Winkler-Pasternak Elastic Foundation Research Articles

Chebyshev–Ritz and Navier's methods for hygro‐magneto vibration of Euler–Bernoulli nanobeam resting on Winkler–Pasternak elastic foundation

AbstractThis research employs Chebyshev–Ritz method along with Navier's method to investigate the vibration characteristics of a nanobeam subject to a longitudinal magnetic field and linear hygroscopic environment. The nanobeam is characterized by a Winkler–Pasternak elastic foundation and follows the nonlocal Euler–Bernoulli beam theory. The governing equation of motion is derived using Hamilton's principle, and non‐dimensional frequency parameters are computed for Simply Supported‐Simply Supported (SS), Clamped‐Clamped (CC), and Clamped‐Free (CF) boundary conditions. The motivation behind this study is to provide a comprehensive and efficient analytical framework for understanding the dynamic behavior of nanobeams in complex environments. By investigating the influence of magnetic and hygroscopic factors on the vibration characteristics of nanobeams, this research aims to offer valuable insights for the design and optimization of nanoscale structures. Employing shifted Chebyshev polynomials as shape functions in Chebyshev–Ritz method offers several advantages in the proposed model. Firstly, these polynomials possess orthogonal properties, which can significantly enhance computational efficiency. The orthogonality of shifted Chebyshev polynomials allow for simpler and more streamlined numerical computations compared to non‐orthogonal basis functions. Additionally, the orthogonality ensures that the resulting system of equations is well‐conditioned, even for higher‐order polynomial approximations. A closed‐form solution for SS boundary condition is obtained through Navier's method. Convergence analysis is performed to validate the accuracy and effectiveness of the proposed model against existing models. The non‐dimensional frequency parameters obtained using both Navier's method and Chebyshev–Ritz method demonstrate strong agreement, further validating the proposed nanobeam model. Additionally, a comprehensive parametric study evaluates the impact of various characteristics, including the small‐scale parameter, Winkler modulus, shear modulus, magnetic parameter, and hygroscopic parameter. The findings contribute to a nuanced understanding of nanobeam vibrations under the influence of a magnetic field and hygroscopic environment, providing valuable insights for the design and optimization of nanoscale structures in practical applications.

EFG meshless-ANN approach for free vibration analysis of functionally graded material plates on elastic foundation in thermal environments

This study focuses on free vibration analysis of functionally graded material (FGM) plates supported by Winkler–Pasternak elastic foundation in thermal environment using element-free Galerkin (EFG) meshless method. Plate kinematics depend on first-order shear deformation theory. Uniform, linear, and nonlinear temperature variations through the thickness direction are considered, along with the temperature-dependent material properties. The numerical outcomes obtained from EFG method are compared with those available in the published literature to validate the proposed method’s accuracy. An artificial neural network (ANN) model that can easily predict the natural frequencies of the plate is constructed from the EFG method outcomes. Further, the effect of foundation parameters, power law index, thickness ratio, temperature variations, and different boundary conditions are investigated; results show that these significantly influence the vibration response of FGM plates supported by the elastic foundation. Increasing the temperature of FGM plates supported by the Winkler–Pasternack foundation causes a decrease in the dimensionless fundamental natural frequency, and the uniform temperature influence is greater than that of linear and nonlinear temperature variation. The proposed EFG-ANN prediction model saves approximately 98.80% computation time when predicting the natural frequency with an accuracy of approximately 98.76% compared to that by EFG meshless method alone.

A novel dynamic stiffness matrix for the nonlocal vibration characteristics of porous functionally graded nanoplates on elastic foundation with small-scale effects

The present paper deals with the development of a dynamic stiffness matrix to evaluate the free vibration response of functionally graded nanoplate (FG-nP) resting on the Winkler-Pasternak elastic foundation. The complete mathematical modeling of the dynamic stiffness matrix for nanostructures is given for the first time. The equation of motion for rectangular FG-nP plates supported on an elastic foundation is derived using Hamilton’s principle in conjunction with nonlocal elasticity theory. The non-local theory is incorporated to account for the size effect in the small-scale plate. The effective material property of the porous FG-nP has been calculated using three recently developed models of porosity. The developed dynamic stiffness matrix is solved using the Wittrick-Williams algorithm to extract the natural frequencies of the FG-nP. The variation of natural frequencies with the change of numerical values, such as nonlocal parameter, aspect ratio, elastic foundation parameters, and porosity volume fraction is analyzed. The validity and accuracy of the results are confirmed through comparison with the available literature. The use of non-local theory in dynamic stiffness analysis is shown to be effective in predicting the natural frequency of the FG-nP on a Winkler-Pasternak elastic foundation, providing new insights into the dynamic behavior of small-scale structures.

