Ground cross-modal impedance as a tool for analyzing ground/plate interaction and ground wave propagation L. Graua) and B. Laulagnet INSA Lyon, Departement Genie Mecanique et Conception, Laboratoire Vibrations et Acoustique, 25 bis av. Jean Capelle, 69100 Villeurbanne, France
(Received 10 July 2014; revised 19 January 2015; accepted 9 April 2015) An analytical approach is investigated to model ground–plate interaction based on modal decomposition and the two-dimensional Fourier transform. A finite rectangular plate subjected to flexural vibration is coupled with the ground and modeled with the Kirchhoff hypothesis. A Navier equation represents the stratified ground, assumed infinite in the x- and y-directions and free at the top surface. To obtain an analytical solution, modal decomposition is applied to the structure and a Fourier Transform is applied to the ground. The result is a new tool for analyzing ground–plate interaction to resolve this problem: ground cross-modal impedance. It allows quantifying the added-stiffness, added-mass, and added-damping from the ground to the structure. Similarity with the parallel acoustic problem is highlighted. A comparison between the theory and the experiment shows good matching. Finally, specific cases are investigated, notably the influence of layer depth on plate C 2015 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4919332] vibration. V [JDM]
Pages: 2901–2914
I. INTRODUCTION
Ground–structure interaction is a significant problem in rail transportation (trains, subways, tramways). Populations living near these infrastructures can be sensitive to the vibration induced by rail vehicles. This has led to the increasing development of tools for analyzing the ground vibration of rail lines. In these cases, a structure made of concrete is often excited directly by a source coupled to the ground via an intermediate layer in contact with the concrete structure. This is the reason why the whole elastic medium is initially considered as a stack of layers when attempting to predict ground vibration generated by the vibration of an external structure. The issue of vibration in a multi-layered ground has been addressed in the past.1–3 It is commonly accepted that the ground vibration problem can be represented by a Navier equation with reasonable accuracy. Consequently, ground vibrations are considered as a linear problem. For a semi-infinite medium, classical solutions based on the potential decomposition of the displacement field, leading to equivalent Helmholtz equations are widely used in research.1,4–7 The Helmholtz decomposition leads to two types of waves that can propagate in an infinite homogeneous medium subjected to general cases of excitation. The dilatational wave and the shear wave described by the potentials are very similar to the corresponding acoustical problem, where the ground is substituted for a fluid. One of the aims of this paper is to exhibit similarities between structural ground coupling and structural acoustic coupling.8–13 The structure interacting with the ground is often made of concrete. Depending on the geometric complexity of the structure considered, the solution of the ground–structure
a)
Also at: ACOUPHEN, 33 Route de Jonage, 69830 Pusignan, France. Electronic mail: [email protected]
J. Acoust. Soc. Am. 137 (5), May 2015
interaction is found using analytical3,13,14 or numerical15–19 solutions. A typical example of the use of an analytical solution concerns the vibration of an infinite beam lying on a multi-layered ground. Here, a solution is obtained by a spatial Fourier transform for both the beam and the ground.1,14,20 An analytical solution was formulated by Karlstrom et al.1 for an infinite beam-plate–ground interaction. Using the Fourier transform to simplify the Navier equation and then considering the infinite beam and plate geometry, the solution is developed in the Fourier domain. On the same basis, the same authors developed an extension of the problem21 by considering trench effects along the railway. In most cases, analytical solutions are difficult to obtain mainly due to the complex geometry involved, which leads researchers to use alternative methods like the finite element method (FEM) or boundary element method.18,22,23 Bonnet24 performed a review of the mathematical basis of methods, which are more appropriate than the FEM for handling linear wave propagation, infinite domain, and mobile boundaries. Although these methods are powerful, long computations and infinite medium must be regularized in some way. Clouteau et al.15 carried out a review of the methods used in structure–environment interaction problems. The authors pointed out that the current methods and results of ground–structure interaction could be used for a wide range of interacting media. Nevertheless, the case of a flexible finite plate coupled to a multi-layered ground has never been addressed analytically in the literature according to our knowledge. It could be very useful to describe analytically the ground vibration caused by tramway tracks, where the track is composed of finite rectangular concrete plates. The acoustical radiation by a finite plate or cylindrical shell has already been described9,10,13,25–30 for cases as different as noise in industrial environments, musical acoustics, and underwater acoustical radiation. The notion of acoustic
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C 2015 Acoustical Society of America V
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modal radiation impedance governed by the fluid addedmass is a useful tool to describe the coupling of structure modes with the surrounding fluid medium. In the case of a light fluid, the fluid coupling is weak and added-mass can be neglected. In the case of a heavy fluid like water, it is no longer possible to neglect the fluids’ influence and the addedmass becomes a dominant effect.31 The aim of this paper is to present new insight on ground–structure interaction and emphasize the influence of the ground on the structural modal behavior, which is of prime importance, as will be demonstrated. It will be shown that ground coupling is extremely strong and we present a new set of cross-modal impedances describing the coupling of a finite plate to a multi-layered ground. The explanation of the phenomena encountered can be highlighted with the concept of the ground cross-modal radiation impedance of a mode plate.32 The frequency dependency of the ground cross-modal impedance exhibits new trends compared to the well-known problem in the acoustical case, linked to the type of wave generated in the ground, that is to say, the added-mass effect in the low frequency range is changed into an added-stiffness effect typical of a ground, which has never been found for a surrounding acoustic fluid. Numerical results reveal that the ground coupling is extremely strong and that the influence of the different layers on the vibratory behavior of the finite plate can be changed. A comparison will be presented between the theoretical presentation and experiment, showing good agreement for plate vibration in the frequency range [10;100] Hz.
FIG. 1. Finite vibrating panel coupled to ground.
•
Dp r4 wðx; yÞ qp hp x2 wðx; yÞ ¼ F0 dðx x0 Þdðy y0 Þ þ rp ðx; yÞ;
An overview of the problem under consideration (see Table I for notations) is presented in Fig. 1. A finite rectangular plate is excited by a harmonic driving force, which generates flexural vibration in the plate. The plate is continuously coupled under its surface to an infinite half-domain. For the sake of brevity, the stratification of the ground is presented in Appendix B.
nm
where anm and /nm ðx; yÞ are the modal amplitude and the modal shape for the mode (n,m), respectively. If the plate is a rectangular plate and the plate boundary conditions are simply supported, the modal shapes are given by np mp x sin y ; /nm ð x; yÞ ¼ sin a b
A. Ground and plate model
8ðx; yÞ 2 ½0; a ½0; b;
The ground and plate model are described separately for the sake of clarity.
Ep h3p 12ð1 p2 Þ Sp ¼ ab
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Ground Characteristics
Length Width Thickness Mass density Young’s modulus Poisson’s ratio Loss factor Flexural rigidity
q E cd gd cs gs k
Mass density Young’s modulus Poisson’s ratio Dilatation wave celerity Dilatational damping ratio Shear wave celerity Shear damping ratio Lame’s first parameter
Surface
l h
Lame’s second parameter Layer thickness
J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
(3)
with n 1 and m 1. If the plate boundary conditions are guided, the modal shapes are given by np mp x cos y ; /nm ð x; yÞ ¼ cos a b
TABLE I. Coefficients of the system.
