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The vibroacoustic response of a cylindrical shell structure with viscoelastic and poroelastic materials
Applied Acoustics 58 (1999) 131±152
The vibroacoustic response of a cylindrical shell structure with viscoelastic and poroelastic materials Sylvain Boily, FrancËois Charron* o
Department of Mechanical Engineering, Universite de Sherbrooke, 2500 Boulevard de l'UniversiteÂ, Sherbrooke, QueÂbec, Canada J1K 2R1 Received 1 November 1997; received in revised form 10 August 1998; accepted 17 October 1998
Abstract The study of the vibroacoustic behavior of an aircraft fuselage requires the generation of very large and complex ®nite element models (FEM) and, furthermore, the numerical results are often very dicult to interpret and to validate. On the other hand, the study of the vibroacoustic behavior of a simple cylindrical shell structure should allow us to validate the results obtained from these simple numerical models and to study the eciency of dierent analysis tools and design solutions for vibration and noise reduction, such as the addition of visco-elastic and porous materials. In order to achieve these objectives, two cylindrical shell structures were studied. The ®rst model was used for the study of the dynamic behavior of the structure with and without visco-elastic material. The second model was used for the study of the vibroacoustic behavior with and without porous material. Results show a good correlation between experimental measurements and numerical predictions. Eciency of the two proposed solutions was evaluated and, ®nally, it has been demonstrated that the eect of the modal truncation of the cavity modal basis is critical for the convergence of the predicted pressure ®eld and that a good understanding of coupling phenomena between structural and acoustic modes is necessary. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Cylindrical shell; Vibroacoustic response; Mechanical excitation; Finite element model; Viscoelastic material; Porous material
1. Introduction Noise inside an airplane has always been a cause of discomfort to passengers and this problem is becoming more important with the increasing use of composite * Corresponding author. Tel.:+1-819-821-7144; fax: +1-819-821-7163; e-mail: francois.charron @gme.usherb.ca 0003-682X/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S000 3-682X(98)0007 0-X
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materials and more powerful engines, which tend to amplify the acoustic problems. In fact, noise and/or vibration reduction inside airplanes is becoming a prime objective for many manufacturers. However, the vibroacoustic behavior of fuselage structures is a very complex problem. In order to tackle this problem and to validate various analysis tools and design solutions, a simpli®ed representation of an airplane fuselage, that are cylindrical shell structures, have been studied. Results from this kind of fuselage-like structures can provide useful insights into the eect of adding visco-elastic and/or porous materials on real airplane structures. This paper is concerned with the study of various cylindrical shell structures regarding vibration and/or noise reduction. The ®nite element method will be used for the modelization of shell structures including the acoustic cavity, and experimental measurements will be done to validate the numerical predictions. Three con®gurations are studied. The ®rst model is a closed cylindrical shell with an interior acoustic cavity. The second model is a cylindrical shell with a narrow patch of visco-constraint material and in the last model, absorbing material is added to the inner shell surface of the ®rst model. FE models for these experimental con®gurations are developed, and modal and dynamic analysis are performed. Numerical predictions are compared to experimental measurements, in order to assess the limitations of the analysis tools. 2. Finite element formulation 2.1. Problem discretization and modal decomposition The ®nite element formulation of the coupled structural-acoustic problem is based on a variational approach, using a displacement ®eld u for the structure and a pressure ®eld p for the cavity. This classical approach [1] is used in the commercial code MSC/NASTRAN [2]. The ®nite element discretization of these equations gives the following linear system of equations:1 fu g 0 fFs g Ks AT 2 Ms 1 ÿ! p A Mf f0 g 0 Kf This system can be solved directly in physical coordinates using direct frequency response analysis, which will be the case for the third model. Unfortunately, the total number of degrees of freedom (sum of structural and acoustic DOF) is often very large and the model needs to be solved for many excitation frequencies. In order to reduce the size of this system, a modal reduction method is often used. The displacement vector fug is projected on the in vacuo structural modes s and the pressure vector fpg is projected on the rigid wall cavity modes f , using the following relations: 1
The eigenvalue problem is unsymmetric.
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fug s qs p f qf
133
2
At the same time, the damping eect is introduced in Ks and Kf 2 using an equivalent structural damping model, classically used in structural analysis [3]: Kf 1 jf Kf Ks 1 js Ks
3
Then, using the two previous equations and premultiplying the ®rst line of Eq. (1) by s T , and the second line by f T , we ®nally obtain: qs ks ÿaT s T fFs g 2 ms 0 4 ÿ! a mf qf 0 kf f0 g This system is now in modal coordinates, with its size being equal to the sum of structural and acoustic modes retained in each modal basis. This modal reduction process is thus advantageous when the number of physical DOF is much larger than the total number of modes . The modes that should be retained in the structural basis s and the acoustic basis f are discussed in the next section. 2.2. Choice of modal basis This section is concerned with the content de®nition of the structural modal basis s and of the acoustic modal basis f for a given excitation frequency range and for a mechanical excitation. In the case of vibration problems, a commonly used rule of thumb is to retain modes outside the excitation frequency range, in order to approximate the contribution of the higher modes. For example, if the frequency range varies from 0 to fmax, all modes between 0 and 1:2 fmax should be retained. But to the authors' knowledge, there is no such general criterion for the case of vibroacoustic problems. The problem of de®ning the proper modal basis for the acoustic cavity is presented in two steps. Firstly, which acoustic modes should be retained, if one wants to compute correctly but also eciently the acoustic response at each structural mode natural frequency in the excitation frequency range? Secondly, which structural modes should be retained, if one wants to compute accurately the acoustic response for each cavity mode natural frequency in the excitation frequency range? In order to answer these questions, Eq. (4) is simpli®ed using the light ¯uid assumption for the cavity, i.e. ÿaT qf s T fFs g: qs 2
s T fFs g ÿ!2 ms ks
5
When this project was realized, it was not possible to directly de®ne f in MSC/NASTRAN. It was thus done using DMAP language.
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ÿ!2 a qs qf ÿ!2 mf kf
6
Eq. (6) gives the relation between the acoustic modal response qf and the structural modal response fqs g. This equation can be rewritten for a given acoustic mode j and a given structural mode i. In the process, if modal masses are normalized to one, the relation between the acoustic modal response of mode j and the structural modal response of mode i can be simpli®ed to the following form: ! T si m 1f Fs gm1 !2 aij 7 q fj 2 ! ÿ !2fj 1 jf ÿ!2 ÿ !2si 1 js Now, Eq. (7) gives the response of acoustic mode j as a function of the response of a structural mode i. Two speci®c cases are of interest. In the ®rst case, the excitation frequency coincides with the structural natural frequency ! !si . Eq. (7) simpli®es to: ! T si m 1f Fs gm1 aij ÿ 8 q fj 2 js !si ÿ !2fj 1 jf In the second case, the excitation frequency coincides with the acoustic natural frequency ! !fj . Eq. (7) simpli®es to: ! T si m1f Fs gm1 aij q fj 2 jf !fj ÿ !2si 1 js
9
From Eqs. (8) and (9), it seems that the following conditions are required in order to obtain an acoustic response for mode j: . the mechanical excitation must be capable of exciting the structural mode or, in others words, mode i needs a good modal participation factor. . the spatial modal coupling must be excellent between the acoustic mode j and the structural mode i. One can also note that for the case where ! !si !fj , maximum acoustic response is achieved. These conditions and remarks will be later used for the evaluation of the content of the structural and acoustic modal basis. 3. Global indicators For dynamic analysis, two global indicators are calculated. The ®rst one is the mean quadratic velocity:
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hv2 !i
n 1 X v vi S i 2S i1 i
135
10
where n is the number of structural DOF. The velocity is weighted by the nodal surface. In dB, the quadratic velocity is calculated using the following relation: 2 v 11 Lv 10 log10 2 ; with v0 5 10ÿ8 m=s v0 The second global indicator is the mean quadratic pressure: hP2 !i
m 1 X p pi Vi 2V i1 i
12
where m is the number of acoustic DOF. The pressure is weighted by the nodal volume. In dB, the quadratic pressure is calculated using the following relation : 2 P 13 Lp 10 log10 2 ; with P0 2 10ÿ5 Pa P0
4. Finite element models 4.1. Closed cylindrical shell 4.1.1. Experimental set-up The shell structure is made of a 1.2 mm thick steel plate. The plate has been rolled and welded at a joint made along the longitudinal axis of the cylinder. The resulting shell is 1.01 m long and has a mean radius of 0.183 m. A steel ring is welded at both ends of the shell and an aluminum 2.54 cm thick circular plate is bolted on each ring, in order to close the cavity. To approximate free±free boundary conditions, the shell is supported by soft springs. 4.1.2. FE model The FE model of the closed shell is made of four parts: a shell, two rigid plates, and an acoustic cavity. Free±free boundary conditions are used. Table 1 gives the physical properties and dimensions of the shell, the plates and the acoustic cavity. The mesh of this FE model is based on a criterion of ®ve to six linear elements per mode shape wavelength. This criterion is applied to both structure and cavity models. The frequency range of interest for the dynamic analysis is 80±480 Hz. Fig. 1 shows the FE model of the shell and the plates. For the cavity, a matching interface is used, which means that the grid points of the wet surface between the structural and acoustic models are coincident. Finally, Table 2 gives element types used for each components of the structure and the cavity.
