DETC98/PTG-5788 VIBRO-ACOUSTIC STUDIES OF TRANSMISSION CASING STRUCTURES

Transcription

1 Proceedings of DETC98: 1998 ASME Design Engineering Technical Conference September 13-16, 1998, Atlanta, GA DETC98/PTG-5788 VIBRO-ACOUSTIC STUDIES O TRANSMISSION CASING STRUCTURES D. Crimaldi Graduate Research Associate Acoustics and Dynamics Laboratory Department of Mechanical Engineering The Ohio State University Columbus, OH crimaldd@elcsci.com R. Singh Professor of Mechanical Engineer Acoustics and Dynamics Laboratory Department of Mechanical Engineering The Ohio State University Columbus, OH singh.3@osu.edu ABSTRACT Automotive transmission casing plates of irregular shape, with complex boundary conditions and non-uniform material properties, are experimentally and computationally studied to acquire a fundamental understanding of their dynamic and acoustic radiation characteristics. A modified flat cover is designed which simplifies the geometry while providing uniform thickness and material properties. Both covers ( reallife and laboratory ) are studied with free and bolted boundary conditions. In particular, the free boundary conditions are useful because they eliminate the cover-housing interaction allowing for a more detailed analysis of the cover plate. inite element models for both covers under the free boundary conditions are developed and refined. Predicted natural frequencies and mode shapes are in excellent agreement with measured modal data. Then the finite element models are coupled with boundary element models to predict acoustic radiation properties. Predictions match well with measured acoustic directivity at resonant frequencies. (a) Cover A 1 INTRODUCTION Casing structures act as receivers of vibration generated within machines and are the major radiators of sound. or example, consider the transfer gear cover from a front wheel drive automatic transmission. Gear mesh frequencies and associated side bands may coincide with one or more natural frequencies of the cover, thereby introducing vibration and noise problems. Due to the complex nature of the real-life cover structure (referred to as cover B in igure 1), nonuniform material properties and bolted boundary conditions, it is somewhat difficult to develop accurate vibro-acoustic (b) Cover B igure 1. Example casings models. Hence, a simplified laboratory cover is studied first: A modified flat cover which simplifies the geometry while 1 Copyright 1998 by ASME

2 providing uniform material properties including thickness (Cover A, see igure 1). Each cover is studied with free and bolted boundary conditions using finite element and boundary element methods to determine the fundamental properties (natural frequency, mode shape and percent damping) and sound directivity patterns. The computational work is validated using structural modal analysis and sound measurements. Specific objectives of this research are as follows: (1) Determine the fundamental properties of cover A and B using experimental modal analyses for both free and bolted boundary conditions. (2) Determine the modal sound directivity and corresponding sound pressure levels from experimental acoustic radiation studies. (3) Develop a dynamic finite element model of each cover that predicts modes shapes and natural frequencies for free boundary conditions. (4) Develop a dynamic finite element model using spring and beam elements to simulates the cover s connection with the transmission housing for the bolted boundary conditions. (5) Develop a boundary element model, based on the finite element models, for predicting modal sound directivity. 2 NOMENCLATURE A cross-sectional area (m 2 ) G ff G, G force spectrum autopower spectrum fx fp cross-power spectrum H 1 A, H P 1 accelerance or pressure transfer function I area moment of inertia (m 4 ) (MAC) ij Assurance Criteria for i th and j th component P X pressure spectrum acceleration spectrum f EXP,r experimental natural frequency (Hz) f EM,r finite element natural frequency (Hz) i value for i th component ij value for i th and j th component j value for j th component k k 1 k 2 r p number of modes of interest distributed bolted boundary model inside spring (N/m) distributed bolted boundary model outside spring (N/m) length (m) microphone location r s, f impulse excitation location ( ) r percent difference of r th mode average percent difference r i r modal damping ratio experimental eigenvector for i th mode natural frequency of r th mode (rad/s) 3 EXPERIMENTAL STUDIES 3.1 Structural Testing Each cover is excited at r s, f by an impulse hammer, where r s, f is the position vector from the center of the cover. Complex valued accelerance transfer functions ( H 1 A ) are measured using the conventional H 1 method given by Equation 1, where i is the spatial mesh location index of the acceleration spectrum ( X ) and j is the spatial mesh location index of the impact force spectrum ( ) at the frequency () of interest. X i j X i G fx ( H 1A( )) ij (1) j j j G ff ij Where * denotes the complex conjugation, denotes a complex number, G fx is the cross-power spectrum and G ff is the autopower spectrum. The data set was transferred to a computer and curve fit using a modal software. This provided mode shapes, natural frequencies and modal damping ratios. 3.2 Experimental Acoustic Radiation Each cover is suspended in an anechoic chamber with the same method used for structural modal analysis. An impulse excitation ( j ) is applied at r s, f and the sound pressure spectrum (P i ) is measured over a hemispherical field mesh ( r p,). The field mesh was oriented to take data over the exposed side of the cover under normal operating conditions. The location of the force ( r s, f ) used to excite the covers is selected based on the modal analysis results. The selected point for each cover provides excitation for the first eight modes. The real and imaginary parts of the transfer function ( H 1 P ), which is calculated by Equation 2, are transferred to a computer where the magnitude and phase are calculated using a software. Note that i is the field mesh location index of the sound pressure spectrum ( P ) and j is the spatial mesh location index of the impact force spectrum ( ) at a frequency of interest. 2 Copyright 1998 by ASME