Frequency Analysis of Asymmetric Circular Organic Solar Cells Embedded in an Elastic Medium under Hygrothermal Conditions

This research represents the first theoretical investigation about the vibration behavior of circular organic solar cells. Therefore, the vibration response of asymmetric circular organic solar cells that represent a perfect renewable energy source is demonstrated. For this purpose, the differential quadrature method (DQM) is employed. The organic solar cell is modeled as a laminated plate consisting of five layers of Al, P3HT:PCBM, PEDOT:PSS, ITO, and Glass. This cell is rested on a Winkler–Pasternak elastic foundation and assumed to be exposed to various types of hygrothermal loadings. There are three different kinds of temperature and moisture variations that are taken into account: uniform, linear, and nonlinear distribution throughout the cell’s thickness. The displacement field is presented based on a new inverse hyperbolic shear deformation theory considering only two unknowns. The motion equations including hygrothermal effect and plate–foundation interaction are established within the framework of Hamilton’s principle. The DQM is utilized to solve these equations. In order to ensure the accuracy of the proposed theory, the present results are compared with those reported by other higher-order theories. A comprehensive parametric illustration is conducted on the impacts of different parameters involving the geometrical configuration, elastic foundation parameters, temperature, and moisture concentration on the deduced eigenfrequency of the circular organic solar cells.

Power flow analysis of the three-beam system on double parameter elastic foundations based on the wave propagation approach

This article investigates the power flow of the three-beam system on Winkler–Pasternak elastic foundations under elastic boundary conditions. The time-averaged power flow of the beam on elastic foundations with elastic boundary conditions is calculated based on the internal shear force and bending moment and the wave propagation approach. The reliability and accuracy of the calculation method are verified by comparing the vibration acceleration calculated by calculation examples and the finite element method. In addition, the effects of structural parameters and boundary conditions on the power flow of the three-beam system are investigated. This contributes to the analysis of the dynamic characteristics of the beam on double parameter elastic foundations under complex boundary conditions.

Vibration analysis of Timoshenko microbeams made of functionally graded materials on a Winkler-Pasternak elastic foundation

In this work, the free vibration analysis of Timoshenko microbeams made of the Functionally Graded Material (FGM) on the Winkler-Paternak elastic foundation based on the Modified Coupled Stress Theory (MCST) is investigated. Material characteristics of the beam vary throughout the thickness according to the power distribution and are estimated through Mori–Tanaka, Hashin-Shtrikman and Voigt homogenization techniques. The Timoshenko microbeam model considering the length scale parameter is applied. The free vibration differential equations of FGM microbeams are established based on the Finite Element Method (FEM) and Kosmatka’s shape functions. The influences of the size-effect, foundation, material, and geometry parameters on the vibration frequency are then analyzed. It is shown that the study can be applied to other FGMs as well as more complex beam structures.

Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets

The nonlinear vibration analysis of functionally graded reinforced with graphene platelet (FG-GRC) porous truncated conical shells surrounded by the Winkler-Pasternak elastic foundation is presented in this paper. An improved model for evaluating the material properties of porous composites is proposed. Three types of porous distribution and three patterns of graphene nanoplatelets (GPLs) dispersion are estimated. Coupled with the effect of the Winkler-Pasternak elastic foundation, the nonlinear governing equations are developed by using the Hamilton principle. The Galerkin integrated technique is employed to obtain the linear and nonlinear frequencies of the shells. After the present method is validated, the effects of the pores, GPLs, the Winkler-Pasternak foundation, and the semi-vertex are investigated in detail. The results show that the linear frequency can be raised by increasing the values of the mass volume of the GPL and foundation parameters. In contrast, the ratio of nonlinear to linear frequency declines as the mass volume of the GPLs and foundation parameters rises. Furthermore, it is found that the minimum ratio of nonlinear to linear frequency can be obtained as the semi-vertex angle is about 55º, and the effect of porosity distribution on the linear and linear frequencies might be neglected.

