq = [freq_ freq_ freq_]; 45

54 46 Appendix B. Matlab Code to import measurement data from PULSE max_resp = [max_ max_ max_]; %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 responses is listed below. clear all close all clc %%%%%% text files with the data % gearbox cover (liquid end) filename = [ 2_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 ; 3_S.txt ; 32_S.txt ; 33_S.txt ; 34_S.txt ; 35_S.txt ; 36_S.txt ; 37_S.txt ; 38_S.txt ; 39_ %%%%%% Extracting the maximum responses with the corresponding frequencies l = length(filename(:,)) % 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 = :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 and direction min_freq = min(min(max_freq(:,:l))); max_freq = max(max(max_freq(:,:l))); min_freq = min(min(max_freq(:,l+:2*l))); max_freq = max(max(max_freq(:,l+:2*l))); min_freq = min(min(max_freq(:,2*+:3*l))); max_freq = max(max(max_freq(:,2*+:3*l))); freq = []; resp = []; freq = []; resp = []; freq = []; resp = []; % Sorting the max response and averaging them q = ; for i = min_freq:2:max_freq % data captured with freq_span = 2 Hz [row,col] = find(max_freq(:,:l) == i); % finding freq between min and max freq (of the 20 frequencies) s = length(row); if s > 0 % if the frequency is found in the max_freq matrix freq(q,) = i; resp(q,) = mean(mean(max_resp(row,col),2)); %average over the number of measurements (@ diff positions) q = q+; end end

55 47 [resp,pos] = sort(resp, descend ); % sort with highest response on top freq = freq(pos); q = ; for i = min_freq:2:max_freq [row,col] = find(max_freq(:,l+:2*l) == i); s = length(row); if s > 0 freq(q,) = i; resp(q,) = mean(mean(max_resp(row,l+col),2)); q = q+; end end [resp,pos] = sort(resp, descend ); freq = freq(pos); q = ; for i = min_freq:2:max_freq [row,col] = find(max_freq(:,2*l+:3*l) == i); s = length(row); if s > 0 freq(q,) = i; resp(q,) = mean(mean(max_resp(row,2*l+col),2)); q = q+; end end [resp,pos] = sort(resp, descend ); freq = freq(pos); %figure with the max (averaged) responses, including bar plots figure( name, averaged ) semilogx(freq,resp, x,freq,resp, o,freq,resp, >, MarkerSize,6, LineWidth,2) hold on bar(freq,resp, k ) bar(freq,resp, k ) bar(freq,resp, k ) xlabel( Frequency [Hz] ) ylabel( Acceleration [m/s^2] ) legend(,, ) grid

56 48 Appendix B. Matlab Code to import measurement data from PULSE

57 Appendix C Ansys input code for the structural dynamic analysis The input file for Ansys in the structural dynamic analysis is listed below. In this file the gearbox cover made out of steel is analyzed. The material properties can be easily adapted when the simulation with plastic material properties has to be done. The lines starting with an exclamation mark are comments. The loadcases for 08 up to 486 Hz are not listed as they are the same as for 54 Hz, only with an other frequency and amplitude input. finish /clear /CWD, S:\all scratch\arjan\gtech\anss_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************** /AU5 IOPTN,MERG,ES!merging of keypoints IOPTN,SOLID,NO!creating of a solid/volume IOPTN,GTOLER,DEFA!tolerance of IGES import IOPTN,SMALL,ES!delete small areas IGESIN, GT50A 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,,shell93!using 8node shell elements 49

58 50 Appendix C. Ansys input code for the structural dynamic analysis MP,E,,E!material property, Modulus MP,PR,,v!material property, poisson ratio MP,DENS,,rho!material property, density!************meshing the geomertry*************** ALLSEL,all ASEL,S,area,,287,290!Select the areas to keep ASEL,A,area,,296 ASEL,A,area,,300,304 ASEL,A,area,,720,722 ASEL,A,area,,725 ASEL,A,area,,73 BOPTN, KEEP, ES!boolean operator to delete the inside areas!lplot!plot the lines LESIE,all, 20.0,,,,,,, 0!Subdivide different lines ALLSEL,all ASEL,S,area,,722,725,3!select the areas on top ASEL,A,area,,300,304 ASEL,A,area,,287,290 ASEL,A,area,,296 SMRTSIE,OFF MSHKE,2 DESIE, 3,,,,,2,20,, mopt,aorder,on mopt,expnd, mopt,trans,2 amesh,all!mesh the areas on top ALLSEL,all LSEL,S,line,,4964 LSEL,A,line,,590 LSEL,A,line,,4976 LSEL,A,line,,578 LSEL,A,line,,8 LSEL,A,line,,3770 LSEL,A,line,,32 LSEL,A,line,,3730 LSEL,A,line,,500 LSEL,A,line,,992 LESIE,all, 5,,,,,,, 0!Subdivide different lines ALLSEL,all ASEL,S,area,,720,73,!select the areas at sides SMRTSIE,OFF MSHKE,2 DESIE, 3,,,,,2,20,, mopt,aorder,on mopt,expnd, mopt,trans,.4 amesh,all ALLSEL,all

