Nonlinear Vibrations Of Plates Research Articles

Some Boundary Problems, in Sobolev Spaces with Constant Exponents, Governed by the Nonlinear Operator of Plate Vibrations

In this paper, we aim to investigate certain nonlinear boundary problems within Sobolev spaces, where the exponents remain constant. We focus on the dynamically modified operator, incorporating a viscosity term into the nonlinear vibrations of plates. Vibrating plates have a broad range of applications. To address user requirements comprehensively, we've taken into account factors such as the geometric configuration, material density, plate thickness, and Poisson's ratio. After formulating the problems, our method involves converting them into hyperbolic-type nonlinear problems. In this study, we examine six boundary value problems, establishing existence and uniqueness theorems for each. Lastly, we establish the existence of a solution for the stationary problem by employing a variation of Brouwer's fixed point theorem.

Large-amplitude vibration analysis of FG rectangular plates under rapid cooling

In the present research, the large-amplitude vibration behavior of rectangular plates subjected to cooling shock is investigated based on a numerical solution approach. It is considered that the plates are made of functionally graded materials as a mixture of stainless steel and low-carbon steel. The thermoelastic properties are also considered temperature-dependent which are estimated based on available experimental data. According to the Mindlin plate theory, the nonlinear governing equations of motion together with corresponding boundary conditions are derived. The temperature profile is also obtained based on a one-dimensional Fourier-type transient heat conduction equation. Moreover, two time-dependent thermal loading scenarios for cooling shock on the plate’s top face are considered. To solve the problem numerically, the well-known generalized differential quadrature technique is used for discretization considering the Chebyshev–Gauss–Lobatto grid. In addition, the governing equations are traced in time by Newmark’s time integration scheme. After showing the validity of developed approach, a parametric study is presented to investigate the stress changes along the thickness and the effects of thermal load rapidity time, geometry, magnitude of thermal load, and material properties on the nonlinear vibrations of plates with various boundary conditions subjected to sudden decrease of temperature. It is concluded that the thermal load rapidity time has a significant role in the vibrational behavior and the stress distribution of the plate.

A new general analytical method for dynamical behaviors of functionally graded graphene nanoplatelet reinforced non-rectangular plates with many different shapes subjected to thermo-electro-mechanical load

This article studied a general analytical method for dynamical behaviors of various FG-GPLRC non-rectangular plates with a piezoelectric layer. For the first time, classical plate theory and Galerkin’s method are used to scrutinize the nonlinear vibration of plates with many different shapes. Profiles of new plates are defined by four edges, specifically two parallel opposite edges, and others are assumed to horizontal change according to arbitrary functions. Linear and nonlinear vibration of plates are confirmed with results obtained by Finite Element Method. Effects of environment, material, and geometrical parameters are studied. Besides, periodic and chaotic motions of plates are investigated.

Assessment of nonlinear vibrations of thin plates undergoing large deflection and moderate rotation using Jacobi elliptic functions

In this article, nonlinear free/forced vibrations of a plate undergoing large deflection and moderate rotation have been investigated based on Jacobi elliptic functions. The formulation has been established using a generalized von-Karman’s strain in which nonlinear terms of deflection and rotations are simultaneously included. The governing equations have been derived with the use of extended Hamilton’s principle assuming that the plate is under an external force of elliptic type. It is proved that conventional approximate solutions of nonlinear vibrations of plates based on single-harmonic assumption are inadequate to consider higher harmonics. Whereas, Jacobi elliptic function method considers the effects of higher-order harmonic leading to a more exact solution. This becomes more important when extra nonlinearity due to nonlinear rotations is included in the calculations.

Nonlinear vibrations and damping of fractional viscoelastic rectangular plates

Damping is largely increasing with the vibration amplitude during nonlinear vibrations of rectangular plates. At the same time, soft materials present an increase in their stiffness with the vibration frequency. These two phenomena appear together and are both explained in the framework of the viscoelasticity. While the literature on nonlinear vibrations of plates is very large, these aspects are rarely touched. The present study applies the fractional linear solid model to describe the viscoelastic material behavior. This allows to capture at the same time (i) the increase in the storage modulus with the vibration frequency and (ii) the frequency-dependent nonlinear damping in nonlinear vibrations of rectangular plates. The solution to the nonlinear vibration problems is obtained through Lagrange equations by deriving the potential energy of the plate and the dissipated energy, both geometrically nonlinear and frequency dependent. The model is then applied to a silicone rubber rectangular plate tested experimentally. The plate was glued to a metal frame and harmonically excited by stepped sine testing at different force levels, and the vibration response was measured by a laser Doppler vibrometer. The comparison of numerical and experimental results was very satisfactorily carried out for: (i) nonlinear vibration responses in the frequency and time domain at different excitation levels, (ii) dissipated energy versus excitation frequency and excitation force, (iii) storage energy and (iv) loss factor, which is particularly interesting to evaluate the plate dissipation versus frequency at different excitation levels. Finally, the linear and nonlinear damping terms are compared.