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Table 1 Physical properties and dimensions of the closed shell model
Mean radius (m) Length (m) Thickness (m) Young modulus (pa)} Density (kg/m3) Poisson ratio Structural damping Speed of sound (m/s)
Shell
Plates
Cavity
0.18256 1.01 0.001219 2.11011 7800 0.3 0.0006 ±
0.18256 ± 0.0254 7.01010 2700 0.3 0.0006 ±
0.18256 1.01 ± ± 1.21 ± 0.0005 340
Table 2 Element types and number of elements for the closed shell model Name
Type
Nodes/element
# of elements
Shell
CQUAD4
Plate (2D)
4
450
Plates
CTRIA3 CQUAD4
Plate (2D) Plate (2D)
3 4
10/plate 290/plate
Cavity
CPENTA CHEXA
Solid (3D) Solid (3D)
6 8
150 4350
4.2. Simply supported shell with a patch of visco-constraint material 4.2.1. Experimental set-up The shell structure is made of a 1.2 mm thick steel plate, as described in Section 4.1.1. The simply supported boundary conditions are approximated using thin perforated rings which are welded at each end of the shell. These rings are bolted to a concrete base. A 4.81 cm wide visco-constraint patch is applied on the shell circumference at the half-length of the shell. Five pieces of material were required to cover the circumference. The visco-constraint material used is the SJ2152X, type 1010, from 3M. The constraint layer is made of stainless steel. 4.2.2. FE model The visco-elastic material behavior is described using a complex shear modulus approach [4]: G G 1 jv ;
14
where G is the shear modulus and v the loss factor. The real part of Eq. (14) is for the elastic behavior of the material, while the imaginary part characterizes the damping behavior. Usually, G and v are frequency dependant. The approach used for the modelization of the visco-constraint layer has been developed by Johnson and Kienholz [5]. The constraint layer is modeled with plate
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elements and the visco-elastic core with solid elements. The FE model is made of three parts: a shell, a visco-elastic core and a constraint layer. Only structural analyses are performed with this model. Again, the frequency range of interest for dynamic analysis is 80±480 Hz. Fig. 2 shows the FE model. The central patch represents the visco-constraint material. Fig. 3 shows a cross-section view of the model. The constraint layer sheet and the shell are modeled with linear quadrilateral elements, while the visco-elastic core is made of linear hexahedral elements. An arrow on Fig. 3 indicates a duplicated grid, which is used to take into account the discontinuity between the ®ve patches of visco-constraint layer applied on the circumference of the experimental shell. Table 3 gives the visco-elastic physical properties
Fig. 1. Finite element model of the shell and plates.
Fig. 2. Finite element model of the shell with a visco-constraint patch.
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and dimensions. For the constraint layer, the physical properties used are those of the shell. For dynamic analysis, no frequency dependence is considered for this material, which is justi®ed for the frequency range of interest. Finally, Table 4 gives the number of elements used for shell, visco-elastic core and constraint layer. 4.3. Closed cylindrical shells with absorbing material 4.3.1. Experimental set-up The shell presented in Section 4.1.1 is used. A 1.3 cm thick foam is installed on the inner surfaces of the shell and of the two end plates. No glue is used, the foam is maintained on the inner surfaces by a tight ®t. 4.3.2. FE model The absorbing material is modeled using an equivalent ¯uid model [6]. This type of model is justi®ed and accurate for very limp porous materials which are used in
Fig. 3. Partial cross-section view of the shell with a visco-constraint patch.
Table 3 Physical properties and dimensions of the viscoelastic material Width (m) Thickness (m) Density (kg/m3) Poisson ratio Shear modulus (Pa) Loss factor
0.0481 2.5410ÿ4 980 0.49 6.0106 1.0
Table 4 Element types and number of elements for the shell with a visco-constraint patch
Shell Viscoelastic Upper face sheet
Name
Type
Nodes/element
# of elements
CQUAD4 CHEXA CQUAD4
Plate (2D) Solid (3D) Plate (2D)
4 8 4
1058 46 46
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many aeronautical applications. The acoustical properties of porous material are de®ned by two acoustical characteristics, the equivalent density eq ! and the equivalent bulk modulus Beq !. For the present study, this approach has been implemented in MSC/NASTRAN using DMAP langage [7]. The basic idea is to use the acoustic elements for the absorbing material and to modify the Mf and Kf matrices corresponding to these elements, using the equivalent ¯uid characteristics eq ! and Beq !. Initially, a FE model based on the one presented in Section 4.1.2 was created. However, the model size exceeds our computer hardware capability because real modal analysis could not be used anymore due to the signi®cant imaginary part of the ¯uid equivalent properties. It was then decided to generate a coarser FE model. This model should be valid from 0 to 250 Hz. For the closed shell, the same physical properties and dimensions are used (see Table 1). Fig. 4 shows this coarser ®nite element model of shell and plates. The evaluation of eq ! and Beq ! for a given absorbing material can be done with many models [8]. However, this subject is beyond the scope of the present study, where these characteristics are considered known for the absorbing material used. Table 5 gives the equivalent ¯uid properties used over the frequency range. In a ®rst approximation, no frequency dependence is considered.
Fig. 4. Finite element model of the shell with the absorbing material.
Table 5 Properties of equivalent ¯uid R Beq I Beq R eq I eq
102 000 Pa 5000 Pa 3.57 kg/m3 ÿ12.0 kg/m3
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5. Results In this section, results obtained from the three ®nite element models described in previous section are presented. These results are compared with experimental results. For the closed shell and the shell with a visco-constraint patch, modal and dynamic analysis results are presented. For the closed shell with absorbing material, only dynamic analysis results are presented, since direct frequency response was used. The excitation is a mechanical harmonic point force and its location is shown on Fig. 5. 5.1. Closed shell 5.1.1. Modal analysis Firstly, a modal analysis was done. Table 6 presents the in vacuo structural modes and compares them with the experimental results for the frequency range of 80 to 480 Hz. For all modes, the plate contribution is negligible and this is due to the signi®cant dierence between the plate and shell thickness. A good agreement between experimental and numerical modes is observed. The error on the natural frequencies is less than 5%, except for modes (3,1) and (4,1). These modes are more sensitive to the modelization of the interface ring between the plates and shell. In reality, the plates are bolted to the ring (discrete interface), whereas in the FE model, plates are welded (continuous interface) to the shell. For the experimental modes, one can also observe that symmetric and anti-symmetric modes are shifted, due to the longitudinal welding joint. Furthermore, an acoustic modal analysis was performed. Table 7 shows the calculated rigid wall cavity modes and compares them with the experimental
Fig. 5. Location of mechanical point force for all FE models.
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measurements. There are only two modes in this range, and an excellent agreement is reached between numerical and experimental results. In conclusion, these structural and acoustic modal results con®rm the validity of the closed shell FE model for the excitation frequency range of interest. 5.1.2. Dynamic analysis The vibration response is ®rst studied using the modal solution approach. Using the 1:2 fmax criterion (Section 2.2), the structural modal basis includes 38 modes between 0 to 600 Hz. Fig. 6 compares the numerical and experimental mean quadratic velocity. Very good agreement is achieved between the two curves. The Table 6 Natural frequencies comparison for the closed shell Mode a n; m (4,1) (4,1) (3,1) (3,1) (5,1) (5,1) (2,1) (2,1) (6,1) (6,1) (5,2) (5,2) (6,2) (6,2) (4,2) (4,2) (7,1) (7,1) (7,2) (7,2) a
Numerical (Hz)
Experimental (Hz)
Error (%)
183.5 183.5 189.9 189.9 237.5 237.5 303.8 303.8 327.4 327.4 344.1 344.1 381.0 381.0 387.5 387.5 441.5 441.5 470.5 470.5
172.7 173.8 178.6 179.1 232.8 233.5 294.8 297.7 326.8 ± 328.3 330.4 368.6 370.3 377.0 ± 444.1 ± 471.1 471.9
6.3 5.6 6.3 6.0 2.0 1.7 3.1 2.0 0.2 ± 4.8 4.1 3.4 2.9 4.6 ± ÿ0.6 ± ÿ0.1 ÿ0.3
n and m are the mode orders with respect to the circumference and length of the shell.