59 5 LSEL,all LSEL,U,line,,0304 LESIE,all, 3,,,,,,, 0!Subdivide remaining lines SMRTSIE,OFF MSHKE,2 DESIE, 3,,,,,2,20,, mopt,aorder,on mopt,expnd, mopt,trans,.4 amesh,72!mesh the areas at back!************boundar CONDITIONS**********!Set all dof=0 except U, 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,,384,3847,2 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,,5657,566,2 D,ALL,U,0 D,ALL,U,0 D,ALL,ROT,0 D,ALL,ROT,0 D,ALL,ROT,0 ALLSEL,ALL!************OBTAIN SOLUTION MODAL ANALSIS************** /solu antype,2!set modal analysis MODOPT,lanb,200!method used is block lanczos, 200 modes expand RESVEC,ON!Calculate residual vector EQSLV,FRONT MPAND,200!200 modes expanded solve finish!************harmonic 54 H************** ALLSEL,ALL /solu LSCLEAR,all!start with no loadsteps NSEL,U,node,,all!Select nodes where harmonic input will be given NSEL,S,node,,555,5539,2

60 52 Appendix C. Ansys input code for the structural dynamic analysis NSEL,A,node,,554 NSEL,A,node,,383,3837,2 NSEL,A,node,,3763 antype,3!set harmonic analysis NSUBST,!number of subsets in this loadcase HARFRQ, 54!Frequency of excitation HROPT, MSUP,200, BETAD,*3.83E-5!Set stiffness damping F, all, F, 5.3!Force in -direction F, all, F, 2.!Force in -direction F, all, F, 0.4!Force in -direction KBC,!stepped loads OUTRES,NSOL!write only the nodal dof solution OUTPR,NSOL!solution printout = dof solu ALLSEL,all LSWRITE...!************HARMONIC 540 H************** ALLSEL,ALL /solu NSEL,U,node,,all!Select nodes where harmonic input will be given NSEL,S,node,,555,5539,2 NSEL,A,node,,554 NSEL,A,node,,383,3837,2 NSEL,A,node,,3763 antype,3!set harmonic analysis NSUBST, HARFRQ, 540!Frequency of excitation HROPT, MSUP,200, BETAD,*3.83E-5!Set stiffness damping F, all, F, 0.!Force in -direction F, all, F, 0.!Force in -direction F, all, F, 0.0!Force in -direction KBC,!stepped loads OUTRES,NSOL!write only the nodal dof solution OUTPR,NSOL!solution printout = dof solu ALLSEL,all LSWRITE finish /solu LSSOLVE,,0, finish /solu ALLSEL,ALL EPASS,on

61 53 NUMEP,all,54,540!BETAD,*3.83E-5!Set stiffness damping OUTPR,nsol,all solve

62 54 Appendix C. Ansys input code for the structural dynamic analysis

63 Appendix D Structural eigenmodes of the gearbox guard that correspond with 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 dominant frequencies from this static impact test are listed. The corresponding mode shapes (first 6) can be found in Figures D., D.2 and D.3. 55

64 STEP= SUB =2 FREQ= RSS=0 DM = S =.639E-03 SM = STEP= SUB =24 FREQ= RSS=0 DM =.8998 S = SM =.8998 STEP= SUB =37 FREQ= RSS=0 DM =.296 S =.567E-03 SM =.296 STEP= SUB =70 FREQ= RSS=0 DM =.05 S =.987E-03 SM =.05 M M APR :09:29 PLOT NO. APR ::8 PLOT NO M M APR :26:06 PLOT NO. APR :25:37 PLOT NO. STEP= SUB =2 FREQ=29.0 RSS=0 DM = S =.860E-04 SM = STEP= SUB =33 FREQ= RSS=0 DM =.064 S =.39E-03 SM =.064 STEP= SUB =4 FREQ= RSS=0 DM = S = SM = STEP= SUB =7 FREQ= RSS=0 DM = S =.833E-04 SM = M M APR :5:23 PLOT NO. APR :6:23 PLOT NO..39E M M APR :32:27 PLOT NO. APR :26:5 PLOT NO. STEP= SUB =9 FREQ=83.62 RSS=0 DM =.547 S =.984E-03 SM =.547 STEP= SUB =38 FREQ=356.0 RSS=0 DM = S =.539E-03 SM = STEP= SUB =54 FREQ= RSS=0 DM =.024 S =.485E-03 SM =.024 STEP= SUB =88 FREQ=660.7 RSS=0 DM =.02 S = SM =.02 M M APR :08:27 PLOT NO. APR :4:07 PLOT NO..539E M M APR :35:8 PLOT NO. APR :3:32 PLOT NO. 56Appendix D. Structural eigenmodes of the gearbox guard that correspond with static impact tests.639e e e (a) Mode 50 Hz (b) Mode 29 Hz (c) Mode 84 Hz (d) Mode 239 Hz (e) Mode 302 Hz (f) Mode 356 Hz Figure D.: Structural eigenmodes of the Gearbox Guard that correspond with the static impact test measurement for -directions..567e e (a) Mode 350 Hz (b) Mode 379 Hz (c) Mode 442 Hz.987E (d) Mode 553 Hz.833E (e) Mode 560 Hz (f) Mode 66 Hz Figure D.2: Structural eigenmodes of the Gearbox Guard that correspond with the static impact test measurement for -directions.