3 P i j P i G fp ( H 1P ( )) ij (2) G j j j ff ij Where G fp is the cross-power spectrum and G ff is the autopower spectrum. The magnitude is then plotted over a hemisphere surface using a spherical coordinates system with (r,) = ( P i ( ), ) where P i ( ) is the magnitude of the sound pressure spectrum for at a point on the acoustic field mesh. This provides a directivity pattern at that could be compared with boundary element directivity patterns and the structural mode shapes. (a) Material property set 1 4 COMPUTATIONAL STUDIES 4.1 Structure Analysis using inite Element Method (EM) Using a commercial code to create and evaluate the dynamic EM models, it is possible to obtain both natural frequencies and mode shapes of each cover. Cover A was modeled using the midplane representation with four node shell elements. A total of 1385 elements are used with approximately 500 master degrees of freedom to extract 15 modes over a range of 10 to 1600 Hz. Cover B is also modeled using the midplane representation with four node elements. Due to the complexity and increased surface area, the cover B model has 2565 elements, 314 point mass elements. Approximately 750 master degrees of freedom are used to extract 15 modes over a range of 10 to 1600 Hz. The shell element allows for a planar model of the cover to be built, thereby simplifying the modeling process and reducing computational time compared to eight node shell element and brick elements. It is assumed that the stamping process used to fabricate the cover B causes a variation in material properties and thickness at the fillets and raised surfaces, along with a concentration of mass on the edge where a flat surface and fillet come together. To account for variations in material property and thickness, fillets and raised surfaces are assigned different material properties sets as shown in igure 2. To simulate the mass concentration at fillet/surface intersections, point mass elements are placed at the nodes shown in igure 2. Note that mass elements are not placed at all of the fillet/surface intersections because the initial EM results provided some information as to which areas may be affected by the mass concentration problem. Using the EM models of each cover, beam and spring elements are added to create the bolted boundary condition model. The boundaries are described by a technique presented in [1] where a set of non-uniform springs and point masses Material set 4 (b) Material property set 2 (c) Material property set 3 Material set 5 (d) Material property sets 4 and 5 igure 2. Material property sets. Shaded gray areas represent the property set. 3 Copyright 1998 by ASME