Bending analysis of functionally graded plates resting on elastic foundation: A Rayleigh-Ritz approach and ANN method

This study presents a comprehensive investigation into the static bending characteristics of rectangular plates constructed from functionally graded materials resting on an elastic foundation. The research examines numerical factors that include central deflection, bending, and normal stresses, in functionally graded rectangular plates (FGRP) subjected to mixed edge constraints under various loading conditions. Classical plate theory is employed, and a computationally efficient mathematical expression for bending analysis is proposed, utilizing the energy principle. The Rayleigh-Ritz method with an algebraic function is applied to solve the governing equation, and the computation is performed using MATLAB. The study investigates the impact of aspect ratios, material property exponents, and the moduli of the Winkler and Pasternak foundations on the numerical factors of embedded FGRPs. The findings reveal that material property exponents and aspect ratios significantly affect numerical factors. The Pasternak foundation moduli demonstrate a more pronounced influence compared to the Winkler foundation moduli. This research provides valuable insights about the static analysis of FGRPs on Winkler-Pasternak elastic foundations, offering practical applications in the design and analysis of structures with similar configurations. Additionally, a salient aspect of this research is in the implementation of an artificial neural network (ANN) for the purpose of predicting the central deformation of FGRP while manipulating the input parameters. This approach is especially advantageous in engineering applications as it facilitates precise predictions for any set of input parameters without using complex computational work.

Nonlinear forced vibration of a nanobeam resting on Winkler-Pasternak elastic foundation and subjected to a mechanical impact

The nonlinear governing equations of nanobeam taking into consideration its curvature, resting on an elastic Winkler-Pasternak foundation and based on non-local Euler-Bernoulli beam theory is analyzed. The equation of motion and the boundary conditions are modeled within the framework of a simple supported nanobeam which accounts the presence of a mechanical impact and nonlinear von-Karman strain. The resulting nonlinear differential equations are reduced to only one differential equation which is studied by means of the Optimal Auxiliary Functions Method (OAFM). An explicit analytical solution is proposed for a complex problem. The main quality of our technique consists in the existence of some auxiliary functions derived from the expressions of the solution for the initial linear equation and the form of nonlinear term calculated from the above solution of the linear equation. The convergence-control parameters present in the auxiliary functions are evaluated by a rigorous mathematical procedure. The obtained solutions are in very good agreement with the numerical solution.

Dynamic stability of rectangular and circular cantilevers on Winkler-Pasternak elastic foundation by a higher-order beam theory

The structural stability of clamped-free beam-columns resting on a two-parameter foundation under a nonconservative force is investigated. The higher-order beam theory is developed by considering the warping of the cross-section, rotary inertia, and axial force’s direction. The characteristic equation of dynamic stability of rectangular/circular cantilevers on a Winkler-Pasternak foundation is obtained for a generalized follower force. Critical loads for divergence instability and flutter instability are evaluated. The influences of foundation stiffness, cross-sectional warping shape, the tangency coefficient on the critical loads are analyzed. The load-frequency interaction-curves are given for different directions of the follower force. The normal/shear stiffness of foundation on divergence and flutter loads has different effects. The normal stiffness increases the first divergence load but decreases the flutter load, reducing to the Herrmann–Smith paradox for Euler columns. The rise of shear stiffness enlarges the first divergence load and lowers the flutter loads. The effects are more sensitive for short beam-columns. Different warping shapes slightly affect the critical loads and the natural frequencies. The exponential warping shape gives the smallest critical divergence loads and the fundamental frequencies. The results of Haringx hypothesis are larger than those of Engesser hypothesis. For shorter columns, the classical results are overestimated and the higher-order beam theory is necessary.

