The nonlinear vibration of an initially stressed laminated plate

Nonlinear partial differential equations of motion for a laminated plate in a general state of non-uniform initial stress are presented in various plate theories. This study uses Lo’s displacement field to derive the governing equations. The higher-order terms in Lo’s theory can be disregarded, to obtain the equations of simpler forms and even other theories for laminated plate. These nonlinear partial equations are transformed to ordinary nonlinear differential equations using the Galerkin method. The Runge–Kutta method is used to obtain the ratio of nonlinear frequency to linear frequency. The numerical solutions of an initially stressed laminate plate based on various plate theories obtained by the Galerkin and Runge–Kutta method are presented herein. Using these equations with various theories, the nonlinear vibration behavior of laminated plate is studied. The results show that apparent discrepancies exist among the various displacement fields, which indicates the transverse shear strain, normal strain and initial stress state have great effect on the vibration behavior of laminate plate under nonlinear vibration.     

Two-dimensional equations for electroelastic plates with relatively large shear deformations

A set of two-dimensional, nonlinear equations for electroelastic plates in moderately large thickness-shear deformations is obtained from the variational formulation of the three-dimensional equations of nonlinear electroelasticity by expanding the mechanical displacement vector and the electric potential into power series in the plate thickness coordinate. As an example, the equations are used to study nonlinear thickness-shear vibrations of a quartz plate driven by an electrical voltage. Nonlinear electrical current amplitude-frequency behavior near resonance is obtained. The equations and results are useful in the study and design of piezoelectric crystal resonators and the measurement of nonlinear material constants of electroelastic materials.     

Finite element analysis for geometrically nonlinear deformations of smart functionally graded plates using vertically reinforced 1-3 piezoelectric composite

In this paper, a finite element model has been developed for the geometrically nonlinear static analysis of simply supported functionally graded (FG) plates integrated with a patch of vertically reinforced 1-3 piezoelectric composite material acting as a distributed actuator. The material properties of the functionally graded substrate plate are assumed to be graded only in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The analysis of the electro-elastic coupled problem includes the transverse deformations of the overall plate to utilize the transverse normal actuation by the distributed actuator for counteracting the nonlinear deformations of smart functionally graded plates. The nonlinear governing equations of equilibrium are solved by using direct iteration method with under-relaxation. The numerical illustrations suggest the potential use of the distributed actuator made of vertically reinforced 1-3 piezoelectric composite material for active control of nonlinear deformations of smart functionally graded plates. The effect of variation of piezoelectric fiber orientation in the distributed actuator on its control authority for counteracting the nonlinear deformations of smart functionally graded plates has also been investigated.     

Nonlinear analysis of smart functionally graded annular sector plates using cylindrically orthotropic piezoelectric fiber reinforced composite

This work deals with the geometrically nonlinear thermo-electro-elastic analysis of functionally graded (FG) annular sector plates integrated with the annular patches of cylindrically orthotropic piezoelectric fiber reinforced composite (PFRC). The annular patches with an external voltage across their thickness act as the distributed actuators and their performance in controlling the nonlinear flexural deformations of the host FG plates is investigated. The temperature field is assumed to be spatially uniform over the plate surfaces and varied through the thickness of the substrate FG plates. The temperature-dependent material properties of the FG plates are assumed to be graded in the thickness direction of the plates according to a power-law distribution while the Poisson’s ratio is assumed to be a constant over the domain of the substrate plate. A finite element model of the overall smart FG annular sector plate is developed based on the first order shear deformation theory and the Von Karman nonlinear strain–displacement relations. The governing nonlinear finite element equations are derived employing the principle of minimum potential energy and solved using direct iteration method. The numerical results illustrate significant control authority of the cylindrically orthotropic PFRC annular patches for active control of nonlinear deformations of the substrate FG annular sector plates. The numerical results also reveal the best radial and circumferential locations of the annular PFRC patches for effective control. For a specified circumferential stretch of the annular PFRC patches, their minimum radial length is numerically estimated in such a way that the performance of the overall smart FG plate is not affected significantly. The effects of the material properties and the temperature of the host FG plate on the performance of the annular PFRC patches are also discussed.     