4 model the clamp-annular disk interaction. Due to the large scale of the cover models only two spring elements, with spring rates of k 1 and k 2, are used to represent the boundaries. This reduces both the models construction and computation times. Spring element group k 1 (k 1 =40000 N/m), designated as Boundary set 1, is located at the inside edge of the coverhousing contact plane. While, Boundary set 2 (k 2 =60000 N/m) is located at the outside edge. The nominal values of the Boundary Sets are determined by an iterative process using the EM model and the experimental results for cover A. Boundary set 3 models the 10 bolt connections as beam elements. Each beam element is assigned the cross sectionalarea (A), moment of inertia (I) and length () of one bolt. A beam is attached to a single master node which is coupled to the other nodes around the bolt hole. The coupling causes all six degrees of freedom for the linked nodes to be identical to that of the master node. EM modes and the differences in frequency is critically examined. The experimental and EM mode shapes correlate well, however the modal order is not identical in each case. Note, that only similar mode shapes are compared to determine the frequency difference. igures 3 and 4 provide simplified views of the modes of each cover. Table I and II summarize the results from the finite element model and compares them with the experimental results. The finite element results differ from the experimental data by an average of 2 % for cover A and 10% cover B. 4.2 Acoustic Radiation Analysis using Boundary Element Method (BEM) The data set and results of the free and bolted boundary condition finite element models for cover A and B are imported into a commercially available acoustic BEM software to calculate the modal directivity patterns. The BEM model is setup to be uncoupled with the fluid medium (air). A sinusoidal force excitation of unity amplitude is applied at the node nearest to the location of the impulsive excitation in the experimental testing. The transitions in pressure are set equal to zero to eliminate any discontinuity along the free edge where two sides of the model come in contact. In each cover model the experimental modal damping ratio, r, is input into the structural damping function of the BEM software. The BEM results are computed only at the natural frequencies for a given cover. These patterns are viewed in terms of a field mesh with a radius r p and 141 field points. Both the EM and BEM predictions are compared with the experimental (EXP) results on a modal basis. The percent difference ( ) and average percent difference ( ) of the natural frequencies (f r in Hz) are calculated as shown. igure 3. ree boundary condition mode shapes and natural frequencies of cover A. EMA = Experimental Analysis, EM = inite Element Analysis ( r =(f EXP,r -f EM,r )/f EXP,r ]x100, % (3) =[ k r=1 () r ] / k (4) Where r is the modal index and k equals the number of modes of interest. 5 RESULTS OR REE BOUNDARY CONDITIONS 5.1 Structural Modes To fully understand the modal analysis results, both the mode shapes and natural frequencies are examined. The experimental mode shapes are compared with the corresponding igure 4. ree boundary condition mode shapes and natural frequencies of cover B. EMA = Experimental Analysis, EM = inite Element Analysis 4 Copyright 1998 by ASME

5 Experimental EM Calculated Difference r r r req. r (r) (Hz) (%) (r) (Hz) (Hz) (%) Average 4 2 Table I. Summary of free boundary condition modal analysis results for cover A. Experimental EM Calculated Difference r r r req. r (r) (Hz) (%) (r) (Hz) (Hz) (%) Average 44 6 Table II. Summary of free boundary condition modal analysis results for cover B. Cover B, when compared with cover A, yielded a different mode shape order due to the irregular shape that adds some stiffness to the structure. or cover B, modes 3 and 4 appear to be the same. Using the Assurance Criteria (MAC) [2], where i is the i th mode experimental eigenvector j is the j th mode experimental eigenvector and H refer to the Hermitian of the vector, (MAC) ij =( H i j ( H i i ( H j j (5) it can be seen that modes 3 and 4 are not identical in shape. The similar mode shape may be caused by the variations in material properties or the presence of residual stresses in cover; also, it may result due to some localized effects. Note that the EM model does not predict mode 3 of cover B. To account for this, both experimental modes 3 and 4 are compared with mode 3 of the EM model. This mode appears to be a localized effect and therefore would be difficult to predicted using EM. The experimental damping ratios ( r ) are given in % of critical damping which implies that 0.08 % is equal to r =0.0008, where r is the modal index. As seen in Tables I and II, covers A and B have relatively low damping as would be expected for a steel cover. The increases in damping present in cover B are believed to be caused by inherent damping of the stamped cover. 5.2 ields Examination of the BEM and experimental results shows single and double lobe patterns; note that these patterns are only for the hemisphere. Comparing the results for cover A, it can be seen that modes 1 through 6, and 8 correlate well with those obtained in the experiments. Mode 7, however, is a single lobe over the hemispherical mesh in BEM while the experimental results show a double lobe pattern. Table III summarizes the results of the acoustic radiation testing and boundary element modeling. Comparing the maximum sound pressure levels of the experimental and BEM results it can be seen that all of the values are within one order of magnitude. or cover B the lobe shapes and maximum sound pressure levels from the BEM model correspond well for all measured modes. Note that cover B typically has higher sound pressure levels at the top of each lobe. This is caused by the location of the larger dimple which was oriented at the top in the experimental and BEM analyses. Tables IV summarize the results of both the experimental and BEM studies for cover B. Comparing the mode shapes with the sound directivity patterns for both covers it can be seen that all torsional modes and the first three bending modes create a single lobe pattern. While higher order bending modes (cover A modes 6 and 7 for example) create double lobe patterns which demonstrate higher order acoustic sources with some cancellation in the radiation field. 6 RESULTS OR BOLTED BOUNDARY CONDITIONS 6.1 Structural Modes The mode shapes and natural frequencies are summarized in igures 5 and 6. The contour lines shown in the mode shapes are simplified and it should be noted that the outside edge of each cover has limited motion relative to the center due to the bolted condition. Through an iterative process using the cover A model the appropriate values of spring constants k 1 and k 2 are developed. To determine k 1 and k 2 both mode shape and natural frequency for the first six mode are correlated. The average percent discrepancy,, for cover A is 4 % while each individual mode had a percent difference, r, less than 5 Copyright 1998 by ASME