Nonlinear vibration and dynamic buckling of eccentrically oblique stiffened FGM plates resting on elastic foundations in thermal environment

This paper presents a semi-analytical approach to investigate the nonlinear dynamic response and vibration of eccentrically oblique stiffened functionally graded plate resting on elastic foundation. The Lekhnitskii's smeared stiffener technique is improved by using a transformation technique for oblique stiffeners. Governing equations are solved by classical shell theory, Galerkin method, stress function with temperature-dependent material effects. The results show the influences of geometrical parameters, material properties, imperfection, the elastic foundations, eccentrically oblique stiffeners, mechanical loads and temperature on the nonlinear dynamic response and nonlinear vibration of plates. The numerical results in this paper are compared with the results reported in other reports.

Nonlinear vibrations of plates in axial pulsating flow

A theoretical approach is presented to study nonlinear vibrations of thin infinitely long and wide rectangular plates subjected to pulsatile axial inviscid flow. The flow is set in motion by a pulsating pressure gradient. The case of plates in axial uniform flow under the action of constant transmural pressure is also addressed for different flow velocities. The plate is assumed to be periodically simply supported in both in-plane directions with immovable edges and the flow channel is bounded by a rigid wall. In this way the system under study is a finite rectangular plate with conditions on the fluid and the plate boundaries coming from the periodicity of the infinite system. The equations of motion are obtained based on the von Kárman nonlinear plate theory retaining in-plane inertia via Lagrangian approach. The fluid model is based on the potential flow theory. The resulting Lagrange equations of motion of the coupled system contain quadratic and cubic nonlinear terms and are studied by using a code based on the pseudo-arc-length continuation and collocation scheme. The effect of different system parameters such as flow velocity, pulsation amplitude, pulsation frequency and channel pressurization on the stability of the plate and its geometrically nonlinear response to pulsating flow are fully discussed. It has been found that the presence of positive transmural uniform pressure and small pulsation frequency would destroy the pitchfork bifurcation (divergence) that flat plates exhibit when subjected to uniform flow. Moreover, in case of zero uniform transmural pressure numerical results show a hardening type behavior for the entire flow velocity range when the pulsation frequency is spanned in the neighborhood of the plate׳s fundamental frequency. On the contrary, a softening type behavior is presented when a uniform transmural pressure is introduced.

Nonlinear vibrations of circular plates with notches. Method of R-functions

We investigate vibrations of geometrically nonlinear circular plates with two notches. For the determination of natural frequencies of vibrations, the method of R-functions is used. Nonlinear vibrations of a plate are expanded in eigenmodes of linear vibrations containing R-functions. As a result of using the Bubnov–Galerkin method, we obtain a dynamic system with three degrees of freedom, which is investigated by the method of multiple scales.

Analysis of Nonlinear Free Vibration of Circular Plates With Cut-Outs Using R-Function Method

Geometrically nonlinear vibrations of circular plate with two cut-outs are simulated by the von Karman equations with respect to displacements. The combination of the Rayleigh–Ritz method and the R-function method, which allows satisfying all boundary conditions, is applied to obtain the vibration modes of the plate. The nonlinear vibrations are expanded using these vibrations modes. The dynamical system with three degrees of freedom is derived by Galerkin method. The influence of cut-outs size on linear and nonlinear vibrations of the plate is analyzed. For different parameters of cut-outs, different internal resonances occur in the plate. The nonlinear vibrations of the system for different internal resonances are analyzed by multiple scale method.

Nonlinear vibration of plates by the hierarchical finite element and continuation methods

The hierarchical finite element (HFEM) and the harmonic balance methods are applied to analyse the geometrically nonlinear vibration of thin, isotropic plates. The von Kármán type of nonlinear strain–displacement relationships are used. Symbolic computation is employed in the derivation of the model. The equations of motion are solved by the Newton and continuation methods. Free and steady-state forced vibration are analysed. The excitations considered are harmonic plane waves at both normal and grazing incidence. The stability of the obtained solutions is investigated by studying the evolution of perturbations to the solutions. The convergence properties of the HFEM and the influence of the middle plane in-plane displacements are discussed and results compared with published experimental and numerical results.