旋转几何非线性复合材料薄壁梁的自由振动分析

研究具有几何非线性的旋转复合材料薄壁梁的自由振动。梁的变形引入了Von Kármán几何非线性, 基于Hamilton原理和变分渐进法 (Variational-Asymptotical Method -VMA),导出旋转复合材料薄壁梁的非线性振动偏微分方程组。采用Galerkin法将振动方程离散化为常微分方程组。借助于谐波平衡法 (Harmonic Balance Method -HBM) 建立自由振动的振幅-非线性固有频率关系方程。将上述方程化为非线性特征值问题,采用迭代算法进行求解。将所建立的旋转复合材料薄壁梁非线性自由振动分析模型和计算方法,应用于周向均匀刚度配置(Circumferentially Uniform Stiffness –CUS) 构型复合材料薄壁梁,通过数值计算揭示了纤维铺层角、旋转速度对非线性振动固有频率-振幅关系的影响。     

横向磁场中机械载荷作用梁式薄板的非线性主共振

研究了磁场环境中受机械载荷作用梁式薄板的非线性主共振问题。在给出薄板的非线性电磁弹性耦合运动基本方程及电磁力表达式的基础上，得到了横向稳恒磁场和机械载荷共同作用下梁式薄板的振动方程。应用伽辽金积分法，并进行无量纲化处理，进一步导出了相应的非线性振动微分方程。采用平均法对主共振问题进行了求解，得到了稳态运动下的幅频响应方程。通过算例给出了几种情况下的幅频响应曲线图和时间历程图，分析了板厚和磁场对系统振动的影响。     

Nonlinear Free Vibration of In-Plane Functionally Graded Rectangular Plates

Nonlinear free vibration of functionally graded (FG) plates with in-plane material inhomogeneity subjected to different boundary conditions is presented. The nonlinear equations of motion and the related boundary conditions are extracted based on the classical plate theory. Green's strain tensor together with von Kármán assumptions is employed to model the geometrical nonlinearity. The differential quadrature method as an efficient and accurate numerical tool is employed to discretize the governing equations in spatial domain. After validating the presented approach, parametric studies are performed to clarify the effects of different parameters on the nonlinear frequency parameters of the in-plane FG plates.     

考虑静水压力的压电弹性层合圆柱壳动力响应的控制模型

基于经典的层合板理论和Navier解法,对静水压作用下压电弹性层合圆柱壳的动力问题的主动控制进行了研究。首先由Hamilton原理导出压电弹性层合壳的非线性动力基本方程。利用压电材料的正、逆压电效应,通过闭环方式,采用速度反馈控制方法得到了任意形式动载作用下带压电感测层/激励层的简支层合圆柱壳动力响应的主动控制模型。数值算例中对于三种不同的外载条件下该控制模型对圆柱壳的动力响应的控制效果进行了研究。结果表明本文中提出的控制模型能够有效抑制动载作用下结构的振动。     

Nonlinear vibration of an initially stressed laminated plate according to a higher-order theory

The nonlinear behavior of laminated plates in a general state of non-uniform initial stress was studied at large vibration amplitudes. The nonlinear governing equations of this study were derived using a higher-order theory approach. The results were compared with the Mindlin plate theory’s results. The results showed that the higher-order shear deformation terms had a significant influence on the plate in a large amplitude vibration when the thickness ratio decreases and the plate was stacked with less layers. In addition, the effect of Young’s modulus in the thickness direction on the frequency ratio was significant for the two-layered plate. However, the results of the four-layered plates were not affected too much.     

含焊接残余应力薄圆板结构自由振动近似解

研究焊接残余应力对薄圆板结构振动特性的影响,解决薄圆板结构振动中存在非均匀分布预应力问题。根据含预应力结构的应变-应力方程,建立含预应力薄圆板结构的运动控制方程。基于Rayleigh-Ritz法构造Lagrange能量泛函方程。将预应力和位移试函数展开成三角级数形式,对含预应力薄圆板结构的自由振动问题进行求解。以周边简支边界薄圆板结构为例,对比焊接残余应力的不同分布形式对薄圆板结构固有频率及振型的影响。数值计算结果验证了所提方法的有效性,可应用于解决任意分布预应力问题。     