6 Experimental (Exp.) Boundary Element Method (BEM) (r) (Pa/N) (r) (Pa/N) Comments 1 Single 9.3x size and orientation are 1 Single 3.6x10 2 Single 8.8x BEM has double lobe with higher 1.1x10 levels in upper lobe 3 Single 4.8x size and orientation are 3 Single 6.7x10 4 Single 4.5x size and orientation are 4 Single 2.0x10 5 Single 4.1x size and orientation are 5 Single 8.5x x size and orientation are 5.7x10 2.7x Poor pattern correlation but 7 Single 2.9x10 maximum levels are similar 8 Single 1.9x Single 3.1x10-2 center. BEM has higher levels Exp. has higher levels above below center igure 5. Bolted boundary condition mode shapes and natural frequencies of cover A. Note, bolting causes the cover edge to have relatively low motion. Table III. Summary of free boundary condition cover acoustic radiation studies for A Experimental (Exp.) Boundary Element Method (BEM) (r) (Pa/N) (r) (Pa/N) Comments 1 Single 1.5x Single 3.5x10-1 center. BEM has higher levels at Exp. has higher levels above center 2 Single 6.3x Single 4.4x10 0 center. BEM has higher levels at Exp. has higher level sabove center 3 % Single 9.5x x x x x Single 2.0x BEM has double lobe with higher 5.0x10 levels in upper lobe 0 size and orientation are 5.0x10 0 size and orientation are 2.6x10 0 size and orientation are 2.7x10 1.3x10 0 Exp. - lobe pattern with one strong and one weak lobe 9.7x10-1 Exp. pattern is noisy and may have double lobe Table IV. Summary of free boundary condition cover acoustic radiation studies for B igure 6. Bolted boundary condition mode shapes and natural frequencies of cover B. Note, bolting causes the cover edge to have relatively low motion. 8 %. These results, presented in Table V, validate the housing model and allow for the extension of these methods to the cover B model. or cover B, the average percent discrepancy 3 % while each individual mode does not exceed a percent difference greater than 6 % as shown in Tables VI The effects of the boundary conditions on each cover are studied in detail by varying the spring constants in the boundary model. The increased stiffness of cover B, due to the stamped shape, decrease it sensitivity to boundary condition changes. 6 Copyright 1998 by ASME

