Multiphysical numerical (FE–BE) solution of sound radiation responses of laminated sandwich shell panel including curvature effect

Abstract

In this paper, a numerical scheme is prepared using the higher-order shear deformation type of kinematic model with the help of a coupled finite and boundary elements (FE–BE) to evaluate the vibroacoustic responses of laminated composite sandwich curved shell panels. The panel is under the influence of a harmonic point load. Further, an FE–BE combined technique is utilized to prepare a generic computer code (MATLAB environment) for the numerical prediction via the proposed mathematical formulation. The structural frequency and the subsequent sound relevant data are obtained by solving the final form of the multiphysics model. In this regard, the structural system equation is derived through Hamilton’s principle and the Helmholtz wave equation for the computation of acoustic responses. The performance of the proposed scheme is established initially through the convergence and the corresponding validation studies. The comparison cases are made with the available published benchmark frequency (free vibration) as well as the acoustic data. Appropriate numbers of numerical examples are solved to draw the meaningful inferences of various factors to show the significant influences on the acoustic radiation responses of the curved sandwich panel type of structural components. The curvature ratio is showing the accentuated influences on the sound radiation responses for the low-frequency ranges whereas the increase in the thickness ratio, i.e. the ratio of core to face leads to an accentuated radiated sound power.

Introduction

Laminated composite sandwich structures are special kind of laminated construction with low strength core and high strength face sheets in the form of composite laminates. This layered pattern renders the sandwich structure with high strength to weight ratio and better energy absorption capacity under dynamic conditions thus making them favourites for high performance applications such as in aerospace, automotive, aircrafts, submarine, trains and civil engineering constructions sectors. Further, (single/doubly) curved laminated composite sandwich panels are widely used as structural components because of their greater stiffness characteristics owing to the spatial curvature besides the general favourable characteristics owing to the face-core-face scheme. However, the sandwich shell/shell panel set-up leads to poor acoustic performance caused by amplified radiation of sound and transmission of acoustic noise [1]. Therefore, careful design of these structures is very much essential for attaining improved sound reduction while maintaining the lightweight properties. It is for this reason that the vibroacoustic response of sandwich structures has been investigated extensively a subject of extensive investigation in the recent past. Several analytical/numerical studies probing the acoustic radiation characteristics (verified with experimental data) of sandwich structures are available in open literature. The numerical modelling leading to computation of acoustic responses catalyses the overall design cycle of a product in contrast to the experimental approach. However, in order to capture the precise influence of the factors such as material, geometry, excitation etc. on the responses, the displacement field describing the motion of the panels need to be judiciously defined [2]. The free vibration responses are largely influenced by the midplane kinematic model chosen and intron influence the vibroacoustic responses of the structure. To address this issue, the bending, buckling and free vibration behaviour of laminated composite and sandwich curved beams/shells/shell panels has been investigated intensely using numerous approaches [3]. Additionally, several equivalent single layer theories including the first-order shear deformation theory (FSDT)  [4], [5], [6], [7], [8], [9], higher-order shear deformation theory (HSDT)  [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], sinusoidal shear-deformation theory [20], [21], layer-wise theory [22], refined theories [23], [24] and theories incorporating nonlinear distribution of transverse shear stresses [25] have been employed to study the influence of design parameters [26] on the free vibration and thermo-mechanical buckling [27] behaviour of composite sandwich shell panels with and without curvature. The free vibration responses thus obtained using the aforementioned theories are then utilized to obtain the acoustic responses of structures. The analysis of vibroacoustic responses of laminated composite [28] and sandwich curved structures has been a matter of intensive probing in the near past. Acoustic radiation responses of thick laminated cylindrical shells with sparse cross stiffeners in wavenumber domain [29], stiffened cylindrical shells with constrained layer damping [30] and the functionally graded (FG) axis-symmetric shells in the surrounding mediums of low and high density utilizing a Kirchhoff–Helmholtz integral formulation [31] have been reported. Further, analytical solutions to the vibroacoustic responses of orthotropic composite cylindrical shell [32] and an orthotropic conical shell using wave propagation approach and Galerkin method [33] in a hygroscopic environment have been presented. Further, influence of elevated temperature environment on the sound radiated by the conical shell panels has been investigated by through commercial software utilizing a hybrid finite-element/statistical-energy-analysis (FE/SEA) formulation [34]. Additionally, a few solutions of the structural as well as the multiphysical problems, i.e. the fluid–structure interaction have also been obtained via the meshfree methods [35] including the iso-geometric type FEM [36], [37], [38] in conjunction with the variational principle [39] which is in direct contrast to the traditional type of theoretical and the experimental approaches [40].