Nonlinear vibration and stability of sandwich functionally graded porous plates reinforced with graphene platelets in subsonic flow on elastic foundation

This paper investigates the nonlinear vibration and stability of a functionally graded porous sandwich plate reinforced with graphene platelets (GPLR-SFGP) interacting with subsonic airflow on elastic foundation. The plate comprises of a functionally graded porous core with graphene platelet reinforcement and two metal face layers. Utilizing Hamilton's principle, the nonlinear equation of the plate is exported and discretized into ordinary equations using the assumed modes method. The influence of porosity, GPL weight fraction, surface thickness ratio and Winkler Pasternak elastic foundation arguments on the critical divergence velocity of the plate under subsonic flow is revealed by calculating the system characteristic values. The Matcont toolbox is occupied to generate nonlinear amplitude frequency resonance curves, allowing for a comprehensive examination of the influence of these parameters on the nonlinear resonance behavior of the system. The GPLR-SFGP plate exhibits outstanding characteristics, including superior stiffness and a reduced mass, rendering it a suitable choice for exterior applications in airplanes, automobiles, and high-speed railways. The findings in this study can provide valuable insight into the key design parameters that significantly affect the performance of GPLR-SFGP plates, enabling future design efforts aimed at enhancing their efficacy and robustness in real-world applications.

Free Vibration and Dynamic Analysis on Free-Constrained Layer of Graphene Based on Composite Conical Shell via Jacobi–Ritz Method

Excessive vibration has always been a serious problem for conical shell structures, while the application of the graphene-based free-constrained layer (GFCL) based on carbon fiber-reinforced composite (CFRC) structure is a novel way to improve structural performance. An analytical model for vibration and dynamic characteristics of the GFCL-CFRC conical shell resting on the Winkler–Pasternak elastic foundation with arbitrary boundary conditions is constructed, and four types of GFCL porosity distribution and GFCL dispersion pattern are considered in this model. The multi-segment technique and virtual spring technique are utilized to simulate arbitrary boundary conditions. Then, the first-order shear deformation theory (FSDT) and Hamilton’s principle are employed to obtain the motion equation of the GFCL-CFRC conical shell, and the motion equation of the GFCL-CFRC conical shell is solved by the Ritz method. In conclusion, the dispersion mode of GFCL, thickness ratio of GFCL, and fiber angle have influence on the dynamic performance. With a reasonable design, the dynamic performance of the GFCL-CFRC conical shell can be further improved.

An analytical closed-form solution for free vibration and stability analysis of curved sandwich panels made of porous metal-foam core and nanocomposite reinforced face-sheets

This paper aims to obtain an analytical solution for free vibration and stability analysis of shallow curved sandwich panels made of a porous metal-foam core and nanocomposites reinforced sheets resting on a Winkler-Pasternak elastic foundation. In this regard, two nanocomposites, carbon nanotubes (CNTs) and graphene nanoplatelets (GNPs) are considered for reinforcing the face-sheets. The governing equations are developed by considering the transversal shear strains based on the first-order shear deformation theory (FSDT). The obtained system of differential equations representing the dynamic equilibrium and compatibility of the proposed curved sandwich panels are solved analytically, using the principal of Galerkin method. In order to examine the results of developed formulation, some benchmark examples are adopted from the existing literature and the obtained results corresponding to the considered examples are compared with those are presented in the references. Using the achieved numerical results in the framework of a comprehensive parametric study, the effects of various material and geometrical factors and also various boundary conditions on the natural frequencies and buckling loads of the proposed sandwich panels are assessed. By performing the extensive parametric study it was observed that several results corresponding to the free vibration and stability of the proposed sandwich panels can be achieved by the present method without spending much time for computations. On the contrary, in order to achieve the similar results using the numerical methods, a time-consuming process is required which can justify the novelty of the present work.

Vibrational characteristics of attached hemispherical-cylindrical ocean shell-structure reinforced by CNT nanocomposite with graded pores on two-parameter elastic foundations applicable in offshore components

One of the essential simulations of offshore components of ocean and marine structures can be counted as a modeling of the interaction between the systems and soil's behavior on the dynamic of the offshore systems. Based on this, for the first time, the fundamental frequencies of the porous nanocomposite attached hemisphere-cylinder shell system (AHCSS), counted as ocean and marine offshore structures, reinforced by carbon nanotube (CNT) rested on the Winkler-Pasternak elastic foundations, added up as soil's behavior simulation, are investigated in this paper. In detail, to expand the effectiveness of the CNT along the matrix, eight distinct functionally graded (FG) forms besides uniform distributions are prepared. Also, two forms of porosity which are FG distributed along the FG-CNT nanocomposite, are realized to add the effect of the porosity on the nanocomposite. Addedly, the first shear deformation theory and Donnell's scheme are coupled to arrange the affiliations of the AHCSS. In addition, Hamilton's principle gets the central motion equations (CMEs) of the AHCSS. Then, the half-analytical procedure, named the generalized differential quadrature method, discretizes the CMEs. Finally, the eigenvalue method finds the frequencies of the porous nanocomposite AHCSS. Remarkably, various numerical samples are prepared and solved to indicate the effects of the Winkler-Pasternak foundations, boundary cases, porosity forms, FG-CNT models, and physical values on the frequencies of the porous nanocomposite AHCSS.