Table 7 Natural frequencies comparison for the cavity Mode a p; q; r
Numerical (Hz)
Experimental (Hz)
Error (%)
(0,0,1) (0,0,2)
168.6 339.1
172.0 343.0
2.0 1.1
a
p, q and r are the mode orders with respect to the circumference, radius and length of the cylindrical cavity.
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frequency shifts are the same as observed in Table 6. The dierences in peak levels are due to the frequency resolution in experimental measurements and to the diculty to properly characterize the modal damping for each modes. This comparison shows that the closed shell FE model gives fairly accurate prediction of the vibration response. Next, the acoustic response is studied. The ®rst step is the choice of the modal basis content. For the structure, s is de®ned using the 0 to 600 Hz frequency band. For the acoustic cavity, a convergence study has been done. Fig. 7 shows the mean
Fig. 6. Mean quadratic velocity comparison for the closed shell.
Fig. 7. Mean quadratic pressure for two acoustic modal basis: 0±600 Hz and 0±2000 Hz.
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quadratic pressure obtained with the 0±600 Hz band (eight modes) and for the 0± 2000 Hz band (110 modes). From these results and the previous discussion in Section 2.2, the following remarks can be made: 1. The 0±600 Hz band is not sucient since there is no peak levels at acoustic resonances. 2. The 0±2000 Hz band is not sucient since some peak levels are missing at structural resonances. In order to correctly choose the acoustic basis, Table 8 shows the modal coupling coecients for each structural mode in the excitation frequency range. It seems that the modal coupling behavior between the acoustic modes and shell modes is in perfect agreement with Lesueur's discussion on this subject [9]. There is a strong coupling between structure and cavity modes when they have the same circumferential order and odd±even (or even±odd) longitudinal order. Unfortunately, no information is provided for order in the radial direction of the cavity. Furthermore, for each shell modes, the term of the third coupled cavity mode is less than 13% of the ®rst, which seems to indicate that in order to obtain an acoustic response at each shell modes, it is necessary to retain at least the ®rst two coupled cavity modes. This con®rmation cannot be readily generalized to other closed shell models without a more thorough study. From the previous comments, the 0±2800 Hz band is chosen for the de®nition of the acoustic modal basis. Now, in order to correctly de®ne the structural basis content, a study in the 0±2500 Hz band was done. In this band, only modes relating to the plate are found to be coupled with the two acoustic modes in the excitation frequency range. However, these modes are not excited by the point force on the shell. The 0±600 Hz frequency band is thus chosen for structural basis. Finally, to complete the acoustic response study, Fig. 8 compares the mean quadratic pressure from the model and from the experimental set-up. A good correlation is obtained, although the experimental curve seems to show a small peak response at cavity resonances. This can be due to a slight misalignment of the shaker in the longitudinal direction of the shell. This misalignment will produce a small force contribution capable of exciting the plate modes, which are strongly coupled to the two cavity modes in the 0±500 Hz frequency range. In conclusion, apart from the discussed problem at cavity resonances, the closed shell model gives an accurate prediction of the vibration and acoustic responses. 5.2. Simply supported shell with a patch of visco-constraint material For this case, no acoustic analysis was performed. The results are obtained from the modal and dynamic analysis of the shell structure. 5.2.1. Modal analysis Table 9 presents the in vacuo modes obtained from the modal analysis and compares them with the experimental results, for the frequency range of 80±480 Hz. It seems that there is very good agreement, with the exception of mode (2,1). This
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mode is more sensitive to simply supported boundary conditions. In general, the model gives an excellent correlation with the experimental modes. 5.2.2. Dynamic analysis The modal basis from 0±600 Hz (33 modes) is used. Fig. 9 shows the mean quadratic velocity for the model with and without the visco-constraint patch. Modes with longitudinal order 1 and 3 show a reduction at peak resonances, up to 30 dB. Physically, this eect is explained by the fact that for these modes, the shear strain Table 8 Numerical modal coupling terms for each shell mode Shell mode n; m
Cavity mode p; q; r
Frequency (Hz)
Numerical coupling term N m
Relative value (%)
(2,0,0) (2,0,2) (2,0,4) (2,0,6) (2,1,0) (2,1,2)
912.5 973.3 1146 1412 2020 2048
432 183 30 8 345 153
100 42.3 6.8 1.7 79.9 35.4
(3,1)
(3,0,0) (3,0,2) (3,0,4) (3,1,4) (3,1,4)
1264 1308 1441 2428 2451
484 223 38 369 172
100 46.0 7.8 76.2 35.5
(4,1)
(4,0,0) (4,0,2) (4,0,4)
1615 1651 1758
510 253 38
100 49.6 7.4
(4.2)
(4,0,1) (4,0,3) (4,0,5)
1624 1695 1840
481 289 59
100 60.1 12.2
(5,1)
(5,0,0) (5,0,2) (5,0,4)
1975 2004 2093
417 222 28
100 53.2 6.7
(5,2)
(5,0,1) (5,0,3) (5,0,5)
1982 2041 2162
481 313 56
100 65.1 11.7
(6,1)
(6,0,0) (6,0,2) (6,0,4)
2347 2372 2448
455 255 27
100 56.0 6.0
(6,2)
(6,0,1) (6,0,3) (6,0,5)
2353 2403 2507
464 323 51
100 69.6 10.9
(7,1)
(7,0,0) (7,0,2)
2736 2757
525 302
100 57.5
(7,2)
(7,0,1) (7,0,3)
2741 2784
474 346
100 73.0
(2,1)
a
aij f s gTi Af fgj m m2 N=m2 N m:
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Fig. 8. Mean quadratic pressure comparison for the closed shell.
Table 9 Natural frequencies comparison for the shell with visco-constraint patch Mode n; m
Numerical (Hz)
Experimental (Hz)
Error (%)
(4,1) (4,1) (3,1) (3,1) (5,1) (5,1) (2,1) (2,1) (5,2) (5,2) (6,1) (6,1) (4,2) (4,2) (6,2) (6,2) (7,1) (7,1) (7,2) (7,2) (6,3) (6,3)
161.4 161.6 163.2 163.3 226.2 226.5 296.1 296.2 312.3 312.3 321.5 322.2 354.8 355.0 359.3 359.6 437.5 438.8 457.1 457.9 469.5 469.6
158.5 160.4 163.3 164.5 223.9 ± 277.0 280.7 312.0 313.8 318.6 ± 351.7 354.2 360.9 361.7 434.4 ± 457.9 459.4 471.1 ±
1.8 0.7 ÿ0.0 ÿ0.7 1.0 ± 6.9 5.5 0.0 ÿ0.5 0.9 ± 0.9 0.2 ÿ0.4 ÿ0.6 0.7 ± ÿ0.2 ÿ0.3 ÿ0.3 ±
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energy in the visco-elastic layer is maximum and thus, result in a high damping eciency. On the other hand, modes of longitudinal order 2 show almost no reduction at peak resonances. These modes produce a minimum shear strain energy in the visco-elastic layer, resulting in low damping eciency. The only exception is mode (2,1), for which no reduction is observed at peak resonances. To understand this exception, the proportion of modal strain energy in the visco-elastic elements is now studied. To predict the damping eciency of a visco-constraint patch on speci®c modes, an interesting criterion is the modal strain energy (MSE), where the proportion of the modal strain energy in the visco-elastic elements with respect to the total strain energy is used to evaluate the eciency of the visco-elastic treatment. Table 10 gives this ratio for each mode in the frequency range of 80±480 Hz. From this table, it
Fig. 9. Predicted mean quadratic velocity, shell without and with a visco-constraint patch. Table 10 Relative modal strain energy in viscoelastic elements for each mode Mode n; m
MSE in viscoelastic elements (%)
(4,1) (3,1) (5,1) (2,1) (5,2) (6,1) (4,2) (6,2) (7,1) (7,2) (6,3)
1.295 0.256 1.780 0.0002 0.017 1.722 0.002 0.035 1.532 0.036 0.674
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seems that the ratio of MSE is 10 times higher for modes with longitudinal order 1 and 3 than for modes with longitudinal order 2. The only exception is mode (2,1), which has the lowest proportion, 10 times lower than mode (4,2). This method is very useful for the identi®cation of modes which are aected by the visco-constraint treatment. For example, Fig. 9 shows that mode (5,1) is the most aected, while mode (3,1) is the least (with the exception of mode (2,1)). The same conclusions can be drawn by looking at Table 10. In conclusion, the MSE method is an interesting tool for initial parametric studies of visco-constraint treatment. The last result shows the measured mean quadratic velocity (Fig. 10). The same tendencies are obtained for modes with longitudinal order 1 and 3 (reduction at peak resonances) and for modes with longitudinal order 2 (almost no reduction at peak resonances). On the experimental side, mode (2,1) is the least aected as predicted. In conclusion, very good agreement between experimental and numerical results is achieved, which indicates that the model gives an accurate prediction of the vibration response for a cylindrical shell treated with a visco-constraint patch. Also, the good correlation between the predictions and the measurements seems to justify our assumption of using a constant shear modulus. Finally, we can observe that even a localized treatment gives a global eect on vibration response, which indicates that the patch geometry and position is a critical parameter for optimum eciency. 