Nonlinear Vibrations of Plates: An Update of Recent Research Developments

This paper comprises a survey on the nonlinear vibration analysis of plates, with emphasis on research carried out since 1987. Most of the research reviewed here deals with the effects of geometric nonlinearity on the vibration behavior of plates. Complicating factors such as material nonlinearity, geometric imperfections, transverse shear and rotatory inertia effects, and magnetic fields on the vibration behavior are included. Recent developments in the analytical and numerical methods of solution of isotropic, orthotropic as well as laminated, composite plates are presented. Experimental, analytical, and numerical investigations are included for all the cases reviewed and some general remarks are presented along with suggestions for future research directions.

Numerical study of nonlinear vibrations of plates and shells with two-frequency excitation

Numerical study of nonlinear vibrations of plates and shells with two-frequency excitation

Forced steady nonlinear vibrations of plates compressed by supercritical longitudinal forces while under a transverse harmonic load

Forced steady nonlinear vibrations of plates compressed by supercritical longitudinal forces while under a transverse harmonic load

Stability and bifurcations of the steady-state modes in nonlinear vibrations of plates and shells

A method is proposed for numerical investigation of stability and bifurcations of nonlinear vibrations of plates and shells under the action of periodic loads. A system of equations of motion are formed by the methods of finite elements and generalized coordinates by accounting for the geometrical nonlinearity of the objects. The obtained system of differential equations is solved by the parameter extension method. As a result, steady-state curves of the nonlinear vibrations are constructed. Branching solutions at the points of bifurcation are obtained by the method of projections. The proposed method makes it possible to determine the subharmonic solutions and investigate their stability over a wide range of load parameters. Nonlinear vibrations of a series of plates and shells are investigated.

Nonlinear forced vibration of orthotropic rectangular plates using the method of multiple scales

The method of multiple scales (MMS) in conjunction with the Galerkin method is used to analyze the nonlinear forced and damped response of a rectangular Orthotropic plate subjected to a uniformly distributed harmonic transverse loading. The effects of damping and in-plane loads are considered. The analysis considers simply supported as well as clamped plates. For each case, both movable and immovable edge conditions are considered. By using MMS, all possible resonances such as primary, subharmonic, and superharmonic reso- nances are studied. For the undamped response without in-plane loading, comparisons of the MMS results with those obtained by the finite-element method show excellent agreement. EVELOPMENT of composite materials comprising lam- inates of Orthotropic or multilayered anisotropic materi- als Recently has been receiving substantially growing research efforts. Due to the increasing demands for energy-efficient, high strength, minimum weight aircraft designs, many re- searchers believe that the use of composite materials offers promising alternatives for aircraft designs. Thin, laminated, composite plates subjected to transverse periodic loadings could encounter deflections of the order of plate thickness or even higher. Responses of this kind cannot be predicted by using the linear theory. Consequently, the need to study large- amplitude-deflection vibrations of composite structures is of paramount importance. The literature survey shows that the equations of motion for the large deflection analysis of heterogeneous anisotropic plates using the von Karman geometrical nonlinearity were first considered by Whitney and Leissa.1 Based on these equa- tions, different methods of analysis have been developed by several researchers. An excellent number of collections on nonlinear free and forced vibrations of composite plates cov- ering the work through 1979 can be found in the comprehen- sive book by Chia. 2 Bert3 has conducted a survey on the dynamics of composite plates for the period 1979-81. A re- view of the literature on nonlinear vibrations of plates can be found in the review paper by Sathyamoorthy4 and the book by Nayfeh and Mook.5 Large deflection analysis of symmetrically laminated rec- equations of motion are presented in terms of the lateral displacement and stress function. The equations are nondi- mensionalized following the transformation introduced by Brunelle and Oyibo.9 Though multimode analysis can be treated, the present study is focused on single-mode analysis. A deflection function representing the first mode and satisfy- ing the boundary conditions is assumed, and subsequently the stress function is found. Next, the Galerkin method is applied to obtain the modal equation, which is solved analytically by using the method of multiple scales (MMS).10 The MMS also provides solutions for subharmonic and superharmonic reso- nances. The effects of damping ratio, plate aspect ratio, and in-plane loading then are studied.