a b hp qp Ep p gp Dp ¼
(1)
where Dp ¼ Dp ð1 þ jgp Þ is the complex flexural stiffness, F0 dðx x0 Þdðy y0 Þ is the harmonic driving normal force located in (x0,y0), rp ðx; yÞ is the stress along ~ z, which characterizes the coupling between the ground and the plate, and w(x,y) is the transverse deflection of the plate at point (x,y). The plate displacement w(x,y) can be expanded in a series of plate modes X anm /nm ðx; yÞ; (2) wðx; yÞ ¼
II. PROBLEM FORMULATION
Plate Characteristics
Plate model: The plate is modeled using the Kirchhoff hypothesis so the shear deformation and the rotary inertia can be neglected in the low frequency range. The wellknown equation of motion in the frequency domain reads as follows:33
8ðx; yÞ 2 ½0; a ½0; b;
•
(4)
with n 0 and m 0. The guided boundary conditions take into account the rigid body mode. Ground model: The ground is assumed to be a continuous, homogeneous, and isotropic medium so the elastodynamic linear equations of motion in terms of displacements in the frequency domain for a semi-infinite domain apply L. Grau and B. Laulagnet: The ground cross-modal impedance
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l$2 uðx;y;zÞ þ ðl þ kÞ$ð$uðx;y;zÞÞ þ qx2 uðx;y;zÞ ¼ 0; (5) where uT ðx; y; zÞ ¼ fux ðx; y; zÞ; uy ðx; y; zÞ; uz ðx; y; zÞg is the displacement vector of the ground and where k and l are the Lame coefficients. Appendix B summarizes the bases of the theory of stratified ground. On the top surface (z ¼ 0) the ground is free, meaning that the stress must vanish everywhere except in the region under the plate. However, because of the flexural motion considered here, the stress components in the directions x and y must also vanish under the plate. Finally, the stress boundary conditions on the top surface are 8 rxz ðx; y; z ¼ 0Þ ¼ 0 8ðx; yÞ 2 > > > > > < ryz ðx; y; z ¼ 0Þ ¼ 0 8ðx; yÞ 2 > rp ðx; yÞ 8ðx; yÞ 2 ½0; a ½0; b; > > r ðx; y; z ¼ 0Þ ¼ > > zz : 0 8ðx; yÞ 2 d2 / > > ~ ðkx ; ky ; zÞ ¼ 0; ðkx ; ky Þ 2 d2 w > > ~ ðkx ; ky ; zÞ ¼ 0; ðkx ; ky Þ 2 kmn , the ground contributes mass to the plate mode. At the frequency where kmn kd , a peak of added-mass is observed, and for kd kmn , the addedmass becomes negligible. It is noteworthy that the dominant wave is the shear wave, which separates two ground influence domains for each plate mode. The stiffness domain does not exist in acoustics. On the basis of what was said regarding uncoupling, i.e., kmn ¼ ks , the plate is not totally uncoupled from the ground L. Grau and B. Laulagnet: The ground cross-modal impedance
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mainly due to the ground added-damping shown in Fig. 6(a) for different modes. At this stage, it is important to point out that the structural modal damping, kmn, is extremely small compared to the modal ground added-damping, kgmn , so kmn could even be neglected. The real part of the ground crossmodal impedance represents the radiation impedance.11 From Fig. 6(a), as in acoustics, all the curves tend to unity after dilatational wavenumber coincidence, meaning that the power exchange between the plate and the ground is extremely high when kd knm . Below this dilatational wavenumber coincidence, and contrary to the acoustical case, i.e., kd knm , their values are still high ( 101 ), meaning that the modal coupling remains strong. C. Effect of ground coupling on plate vibration
It is well known that ground-structure coupling is very strong. By drawing a parallel with the equivalent acoustic problem, it is often assumed that in the case of a light fluid, like air, the coupling effect can be neglected. On the contrary, in the case of a heavy fluid, like water, this is not the case. Obviously such a trend is still true in the case of ground. The modal series must be truncated to solve the numerical computation. For the frequency range of interest, limited to 100 Hz, and for the excitation applied to the center, only modes (1,1), (1,3), and (3,3) need to be considered to ensure good convergence of the result. The ground characteristics are q ¼ 1200 kg m3 , cd ¼ 600 m s1 , cs ¼ 160 m s1 , and gs ¼ gd ¼ 0:02. Figure 7 shows the RMS surface averaged plate velocity without coupling media and coupled with the ground. The free, simply supported plate vibration (round markers) shows two structural resonance frequencies at 17 Hz and 84 Hz, which correspond to modes 11 and 13, respectively. Here, only the oddodd modes participate in the vibration because the driving force is located at the center of the plate. First, the coupling is very strong, leading to a decrease in vibration from 20 dB to 40 dB. It is even stronger than heavy fluid coupling. In comparison with a heavy fluid,31 we still observed a major difference with ground coupling.
FIG. 7. Influence of ground coupling on plate vibration depending on frequency. 2908
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Through the modal decomposition used to solve the equation of motion, it is possible to analyze the contribution of each modal amplitude independently. This will be discussed below to provide a better overview of the plate–structure interaction. Figures 8(a) and 8(b) show the contribution of the first two modes of the quadratic surface average plate velocity for a 1N excitation. This decomposition is interesting because it allows the identification of the resonance frequency from the peak amplitude, which moves due to the coupling to either a higher or lower frequency. Indeed, Figs. 8(a) and 8(b) represent the modal RMS surface average plate velocity of modes 11 and 13 without ground added-damping, i.e., Rmnmn ðxÞ ¼ 0. The objective of eliminating the addeddamping is to highlight only the effect of added-mass and stiffness. The modal amplitude must be compared with the uncoupled modal amplitude of the plate that will be used as a reference to understand whether it is an added-mass contribution or an added-stiffness contribution. First, Fig. 8(a) shows that the modal amplitude coupled to the ground shifts slightly to a higher frequency up to 20 Hz, as a result of the added-stiffness, identified in Fig. 6(b). Figure 8(b) shows an example of added-mass with a modal amplitude that moves to lower frequency. This shift is stronger than for mode 11 because the added-mass starts at 46 Hz [cf. Fig. 6(b)], which is far from the resonance frequency of mode 13 (85 Hz),
FIG. 8. (a) Modal amplitude of the mode 11 for a plate uncoupled and coupled to ground without ground damping (Rmnmn ðxÞ ¼ 0). (b) Modal amplitude of the mode 13 for a plate uncoupled and coupled to ground without ground damping (Rmnmn ðxÞ ¼ 0). L. Grau and B. Laulagnet: The ground cross-modal impedance
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whereas the added-stiffness of mode 11 stops at 20 Hz, close to the resonance frequency of the plate. These figures also show that the considerable added-mass and -stiffness contribute to reducing the modal amplitude by 10 dB. This contribution shifts the resonance frequency of the plate, more or less depending on the zero of Xmnmn, which is the point where the added-stiffness becomes the added-mass, but it also contributes to reducing the vibration of the plate. Figures 9(a) and 9(b) show the contribution of the added ground damping to the plate due to coupling, i.e., Xmnmn ðxÞ ¼ 0. The objective is to remove the contribution of the added-mass and -stiffness to the plate and to highlight the effect of the added-damping only. By taking the free plate vibration as a reference, the difference with the uncoupled modal amplitude is 20 dB and rises to 40 dB at the resonance frequency of the plate. With regard to damping, that of the plate and the ground is intrinsic, which attenuates the vibration. However, the considerable transfer of energy from the plate to the ground due to strong coupling gives rise to high ground added-damping. These results validate the conclusion based on the results of Fig. 7: the added ground damping is very significant and strengthens the ground/plate coupling. It can even be assumed that the reactive effects of the ground to plate vibration could be neglected compared to the damping effects! This is a major conclusion presented by Figs. 9(a) and 9(b) in which the
FIG. 9. (a) Modal amplitude of the mode 11 for a plate uncoupled and coupled to ground without ground added-mass and stiffness (Xmnmn ðxÞ ¼ 0). (b) Modal amplitude of the mode 13 for a plate uncoupled and coupled to ground without ground added-mass and stiffness (Xmnmn ðxÞ ¼ 0). J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
reactive effects are removed. It is now possible to provide a physical explanation when considering the plate vibration coupled with the ground. Because of the complexity of a medium such as the ground, the cross-modal impedance of the latter is an interesting tool for assessing its impact. IV. COMPARISON WITH MEASUREMENT
One of the aims of this modeling is to predict plate vibration in the case of ground plate coupling. That is why the comparison between theory and experiment was conducted on a tramway slab. In this section, we will compare an experiment with the numerical results obtained from simulations. This theory/experiment comparison is focused on plate vibration, something that has not yet been done. The plate under consideration is a concrete slab made for a tramway with the following characteristics: a ¼ 10 m, b ¼ 6 m, hp ¼ 0:6 m, Ep ¼ 2:5 1010 Pa, p ¼ 0:3, qp ¼ 2500 kg m3 , and gp ¼ 2%. Figure 11 shows a picture of the slab under consideration. This picture shows the four different locations of the accelerometers shown by numbered circles. The circles, called impact, correspond to the impact on the slab. Accelerometers 1 and 4 are located in the 6 m wide slab and accelerometers 2 and 3 are located in the 10 m long slab. Simply supported boundary conditions do not apply for such a plate: the modal shape may be changed from a sinus basis to a cosine basis. This represents the guided boundary conditions that take into account the rigid body mode. In Sec. III, the simply supported case was studied as the problem was better regularized. However, the previous interpretation still holds in the guided case. The calculation of the ground cross-modal impedance must be done by taking this change into account. The aim of this experimental comparison is to verify if the main phenomena described in the previous section are realistic, especially if addedstiffness or -mass and strong added-damping can be demonstrated. The tramway rails can be neglected in the model in the first approximation In the experiment, four “piezoelectric” accelerometers (sensitivity: 1 V/g) were attached to the slab to measure the vibrations: two 0.3 m from the edge and spaced 3 m apart, and two 3 m from the edge and also spaced 3 m apart. A transient hammer excitation force (Kistler Instrument Corporation, Les Ulis, France, sensitivity: 0.000 222 V/g) was used to generated a force 1.5 m from the edge of the slab and 5 m from its end. The post-treatement software from Acoem company (Limonest, France) was used. Ten impacts were generated between the tramway rail and the plate acceleration, taking into account coherence >90% in the frequency range of interest. Regarding the ground characteristics, the SASW method5,34 was applied in order to find out the wave celerities and the thickness of the layers. Ground damping is a difficult parameter to estimate, so a typical value for sand was considered. The tramway slab was coupled with a two layer ground with the following characteristics: q1 ¼ 1500 kg m3 , cs1 ¼ 200 m s1 , qcd1 ¼ 500 m s1 , h ¼ 5 m, and gs1 ¼ gd1 ¼ 0:02, q2 ¼ 3000 kgm3 , cs2 ¼ 3000 ms1 , cd2 ¼ 7000 ms1 , and gs2 ¼ gd2 ¼ 0:02. The case of a double sand layer is studied L. Grau and B. Laulagnet: The ground cross-modal impedance
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here. Subscript 1 refers to the first layer at the top surface, while subscript 2 refers to layer 2 inside the ground. The tramway slab was located at the “Decines Grand Stade” site at the terminus of the Lyon T3 tram line. Figure 10 presents the transfer function between the injected force and the acceleration on the slab. The round dots represent the measured transfer function on the plate and the crossed dots represent the simulated transfer function. A good match is shown between the theory and experiment up to 200 Hz in trend. An average difference of 3 dB can be noticed, which is low compared to the considerable simplification performed on the plate model. For the sake of brevity, the theory/experiment comparisons are not presented for the other accelerometers. However, the comparison also matches well, following the same trend as that presented in Fig. 10. This theory/experiment comparison emphasizes the result of the simulation presented above: strong attenuation of the plate vibration coupled to the ground, and more especially the disappearance of the resonance peak, even for a slab of 60 cm thick. V. GROUND CROSS-MODAL IMPEDANCE IN THE CASE OF A STRATIFIED GROUND
The aim of this section is to investigate the influence of a stratified ground on the frequency dependents of ground cross-modal impedances. The formulation of the stratified ground model is presented in Appendix B. The features of the ground composed of two layers are discussed in this paragraph. The plate characteristics remain unchanged, with dimensions a ¼ b ¼ 6 m and hp ¼ 0:2 m. A few cases are considered in order to show the modification of the ground crossmodal impedance due to ground stratification. The plate boundary conditions are simply supported. The case of a double sand layer is studied here. Subscript 1 refers to the first layer at the top surface and subscript 2 refers to layer 2 inside the ground, the first layer being light sand (q1 ¼ 1200 kg m3 ; cd1 ¼ 600 m s1 ; cs1 ¼ 160 m s1 ; gd1 ¼ gs1 ¼ 0:02; h ¼ 1 m) and the second layer heavy sand (q2 ¼ 1800 kgm3 ; cd2 ¼ 1200 ms1 ; cs2 ¼ 600 ms1 ; gd2 ¼ gs2
FIG. 10. Comparison of the experimental and simulated transfer function between the injected force and the plate acceleration on the site “Decines Grand Stade.” 2910
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FIG. 11. (Color online) Picture of the excited plate coupled to the ground with four accelerometers on the plate.