5.3. Closed shell with absorbing material In this section, results from the closed cylindrical shell with absorbing material are presented. Our main objective is to verify if the ¯uid equivalent approach predicts the global tendencies observed on the experimental set-up. For this study, the upper limit of the frequency range is set to 250 Hz. 5.3.1. Vibration response Fig. 11 compares the predicted mean quadratic velocity for the shell with and without absorbing material. There is a signi®cant reduction (over 20 dB) at peak resonances, while the vibration level stays the same at o resonances. Also, no frequency shift is observed at resonances, because the ¯uid equivalent approach does not take into account the eect of the structural mass of the absorbing material. These results suggest a strong coupling between the absorbing material and the structure. It is interesting to observe that with the presence of absorbing material, the structural response cannot be decoupled anymore from the acoustic response, as it was the case with no absorbing material in the cavity. Next, Fig. 12 compares the measured mean quadratic velocity for the shell with and without absorbing material. There is a reduction observed at peak resonances. A frequency shift is also observed at resonances caused by the added mass eect of the absorbing material. At o resonances, the vibration level remains the same. To compare predicted and measured results, Table 11 shows the level reduction observed at each resonance for the FE model and the experimental set-up. The same global tendencies are observed, with a 25 dB reduction for the FE model and a 15 dB reduction for the experimental set-up. This 10 dB discrepancy between the model
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and the set-up is explained in part by the limited frequency resolution in experimental results, which does not necessarily give peak levels for the case without the acoustic material. Globally, the equivalent ¯uid model gives the tendencies of the experimental set-up for the mean quadratic velocity. Also, the use of constant equivalent ¯uid properties seems to be an adequate choice for this frequency range. 5.3.2. Acoustic response Fig. 13 shows the predicted quadratic pressure, with and without the absorbing material. For the curve without absorbing material, there is a peak level at cavity
Fig. 10. Mean quadratic velocity comparison, shell with a visco-constraint patch.
Fig. 11. Predicted mean quadratic velocity, shell without and with absorbing material.
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Fig. 12. Measured mean quadratic velocity, shell without and with absorbing material.
Table 11 Mean quadratic velocity reduction at peak resonances for shell with absorbing material Mode n; m
Numerical (dB)
Experimental (dB)
(4,1) (3,1) (5,1)
25.7 24.6 23.9
14.7 15.7 14.7
Fig. 13. Predicted mean quadratic pressure, shell without and with absorbing material.
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resonance (170 Hz), which was not the case with the modal approach. This problem is discussed in detail in the next section. The presence of absorbing material reduces peak levels at structural resonances, and totally removes the peak level at cavity resonances. Finally, there is no frequency shift at structural resonances due to the eect previously discussed. Next, Fig. 14 presents the experimental quadratic pressure, for the case with and without absorbing material. A reduction is observed at structural resonances. For the cavity resonances, it is hard to tell if the peak is removed, since the frequencies are too close to structural resonances. A frequency shift is observed at structural resonances, because of the added mass eect of the absorbing material. Table 12 compares the reduction at peak resonances for the FE model with the experimental set-up. The same tendencies are observed, except for mode (3,1). Again, the limited frequency resolution explains partially the discrepancies. Globally, the FE model gives the same tendencies than the experimental set-up for the mean quadratic pressure. In conclusion, the equivalent ¯uid model, applied to a closed shell with absorbing material, is a simple approach which gives an acceptable evaluation of the absorbing material eciency on vibration and acoustic responses.
Fig. 14. Measured mean quadratic pressure, shell without and with absorbing material.
Table 12 Mean quadratic pressure reduction at peak resonances for shell with absorbing material Mode n; m
Numerical (dB)
Experimental (dB)
(4,1) (3,1) (5,1)
19.6 20.5 15.9
26.6 12.0 19.2
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Fig. 15. Predicted mean quadratic pressure comparison around the ®rst cavity resonance between modal and direct solution for the shell without absorbing material. For modal solution, three structural basis s are shown: 0±1100, 0±5000 and 0±20 000 Hz.
6. Convergence study In this section, a convergence study is done to determine the content of the structural modal basis s for the case with absorbing material. Fig. 15 shows the mean quadratic pressure obtained around the ®rst cavity mode, with the direct solution and the modal solution. For the modal solution, three structural basis are shown: 0± 1100, 0±5000 and 0±20 000 Hz. For acoustic modal basis, the 0±2800 Hz band is always used. It seems that with the 0±1100 Hz band, no peak level is observed. This frequency band contains plate modes, which are coupled to this cavity mode, thus con®rming that the point force does not excite plate modes. However, with the 0± 5000 Hz band, a peak begins to appear. With the structural basis 0-20 000 Hz, the peak level is higher, but this peak has not yet converged to one obtained from the direct response approach, even if 572 modes are more than one-quarter of the total number of DOF in the model. The convergence is extremely slow. For this problem, no structural mode seems to be well excited by the point force and coupled to the cavity mode. To converge, almost every structural modes should be retained in the modal basis, rendering ineective the modal solution. Further studies will be done to investigate this particular problem. 7. Conclusion In this paper, three shell structures were studied regarding noise and/or vibration reduction, using a ®nite element approach. Modal and dynamic results obtained from these models were compared with experimental results. In conclusion:
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. For the closed shell, it was shown that the use of the 1:2 fmax criterion was not sucient to correctly choose the structural and acoustic basis for dynamic analysis. New criteria were proposed, based on the study of the modal coupling terms. These criteria proved to be eective. Very good agreement with experiment was observed for modal and dynamic response analysis. However, for the acoustic response, a modal convergence problem was observed at acoustic natural frequencies. This problem will be further investigated and a more eective criterion for modal basis de®nition is presently under development. . For the shell with visco-constraint material, the simple approach used to model the visco-elastic behavior proved to be eective in the prediction of vibration response, with an excellent comparison with experimental results. . For the closed shell with absorbing material, the equivalent ¯uid approach was implemented in MSC/NASTRAN using DMAP language. It has been shown that this simple approach provides acceptable vibration and acoustic response results.
Acknowledgements The authors express their sincere thanks to Professor Jean Nicolas, to Raymond Panneton and to Jean-Marie Guertin for their help and to Canadair and NSERC for their sponsorship of this project. Thanks also go to Leslie R. Dorsett from Canadair, for his valuable discussions during this work. References [1] Morand HJ-P, Ohayon R. Interactions ¯uides-structures. Masson, Paris, France 1992. [2] Izadpanah K, Kansakar R, Reymond M, Wallerstein DV. Coupled ¯uid/structure interaction analysis using MSC/NASTRAN. MSC European User's conference, 1989. [3] Imbert JF. Analyse des structures par eÂleÂments ®nis. 3rd Cepadues, Paris, France 1991. [4] Nashif AD, Jones IGJ, Henderson JP. Vibration damping. New York: John Wiley & Sons, 1985. [5] Johnson CD, Kienholz DA. Finite element prediction of damping in structures with constrained viscoelastic layers. AIAA Journal 1982;20(9):1284±90. [6] Atalla N, Charron F, Panneton R. A ®nite-element formation for the vibroacoustic behavior of double-plate structures with cavity absorption. Canadian Aeronautics and Space Journal 1995;41(1):5±12. [7] Boily, S. EÂtude numeÂrique du comportement vibroacoustique de coques semi-complexes pour une execitation solidienne. Master's thesis, Universite de Sherbrooke, Canada, 1996. [8] Panneton R. ModeÂlisation par eÂleÂments ®nis de mateÂriaux poreux. Ph.D. thesis, Universite de Sherbrooke, Canada, 1996. [9] Lesueur C. Rayonnement acoustique des structures. Collection DeÂpartment Etudes et Recherches EDF, Eyrolles, France 1988.