Nano-Clay Platelet Integration for Enhanced Bending Performance of Concrete Beams Resting on Elastic Foundation: An Analytical Investigation.

Acknowledging the growing impact of nanotechnologies across various fields, this engaging research paper focuses on harnessing the potential of nano-sized materials as enhancers for concretes. The paper emphasizes the strategic integration of these entities to comprehensively improve the strength and performance of concrete matrixes. To achieve this, an analytical study is conducted to investigate the static behavior of concrete beams infused with different types of clay nano-platelets (NC's), employing quasi-3D beam theory. The study leverages the effective Eshelby's homogenization model to determine the equivalent elastic characteristics of the nanocomposite. The intricate interactions of the soil medium are captured through the use of a Winkler-Pasternak elastic foundation. By employing virtual work principles, the study derives equations of motion and proposes analytical solutions based on Navier's theory to unravel the equilibrium equations of simply supported concrete beams. The results shed light on influential factors, such as the clay nano-platelet type, volume percentage, geometric parameters, and soil medium, providing insights into the static behavior of the beams. Moreover, this research presents previously unreported referential results, highlighting the potential of clay nano-platelets as reinforcements for enhancing structural mechanical resistance.

Geometrically nonlinear dynamic analysis of a damped porous microplate resting on elastic foundations under in-plane nonuniform excitation

This article uses the semi-analytical approach to study the combined nonlinear vibration and nonlinear response of a damped porous microplate under nonuniform periodic parametric excitation to understand the complete nonlinear dynamic behavior of the plate. The plate is supported by a Winkler-Pasternak elastic foundation and modeled using modified strain gradient and third-order shear deformation theories to simulate the small-scale effects and shear deformation, respectively. Using Hamilton’s principle, the governing partial differential equations of motion are derived and solved using Galerkin’s method to convert them into ordinary differential equations (ODEs). These ODEs are solved using a combined incremental harmonic balance (IHB) and arc-length continuation approaches to get the nonlinear vibration (frequency–amplitude curves). The same ODEs are solved using the Newmark-β technique to obtain the nonlinear response (time–amplitude curves). The effect of elastic foundation parameters and aspect ratio on mode shape is presented. The effect of parameters such as the porosity coefficient, type of porosity, Winkler-Pasternak elastic foundation parameters, different size-dependent theories, plate thickness, size of plate, damping coefficient, different loading profiles, and loading concentrations on the nonlinear vibration and nonlinear response is examined. Also, the dependence of initial displacements on the frequency–amplitude curves with respect to the excitation frequency is demonstrated with the help of time-amplitude curves.

Nonlocal Strain Gradient Model for the Nonlinear Static Analysis of a Circular/Annular Nanoplate.

A nonlinear static analysis of a circular/annular nanoplate on the Winkler-Pasternak elastic foundation based on the nonlocal strain gradient theory is presented in the paper. The governing equations of the graphene plate are derived using first-order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT) with nonlinear von Karman strains. The article analyses a bilayer circular/annular nanoplate on the Winkler-Pasternak elastic foundation. HSDT while providing a suitable distribution of shear stress along the thickness of the FSDT plate, eliminating the defects of the FSDT and providing good accuracy without using a shear correction factor. To solve the governing equations of the present study, the differential quadratic method (DQM) has been used. Moreover, to validate numerical solutions, the results were compared with the results from other papers. Finally, the effect of the nonlocal coefficient, strain gradient parameter, geometric dimensions, boundary conditions, and foundation elasticity on maximum non-dimensional deflection are investigated. In addition, the deflection results obtained by HSDT have been compared with the results of FSDT, and the importance of using higher-order models has been investigated. From the results, it can be observed that both strain gradient and nonlocal parameters have significant effects on reducing or increasing the dimensionless maximum deflection of the nanoplate. In addition, it is observed that by increasing load values, the importance of considering both strain gradient and nonlocal coefficients in the bending analysis of nanoplates is highlighted. Furthermore, replacing a bilayer nanoplate (considering van der Waals forces between layers) with a single-layer nanoplate (which has the same equivalent thickness as the bilayer nanoplate) is not possible when attempting to obtain exact deflection results, especially when reducing the stiffness of elastic foundations (or in higher bending loads). In addition, the single-layer nanoplate underestimates the deflection results compared to the bilayer nanoplate. Because performing the experiment at the nanoscale is difficult and molecular dynamics simulation is also time-consuming, the potential application of the present study can be expected for the analysis, design, and development of nanoscale devices, such as circular gate transistors, etc.