7 Experimental EM Calculated Difference r r r req. r (r) (Hz) (%) (r) (Hz) (Hz) (%) Average 36 4 Table V. Summary of bolted boundary condition modal analysis results for cover A. predicted sound directivity patterns are parallel to the cover mode shapes. The difference may occur due to two possible reasons: (1) The experimental results may be skewed by noise radiated from the housing or (2) because the BEM model does not account for the acoustic space filled by the simplified housing in the experimental studies. Cover B has both single and double lobe shapes, however, they are not as well defined as those of cover A. The experimental results for modes 1, 2, 4, and 5 produce double lobe type directivity patterns that are oriented 90 degrees to the cover surface motion. While modes 3 and 6 show single lobe directivity patterns (or nearly single lobe). BEM patterns correlate well for modes 1, 3, 5, and 6. However similar to cover A the mode 2 and 4 have similar shapes, but each is rotated 90 degrees about the z-axis. The results of the bolted sound directivity studies are summarized in Tables VII and VIII. Experimental EM Calculated Difference r r r requency r (r) (Hz) (%) (r) (Hz) (Hz) (%) Average 35 3 Table VI. Summary of bolted boundary condition modal analysis results for cover B. 6.2 ields Similar to the free cover studies, single and double lobe patterns occur in the bolted cover study. However, the bolted cover studies also have a four lobe patterns present for certain modes. Examining the cover A mode shapes and the corresponding sound directivity patterns it can be seen that modes 1, 2, 4, and 5 have an identical number of nodal line areas and sound directivity lobes. Therefore, it is assumed that each surface radiates noise to create a lobe in the directivity pattern. or modes 3 and 6 there must be a cancellation effect occurring to create the double lobe directivity. Comparing the predicted directivity patterns with the measured results shows that similar patterns occur for all modes. However, the BEM double lobe patterns of mode 2, 4 and 6 are rotated 90 degrees about the z-axis. Examining the corresponding mode shapes it can be seen that the measured directivity patterns are perpendicular to the cover mode shape motion while the Experimental (Exp.) Boundary Element Method (BEM) (r) (Pa/N) (r) (Pa/N) Comments 1 Single 9.7x Exp. results have higher sound 1 Single 2.5x10 levels 2 1.1x x10-1 pattern is rotated 90 about z- axis compared BEM predict lower levels and to Exp. results 3 Single 6.7x Single 1.1x10 0 equal size and orientation are for Exp. and BEM 4 1.6x x10-1 about BEM pattern is rotated 90 z-axis compared to Exp. results 5 our 7.2x our 5.1x10-1 levels are order of magnitude higher in shape model correct but BEM results 6 3.3x Difficult to compare since Exp x10 results are noisy. Table VII. Summary of bolted boundary condition cover acoustic radiation studies for A 7 Copyright 1998 by ASME

8 Experimental (Exp.) Boundary Element Method (BEM) (r) (Pa/N) (r) (Pa/N) Comments 1 Single 3.4x Exp. results have higher sound 1 Single 3.0 x10 levels the BEM model and experimental setup. urther research is needed to resolve this issue. ACKNOWLEDGMENTS We are grateful for the financial support provided by the Chrysler Corporation that made this study possible x x10 0 BEM pattern is rotated 90 about z-axis compared to Exp. results 2.1x Single 1.2x10 0 Exp. and BEM however BEM orientation are equal for predicts higher levels 1.6x x x x10 0 BEM predict higher levels and pattern is rotated 90 about z- axis compared to Exp. results 1.1x10 0 Both lobe are irregular shapes with high levels in same areas but BEM predicts higher levels 4.7x10 0 Both lobe are irregular shapes with high levels in same areas but BEM predicts higher levels REERENCES 1 Harsh, V., and Singh, R., Eigensolutions of Annular-like Elastic Disks with Intentionally Removed or Added Material, Journal of and Vibration, 192(4), , Allemang, R. J., and Brown, D. L., Correlation Coefficient for Vector Analysis, Proceedings of International Analysis Conference, , Table VIII. Summary of bolted boundary condition cover acoustic radiation studies for B 7 CONCLUSION Experimental and computational modal analysis and acoustic radiation techniques are used to examine the transfer gear cover. Because of the irregular shape, and non-uniform material properties, the cover causes difficulties when evaluating the is fundamental properties and the bolted boundary conditions. Therefore a simplified flat cover is also studied. Each cover is examined with both free and bolted boundary conditions. In the free boundary condition studies the fundamental properties such as natural frequencies, mode shapes and modal damping ratios and the acoustic directivity patterns are determined. Note that the experimental work has provided a baseline understanding of the covers which is critical to the development of the accurate models. The EM results correlate well with the experimental results. The BEM models, which are developed based on the EM models, also provided satisfactory results. The experimental and computational methodology of the free boundary studies is then applied to the bolted boundary conditions. Again, the fundamental modal and acoustic radiation properties of the covers are determined. The distributed boundary model is developed using two springs and each spring rate is estimated by an iterative process within EM. Again, the EM model provides satisfactory results. The BEM models show similar patterns to the measured results but in some modes the orientation is incorrect. Errors may occur because of radiated noise from the housing present in the experimental result or differences in acoustic baffles between 8 Copyright 1998 by ASME