The vibroacoustic response of a cylindrical shell structure with viscoelastic and poroelastic materials Sylvain Boily, FrancËois Charron* o
Department of Mechanical Engineering, Universite de Sherbrooke, 2500 Boulevard de l'UniversiteÂ, Sherbrooke, QueÂbec, Canada J1K 2R1 Received 1 November 1997; received in revised form 10 August 1998; accepted 17 October 1998
Abstract The study of the vibroacoustic behavior of an aircraft fuselage requires the generation of very large and complex ®nite element models (FEM) and, furthermore, the numerical results are often very dicult to interpret and to validate. On the other hand, the study of the vibroacoustic behavior of a simple cylindrical shell structure should allow us to validate the results obtained from these simple numerical models and to study the eciency of dierent analysis tools and design solutions for vibration and noise reduction, such as the addition of visco-elastic and porous materials. In order to achieve these objectives, two cylindrical shell structures were studied. The ®rst model was used for the study of the dynamic behavior of the structure with and without visco-elastic material. The second model was used for the study of the vibroacoustic behavior with and without porous material. Results show a good correlation between experimental measurements and numerical predictions. Eciency of the two proposed solutions was evaluated and, ®nally, it has been demonstrated that the eect of the modal truncation of the cavity modal basis is critical for the convergence of the predicted pressure ®eld and that a good understanding of coupling phenomena between structural and acoustic modes is necessary. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Cylindrical shell; Vibroacoustic response; Mechanical excitation; Finite element model; Viscoelastic material; Porous material
1. Introduction Noise inside an airplane has always been a cause of discomfort to passengers and this problem is becoming more important with the increasing use of composite * Corresponding author. Tel.:+1-819-821-7144; fax: +1-819-821-7163; e-mail: francois.charron @gme.usherb.ca 0003-682X/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S000 3-682X(98)0007 0-X
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materials and more powerful engines, which tend to amplify the acoustic problems. In fact, noise and/or vibration reduction inside airplanes is becoming a prime objective for many manufacturers. However, the vibroacoustic behavior of fuselage structures is a very complex problem. In order to tackle this problem and to validate various analysis tools and design solutions, a simpli®ed representation of an airplane fuselage, that are cylindrical shell structures, have been studied. Results from this kind of fuselage-like structures can provide useful insights into the eect of adding visco-elastic and/or porous materials on real airplane structures. This paper is concerned with the study of various cylindrical shell structures regarding vibration and/or noise reduction. The ®nite element method will be used for the modelization of shell structures including the acoustic cavity, and experimental measurements will be done to validate the numerical predictions. Three con®gurations are studied. The ®rst model is a closed cylindrical shell with an interior acoustic cavity. The second model is a cylindrical shell with a narrow patch of visco-constraint material and in the last model, absorbing material is added to the inner shell surface of the ®rst model. FE models for these experimental con®gurations are developed, and modal and dynamic analysis are performed. Numerical predictions are compared to experimental measurements, in order to assess the limitations of the analysis tools. 2. Finite element formulation 2.1. Problem discretization and modal decomposition The ®nite element formulation of the coupled structural-acoustic problem is based on a variational approach, using a displacement ®eld u for the structure and a pressure ®eld p for the cavity. This classical approach [1] is used in the commercial code MSC/NASTRAN [2]. The ®nite element discretization of these equations gives the following linear system of equations:1 fu g 0 fFs g Ks AT 2 Ms 1 ÿ! p A Mf f0 g 0 Kf This system can be solved directly in physical coordinates using direct frequency response analysis, which will be the case for the third model. Unfortunately, the total number of degrees of freedom (sum of structural and acoustic DOF) is often very large and the model needs to be solved for many excitation frequencies. In order to reduce the size of this system, a modal reduction method is often used. The displacement vector fug is projected on the in vacuo structural modes s and the pressure vector fpg is projected on the rigid wall cavity modes f , using the following relations: 1
The eigenvalue problem is unsymmetric.
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fug s qs p f qf
133
2
At the same time, the damping eect is introduced in Ks and Kf 2 using an equivalent structural damping model, classically used in structural analysis [3]: Kf 1 jf Kf Ks 1 js Ks
3
Then, using the two previous equations and premultiplying the ®rst line of Eq. (1) by s T , and the second line by f T , we ®nally obtain: qs ks ÿaT s T fFs g 2 ms 0 4 ÿ! a mf qf 0 kf f0 g This system is now in modal coordinates, with its size being equal to the sum of structural and acoustic modes retained in each modal basis. This modal reduction process is thus advantageous when the number of physical DOF is much larger than the total number of modes . The modes that should be retained in the structural basis s and the acoustic basis f are discussed in the next section. 2.2. Choice of modal basis This section is concerned with the content de®nition of the structural modal basis s and of the acoustic modal basis f for a given excitation frequency range and for a mechanical excitation. In the case of vibration problems, a commonly used rule of thumb is to retain modes outside the excitation frequency range, in order to approximate the contribution of the higher modes. For example, if the frequency range varies from 0 to fmax, all modes between 0 and 1:2 fmax should be retained. But to the authors' knowledge, there is no such general criterion for the case of vibroacoustic problems. The problem of de®ning the proper modal basis for the acoustic cavity is presented in two steps. Firstly, which acoustic modes should be retained, if one wants to compute correctly but also eciently the acoustic response at each structural mode natural frequency in the excitation frequency range? Secondly, which structural modes should be retained, if one wants to compute accurately the acoustic response for each cavity mode natural frequency in the excitation frequency range? In order to answer these questions, Eq. (4) is simpli®ed using the light ¯uid assumption for the cavity, i.e. ÿaT qf s T fFs g: qs 2
s T fFs g ÿ!2 ms ks
5
When this project was realized, it was not possible to directly de®ne f in MSC/NASTRAN. It was thus done using DMAP language.
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ÿ!2 a qs qf ÿ!2 mf kf
6
Eq. (6) gives the relation between the acoustic modal response qf and the structural modal response fqs g. This equation can be rewritten for a given acoustic mode j and a given structural mode i. In the process, if modal masses are normalized to one, the relation between the acoustic modal response of mode j and the structural modal response of mode i can be simpli®ed to the following form: ! T si m 1f Fs gm1 !2 aij 7 q fj 2 ! ÿ !2fj 1 jf ÿ!2 ÿ !2si 1 js Now, Eq. (7) gives the response of acoustic mode j as a function of the response of a structural mode i. Two speci®c cases are of interest. In the ®rst case, the excitation frequency coincides with the structural natural frequency ! !si . Eq. (7) simpli®es to: ! T si m 1f Fs gm1 aij ÿ 8 q fj 2 js !si ÿ !2fj 1 jf In the second case, the excitation frequency coincides with the acoustic natural frequency ! !fj . Eq. (7) simpli®es to: ! T si m1f Fs gm1 aij q fj 2 jf !fj ÿ !2si 1 js
9
From Eqs. (8) and (9), it seems that the following conditions are required in order to obtain an acoustic response for mode j: . the mechanical excitation must be capable of exciting the structural mode or, in others words, mode i needs a good modal participation factor. . the spatial modal coupling must be excellent between the acoustic mode j and the structural mode i. One can also note that for the case where ! !si !fj , maximum acoustic response is achieved. These conditions and remarks will be later used for the evaluation of the content of the structural and acoustic modal basis. 3. Global indicators For dynamic analysis, two global indicators are calculated. The ®rst one is the mean quadratic velocity:
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hv2 !i
n 1 X v vi S i 2S i1 i
135
10
where n is the number of structural DOF. The velocity is weighted by the nodal surface. In dB, the quadratic velocity is calculated using the following relation: 2 v 11 Lv 10 log10 2 ; with v0 5 10ÿ8 m=s v0 The second global indicator is the mean quadratic pressure: hP2 !i
m 1 X p pi Vi 2V i1 i
12
where m is the number of acoustic DOF. The pressure is weighted by the nodal volume. In dB, the quadratic pressure is calculated using the following relation : 2 P 13 Lp 10 log10 2 ; with P0 2 10ÿ5 Pa P0
4. Finite element models 4.1. Closed cylindrical shell 4.1.1. Experimental set-up The shell structure is made of a 1.2 mm thick steel plate. The plate has been rolled and welded at a joint made along the longitudinal axis of the cylinder. The resulting shell is 1.01 m long and has a mean radius of 0.183 m. A steel ring is welded at both ends of the shell and an aluminum 2.54 cm thick circular plate is bolted on each ring, in order to close the cavity. To approximate free±free boundary conditions, the shell is supported by soft springs. 4.1.2. FE model The FE model of the closed shell is made of four parts: a shell, two rigid plates, and an acoustic cavity. Free±free boundary conditions are used. Table 1 gives the physical properties and dimensions of the shell, the plates and the acoustic cavity. The mesh of this FE model is based on a criterion of ®ve to six linear elements per mode shape wavelength. This criterion is applied to both structure and cavity models. The frequency range of interest for the dynamic analysis is 80±480 Hz. Fig. 1 shows the FE model of the shell and the plates. For the cavity, a matching interface is used, which means that the grid points of the wet surface between the structural and acoustic models are coincident. Finally, Table 2 gives element types used for each components of the structure and the cavity.