¼ 0:02). In Fig. 12, the RMS plate velocity exhibits a strong attenuation at low frequency and a resonance frequency at 78Hz. The plate velocity peak at this frequency is 10 dB compared with the case h ¼1. This makes analysis interesting with the corresponding ground cross-modal impedance in Figs. 13 and 14. At low frequency [10;40] Hz, the RMS plate velocity coupled to a stratified ground is strongly attenuated compared to the non-stratified one. This is due to the high ground added-stiffness due to stratification, shown in Fig. 13. At low frequency, i.e., [10;40] Hz, plate vibration is mainly due to mode 11. Figure 13 shows that the ground added-stiffness of this mode is higher for the stratified ground. Namely, at 5 Hz, the imaginary part changes from 35.75 to 244, meaning that the ground added-stiffness effect is stronger and contributes to the decrease of the plate vibration level observed. At higher frequency, i.e., [60;100] Hz, vibration was seen to increase at 80 Hz. This “resonance” peak can be explained by the ground stratification and especially by the real part of the ground cross-modal impedance responsible for the energy transfer. Plate vibration was mainly due to mode (1,3) in this frequency range. The real part of the ground cross-modal impedance for mode (1,3) is presented in Fig. 14. Due to the ground stratification, part of the energy was reflected from the ground to the plate, leading to a
FIG. 12. Comparison of the quadratic surface average plate velocity in double ground layer and one ground layer. L. Grau and B. Laulagnet: The ground cross-modal impedance
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In brief, the results show that the ground can change the vibration of a concrete plate of 6 m 6 m 0.2 m for a 1 m ground thickness. Although the added-damping flattens the plate velocity, it was still possible to identify the resonance frequency, especially when the ground resonance frequency matched the structural resonance. VI. CONCLUSION
FIG. 13. (Color online) x imaginary part of the normalized ground crossmodal impedance of the plate with a double ground layer with a first layer thickness of 1 m.
minimized energy transfer. Consequently, the real part of the ground cross-modal impedance was lower in the stratified ground than in the non-stratified one, typically 0.06 instead of 0.34 at 81 Hz. It should be remembered that for this layer thickness, coincidence occurred between the structural resonance frequency of mode 13 and the ground resonance frequency equal to fground ¼ cs1 =2h, where h represents the thickness of the first layer, and cs1 is the shear celerity of the first layer. In this case, the coincidence frequency was 80 Hz, which corresponds to the resonance peak observed in the plate vibration. Moreover, a specific trend appeared on the imaginary part of the corresponding impedance. The ground crossmodal impedance changed sign several times, contrary to that of a single ground layer. For a double ground layer like that presented in Fig. 13, the ground cross-modal impedance had two ranges of negative values, i.e., [5;40] Hz and [82;170] Hz, indicating added-stiffness and two positive value ranges, i.e., [40;82] Hz and [170;200] Hz, indicating added-mass. The extension to three layers would have produced several negative and positive value ranges, and so on.
A model for the ground/structure interaction of a simply supported and guided rectangular plate coupled to a semiinfinite ground was presented. The considerable importance of coupling, capable of contributing an attenuation of up to 40 dB at the resonance frequency of the structure, was shown. We also developed a new analysis tool, ground cross-modal impedance, which provided a novel means of analyzing ground–structure interaction. It was shown that ground/structure coupling can be expressed in terms of added-stiffness and mass. They can both be very high and, moreover, frequency dependent. At a specific frequency, the added-stiffness can reach 100 times that of the plate and the added-mass twice that of the physical plate. Also, the imaginary part of the impedance exhibited negative values, which do not exist in equivalent structure-fluid problems, due to the shear waves in the ground responsible for modal added-stiffness below the shear wavenumber coincidence. With regard to the ground cross-modal impedance, similarities can be pointed out with the equivalent acoustical cross-modal impedance. Several experiments were carried out and presented good matching with the model results. Finally, the effect of a stratified ground on plate vibration led to resonance due to reflection in the ground layer. This behavior of this reflection affects the ground cross-modal impedance. ACKNOWLEDGMENTS
The authors would like to thank the company ACOUPHEN (Pusignan, France) for its support and its experimental assistance, which helped to provide a better overview of the work. We would also like to thank the company SYSTRA (Paris, France) for valuable discussions. APPENDIX A: RELATION BETWEEN STRESS AND DISPLACEMENT FOR ISOTROPIC MEDIA
The aim of this section is to present the relation between the ground displacement and the ground stress in the Fourier domain. The ground displacement can be decomposed into a scalar potential and a vector potential, that is to say, a Helmholtz decomposition, which gives for each component
FIG. 14. (Color online) Real part of the normalized ground cross-modal impedance of the plate with a double ground layer with a first layer thickness of 1 m. J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
8 @/ð x; y; zÞ @wz ð x; y; zÞ @wy ð x; y; zÞ > > > þ ; ux ð x; y; zÞ ¼ > > @x @y @z > > > < @/ð x; y; zÞ @wx ðx; y; zÞ @wz ð x; y; zÞ þ ; uy ð x; y; zÞ ¼ > @y @z @x > > > > @/ð x; y; zÞ @wy ð x; y; zÞ @wx ð x; y; zÞ > > > þ : : uz ðx; y; zÞ ¼ @z @y @x (A1) L. Grau and B. Laulagnet: The ground cross-modal impedance
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Considering the 2D Fourier transform for the ground displacement in directions x and y, we obtain 8 ~ > > > ~ ðkx ; ky ; zÞ þ jky w ~ ðk x ; k y ; zÞ d w y ðk x ; k y ; zÞ ; > u~x ðkx ; ky ; zÞ ¼ jkx / > z > dz > > < ~ ðk x ; k y ; zÞ d w ~ ðkx ; ky ; zÞ jkx w ~ ðk x ; k y ; zÞ þ x ; u~y ðkx ; ky ; zÞ ¼ jky / z > > dx > > > ~ ð k x ; k y ; zÞ > d/ > > ~ ðkx ; ky ; zÞ jky w ~ ðk x ; k y ; zÞ: : u~z ðkx ; ky ; zÞ ¼ þ jkx w y x dz
The relation between the stress tensor and the strain tensor is given by Hook’s law for an isotropic media through rij ¼
lE E kk dij þ ij : 1þl ð1 þ lÞð1 2lÞ
(A3)
Assuming that the strain is infinitesimal, meaning kuk 1 and kruk 1, the displacement expression depending on the stress is rij ð x; y; zÞ ¼
lE ðux;x þ uy;y þ uz;z Þdij ð1 þ lÞð1 2lÞ E 1 þ ðui;j þ uj;i Þ: 1 þ l2
(A4)
(A2)
The shear and dilatation velocities in the nth ground layer are linked to Lames coefficient 8 > > > c2dn ¼ kn þ 2ln ; < qn (B4) l > n 2 >c ¼ : > : sn q n Ground damping can also occur through dilatational and shear waves by defining a complex wave as follows, assuming gn is small compared to unity: cdn ¼ cdn ð1 þ jgdn Þ; (B5) csn ¼ csn ð1 þ jgsn Þ:
In the Fourier domain, we obtain 8 > @ u~x > > >r ~ xz ðkx ; ky ; zÞ ¼ l þ jkx uz : > > @z > > < @ u~y ~ yz ðkx ; ky ; zÞ ¼ l r þ jky uz : > @z > > > > > @ u~z > > ~ zz ðkx ; ky ; zÞ ¼ jkx k~ : u x þ kky u~y þ ðk þ lÞ :r @z
APPENDIX C: POTENTIAL’S COEFFICIENT FOR THE SINGLE LAYER GROUND
(A5)
APPENDIX B: THEORY OF STRATIFIED GROUND
An analytical solution to the equation of motion can be found using the Helmholtz decomposition. The theorem states that every smooth vector field, un , can be decomposed into a rotational part (rot) and an irrotational part (grad), meaning ln $2 un ðx; y; zÞ þ ðln þ kn Þ$ð$un ðx; y; zÞÞ
