Vibro-acoustics response of an isotropic plate under non-uniform edge loading: An analytical investigation

Abstract

Analytical studies carried out on the vibro-acoustic response behavior of an isotropic plate under non-uniform edge loads subjected to steady-state mechanical excitation is presented. An analytical method based on the energy approach is used to calculate the buckling load ($P_{cr}$). Free and forced vibration responses of the plate are obtained using an analytical method based on Reddy's third-order shear deformation theorem (TSDT) while sound radiation behavior is analyzed using Rayleigh Integral. Results revealed that $P_{cr}$ is significantly influenced by the nature of non-uniform edge load. Similarly, natural frequencies reduce with an increase in axial compressive load due to a reduction in structural stiffness. Vibration and acoustic resonant amplitudes are affected by the intensity of the compressive load. Sound transmission loss reduces with an increase in compressive load magnitude and the effect is significant in the stiffness dominant region.

Introduction

Buckling and dynamic response are common phenomenal occurrence in the mechanical structures which are widely used in aerospace, marine and automobile industries. These structures experience in-plane stress during their service life, which in-turn influence the stiffness of the structure and causes the buckling and dynamic instability. This leads to structure born noise issues and acoustic discomfort in transportation vehicles. So, the vibro-acoustics properties of structures under the buckling phase are still a very interesting topic.

In most of the practical cases the structures are subjected to non-uniform uni-axial stresses either along longitudinal or transverse direction. Examples are local instability of stiffener, walls, floor and skin panels of wings of an aircraft are widely made up of isotropic structures. During the on-flight conditions, these structures experience normal compression with uniform pressure. Due to this the panel edges are subjected to combined tensile and compressive stresses. So, it is important to investigate effect of non-uniform edge loadings on buckling and dynamic characteristics of a structure.

Baker et al. [1] have benchmarked the Mathieu's approach for the arbitrary edgewise compressive force into modest fundamental stress problems, by obtaining the exact analytical solution for the elasticity problem as a stress input factor. Mijuskovie et al. [2]-[3] have derived analytical results for the buckling load, under the arbitrary in-plane load with the variable boundary condition by using Ritz energy method. Kang and Leissa [4] expressed the accurate solution method for the buckling study of simply supported isotropic plate subjected to the linearly varying load using the method of Frobenius. Timoshenko and Gere [5] and Jones [6] analyzed the buckling of the rectangular plate under non-uniform edge loading using analytical method to attain the precise buckling strength. Bert and Devarakonda [7] examined the buckling characteristic of Kirchhoff plate under the simply supported condition exposed to non-uniform edge stress using an Galerkin method. Zhong and Gu [8] studied the impact of load intensity/configurations on the buckling strength of simply supported plate exposed to non-uniform edgewise load. Kang and Leissa [9] examined the dynamic and buckling characteristic of the rectangular plate to the non-uniform edge load by using analytical method of Frobenius. Leissa and Kang [10]-[11] studied the effects of compressive load on the free vibrational features of the plate with simply supported edges, exposed to a linearly varying load and noted that increase in load results in the shift of natural frequencies towards zero. Liew et al. [12] examined the vibration and buckling characteristics of FSDT (First-order shear deformation theory) plate by employing the mesh-free procedure and compared the results with the numerical solutions. Neves et al. [13] used HSDT (Higher-order shear deformation theory) to analyze deflection, stresses, free vibration and buckling characteristics. Jeyaraj [14] has done a detailed numerical study on buckling strength and dynamic response of metal plate by using the FEM approach, to show the instability of the nodal and anti-nodal lines when the plate exposes to high temperature. Ferreira et al. [15] numerically examined buckling and free vibration based on FSDT, which produces highly accurate buckling load and natural frequencies with their respective mode shapes by using collocation approach with radial basis function.