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Table 1 Physical properties and dimensions of the closed shell model
Mean radius (m) Length (m) Thickness (m) Young modulus (pa)} Density (kg/m3) Poisson ratio Structural damping Speed of sound (m/s)
Shell
Plates
Cavity
0.18256 1.01 0.001219 2.11011 7800 0.3 0.0006 ±
0.18256 ± 0.0254 7.01010 2700 0.3 0.0006 ±
0.18256 1.01 ± ± 1.21 ± 0.0005 340
Table 2 Element types and number of elements for the closed shell model Name
Type
Nodes/element
# of elements
Shell
CQUAD4
Plate (2D)
4
450
Plates
CTRIA3 CQUAD4
Plate (2D) Plate (2D)
3 4
10/plate 290/plate
Cavity
CPENTA CHEXA
Solid (3D) Solid (3D)
6 8
150 4350
4.2. Simply supported shell with a patch of visco-constraint material 4.2.1. Experimental set-up The shell structure is made of a 1.2 mm thick steel plate, as described in Section 4.1.1. The simply supported boundary conditions are approximated using thin perforated rings which are welded at each end of the shell. These rings are bolted to a concrete base. A 4.81 cm wide visco-constraint patch is applied on the shell circumference at the half-length of the shell. Five pieces of material were required to cover the circumference. The visco-constraint material used is the SJ2152X, type 1010, from 3M. The constraint layer is made of stainless steel. 4.2.2. FE model The visco-elastic material behavior is described using a complex shear modulus approach [4]: G G 1 jv ;
14
where G is the shear modulus and v the loss factor. The real part of Eq. (14) is for the elastic behavior of the material, while the imaginary part characterizes the damping behavior. Usually, G and v are frequency dependant. The approach used for the modelization of the visco-constraint layer has been developed by Johnson and Kienholz [5]. The constraint layer is modeled with plate
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elements and the visco-elastic core with solid elements. The FE model is made of three parts: a shell, a visco-elastic core and a constraint layer. Only structural analyses are performed with this model. Again, the frequency range of interest for dynamic analysis is 80±480 Hz. Fig. 2 shows the FE model. The central patch represents the visco-constraint material. Fig. 3 shows a cross-section view of the model. The constraint layer sheet and the shell are modeled with linear quadrilateral elements, while the visco-elastic core is made of linear hexahedral elements. An arrow on Fig. 3 indicates a duplicated grid, which is used to take into account the discontinuity between the ®ve patches of visco-constraint layer applied on the circumference of the experimental shell. Table 3 gives the visco-elastic physical properties
Fig. 1. Finite element model of the shell and plates.
Fig. 2. Finite element model of the shell with a visco-constraint patch.
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and dimensions. For the constraint layer, the physical properties used are those of the shell. For dynamic analysis, no frequency dependence is considered for this material, which is justi®ed for the frequency range of interest. Finally, Table 4 gives the number of elements used for shell, visco-elastic core and constraint layer. 4.3. Closed cylindrical shells with absorbing material 4.3.1. Experimental set-up The shell presented in Section 4.1.1 is used. A 1.3 cm thick foam is installed on the inner surfaces of the shell and of the two end plates. No glue is used, the foam is maintained on the inner surfaces by a tight ®t. 4.3.2. FE model The absorbing material is modeled using an equivalent ¯uid model [6]. This type of model is justi®ed and accurate for very limp porous materials which are used in
Fig. 3. Partial cross-section view of the shell with a visco-constraint patch.
Table 3 Physical properties and dimensions of the viscoelastic material Width (m) Thickness (m) Density (kg/m3) Poisson ratio Shear modulus (Pa) Loss factor
0.0481 2.5410ÿ4 980 0.49 6.0106 1.0
Table 4 Element types and number of elements for the shell with a visco-constraint patch
Shell Viscoelastic Upper face sheet
Name
Type
Nodes/element
# of elements
CQUAD4 CHEXA CQUAD4
Plate (2D) Solid (3D) Plate (2D)
4 8 4
1058 46 46
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many aeronautical applications. The acoustical properties of porous material are de®ned by two acoustical characteristics, the equivalent density eq ! and the equivalent bulk modulus Beq !. For the present study, this approach has been implemented in MSC/NASTRAN using DMAP langage [7]. The basic idea is to use the acoustic elements for the absorbing material and to modify the Mf and Kf matrices corresponding to these elements, using the equivalent ¯uid characteristics eq ! and Beq !. Initially, a FE model based on the one presented in Section 4.1.2 was created. However, the model size exceeds our computer hardware capability because real modal analysis could not be used anymore due to the signi®cant imaginary part of the ¯uid equivalent properties. It was then decided to generate a coarser FE model. This model should be valid from 0 to 250 Hz. For the closed shell, the same physical properties and dimensions are used (see Table 1). Fig. 4 shows this coarser ®nite element model of shell and plates. The evaluation of eq ! and Beq ! for a given absorbing material can be done with many models [8]. However, this subject is beyond the scope of the present study, where these characteristics are considered known for the absorbing material used. Table 5 gives the equivalent ¯uid properties used over the frequency range. In a ®rst approximation, no frequency dependence is considered.
Fig. 4. Finite element model of the shell with the absorbing material.
Table 5 Properties of equivalent ¯uid R Beq I Beq R eq I eq
102 000 Pa 5000 Pa 3.57 kg/m3 ÿ12.0 kg/m3
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5. Results In this section, results obtained from the three ®nite element models described in previous section are presented. These results are compared with experimental results. For the closed shell and the shell with a visco-constraint patch, modal and dynamic analysis results are presented. For the closed shell with absorbing material, only dynamic analysis results are presented, since direct frequency response was used. The excitation is a mechanical harmonic point force and its location is shown on Fig. 5. 5.1. Closed shell 5.1.1. Modal analysis Firstly, a modal analysis was done. Table 6 presents the in vacuo structural modes and compares them with the experimental results for the frequency range of 80 to 480 Hz. For all modes, the plate contribution is negligible and this is due to the signi®cant dierence between the plate and shell thickness. A good agreement between experimental and numerical modes is observed. The error on the natural frequencies is less than 5%, except for modes (3,1) and (4,1). These modes are more sensitive to the modelization of the interface ring between the plates and shell. In reality, the plates are bolted to the ring (discrete interface), whereas in the FE model, plates are welded (continuous interface) to the shell. For the experimental modes, one can also observe that symmetric and anti-symmetric modes are shifted, due to the longitudinal welding joint. Furthermore, an acoustic modal analysis was performed. Table 7 shows the calculated rigid wall cavity modes and compares them with the experimental
Fig. 5. Location of mechanical point force for all FE models.
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141
measurements. There are only two modes in this range, and an excellent agreement is reached between numerical and experimental results. In conclusion, these structural and acoustic modal results con®rm the validity of the closed shell FE model for the excitation frequency range of interest. 5.1.2. Dynamic analysis The vibration response is ®rst studied using the modal solution approach. Using the 1:2 fmax criterion (Section 2.2), the structural modal basis includes 38 modes between 0 to 600 Hz. Fig. 6 compares the numerical and experimental mean quadratic velocity. Very good agreement is achieved between the two curves. The Table 6 Natural frequencies comparison for the closed shell Mode a n; m (4,1) (4,1) (3,1) (3,1) (5,1) (5,1) (2,1) (2,1) (6,1) (6,1) (5,2) (5,2) (6,2) (6,2) (4,2) (4,2) (7,1) (7,1) (7,2) (7,2) a
Numerical (Hz)
Experimental (Hz)
Error (%)
183.5 183.5 189.9 189.9 237.5 237.5 303.8 303.8 327.4 327.4 344.1 344.1 381.0 381.0 387.5 387.5 441.5 441.5 470.5 470.5
172.7 173.8 178.6 179.1 232.8 233.5 294.8 297.7 326.8 ± 328.3 330.4 368.6 370.3 377.0 ± 444.1 ± 471.1 471.9
6.3 5.6 6.3 6.0 2.0 1.7 3.1 2.0 0.2 ± 4.8 4.1 3.4 2.9 4.6 ± ÿ0.6 ± ÿ0.1 ÿ0.3
n and m are the mode orders with respect to the circumference and length of the shell.