In this section, the analytical expression in the Fourier domain between displacement and stress in Eq. (16) is justified. In the case of a single ground layer, the solution of Eq. (B3) is as follows after performing a 2D Fourier transform: 8 ~ x ; ky ; zÞ ¼ Aejk1 z ; /ðk > > > > > > ~ ðkx ; ky ; zÞ ¼ Bejk2 z ; ~ ðkx ; ky ; zÞ ¼ Cejk2 z ; > w > y > > > :~ w z ðkx ; ky ; zÞ ¼ Dejk2 z ; where k12 ¼ kd2 kx2 ky2 and k22 ¼ ks2 kx2 ky2 . Using Eqs. (A2), (A5), and the gauge condition with the boundary conditions (6), this leads to
þ qn x2 un ðx; y; zÞ ¼ 0;
(B1)
0
0
un ¼ gradð/n Þ þ rotðwn Þ:
(B2)
B B B @
2kx k1 2ky k1
By introducing Eq. (B2) in Eq. (B1), we obtain four uncoupled equations from the unknown functions /n and wn , ( 2 /n ðx; y; zÞ ¼ 0; r2 /n ðx; y; zÞ þ kdn (B3) 2 2 r wn ðx; y; zÞ þ ksn nn ðx; y; zÞ ¼ 0;
ðkkd2 2lk12 Þ 0 1 0 B 0 C B C ¼ B C: @ 0 A
10 1 A 2 2 C B kx ky ðk2 kx Þ ky k2 CB B C C CB C ðk22 ky2 Þ kx ky kx k2 A@ C A kx
ky
k2
2lky k2
2lkx k2
0
D
(C2)
rzz where kdn ¼ x=cdn and ksn ¼ x=csn denote the dilatational and shear wavenumbers, respectively. 2912
J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
Inverting this matrix leads to L. Grau and B. Laulagnet: The ground cross-modal impedance
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8 > k22 þ kx2 þ ky2 > > > ~ zz ; A ¼ r > > gðkx ; ky Þ > > > > > 2k1 ky < ~ zz ; B¼ r gðkx ; ky Þ > > > 2k1 kx > > > ~ zz ; C¼ r > > gðkx ; ky Þ > > > : D ¼ 0;
(C3)
where gðkx ; ky Þ was defined in Eq. (16). It is now possible to obtain an expression of u~z ðkx ; ky ; 0Þ using Eq. (A2), u~z ðkx ; ky ; 0Þ ¼
jk1 ks2 ~ zz ðkx ; ky ; 0Þ: r gðkx ; ky Þ
(C4)
APPENDIX D: CONVERGENCE STUDY OF GROUND CROSS-MODAL IMPEDANCE
In this section, we discuss the criteria used to ensure good convergence of the double integral below: ð ð þ1 jk1 ks2 ~ ~ / / dkx dky ; (D1) cpqmn ¼ 4 gðkx ; ky Þ mn pq 0
jk1 ks2 =gðkx ; ky Þ term behaves at infinity like Oððkx2 þ ky2 Þ3=2 Þ. 4 4 ~ ~ / Regarding / mn pq , the behavior at infinity is Oðkx ky Þ for the simply supported plate and Oðkx2 ky2 Þ for the guided plate. Finally, the behavior of the function to integrate is Oððkx2 þ ky2 Þ3=2 kx4 ky4 Þ for the simply supported plate and Oððkx2 þ ky2 Þ3=2 kx2 ky2 Þ for the guided plate, which renders the integral convergent. Regarding the singularity at the de~ , such as kx ¼ np=a, we refer nominator of the function / nm to the work of Laulagnet,31 which gives an asymptotic solution near the singularities. In the numerical results we present, the limit for the domain of integration is truncated at a finite value, such as kmax ¼ 100=f if kmax < 5jks j and kmax ¼ 5jks j if kmax > 5jks j. In this paper, the discretization of the domain is based on a wavelength criterion. The criterion is fixed at 24 points per wavelength, i.e., Dkx ¼ ð2p=aÞð1=24Þ and Dky ¼ ð2p=bÞð1=24Þ. This criterion must be more accurate for a stratified ground and leads to 48 points per wavelength to ensure good convergence of the integral. It can be seen that the integrand is odd with respect to kx if the orders (n,p) versus the x direction have different parities, and that the integrand is odd with respect to ky for the same reasons versus the mode orders (q,m). It can be seen that the integrand is even in the remaining cases of parity. This leads to the following properties:
8 ð ð þ1 > jk1 ks2 ~ ~ > < cnmpq ¼ 4 / / dkx dky ; when p and n have the same parity; and q and m have the same parity; gðkx ; ky Þ mn pq 0 > > :c ¼ 0; when p and n have different parities; and q and m have different parities:
(D2)
nmpq
1
A. Karlstrom and A. Bostrom, “An analytical model for train-induced ground vibrations from railways,” J. Sound Vib. 292, 221–241 (2006). 2 X. Sheng, C. J. C. Jones, and M. Petyt, “Ground vibrations generated by load moving along a railway track,” J. Sound Vib. 228, 129–156 (1999). 3 A. Karlstrom, “An analytical model for ground vibrations from accelerating trains,” J. Sound Vib. 293, 587–598 (2006). 4 A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, Oxford, UK, 1944), pp. 1–662. 5 G. Kourousis, “Modelisation des effets vibratoires du traffic frroviaire sur l’environnement” (“Modeling of railway traffic effect on the environment”), Ph.D. thesis, Faculte polytechnique de Mons, 2009. 6 M. Maldonado, “Vibrations dues au passage d’un tramway: Mesures exprimentales et simulations numriques” (“Vibration induced by tramway: Experimental measurements and simulations”), Ph.D. thesis, Ecole Centrale de Nantes, 2008. 7 G. Lefeuve-Mesgouez, D. Le Houedec, and A. T. Peplow, “Ground vibrations in the vicinity of a high-speed moving harmonic strip load,” J. Sound Vib. 231, 1289–1309 (2000). 8 C. Lesueur, Rayonnement Acoustique des Structures: Vibroacoustique, Interactions Fluide-Structure (Acoustical Radiation of Structure: Vibroacoustics, Fluid-Structure Interaction) (Eyrolles, Villeurbanne, France, 1988), pp. 1–591. 9 R. Pico, F. Gautier, and J. Redondo, “Acoustic input impedance of a vibrating cylindrical tube,” J. Sound Vib. 301, 649–664 (2007). 10 C. E. Wallace, “Radiation resistance of a baffled beam,” J. Acoust. Soc. Am. 51, 936–945 (1972). J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
11
E. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography (Academic, Washington, DC, 1999), pp. 1–306. 12 D. Thompson, Railway Noise and Vibration: Mechanisms, Modelling and Means of Control (Elsevier Science, Southampton, UK, 2008), pp. 1–536. 13 G. Maidanik, “Response of ribbed panels to reverberant acoustic fields,” J. Acoust. Soc. Am. 34, 809–826 (1962). 14 D. V. Jones, D. Le Houedec, and M. Petyt, “Ground vibrations due to a rectangular harmonic load,” J. Sound Vib. 212, 61–74 (1998). 15 D. Clouteau, R. Cottereau, and G. Lombaert, “Dynamics of structures coupled with elastic media—A review of numerical models and methods,” J. Sound Vib. 332, 2415–2436 (2013). 16 L. Auersch, “Response to harmonic wave excitation of finite or infinite elastic plates on a homogeneous or layered half-space,” Comput. Geotech. 51, 50–59 (2013). 17 B. B. Guzina, S. N. Fata, and M. Bonnet, “On the stress-wave imaging of cavities in a semi-infinite solid,” Computers and geotechnics 40, 1505–1523 (2003). 18 H. R. Yerli, S. Kacin, and S. Kocak, “A parallel finite-infinite element model for a two-dimensional soil-structure interaction problems,” Soil Dyn. Earthquake Eng. 23, 249–253 (2003). 19 P. Roussillon, “Interaction sol-structure et site-ville: Aspects fondamentaux et modlisation” (“Ground structure interaction: Theoretical aspect and modelling”), Ph.D. thesis, Vaulx en Velin CNRS, 2006. 20 X. Sheng, C. J. C. Jones, and D. J. Thompson, “A theoretical study on the influence of the track on train-induced ground vibration,” J. Sound Vib. 272, 909–936 (2004). L. Grau and B. Laulagnet: The ground cross-modal impedance
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21
A. Karlstrom and A. Bostrom, “Efficiency of trenches along railways for trains moving at sub- or supersonic speeds,” Soil Dyn. Earthquake Eng. 27, 625–641 (2007). 22 H. Chebli, D. Clouteau, and L. Schmitt, “Dynamic response of high-speed ballasted railway tracks: 3D periodic model and in situ measurements,” Soil Dyn. Earthquake Eng. 28, 118–131 (2008). 23 S. Guenfoud, S. V. Bosakov, and D. F. Laefer, “A Ritz’s method based solution for the contact problem of a deformable rectangular plate on an elastic quarter-space,” Int. J. Solids Struct. 47, 1822–1829 (2010). 24 M. Bonnet, Boundary Integral Equation Methods for Solids and Fluids (Wiley, Paris, France, 1999), pp. 1–412. 25 C. E. Wallace, “Radiation resistance of a rectangular panel,” J. Acoust. Soc. Am. 51, 946–952 (1972). 26 C. B. Burroughs, “Acoustic radiation from fluid-loaded infinite circular cylinders with doubly periodic ring supports,” J. Acoust. Soc. Am. 75, 715–722 (1984). 27 B. Laulagnet and J. L. Guyader, “Sound radiation from finite cylindrical shells, partially covered with longitudinal strips of compliant layer,” J. Acoust. Soc. Am. 186, 723–742 (1995).