65 STEP= SUB =0 FREQ=4.699 RSS=0 DM =.442 S =.230E-03 SM =.442 STEP= SUB =52 FREQ= RSS=0 DM = S =.9E-03 SM = M M APR :43: PLOT NO. APR :4:00 PLOT NO. STEP= SUB =38 FREQ=356.0 RSS=0 DM = S =.539E-03 SM = STEP= SUB =55 FREQ= RSS=0 DM =.05 S = SM =.05 M M APR :4:52 PLOT NO. APR :4:2 PLOT NO. STEP= SUB =5 FREQ= RSS=0 DM =. S =.0023 SM =. STEP= SUB =58 FREQ= RSS=0 DM =2.378 S =.0082 SM =2.378 M M APR :40:37 PLOT NO. APR :37:6 PLOT NO E E (a) Mode 5 Hz (b) Mode 356 Hz (c) Mode 437 Hz.9E (d) Mode 438 Hz (e) Mode 464 Hz (f) Mode 494 Hz Figure D.3: Structural eigenmodes of the Gearbox Guard that correspond with the static impact test measurement for -directions.

66 58Appendix D. Structural eigenmodes of the gearbox guard that correspond with static impact tests

67 Appendix E Results of the forced response of the gearbox guard made out of UHMWPE without damping The results of the deformed shape of the gearbox guard made out of UHMWPE without damping due to applied harmonic forces are presented in Figure E.. 59

68 M.38E E M.656E E-04.36E E-03.27E E E E-03.54E E-03 M.860E-07.69E-05.37E E E E-04.40E E E-04.65E-04 M M M.895E E M.224E E-04.38E E E-03.34E E E E-03.6E E E-05.87E E E E E E E E-04 M M M 60Appendix E. Results of the forced response of the gearbox guard made out of UHMWPE without damping STEP= SUB = FREQ=54 RSS=0 DM = S =.38E-04 SM = APR :55:2 PLOT NO. STEP= SUB =2 FREQ=08 RSS=0 DM = S =.895E-04 SM = APR :57:9 PLOT NO. Modalanalysis of Plastic Gearbox Guard (a) Deformed 54 Hz. Modalanalysis of Plastic Gearbox Guard (b) Deformed 08 Hz. STEP= SUB =3 FREQ=62 RSS=0 DM = S =.803E-04 SM = APR :59:22 PLOT NO. STEP= SUB =4 FREQ=26 RSS=0 DM = S =.340E-04 SM = APR :00:33 PLOT NO. Modalanalysis of Plastic Gearbox Guard (c) Deformed 62 Hz. Modalanalysis of Plastic Gearbox Guard (d) Deformed 26 Hz. STEP= SUB =5 FREQ=270 RSS=0 DM = S =.507E-05 SM = APR :0:25 PLOT NO. STEP= SUB =6 FREQ=324 RSS=0 DM =.450E-03 S =.278E-05 SM =.450E-03 APR :02:8 PLOT NO..507E E Modalanalysis of Plastic Gearbox Guard (e) Deformed 270 Hz..278E E-04.02E-03.52E-03.20E-03.25E-03.30E E E E-03 Modalanalysis of Plastic Gearbox Guard (f) Deformed 324 Hz. STEP= SUB =7 FREQ=378 RSS=0 DM =.609E-03 S =.656E-06 SM =.609E-03 APR :03:25 PLOT NO. STEP= SUB =8 FREQ=432 RSS=0 DM =.6E-03 S =.224E-05 SM =.6E-03 APR :04:36 PLOT NO. Modalanalysis of Plastic Gearbox Guard (g) Deformed 378 Hz. Modalanalysis of Plastic Gearbox Guard (h) Deformed 432 Hz. STEP= SUB =9 FREQ=486 RSS=0 DM =.65E-04 S =.860E-07 SM =.65E-04 APR :05:29 PLOT NO. STEP= SUB =0 FREQ=540 RSS=0 DM =.834E-04 S =.206E-06 SM =.834E-04 APR :07:04 PLOT NO. Modalanalysis of Plastic Gearbox Guard (i) Deformed 486 Hz. Modalanalysis of Plastic Gearbox Guard (j) Deformed 540 Hz. Figure E.: Nodal displacements of the gearbox cover for different frequencies of excitation.