Table 7 Natural frequencies comparison for the cavity Mode a p; q; r
Numerical (Hz)
Experimental (Hz)
Error (%)
(0,0,1) (0,0,2)
168.6 339.1
172.0 343.0
2.0 1.1
a
p, q and r are the mode orders with respect to the circumference, radius and length of the cylindrical cavity.
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frequency shifts are the same as observed in Table 6. The dierences in peak levels are due to the frequency resolution in experimental measurements and to the diculty to properly characterize the modal damping for each modes. This comparison shows that the closed shell FE model gives fairly accurate prediction of the vibration response. Next, the acoustic response is studied. The ®rst step is the choice of the modal basis content. For the structure, s is de®ned using the 0 to 600 Hz frequency band. For the acoustic cavity, a convergence study has been done. Fig. 7 shows the mean
Fig. 6. Mean quadratic velocity comparison for the closed shell.
Fig. 7. Mean quadratic pressure for two acoustic modal basis: 0±600 Hz and 0±2000 Hz.
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143
quadratic pressure obtained with the 0±600 Hz band (eight modes) and for the 0± 2000 Hz band (110 modes). From these results and the previous discussion in Section 2.2, the following remarks can be made: 1. The 0±600 Hz band is not sucient since there is no peak levels at acoustic resonances. 2. The 0±2000 Hz band is not sucient since some peak levels are missing at structural resonances. In order to correctly choose the acoustic basis, Table 8 shows the modal coupling coecients for each structural mode in the excitation frequency range. It seems that the modal coupling behavior between the acoustic modes and shell modes is in perfect agreement with Lesueur's discussion on this subject [9]. There is a strong coupling between structure and cavity modes when they have the same circumferential order and odd±even (or even±odd) longitudinal order. Unfortunately, no information is provided for order in the radial direction of the cavity. Furthermore, for each shell modes, the term of the third coupled cavity mode is less than 13% of the ®rst, which seems to indicate that in order to obtain an acoustic response at each shell modes, it is necessary to retain at least the ®rst two coupled cavity modes. This con®rmation cannot be readily generalized to other closed shell models without a more thorough study. From the previous comments, the 0±2800 Hz band is chosen for the de®nition of the acoustic modal basis. Now, in order to correctly de®ne the structural basis content, a study in the 0±2500 Hz band was done. In this band, only modes relating to the plate are found to be coupled with the two acoustic modes in the excitation frequency range. However, these modes are not excited by the point force on the shell. The 0±600 Hz frequency band is thus chosen for structural basis. Finally, to complete the acoustic response study, Fig. 8 compares the mean quadratic pressure from the model and from the experimental set-up. A good correlation is obtained, although the experimental curve seems to show a small peak response at cavity resonances. This can be due to a slight misalignment of the shaker in the longitudinal direction of the shell. This misalignment will produce a small force contribution capable of exciting the plate modes, which are strongly coupled to the two cavity modes in the 0±500 Hz frequency range. In conclusion, apart from the discussed problem at cavity resonances, the closed shell model gives an accurate prediction of the vibration and acoustic responses. 5.2. Simply supported shell with a patch of visco-constraint material For this case, no acoustic analysis was performed. The results are obtained from the modal and dynamic analysis of the shell structure. 5.2.1. Modal analysis Table 9 presents the in vacuo modes obtained from the modal analysis and compares them with the experimental results, for the frequency range of 80±480 Hz. It seems that there is very good agreement, with the exception of mode (2,1). This
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mode is more sensitive to simply supported boundary conditions. In general, the model gives an excellent correlation with the experimental modes. 5.2.2. Dynamic analysis The modal basis from 0±600 Hz (33 modes) is used. Fig. 9 shows the mean quadratic velocity for the model with and without the visco-constraint patch. Modes with longitudinal order 1 and 3 show a reduction at peak resonances, up to 30 dB. Physically, this eect is explained by the fact that for these modes, the shear strain Table 8 Numerical modal coupling terms for each shell mode Shell mode n; m
Cavity mode p; q; r
Frequency (Hz)
Numerical coupling term N m
Relative value (%)
(2,0,0) (2,0,2) (2,0,4) (2,0,6) (2,1,0) (2,1,2)
912.5 973.3 1146 1412 2020 2048
432 183 30 8 345 153
100 42.3 6.8 1.7 79.9 35.4
(3,1)
(3,0,0) (3,0,2) (3,0,4) (3,1,4) (3,1,4)
1264 1308 1441 2428 2451
484 223 38 369 172
100 46.0 7.8 76.2 35.5
(4,1)
(4,0,0) (4,0,2) (4,0,4)
1615 1651 1758
510 253 38
100 49.6 7.4
(4.2)
(4,0,1) (4,0,3) (4,0,5)
1624 1695 1840
481 289 59
100 60.1 12.2
(5,1)
(5,0,0) (5,0,2) (5,0,4)
1975 2004 2093
417 222 28
100 53.2 6.7
(5,2)
(5,0,1) (5,0,3) (5,0,5)
1982 2041 2162
481 313 56
100 65.1 11.7
(6,1)
(6,0,0) (6,0,2) (6,0,4)
2347 2372 2448
455 255 27
100 56.0 6.0
(6,2)
(6,0,1) (6,0,3) (6,0,5)
2353 2403 2507
464 323 51
100 69.6 10.9
(7,1)
(7,0,0) (7,0,2)
2736 2757
525 302
100 57.5
(7,2)
(7,0,1) (7,0,3)
2741 2784
474 346
100 73.0
(2,1)
a
aij f s gTi Af fgj m m2 N=m2 N m:
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Fig. 8. Mean quadratic pressure comparison for the closed shell.
Table 9 Natural frequencies comparison for the shell with visco-constraint patch Mode n; m
Numerical (Hz)
Experimental (Hz)
Error (%)
(4,1) (4,1) (3,1) (3,1) (5,1) (5,1) (2,1) (2,1) (5,2) (5,2) (6,1) (6,1) (4,2) (4,2) (6,2) (6,2) (7,1) (7,1) (7,2) (7,2) (6,3) (6,3)
161.4 161.6 163.2 163.3 226.2 226.5 296.1 296.2 312.3 312.3 321.5 322.2 354.8 355.0 359.3 359.6 437.5 438.8 457.1 457.9 469.5 469.6
158.5 160.4 163.3 164.5 223.9 ± 277.0 280.7 312.0 313.8 318.6 ± 351.7 354.2 360.9 361.7 434.4 ± 457.9 459.4 471.1 ±
1.8 0.7 ÿ0.0 ÿ0.7 1.0 ± 6.9 5.5 0.0 ÿ0.5 0.9 ± 0.9 0.2 ÿ0.4 ÿ0.6 0.7 ± ÿ0.2 ÿ0.3 ÿ0.3 ±
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energy in the visco-elastic layer is maximum and thus, result in a high damping eciency. On the other hand, modes of longitudinal order 2 show almost no reduction at peak resonances. These modes produce a minimum shear strain energy in the visco-elastic layer, resulting in low damping eciency. The only exception is mode (2,1), for which no reduction is observed at peak resonances. To understand this exception, the proportion of modal strain energy in the visco-elastic elements is now studied. To predict the damping eciency of a visco-constraint patch on speci®c modes, an interesting criterion is the modal strain energy (MSE), where the proportion of the modal strain energy in the visco-elastic elements with respect to the total strain energy is used to evaluate the eciency of the visco-elastic treatment. Table 10 gives this ratio for each mode in the frequency range of 80±480 Hz. From this table, it
Fig. 9. Predicted mean quadratic velocity, shell without and with a visco-constraint patch. Table 10 Relative modal strain energy in viscoelastic elements for each mode Mode n; m
MSE in viscoelastic elements (%)
(4,1) (3,1) (5,1) (2,1) (5,2) (6,1) (4,2) (6,2) (7,1) (7,2) (6,3)
1.295 0.256 1.780 0.0002 0.017 1.722 0.002 0.035 1.532 0.036 0.674
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seems that the ratio of MSE is 10 times higher for modes with longitudinal order 1 and 3 than for modes with longitudinal order 2. The only exception is mode (2,1), which has the lowest proportion, 10 times lower than mode (4,2). This method is very useful for the identi®cation of modes which are aected by the visco-constraint treatment. For example, Fig. 9 shows that mode (5,1) is the most aected, while mode (3,1) is the least (with the exception of mode (2,1)). The same conclusions can be drawn by looking at Table 10. In conclusion, the MSE method is an interesting tool for initial parametric studies of visco-constraint treatment. The last result shows the measured mean quadratic velocity (Fig. 10). The same tendencies are obtained for modes with longitudinal order 1 and 3 (reduction at peak resonances) and for modes with longitudinal order 2 (almost no reduction at peak resonances). On the experimental side, mode (2,1) is the least aected as predicted. In conclusion, very good agreement between experimental and numerical results is achieved, which indicates that the model gives an accurate prediction of the vibration response for a cylindrical shell treated with a visco-constraint patch. Also, the good correlation between the predictions and the measurements seems to justify our assumption of using a constant shear modulus. Finally, we can observe that even a localized treatment gives a global eect on vibration response, which indicates that the patch geometry and position is a critical parameter for optimum eciency. 