Jeyaraj [16] numerically investigated vibro-acoustic characteristics of an isotropic plate with different taper configurations by combining FEM and BEM. Khorshidi [17] derived a dimensionless equation of motion based on FSDT to attain the transverse dynamic and acoustic radiation characterization of a metal plate with altered boundary conditions. The influence of boundary conditions, types of load, aspect ratio and thickness on sound radiation has been explained in detail. Chandra et al. [18] examined analytically vibro-acoustical behavior of the functionally graded plates by following FSDT and shown that sound power level decrease by increasing power-law index. Geng et al. [19] investigated both experimentally and numerically the vibroacoustic performance of a metal plate exposed to the thermal load. They clearly have shown that the increase in thermal loading decrease the natural frequency and dynamic response of the plate is carried out by both mechanical and acoustical excitation. Li and Li [20] investigate analytically the impact of disseminated mass loading on the acoustical radiation behavior of isotropic plate in both air and liquid medium and concluded that the acoustical characteristics charges at a high rate with respective medium. Geng and Li [21] examined the impact of thermal load on vibration and acoustical performance of an isotropic plate and shown the reduction of radiation efficiency of the plate at mid-frequency band. They also suggest that combine Finite element analysis and the Boundary element method is more favorable for sound radiation calculation. Vijay et al. [22] examined numerically the impact of buckling load fraction on the vibroacoustic response of a mild steel plate by employing a combined finite/boundary element method and noted that nature of the compression load influences the buckling and dynamic characteristics of the plate significantly. Reddy [23] analyzed the dynamic response of the plate by obtaining various shear deformation theories to attain the dynamic behavior of the plate under various boundary conditions. Arunkumar et al. [24] investigated the vibro-acoustics behavior of the sandwich plate by implementing a finite element model for vibration response and Rayleigh integral method for predicting acoustic characteristics. Fu et al. [25] done a detailed analytically study on sound radiation and sound transmission behavior of the stiffened plate under different boundary conditions. Talebitooti et al. [26] derived the analytical model based on the HSDT method to obtain the sound transmission loss characteristics of the composite structure. Zhou and Zarastvand [27] investigated analytically the dynamic and acoustic behavior of the porous functionally graded material under the thermal environment by implementing the FSDT method. Zhang et al. [28]-[29] proposed the vibro-acoustic coupling model to investigate the sound radiation behavior of the structure under different elastic boundary condition and impedance wall boundary conditions. Sarigul and Karagazlu [30] by using finite element method investigated the vibro-acoustic coupling analysis of the composite plate and shown that the plate behavior has a significant effect on material, ply orientation and number of layers.

Li et al. [31] analytically studied the vibro-acoustics of plates and shells by using efficient scaled boundary finite element method and numerical results are presented to validate their results. Sharma et al. [32], [33], [34], [35], [36], [37], [38] numerically studied the vibro-acoustic behavior of flat and curved laminated composite panel under different boundary conditions by using combined higher order FEM/BEM method. To highlight the effects of thermal loading, aspect ratio, slenderness and lay-up scheme on sound radiation characteristics. Zhang et al. [39] formed an analytical method to study vibro-acoustic behavior of FRP plate coupling with an impedance cavity by merging simplified plate theory and Fourier-Ritz method. Wang et al. [40] numerically studied the vibro-acoustic behavior of double walled cylindrical shell by using precise transfer matrix method for calculating the sound pressure coefficients and the predicted results are validated with the experimental results. Xu and Huang [41] investigated the vibro-acoustic behavior of FG-GRC plate exposed to thermo-mechanical loads and results shown that graphene distribution and weight fraction have significant effect on acoustic behavior. Assaf et al. [42] numerically studied the prediction of transmission loss of the sandwich structure exposed to acoustic plane waves by using combined finite and boundary element method. Fuller [43] analytically studied the sound radiation of vibrating plates subjected to the direct oscillating forces and demonstrated that the radiation efficiency is related to the nature of coupling between the plate modes of response and radiated field. Thompson et al. [44] done a detailed study on sound transmission loss through panels based on the model of infinite panel and highlights that transmission coefficients greater than unity should not necessarily be seen as error in calculation. Li and Yang [45] numerically investigated the vibro-acoustic behavior of meta material with the negative Poisson's ratios by using spectral element method. Natarajan and Padmanabhan [46] proposed a better semi-analytical method to predict the acoustic characteristics by implementing the scaled boundary finite element method. Fhay and Gardonio [47] proposed the powerful computational methods and procedure for numerical analysis of structural vibration, acoustic fields related to sound radiation and transmission loss.

From the above-detailed literature review, it is found that no study has been conceded to investigate the influence of static non-uniform edge loading on vibration and acoustical characteristics of structures subjected to steady state excitation. The pre-stress developed due to the non-uniform edge loads will influence the dynamic response of a structure subjected to the transverse harmonic excitation. However, it is essential to examine the vibration and acoustical performance of the structure under the influence of non-uniform edge loads subjected to steady-state mechanical excitation for a better design. In this work, an analytical approach based on strain energy approach and TDST approach has been used to investigate the dynamic and acoustical behavior of a metal plate exposed to the non-uniform edge loading. The flow chart shown in Fig. 1 clarifies the inspiration for the proposed study.