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B. Laulagnet and J. L. Guyader, “Sound radiation from finite cylindrical coated shells, by means of asymptotic expansion of three-dimensional equations for coating,” J. Acoust. Soc. Am. 96, 277–286 (1994). 29 L. Cheng and M. Hatam, “Further assessment of the coupling load decomposition technique for the vibration analyses of planar coupled structures,” Appl. Acoust. 62, 411428 (2001). 30 L. Cheng and J. Nicolas, “Radiation of sound into a cylindrical enclosure from a point-driven end plate with general boundary conditions,” J. Acoust. Soc. Am. 91, 1504–1513 (1992). 31 B. Laulagnet, “Sound radiation by a simply supported unbaffled plate,” J. Acoust. Soc. Am. 103, 2451–2462 (1998). 32 B. Laulagnet, “Soil effect on the finite simply supported plate vibration,” in Internoise, Proceedings of the American Society of Mechanical Engineering/Noise Control and Acoustics Division Meeting (2013). 33 J. L. Guyader, Vibrations des Milieux Continus (Vibration in Continuous Media) (Hermes Science Publications, Villeurbanne, France, 2002), pp. 1–445. 34 S. Foti, “Multistation methods for geotechnical characterization using surface waves,” Ph.D. thesis, Politecnico di Torino, 2000.
L. Grau and B. Laulagnet: The ground cross-modal impedance
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(Received 10 July 2014; revised 19 January 2015; accepted 9 April 2015) An analytical approach is investigated to model ground–plate interaction based on modal decomposition and the two-dimensional Fourier transform. A finite rectangular plate subjected to flexural vibration is coupled with the ground and modeled with the Kirchhoff hypothesis. A Navier equation represents the stratified ground, assumed infinite in the x- and y-directions and free at the top surface. To obtain an analytical solution, modal decomposition is applied to the structure and a Fourier Transform is applied to the ground. The result is a new tool for analyzing ground–plate interaction to resolve this problem: ground cross-modal impedance. It allows quantifying the added-stiffness, added-mass, and added-damping from the ground to the structure. Similarity with the parallel acoustic problem is highlighted. A comparison between the theory and the experiment shows good matching. Finally, specific cases are investigated, notably the influence of layer depth on plate C 2015 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4919332] vibration. V [JDM]
Pages: 2901–2914
I. INTRODUCTION
Ground–structure interaction is a significant problem in rail transportation (trains, subways, tramways). Populations living near these infrastructures can be sensitive to the vibration induced by rail vehicles. This has led to the increasing development of tools for analyzing the ground vibration of rail lines. In these cases, a structure made of concrete is often excited directly by a source coupled to the ground via an intermediate layer in contact with the concrete structure. This is the reason why the whole elastic medium is initially considered as a stack of layers when attempting to predict ground vibration generated by the vibration of an external structure. The issue of vibration in a multi-layered ground has been addressed in the past.1–3 It is commonly accepted that the ground vibration problem can be represented by a Navier equation with reasonable accuracy. Consequently, ground vibrations are considered as a linear problem. For a semi-infinite medium, classical solutions based on the potential decomposition of the displacement field, leading to equivalent Helmholtz equations are widely used in research.1,4–7 The Helmholtz decomposition leads to two types of waves that can propagate in an infinite homogeneous medium subjected to general cases of excitation. The dilatational wave and the shear wave described by the potentials are very similar to the corresponding acoustical problem, where the ground is substituted for a fluid. One of the aims of this paper is to exhibit similarities between structural ground coupling and structural acoustic coupling.8–13 The structure interacting with the ground is often made of concrete. Depending on the geometric complexity of the structure considered, the solution of the ground–structure
a)
Also at: ACOUPHEN, 33 Route de Jonage, 69830 Pusignan, France. Electronic mail: [email protected]
J. Acoust. Soc. Am. 137 (5), May 2015
interaction is found using analytical3,13,14 or numerical15–19 solutions. A typical example of the use of an analytical solution concerns the vibration of an infinite beam lying on a multi-layered ground. Here, a solution is obtained by a spatial Fourier transform for both the beam and the ground.1,14,20 An analytical solution was formulated by Karlstrom et al.1 for an infinite beam-plate–ground interaction. Using the Fourier transform to simplify the Navier equation and then considering the infinite beam and plate geometry, the solution is developed in the Fourier domain. On the same basis, the same authors developed an extension of the problem21 by considering trench effects along the railway. In most cases, analytical solutions are difficult to obtain mainly due to the complex geometry involved, which leads researchers to use alternative methods like the finite element method (FEM) or boundary element method.18,22,23 Bonnet24 performed a review of the mathematical basis of methods, which are more appropriate than the FEM for handling linear wave propagation, infinite domain, and mobile boundaries. Although these methods are powerful, long computations and infinite medium must be regularized in some way. Clouteau et al.15 carried out a review of the methods used in structure–environment interaction problems. The authors pointed out that the current methods and results of ground–structure interaction could be used for a wide range of interacting media. Nevertheless, the case of a flexible finite plate coupled to a multi-layered ground has never been addressed analytically in the literature according to our knowledge. It could be very useful to describe analytically the ground vibration caused by tramway tracks, where the track is composed of finite rectangular concrete plates. The acoustical radiation by a finite plate or cylindrical shell has already been described9,10,13,25–30 for cases as different as noise in industrial environments, musical acoustics, and underwater acoustical radiation. The notion of acoustic
0001-4966/2015/137(5)/2901/14/$30.00
C 2015 Acoustical Society of America V
2901
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modal radiation impedance governed by the fluid addedmass is a useful tool to describe the coupling of structure modes with the surrounding fluid medium. In the case of a light fluid, the fluid coupling is weak and added-mass can be neglected. In the case of a heavy fluid like water, it is no longer possible to neglect the fluids’ influence and the addedmass becomes a dominant effect.31 The aim of this paper is to present new insight on ground–structure interaction and emphasize the influence of the ground on the structural modal behavior, which is of prime importance, as will be demonstrated. It will be shown that ground coupling is extremely strong and we present a new set of cross-modal impedances describing the coupling of a finite plate to a multi-layered ground. The explanation of the phenomena encountered can be highlighted with the concept of the ground cross-modal radiation impedance of a mode plate.32 The frequency dependency of the ground cross-modal impedance exhibits new trends compared to the well-known problem in the acoustical case, linked to the type of wave generated in the ground, that is to say, the added-mass effect in the low frequency range is changed into an added-stiffness effect typical of a ground, which has never been found for a surrounding acoustic fluid. Numerical results reveal that the ground coupling is extremely strong and that the influence of the different layers on the vibratory behavior of the finite plate can be changed. A comparison will be presented between the theoretical presentation and experiment, showing good agreement for plate vibration in the frequency range [10;100] Hz.
FIG. 1. Finite vibrating panel coupled to ground.