5.3. Closed shell with absorbing material In this section, results from the closed cylindrical shell with absorbing material are presented. Our main objective is to verify if the ¯uid equivalent approach predicts the global tendencies observed on the experimental set-up. For this study, the upper limit of the frequency range is set to 250 Hz. 5.3.1. Vibration response Fig. 11 compares the predicted mean quadratic velocity for the shell with and without absorbing material. There is a signi®cant reduction (over 20 dB) at peak resonances, while the vibration level stays the same at o resonances. Also, no frequency shift is observed at resonances, because the ¯uid equivalent approach does not take into account the eect of the structural mass of the absorbing material. These results suggest a strong coupling between the absorbing material and the structure. It is interesting to observe that with the presence of absorbing material, the structural response cannot be decoupled anymore from the acoustic response, as it was the case with no absorbing material in the cavity. Next, Fig. 12 compares the measured mean quadratic velocity for the shell with and without absorbing material. There is a reduction observed at peak resonances. A frequency shift is also observed at resonances caused by the added mass eect of the absorbing material. At o resonances, the vibration level remains the same. To compare predicted and measured results, Table 11 shows the level reduction observed at each resonance for the FE model and the experimental set-up. The same global tendencies are observed, with a 25 dB reduction for the FE model and a 15 dB reduction for the experimental set-up. This 10 dB discrepancy between the model
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and the set-up is explained in part by the limited frequency resolution in experimental results, which does not necessarily give peak levels for the case without the acoustic material. Globally, the equivalent ¯uid model gives the tendencies of the experimental set-up for the mean quadratic velocity. Also, the use of constant equivalent ¯uid properties seems to be an adequate choice for this frequency range. 5.3.2. Acoustic response Fig. 13 shows the predicted quadratic pressure, with and without the absorbing material. For the curve without absorbing material, there is a peak level at cavity
Fig. 10. Mean quadratic velocity comparison, shell with a visco-constraint patch.
Fig. 11. Predicted mean quadratic velocity, shell without and with absorbing material.
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Fig. 12. Measured mean quadratic velocity, shell without and with absorbing material.
Table 11 Mean quadratic velocity reduction at peak resonances for shell with absorbing material Mode n; m
Numerical (dB)
Experimental (dB)
(4,1) (3,1) (5,1)
25.7 24.6 23.9
14.7 15.7 14.7
Fig. 13. Predicted mean quadratic pressure, shell without and with absorbing material.
149
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resonance (170 Hz), which was not the case with the modal approach. This problem is discussed in detail in the next section. The presence of absorbing material reduces peak levels at structural resonances, and totally removes the peak level at cavity resonances. Finally, there is no frequency shift at structural resonances due to the eect previously discussed. Next, Fig. 14 presents the experimental quadratic pressure, for the case with and without absorbing material. A reduction is observed at structural resonances. For the cavity resonances, it is hard to tell if the peak is removed, since the frequencies are too close to structural resonances. A frequency shift is observed at structural resonances, because of the added mass eect of the absorbing material. Table 12 compares the reduction at peak resonances for the FE model with the experimental set-up. The same tendencies are observed, except for mode (3,1). Again, the limited frequency resolution explains partially the discrepancies. Globally, the FE model gives the same tendencies than the experimental set-up for the mean quadratic pressure. In conclusion, the equivalent ¯uid model, applied to a closed shell with absorbing material, is a simple approach which gives an acceptable evaluation of the absorbing material eciency on vibration and acoustic responses.
Fig. 14. Measured mean quadratic pressure, shell without and with absorbing material.
Table 12 Mean quadratic pressure reduction at peak resonances for shell with absorbing material Mode n; m
Numerical (dB)
Experimental (dB)
(4,1) (3,1) (5,1)
19.6 20.5 15.9
26.6 12.0 19.2
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Fig. 15. Predicted mean quadratic pressure comparison around the ®rst cavity resonance between modal and direct solution for the shell without absorbing material. For modal solution, three structural basis s are shown: 0±1100, 0±5000 and 0±20 000 Hz.
6. Convergence study In this section, a convergence study is done to determine the content of the structural modal basis s for the case with absorbing material. Fig. 15 shows the mean quadratic pressure obtained around the ®rst cavity mode, with the direct solution and the modal solution. For the modal solution, three structural basis are shown: 0± 1100, 0±5000 and 0±20 000 Hz. For acoustic modal basis, the 0±2800 Hz band is always used. It seems that with the 0±1100 Hz band, no peak level is observed. This frequency band contains plate modes, which are coupled to this cavity mode, thus con®rming that the point force does not excite plate modes. However, with the 0± 5000 Hz band, a peak begins to appear. With the structural basis 0-20 000 Hz, the peak level is higher, but this peak has not yet converged to one obtained from the direct response approach, even if 572 modes are more than one-quarter of the total number of DOF in the model. The convergence is extremely slow. For this problem, no structural mode seems to be well excited by the point force and coupled to the cavity mode. To converge, almost every structural modes should be retained in the modal basis, rendering ineective the modal solution. Further studies will be done to investigate this particular problem. 7. Conclusion In this paper, three shell structures were studied regarding noise and/or vibration reduction, using a ®nite element approach. Modal and dynamic results obtained from these models were compared with experimental results. In conclusion:
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. For the closed shell, it was shown that the use of the 1:2 fmax criterion was not sucient to correctly choose the structural and acoustic basis for dynamic analysis. New criteria were proposed, based on the study of the modal coupling terms. These criteria proved to be eective. Very good agreement with experiment was observed for modal and dynamic response analysis. However, for the acoustic response, a modal convergence problem was observed at acoustic natural frequencies. This problem will be further investigated and a more eective criterion for modal basis de®nition is presently under development. . For the shell with visco-constraint material, the simple approach used to model the visco-elastic behavior proved to be eective in the prediction of vibration response, with an excellent comparison with experimental results. . For the closed shell with absorbing material, the equivalent ¯uid approach was implemented in MSC/NASTRAN using DMAP language. It has been shown that this simple approach provides acceptable vibration and acoustic response results.
Acknowledgements The authors express their sincere thanks to Professor Jean Nicolas, to Raymond Panneton and to Jean-Marie Guertin for their help and to Canadair and NSERC for their sponsorship of this project. Thanks also go to Leslie R. Dorsett from Canadair, for his valuable discussions during this work. References [1] Morand HJ-P, Ohayon R. Interactions ¯uides-structures. Masson, Paris, France 1992. [2] Izadpanah K, Kansakar R, Reymond M, Wallerstein DV. Coupled ¯uid/structure interaction analysis using MSC/NASTRAN. MSC European User's conference, 1989. [3] Imbert JF. Analyse des structures par eÂleÂments ®nis. 3rd Cepadues, Paris, France 1991. [4] Nashif AD, Jones IGJ, Henderson JP. Vibration damping. New York: John Wiley & Sons, 1985. [5] Johnson CD, Kienholz DA. Finite element prediction of damping in structures with constrained viscoelastic layers. AIAA Journal 1982;20(9):1284±90. [6] Atalla N, Charron F, Panneton R. A ®nite-element formation for the vibroacoustic behavior of double-plate structures with cavity absorption. Canadian Aeronautics and Space Journal 1995;41(1):5±12. [7] Boily, S. EÂtude numeÂrique du comportement vibroacoustique de coques semi-complexes pour une execitation solidienne. Master's thesis, Universite de Sherbrooke, Canada, 1996. [8] Panneton R. ModeÂlisation par eÂleÂments ®nis de mateÂriaux poreux. Ph.D. thesis, Universite de Sherbrooke, Canada, 1996. [9] Lesueur C. Rayonnement acoustique des structures. Collection DeÂpartment Etudes et Recherches EDF, Eyrolles, France 1988.