This paper investigates the nonlinear vibration control and dynamic instability of cantilevered piezoelectric microlaminates with geometric imperfections, taking into account an unknown time-varying delay and external disturbances. The dynamic equations for the microlaminates are derived using the MCST, FSDT, nonlinear von Karman equation, and Hamilton’s principle. The Galerkin method is applied to discretize these nonlinear partial differential equations. It is found that initial imperfections can trigger the dynamic phenomenon. To control the vibrations, two control strategies are implemented: The adaptive radial basis function (ARBF) feedback linearization controller and the ARBF sliding mode controller. A comparative analysis of the time histories, control voltage, and adaptive parameter responses of the two controllers under four different external conditions is conducted. The results indicate that both controllers effectively suppress the vibration of the system, thereby mitigating the risk of dynamic instability. Additionally, the findings suggest that, although both controllers meet the targeted control objectives, the ARBF sliding mode controller not only demonstrates enhanced control performance but also necessitates lower energy consumption in comparison to the ARBF feedback linearization controller.

This paper deals with evolutionary von Karman equation with rotational forces not clamped boundary conditions and interior nonlinear damping for shallow shell. We study the existence and uniqueness of a weak solution under weaker conditions to the damping and lower regularity of the internal force. Finally, we employ the finite difference method to approximate the solution to the problem.

Nonlinear statics of magneto-electro-elastic nanoplates considering flexomagnetoelectric effect based on nonlocal strain gradient theory

Stability and nonlinear vibration of a fuel rod in axial flow with geometric nonlinearity and thermal expansion

In this paper, we derive a general stability result for a viscoelastic von Karman equation without the condition that involves differential inequalities on the kernel. The kernel just stays below a certain known function as like where is a known function satisfying some specific properties. By considering such kernel functions, we expand earlier results for viscoelastic von Karman equations in the literature.

This paper is concerned with mathematical analysis for the viscoelastic von Karman equation with memory and time‐varying delay. Based on the existence and uniqueness of the strong solution established by Galerkin method, the finite dimensional global attractor for the system without time‐varying delay has been shown by using the quasi‐stability theory. The difficulties for our desired well‐posedness are the limiting of fourth‐order derivative terms and the von Karman bracket term, which need some delicate estimates to achieve. The key point for the dynamics is to obtain the quasi‐stability for gradient system, which needs the higher regularity of trajectory especially the von Karman bracket.

In this paper, by using invariants of Riemann and general solutions of Euler-Poisson-Darboux-Riemann equations, a new class of exact solutions of von Karman's equation in the nonlinear theory of gas dynamics is given.

In this paper we are interested to the dynamic von Karman equations coupled with viscous damping and without rotational forces, (α=0) [Chueshov I., Lasiecka I. (2010)], this problem describes the buckling and flexible phenomenon of small nonlinear vibration of vertical displacement to the elastic plates. Our fundamental goal is to establish the existence and the uniqueness to the weak solution for the so-called global energy, under assumption F0∈H3+ϵ(ω). Finally for illustrate our theoretical results we use the finite difference method.

Frequency response analysis of edge-cracked magneto-electro-elastic functionally graded plates using extended finite element method

In this paper, we consider a viscoelastic von Karman equation with damping, delay, and source effects of variable exponent type. Firstly, we show the global existence of solution applying the potential well method. Then, by making use of the perturbed energy method and properties of convex functions, we derive general decay results for the solution under more general conditions of a relaxation function. General decay results of solutions for viscoelastic von Karman equations with variable exponent nonlinearities have not been discussed before. Our results extend and complement many results for von Karman equations in the literature.

The nonlinear dynamic response of magnetoelectroelastic (MEE) functionally graded plates (MEEFG) subjected to blast loads and operating in a temperature environment is examined in this study. The goal of this research is to investigate the impact of pyrocoupling along with the blast loads on the coupled nonlinear behavior of smart MEEFG plates. The material properties of MEEFG plates are calculated using the well-known modified power law. The kinematics of the plate is supposed to follow Reddy's higher-order shear deformation theory (HSDT). Also, von-Karman's equations are used to bring nonlinearity into the model. The blast loads are calculated using an exponential function and are expected to be dispersed evenly across the plate's surface. The principle of virtual work is used to develop the governing equations using finite element (FE) methods, which is then solved using the direct iterative technique. The numerical examples are offered to show how gradient patterns, gradient index, and blast loading parameters affect the nonlinear dynamic response.

<abstract> We study a viscoelastic von Karman equation of memory type with acoustic boundary conditions. Utilizing some properties of convex functions and the perturbed energy method, we build a general decay result when the kernel function $ k $ is a very general type. This work extends and complements some previous decay results of solutions for von von Karman equations. </abstract>

Abstract The Foppl-von Karman equations describe the highly nonlinear postbuckling behavior of elastic plates but are notorious for their unwieldiness. Owing to the lack of a sufficiently general so...

In this article, we deal with a strongly damped von Karman equation with variable exponent source and memory effects. We investigate blow-up results of solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy. Furthermore, we estimate not only the upper bound but also the lower bound of the blow-up time.

General decay for a von Karman equation with memory and time-varying delay

Linking the von Karman equations to the design of steel plates

For a thin plate, if the deformation is on the order of the thickness and stay elastic, linear theory might not turn out correct results because it does not predict the in plane movement of the member. Therefore, to account for the inconsistencies of geometric nonlinearity, large deflection theory is required [1]. This report pertains to the analytical study dispensed to check the behavior of thin plate under fixed and pinned edge conditions, and for diverse thicknesses, under the small and large deflection theories. The deformation is additionally studied, supported by Von-Karman equations. Non linear analysis has been performed on FE model using the ANSYS software. The consequences of geometric nonlinearities are mentioned. Outline on conclusion of the theoretical and experimental results obtained, are compared so as to review the similarity of the modeling and theory.

The elastic post-buckling behavior of thin plates covers a relatively vast region in which geometry nonlinearity (large deflection) and material linearity (Hooke's low) are realized. In this region, a thin rectangular plate has constant stiffnesses in both orthogonal directions. Few simplified analysis guidelines have been analytically represented for in-plane stiffnesses of an elastic post-buckled thin plate subjected to biaxial loads. In this study, Marguerre's equations (the generalized form of von Karman equations), which describe the elastic post-buckling behavior of imperfect thin plates, are solved. Galerkin's method is used to solve these equations in a semi-analytical procedure. Simply supported imperfect thin rectangular plates are considered, and the stresses and displacements functions are obtained in two orthogonal directions to determine corresponding in-plane stiffnesses of the plate. Also, the maximum applicable load is obtained so that the material's linear behavior is maintained. The semi-analytical procedure has accuracy enough to predict the in-plane stiffness of post-buckled plates and can be easily used for practical purposes.

In this article, we consider a von Karman equation with variable exponent nonlinearities in a bounded domain . We firstly discuss an existence result of solutions by utilizing Faedo‐Galerkin approximation technique. Then, we undertake an investigation of asymptotic stability to the solutions by making use of the multiplier method.

This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Karman equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.