•
Dp r4 wðx; yÞ qp hp x2 wðx; yÞ ¼ F0 dðx x0 Þdðy y0 Þ þ rp ðx; yÞ;
An overview of the problem under consideration (see Table I for notations) is presented in Fig. 1. A finite rectangular plate is excited by a harmonic driving force, which generates flexural vibration in the plate. The plate is continuously coupled under its surface to an infinite half-domain. For the sake of brevity, the stratification of the ground is presented in Appendix B.
nm
where anm and /nm ðx; yÞ are the modal amplitude and the modal shape for the mode (n,m), respectively. If the plate is a rectangular plate and the plate boundary conditions are simply supported, the modal shapes are given by np mp x sin y ; /nm ð x; yÞ ¼ sin a b
A. Ground and plate model
8ðx; yÞ 2 ½0; a ½0; b;
The ground and plate model are described separately for the sake of clarity.
Ep h3p 12ð1 p2 Þ Sp ¼ ab
2902
Ground Characteristics
Length Width Thickness Mass density Young’s modulus Poisson’s ratio Loss factor Flexural rigidity
q E cd gd cs gs k
Mass density Young’s modulus Poisson’s ratio Dilatation wave celerity Dilatational damping ratio Shear wave celerity Shear damping ratio Lame’s first parameter
Surface
l h
Lame’s second parameter Layer thickness
J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
(3)
with n 1 and m 1. If the plate boundary conditions are guided, the modal shapes are given by np mp x cos y ; /nm ð x; yÞ ¼ cos a b
TABLE I. Coefficients of the system.
a b hp qp Ep p gp Dp ¼
(1)
where Dp ¼ Dp ð1 þ jgp Þ is the complex flexural stiffness, F0 dðx x0 Þdðy y0 Þ is the harmonic driving normal force located in (x0,y0), rp ðx; yÞ is the stress along ~ z, which characterizes the coupling between the ground and the plate, and w(x,y) is the transverse deflection of the plate at point (x,y). The plate displacement w(x,y) can be expanded in a series of plate modes X anm /nm ðx; yÞ; (2) wðx; yÞ ¼
II. PROBLEM FORMULATION
Plate Characteristics
Plate model: The plate is modeled using the Kirchhoff hypothesis so the shear deformation and the rotary inertia can be neglected in the low frequency range. The wellknown equation of motion in the frequency domain reads as follows:33
8ðx; yÞ 2 ½0; a ½0; b;
•
(4)
with n 0 and m 0. The guided boundary conditions take into account the rigid body mode. Ground model: The ground is assumed to be a continuous, homogeneous, and isotropic medium so the elastodynamic linear equations of motion in terms of displacements in the frequency domain for a semi-infinite domain apply L. Grau and B. Laulagnet: The ground cross-modal impedance
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l$2 uðx;y;zÞ þ ðl þ kÞ$ð$uðx;y;zÞÞ þ qx2 uðx;y;zÞ ¼ 0; (5) where uT ðx; y; zÞ ¼ fux ðx; y; zÞ; uy ðx; y; zÞ; uz ðx; y; zÞg is the displacement vector of the ground and where k and l are the Lame coefficients. Appendix B summarizes the bases of the theory of stratified ground. On the top surface (z ¼ 0) the ground is free, meaning that the stress must vanish everywhere except in the region under the plate. However, because of the flexural motion considered here, the stress components in the directions x and y must also vanish under the plate. Finally, the stress boundary conditions on the top surface are 8 rxz ðx; y; z ¼ 0Þ ¼ 0 8ðx; yÞ 2 > > > > > < ryz ðx; y; z ¼ 0Þ ¼ 0 8ðx; yÞ 2 > rp ðx; yÞ 8ðx; yÞ 2 ½0; a ½0; b; > > r ðx; y; z ¼ 0Þ ¼ > > zz : 0 8ðx; yÞ 2 d2 / > > ~ ðkx ; ky ; zÞ ¼ 0; ðkx ; ky Þ 2 d2 w > > ~ ðkx ; ky ; zÞ ¼ 0; ðkx ; ky Þ 2 kmn , the ground contributes mass to the plate mode. At the frequency where kmn kd , a peak of added-mass is observed, and for kd kmn , the addedmass becomes negligible. It is noteworthy that the dominant wave is the shear wave, which separates two ground influence domains for each plate mode. The stiffness domain does not exist in acoustics. On the basis of what was said regarding uncoupling, i.e., kmn ¼ ks , the plate is not totally uncoupled from the ground L. Grau and B. Laulagnet: The ground cross-modal impedance
2907
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mainly due to the ground added-damping shown in Fig. 6(a) for different modes. At this stage, it is important to point out that the structural modal damping, kmn, is extremely small compared to the modal ground added-damping, kgmn , so kmn could even be neglected. The real part of the ground crossmodal impedance represents the radiation impedance.11 From Fig. 6(a), as in acoustics, all the curves tend to unity after dilatational wavenumber coincidence, meaning that the power exchange between the plate and the ground is extremely high when kd knm . Below this dilatational wavenumber coincidence, and contrary to the acoustical case, i.e., kd knm , their values are still high ( 101 ), meaning that the modal coupling remains strong. C. Effect of ground coupling on plate vibration
It is well known that ground-structure coupling is very strong. By drawing a parallel with the equivalent acoustic problem, it is often assumed that in the case of a light fluid, like air, the coupling effect can be neglected. On the contrary, in the case of a heavy fluid, like water, this is not the case. Obviously such a trend is still true in the case of ground. The modal series must be truncated to solve the numerical computation. For the frequency range of interest, limited to 100 Hz, and for the excitation applied to the center, only modes (1,1), (1,3), and (3,3) need to be considered to ensure good convergence of the result. The ground characteristics are q ¼ 1200 kg m3 , cd ¼ 600 m s1 , cs ¼ 160 m s1 , and gs ¼ gd ¼ 0:02. Figure 7 shows the RMS surface averaged plate velocity without coupling media and coupled with the ground. The free, simply supported plate vibration (round markers) shows two structural resonance frequencies at 17 Hz and 84 Hz, which correspond to modes 11 and 13, respectively. Here, only the oddodd modes participate in the vibration because the driving force is located at the center of the plate. First, the coupling is very strong, leading to a decrease in vibration from 20 dB to 40 dB. It is even stronger than heavy fluid coupling. In comparison with a heavy fluid,31 we still observed a major difference with ground coupling.
FIG. 7. Influence of ground coupling on plate vibration depending on frequency. 2908
J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
Through the modal decomposition used to solve the equation of motion, it is possible to analyze the contribution of each modal amplitude independently. This will be discussed below to provide a better overview of the plate–structure interaction. Figures 8(a) and 8(b) show the contribution of the first two modes of the quadratic surface average plate velocity for a 1N excitation. This decomposition is interesting because it allows the identification of the resonance frequency from the peak amplitude, which moves due to the coupling to either a higher or lower frequency. Indeed, Figs. 8(a) and 8(b) represent the modal RMS surface average plate velocity of modes 11 and 13 without ground added-damping, i.e., Rmnmn ðxÞ ¼ 0. The objective of eliminating the addeddamping is to highlight only the effect of added-mass and stiffness. The modal amplitude must be compared with the uncoupled modal amplitude of the plate that will be used as a reference to understand whether it is an added-mass contribution or an added-stiffness contribution. First, Fig. 8(a) shows that the modal amplitude coupled to the ground shifts slightly to a higher frequency up to 20 Hz, as a result of the added-stiffness, identified in Fig. 6(b). Figure 8(b) shows an example of added-mass with a modal amplitude that moves to lower frequency. This shift is stronger than for mode 11 because the added-mass starts at 46 Hz [cf. Fig. 6(b)], which is far from the resonance frequency of mode 13 (85 Hz),
FIG. 8. (a) Modal amplitude of the mode 11 for a plate uncoupled and coupled to ground without ground damping (Rmnmn ðxÞ ¼ 0). (b) Modal amplitude of the mode 13 for a plate uncoupled and coupled to ground without ground damping (Rmnmn ðxÞ ¼ 0). L. Grau and B. Laulagnet: The ground cross-modal impedance
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whereas the added-stiffness of mode 11 stops at 20 Hz, close to the resonance frequency of the plate. These figures also show that the considerable added-mass and -stiffness contribute to reducing the modal amplitude by 10 dB. This contribution shifts the resonance frequency of the plate, more or less depending on the zero of Xmnmn, which is the point where the added-stiffness becomes the added-mass, but it also contributes to reducing the vibration of the plate. Figures 9(a) and 9(b) show the contribution of the added ground damping to the plate due to coupling, i.e., Xmnmn ðxÞ ¼ 0. The objective is to remove the contribution of the added-mass and -stiffness to the plate and to highlight the effect of the added-damping only. By taking the free plate vibration as a reference, the difference with the uncoupled modal amplitude is 20 dB and rises to 40 dB at the resonance frequency of the plate. With regard to damping, that of the plate and the ground is intrinsic, which attenuates the vibration. However, the considerable transfer of energy from the plate to the ground due to strong coupling gives rise to high ground added-damping. These results validate the conclusion based on the results of Fig. 7: the added ground damping is very significant and strengthens the ground/plate coupling. It can even be assumed that the reactive effects of the ground to plate vibration could be neglected compared to the damping effects! This is a major conclusion presented by Figs. 9(a) and 9(b) in which the