Porosity-dependent impact study of a plate with Winkler-Pasternak elastic foundations reinforced with agglomerated CNTs

PurposeThe purpose of this article is to investigate the porosity-dependent impact study of a plate with Winkler–Pasternak elastic foundations reinforced with agglomerated carbon nanotubes (CNTs).Design/methodology/approachBased on the first-order shear deformation plate theory, the strain energy related to elastic foundations is added to system strain energy. Using separation of variables and Lagrangian generalized equations, the nonlinear and time-dependent motion equations are extracted.FindingsVerification examples are fulfilled to prove the precision and effectiveness of the presented model. The impact outputs illustrate the effects of various distribution of CNTs porosity functions along the plate thickness direction, Winkler–Pasternak elastic foundations and different boundary conditions on the Hertz contact law, the plate center displacement, impactor displacement and impactor velocity.Originality/valueThis paper investigates the effect of Winkler–Pasternak elastic foundations on the functionally graded porous plate reinforced with agglomerated CNTs under impact loading.

Higher-order model with interlaminar stress continuity for multi-directional FG-GRC porous multilayer panels resting on elastic foundation

In this paper, the bending behaviors of multilayer panels composed of multi-directional functionally graded porous graphene reinforced composites (GRCs) resting on a two-parameter Winkler-Pasternak elastic foundation are investigated for the first time based on a higher-order layerwise model with interlaminar stress continuity. Since the material properties are discontinuously distributed through the thickness of specific multilayer structure and multi-directional graded in each layer of the panels caused by the distribution of both graphene and porosities, traditional higher-order plate theories have difficulty in predicting continuous stress fields efficiently. For this reason, the proposed model adopts higher-order interpolation functions and piecewise descriptions for transverse shear-stress fields to obtain accurate results satisfying continuity of interlaminar stresses, accounting for the distribution of material properties along the thickness direction and multilayer structure of the panel. Furthermore, to perform efficient numerical results for multi-directional functionally graded materials, the proposed higher-order layerwise model is combined with graded finite element method (GFEM) by sampling material properties directly at the Gaussian points within each element. Lastly, numerical examples are presented to verify the accuracy and efficiency of the proposed model, and the effects of porosity and graphene distribution patterns, graphene weight fraction, porosity coefficient, and elastic foundation parameters are investigated.

Thermoacoustic response of polymethyl methacrylate composite shells reinforced by carbon nanotube resting on Winkler–Pasternak elastic foundation

This paper presented an analytical method for investigating the vibroacoustic response of polymethyl methacrylate composite shells reinforced by CNTs resting on an elastic foundation under thermal loading. This structure is submerged in a moving fluid and excited by an acoustics plane wave. The CNTs are assumed to have a graded distribution in the polymethyl methacrylate matrix. The properties of the CNTs are assumed to be temperature-dependent. Different types of CNTs distributions such as UD, X, V, and O are considered. The Winkler–Pasternak model is applied to describe the vibrational behavior of the elastic foundation. The equations of motion of the structure are based on Hamilton’s principle using the TSDT, considering the effects of shear and rotation. To validate the present analytical method, the obtained results are compared with those of other researchers. Next, the effects of various significant parameters such as elastic foundation, temperature gradient, CNTs distributions, and Mach number on the TL are studied. The results show increasing the temperature reduces the TL because the stiffness bending of the structure decreases. It can be seen that the addition of CNTs to the matrix, with an appropriate volume fraction, improves the sound transmission loss due to the enhancement of stiffness.