The sound radiation responses of sandwich panels have been investigated by various researchers. Liu and Li [41] analysed the thermal acoustic radiation responses of sandwich plates having isotropic core and face sheets, whereas Li and Yu [42] studied the vibroacoustic responses of sandwich plates with orthotropic face sheets exposed to elevated temperature environment utilizing a piecewise low order shear deformation midplane kinematics. Moreover, studies utilizing commercial finite-element/boundary-element (FE/BE) tools to study the sound radiation characteristics of flat [43] and thermally stressed cylindrical sandwich shell panels with viscoelastic core [44] have also been reported. Petrone et al. [45] utilized Rayleigh’s integral to compute the acoustic power of the flat sandwich panels with aluminium foam in NASTRAN and compared the results with those obtained via experiments. The alteration caused by the temperature loading in the sound insulation characteristics of curved sandwich shell panels has also been probed experimentally by Tang [46]. Ghinet et al. utilized SEA [47] whereas Chronopoulos implemented wave finite element method (WFEM) [48] to study the transmission loss characteristics of curved composite sandwich shell panels. It has been established that the structural and the acoustic radiation characteristics of sandwich laminate cannot be accurately predicted by the FSDT as it assumes uniform transverse shear strain over the entire thickness [10]. Also, it has also been recognized that the HSDT not only leads to an adequately accurate approximation of the flexure of the shear deformable composite shell panels but it is also relatively easier to implement [49], [50], [51].

The review of the literature mentioned above clearly suggests that the vibroacoustic responses of curved sandwich panels have been widely studied experimentally as well as numerically utilizing various displacement models. It is also evident that the sound absorption and transmission characteristics of sandwich structure have been probed more in contrast to the sound radiation characteristics. However, the acoustic radiation responses of the doubly curved composite sandwich shell panels of cylindrical, spherical, elliptic, hyperboloid geometries subjected to harmonic point excitation are yet to be reported. Moreover, the study using the HSDT type of kinematic theory for the sandwich structural modelling followed by the vibroacoustic analysis with the help of a coupled FE–BE formulation is far from existence. Therefore, the present work is aimed at evaluating the vibroacoustic behaviour of composite sandwich shell panels having curvature in longitudinal as well as transverse directions (doubly curved) using a coupled FE–BE scheme. The sandwich shell panel is modelled to have an isotropic core and laminated composite face sheets. The structural model has been developed in the framework of the HSDT mid-plane kinematic including the curvature parameters in longitudinal and transverse direction. The FE model of the curved shell panels is discretized using a Lagrangian nine noded isoparametric element having nine nodal degrees of freedom. The Hamilton’s principle is employed to obtain the panel stiffness and mass matrices and the BE method is used to model the surrounding medium. The coupling between the vibrating curved panels and the surrounding medium is established, and responses are obtained by solving the coupled fluid–structure equation. The close conformance of the exemplary results of free vibration and acoustic responses with the available benchmark solutions ascertain the efficacy and accuracy of the present model. Finally, the model is extended to solve numerous numerical illustrations to understand the influence of thickness ratio of core and face, curvature ratio, modular ratio of core and face, geometry on the acoustic radiation responses of composite sandwich curved shell panels.