FIG. 9. (a) Modal amplitude of the mode 11 for a plate uncoupled and coupled to ground without ground added-mass and stiffness (Xmnmn ðxÞ ¼ 0). (b) Modal amplitude of the mode 13 for a plate uncoupled and coupled to ground without ground added-mass and stiffness (Xmnmn ðxÞ ¼ 0). J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
reactive effects are removed. It is now possible to provide a physical explanation when considering the plate vibration coupled with the ground. Because of the complexity of a medium such as the ground, the cross-modal impedance of the latter is an interesting tool for assessing its impact. IV. COMPARISON WITH MEASUREMENT
One of the aims of this modeling is to predict plate vibration in the case of ground plate coupling. That is why the comparison between theory and experiment was conducted on a tramway slab. In this section, we will compare an experiment with the numerical results obtained from simulations. This theory/experiment comparison is focused on plate vibration, something that has not yet been done. The plate under consideration is a concrete slab made for a tramway with the following characteristics: a ¼ 10 m, b ¼ 6 m, hp ¼ 0:6 m, Ep ¼ 2:5 1010 Pa, p ¼ 0:3, qp ¼ 2500 kg m3 , and gp ¼ 2%. Figure 11 shows a picture of the slab under consideration. This picture shows the four different locations of the accelerometers shown by numbered circles. The circles, called impact, correspond to the impact on the slab. Accelerometers 1 and 4 are located in the 6 m wide slab and accelerometers 2 and 3 are located in the 10 m long slab. Simply supported boundary conditions do not apply for such a plate: the modal shape may be changed from a sinus basis to a cosine basis. This represents the guided boundary conditions that take into account the rigid body mode. In Sec. III, the simply supported case was studied as the problem was better regularized. However, the previous interpretation still holds in the guided case. The calculation of the ground cross-modal impedance must be done by taking this change into account. The aim of this experimental comparison is to verify if the main phenomena described in the previous section are realistic, especially if addedstiffness or -mass and strong added-damping can be demonstrated. The tramway rails can be neglected in the model in the first approximation In the experiment, four “piezoelectric” accelerometers (sensitivity: 1 V/g) were attached to the slab to measure the vibrations: two 0.3 m from the edge and spaced 3 m apart, and two 3 m from the edge and also spaced 3 m apart. A transient hammer excitation force (Kistler Instrument Corporation, Les Ulis, France, sensitivity: 0.000 222 V/g) was used to generated a force 1.5 m from the edge of the slab and 5 m from its end. The post-treatement software from Acoem company (Limonest, France) was used. Ten impacts were generated between the tramway rail and the plate acceleration, taking into account coherence >90% in the frequency range of interest. Regarding the ground characteristics, the SASW method5,34 was applied in order to find out the wave celerities and the thickness of the layers. Ground damping is a difficult parameter to estimate, so a typical value for sand was considered. The tramway slab was coupled with a two layer ground with the following characteristics: q1 ¼ 1500 kg m3 , cs1 ¼ 200 m s1 , qcd1 ¼ 500 m s1 , h ¼ 5 m, and gs1 ¼ gd1 ¼ 0:02, q2 ¼ 3000 kgm3 , cs2 ¼ 3000 ms1 , cd2 ¼ 7000 ms1 , and gs2 ¼ gd2 ¼ 0:02. The case of a double sand layer is studied L. Grau and B. Laulagnet: The ground cross-modal impedance
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here. Subscript 1 refers to the first layer at the top surface, while subscript 2 refers to layer 2 inside the ground. The tramway slab was located at the “Decines Grand Stade” site at the terminus of the Lyon T3 tram line. Figure 10 presents the transfer function between the injected force and the acceleration on the slab. The round dots represent the measured transfer function on the plate and the crossed dots represent the simulated transfer function. A good match is shown between the theory and experiment up to 200 Hz in trend. An average difference of 3 dB can be noticed, which is low compared to the considerable simplification performed on the plate model. For the sake of brevity, the theory/experiment comparisons are not presented for the other accelerometers. However, the comparison also matches well, following the same trend as that presented in Fig. 10. This theory/experiment comparison emphasizes the result of the simulation presented above: strong attenuation of the plate vibration coupled to the ground, and more especially the disappearance of the resonance peak, even for a slab of 60 cm thick. V. GROUND CROSS-MODAL IMPEDANCE IN THE CASE OF A STRATIFIED GROUND
The aim of this section is to investigate the influence of a stratified ground on the frequency dependents of ground cross-modal impedances. The formulation of the stratified ground model is presented in Appendix B. The features of the ground composed of two layers are discussed in this paragraph. The plate characteristics remain unchanged, with dimensions a ¼ b ¼ 6 m and hp ¼ 0:2 m. A few cases are considered in order to show the modification of the ground crossmodal impedance due to ground stratification. The plate boundary conditions are simply supported. The case of a double sand layer is studied here. Subscript 1 refers to the first layer at the top surface and subscript 2 refers to layer 2 inside the ground, the first layer being light sand (q1 ¼ 1200 kg m3 ; cd1 ¼ 600 m s1 ; cs1 ¼ 160 m s1 ; gd1 ¼ gs1 ¼ 0:02; h ¼ 1 m) and the second layer heavy sand (q2 ¼ 1800 kgm3 ; cd2 ¼ 1200 ms1 ; cs2 ¼ 600 ms1 ; gd2 ¼ gs2
FIG. 10. Comparison of the experimental and simulated transfer function between the injected force and the plate acceleration on the site “Decines Grand Stade.” 2910
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FIG. 11. (Color online) Picture of the excited plate coupled to the ground with four accelerometers on the plate.
¼ 0:02). In Fig. 12, the RMS plate velocity exhibits a strong attenuation at low frequency and a resonance frequency at 78Hz. The plate velocity peak at this frequency is 10 dB compared with the case h ¼1. This makes analysis interesting with the corresponding ground cross-modal impedance in Figs. 13 and 14. At low frequency [10;40] Hz, the RMS plate velocity coupled to a stratified ground is strongly attenuated compared to the non-stratified one. This is due to the high ground added-stiffness due to stratification, shown in Fig. 13. At low frequency, i.e., [10;40] Hz, plate vibration is mainly due to mode 11. Figure 13 shows that the ground added-stiffness of this mode is higher for the stratified ground. Namely, at 5 Hz, the imaginary part changes from 35.75 to 244, meaning that the ground added-stiffness effect is stronger and contributes to the decrease of the plate vibration level observed. At higher frequency, i.e., [60;100] Hz, vibration was seen to increase at 80 Hz. This “resonance” peak can be explained by the ground stratification and especially by the real part of the ground cross-modal impedance responsible for the energy transfer. Plate vibration was mainly due to mode (1,3) in this frequency range. The real part of the ground cross-modal impedance for mode (1,3) is presented in Fig. 14. Due to the ground stratification, part of the energy was reflected from the ground to the plate, leading to a
FIG. 12. Comparison of the quadratic surface average plate velocity in double ground layer and one ground layer. L. Grau and B. Laulagnet: The ground cross-modal impedance
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In brief, the results show that the ground can change the vibration of a concrete plate of 6 m 6 m 0.2 m for a 1 m ground thickness. Although the added-damping flattens the plate velocity, it was still possible to identify the resonance frequency, especially when the ground resonance frequency matched the structural resonance. VI. CONCLUSION
FIG. 13. (Color online) x imaginary part of the normalized ground crossmodal impedance of the plate with a double ground layer with a first layer thickness of 1 m.
minimized energy transfer. Consequently, the real part of the ground cross-modal impedance was lower in the stratified ground than in the non-stratified one, typically 0.06 instead of 0.34 at 81 Hz. It should be remembered that for this layer thickness, coincidence occurred between the structural resonance frequency of mode 13 and the ground resonance frequency equal to fground ¼ cs1 =2h, where h represents the thickness of the first layer, and cs1 is the shear celerity of the first layer. In this case, the coincidence frequency was 80 Hz, which corresponds to the resonance peak observed in the plate vibration. Moreover, a specific trend appeared on the imaginary part of the corresponding impedance. The ground crossmodal impedance changed sign several times, contrary to that of a single ground layer. For a double ground layer like that presented in Fig. 13, the ground cross-modal impedance had two ranges of negative values, i.e., [5;40] Hz and [82;170] Hz, indicating added-stiffness and two positive value ranges, i.e., [40;82] Hz and [170;200] Hz, indicating added-mass. The extension to three layers would have produced several negative and positive value ranges, and so on.
A model for the ground/structure interaction of a simply supported and guided rectangular plate coupled to a semiinfinite ground was presented. The considerable importance of coupling, capable of contributing an attenuation of up to 40 dB at the resonance frequency of the structure, was shown. We also developed a new analysis tool, ground cross-modal impedance, which provided a novel means of analyzing ground–structure interaction. It was shown that ground/structure coupling can be expressed in terms of added-stiffness and mass. They can both be very high and, moreover, frequency dependent. At a specific frequency, the added-stiffness can reach 100 times that of the plate and the added-mass twice that of the physical plate. Also, the imaginary part of the impedance exhibited negative values, which do not exist in equivalent structure-fluid problems, due to the shear waves in the ground responsible for modal added-stiffness below the shear wavenumber coincidence. With regard to the ground cross-modal impedance, similarities can be pointed out with the equivalent acoustical cross-modal impedance. Several experiments were carried out and presented good matching with the model results. Finally, the effect of a stratified ground on plate vibration led to resonance due to reflection in the ground layer. This behavior of this reflection affects the ground cross-modal impedance. ACKNOWLEDGMENTS
The authors would like to thank the company ACOUPHEN (Pusignan, France) for its support and its experimental assistance, which helped to provide a better overview of the work. We would also like to thank the company SYSTRA (Paris, France) for valuable discussions. APPENDIX A: RELATION BETWEEN STRESS AND DISPLACEMENT FOR ISOTROPIC MEDIA
The aim of this section is to present the relation between the ground displacement and the ground stress in the Fourier domain. The ground displacement can be decomposed into a scalar potential and a vector potential, that is to say, a Helmholtz decomposition, which gives for each component
FIG. 14. (Color online) Real part of the normalized ground cross-modal impedance of the plate with a double ground layer with a first layer thickness of 1 m. J. Acoust. Soc. Am., Vol. 137, No. 5, May 2015
8 @/ð x; y; zÞ @wz ð x; y; zÞ @wy ð x; y; zÞ > > > þ ; ux ð x; y; zÞ ¼ > > @x @y @z > > > < @/ð x; y; zÞ @wx ðx; y; zÞ @wz ð x; y; zÞ þ ; uy ð x; y; zÞ ¼ > @y @z @x > > > > @/ð x; y; zÞ @wy ð x; y; zÞ @wx ð x; y; zÞ > > > þ : : uz ðx; y; zÞ ¼ @z @y @x (A1) L. Grau and B. Laulagnet: The ground cross-modal impedance
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Considering the 2D Fourier transform for the ground displacement in directions x and y, we obtain 8 ~ > > > ~ ðkx ; ky ; zÞ þ jky w ~ ðk x ; k y ; zÞ d w y ðk x ; k y ; zÞ ; > u~x ðkx ; ky ; zÞ ¼ jkx / > z > dz > > < ~ ðk x ; k y ; zÞ d w ~ ðkx ; ky ; zÞ jkx w ~ ðk x ; k y ; zÞ þ x ; u~y ðkx ; ky ; zÞ ¼ jky / z > > dx > > > ~ ð k x ; k y ; zÞ > d/ > > ~ ðkx ; ky ; zÞ jky w ~ ðk x ; k y ; zÞ: : u~z ðkx ; ky ; zÞ ¼ þ jkx w y x dz
The relation between the stress tensor and the strain tensor is given by Hook’s law for an isotropic media through rij ¼
lE E kk dij þ ij : 1þl ð1 þ lÞð1 2lÞ
(A3)
Assuming that the strain is infinitesimal, meaning kuk 1 and kruk 1, the displacement expression depending on the stress is rij ð x; y; zÞ ¼
lE ðux;x þ uy;y þ uz;z Þdij ð1 þ lÞð1 2lÞ E 1 þ ðui;j þ uj;i Þ: 1 þ l2
(A4)
(A2)
The shear and dilatation velocities in the nth ground layer are linked to Lames coefficient 8 > > > c2dn ¼ kn þ 2ln ; < qn (B4) l > n 2 >c ¼ : > : sn q n Ground damping can also occur through dilatational and shear waves by defining a complex wave as follows, assuming gn is small compared to unity: cdn ¼ cdn ð1 þ jgdn Þ; (B5) csn ¼ csn ð1 þ jgsn Þ:
In the Fourier domain, we obtain 8 > @ u~x > > >r ~ xz ðkx ; ky ; zÞ ¼ l þ jkx uz : > > @z > > < @ u~y ~ yz ðkx ; ky ; zÞ ¼ l r þ jky uz : > @z > > > > > @ u~z > > ~ zz ðkx ; ky ; zÞ ¼ jkx k~ : u x þ kky u~y þ ðk þ lÞ :r @z
APPENDIX C: POTENTIAL’S COEFFICIENT FOR THE SINGLE LAYER GROUND
(A5)
APPENDIX B: THEORY OF STRATIFIED GROUND
An analytical solution to the equation of motion can be found using the Helmholtz decomposition. The theorem states that every smooth vector field, un , can be decomposed into a rotational part (rot) and an irrotational part (grad), meaning ln $2 un ðx; y; zÞ þ ðln þ kn Þ$ð$un ðx; y; zÞÞ
In this section, the analytical expression in the Fourier domain between displacement and stress in Eq. (16) is justified. In the case of a single ground layer, the solution of Eq. (B3) is as follows after performing a 2D Fourier transform: 8 ~ x ; ky ; zÞ ¼ Aejk1 z ; /ðk > > > > > > ~ ðkx ; ky ; zÞ ¼ Bejk2 z ; ~ ðkx ; ky ; zÞ ¼ Cejk2 z ; > w > y > > > :~ w z ðkx ; ky ; zÞ ¼ Dejk2 z ; where k12 ¼ kd2 kx2 ky2 and k22 ¼ ks2 kx2 ky2 . Using Eqs. (A2), (A5), and the gauge condition with the boundary conditions (6), this leads to
þ qn x2 un ðx; y; zÞ ¼ 0;
(B1)
0
0
un ¼ gradð/n Þ þ rotðwn Þ:
(B2)
B B B @
2kx k1 2ky k1
By introducing Eq. (B2) in Eq. (B1), we obtain four uncoupled equations from the unknown functions /n and wn , ( 2 /n ðx; y; zÞ ¼ 0; r2 /n ðx; y; zÞ þ kdn (B3) 2 2 r wn ðx; y; zÞ þ ksn nn ðx; y; zÞ ¼ 0;
ðkkd2 2lk12 Þ 0 1 0 B 0 C B C ¼ B C: @ 0 A
10 1 A 2 2 C B kx ky ðk2 kx Þ ky k2 CB B C C CB C ðk22 ky2 Þ kx ky kx k2 A@ C A kx
ky
k2
2lky k2
2lkx k2
0
D
(C2)
rzz where kdn ¼ x=cdn and ksn ¼ x=csn denote the dilatational and shear wavenumbers, respectively. 2912
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Inverting this matrix leads to L. Grau and B. Laulagnet: The ground cross-modal impedance
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8 > k22 þ kx2 þ ky2 > > > ~ zz ; A ¼ r > > gðkx ; ky Þ > > > > > 2k1 ky < ~ zz ; B¼ r gðkx ; ky Þ > > > 2k1 kx > > > ~ zz ; C¼ r > > gðkx ; ky Þ > > > : D ¼ 0;
(C3)
where gðkx ; ky Þ was defined in Eq. (16). It is now possible to obtain an expression of u~z ðkx ; ky ; 0Þ using Eq. (A2), u~z ðkx ; ky ; 0Þ ¼
jk1 ks2 ~ zz ðkx ; ky ; 0Þ: r gðkx ; ky Þ
(C4)
APPENDIX D: CONVERGENCE STUDY OF GROUND CROSS-MODAL IMPEDANCE
In this section, we discuss the criteria used to ensure good convergence of the double integral below: ð ð þ1 jk1 ks2 ~ ~ / / dkx dky ; (D1) cpqmn ¼ 4 gðkx ; ky Þ mn pq 0
jk1 ks2 =gðkx ; ky Þ term behaves at infinity like Oððkx2 þ ky2 Þ3=2 Þ. 4 4 ~ ~ / Regarding / mn pq , the behavior at infinity is Oðkx ky Þ for the simply supported plate and Oðkx2 ky2 Þ for the guided plate. Finally, the behavior of the function to integrate is Oððkx2 þ ky2 Þ3=2 kx4 ky4 Þ for the simply supported plate and Oððkx2 þ ky2 Þ3=2 kx2 ky2 Þ for the guided plate, which renders the integral convergent. Regarding the singularity at the de~ , such as kx ¼ np=a, we refer nominator of the function / nm to the work of Laulagnet,31 which gives an asymptotic solution near the singularities. In the numerical results we present, the limit for the domain of integration is truncated at a finite value, such as kmax ¼ 100=f if kmax < 5jks j and kmax ¼ 5jks j if kmax > 5jks j. In this paper, the discretization of the domain is based on a wavelength criterion. The criterion is fixed at 24 points per wavelength, i.e., Dkx ¼ ð2p=aÞð1=24Þ and Dky ¼ ð2p=bÞð1=24Þ. This criterion must be more accurate for a stratified ground and leads to 48 points per wavelength to ensure good convergence of the integral. It can be seen that the integrand is odd with respect to kx if the orders (n,p) versus the x direction have different parities, and that the integrand is odd with respect to ky for the same reasons versus the mode orders (q,m). It can be seen that the integrand is even in the remaining cases of parity. This leads to the following properties:
8 ð ð þ1 > jk1 ks2 ~ ~ > < cnmpq ¼ 4 / / dkx dky ; when p and n have the same parity; and q and m have the same parity; gðkx ; ky Þ mn pq 0 > > :c ¼ 0; when p and n have different parities; and q and m have different parities:
(D2)
nmpq
1
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L. Grau and B. Laulagnet: The ground cross-modal impedance
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