Nonlinear behavior of functionally graded circular plates with various boundary supports under asymmetric thermo-mechanical loading

The equilibrium equations of the first-order nonlinear von Karman theory for FG circular plates under asymmetric transverse loading and heat conduction through the plate thickness are reformulated into those describing the interior and edge-zone problems of the plate. A two parameter perturbation technique, in conjunction with Fourier series method is used to obtain analytical solutions for nonlinear behavior of functionally graded circular plates with various clamped and simply-supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified with known results in the literature. The load–deflection curves for different loadings, boundary conditions, and material constant in a solid circular plate are studied and discussed. It is shown that the behavior of FG plates with clamped or simply-supported boundary conditions are completely different. Under thermo-mechanical loading, snap-through buckling behavior is observed in simply-supported FG plates which are immovable in radial direction. Moreover, it is found that linear theory is inadequate for analyzing FG and also homogenous plates with immovable boundary supports in radial direction and subjected to thermal loading, even for deflections that are normally considered small.     

Imperfection sensitivity in the nonlinear vibration of initially stresses functionally graded plates

In this paper, the nonlinear partial differential equations of nonlinear vibration for an imperfect functionally graded plate (FGP) in a general state of arbitrary initial stresses are presented. The derived equations include the effects of initial stresses and initial imperfections size. The material properties of a FGP are graded continuously in the direction of thickness. The variation of the properties follows a simple power-law distribution in terms of the volume fractions of the constituents. Using these derived governing equations, the nonlinear vibration of initially stressed FGPs with geometric imperfection was studied. The present approach employed a perturbation technique, the Galerkin method and the Runge–Kutta method. The perturbation technique was used to derive the nonlinear governing equations. The motion of imperfect FGPs was obtained by performing the Galerkin method and then solved by the Runge–Kutta method. Numerical solutions are presented for the performances of perfect and imperfect FGPs. The nonlinear vibration of a simply supported ceramic/metal FGP was solved. It is found that the initial stress, geometric imperfection and volume fraction index greatly change the behavior of nonlinear vibration.     

Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.     

Geometrically nonlinear rectangular simply supported plates subjected to a moving mass

The dynamic deformation of a geometrically nonlinear rectangular simply supported plate under a moving lumped mass is evaluated using mode expansion method. The governing differential equations of motion for a largely deformable rectangular plate are derived using Lagrange method based on appropriate in and out-of-plane spatial functions which satisfy the proposed boundary conditions. Although the proposed procedure is applicable for any arbitrary edge boundary conditions, only the simply supported plates are addressed in the present work. On the other hand, all inertial components of the moving mass are included in the derivation of the equations of motion. A numerical example is used to study the dynamic behavior of the plate, considering large deformations. The obtained results indicate that ignoring the geometric nonlinearity in determining the vertical deformations of plates under the effect of moving masses, especially those of considerable weight and velocity, results in excessively large linear amplitudes leading to an unfavorable conservative structural design.     

Non-linear active control of FG beams in thermal environments subjected to blast loads with integrated FGP sensor/actuator layers

Non-linear active control of dynamic response of functionally graded (FG) beams with rectangular cross-section in thermal environments exposed to blast loadings is presented. Two FG piezoelectric layers are bonded to the beam surfaces to act as sensor and actuator. Non-linear equations of motion of the smart beam are derived based on the first-order shear deformation theory and the von Karman geometrical non-linearity. Constant velocity feedback algorithm is used to control the dynamic response of the FG beam actively through closed loop control. The generalized differential quadrature method together with the Newmark-beta scheme is utilized to solve the non-linear partial differential equations in spatial and time domains. The resulted non-linear algebraic equations are then solved using the modified Newton–Raphson method. A detailed analysis of the influence of the geometric non-linearity, material parameters and temperature field on the active vibration control of FG beams subjected to various impulsive loads is carried out.     

Optimal shapes of PZT actuators for laminated structures subjected to displacement or eigenfrequency constraints

In this paper, dynamics, electromechanical couplings, and control of piezoelectric laminated cylindrical shells and rectangular plates are investigated. It is assumed that the piezoelectric layers are distributed on the top and bottom surfaces of the structures. First of all the governing equations and boundary conditions including elastic and piezoelectric couplings are formulated and solutions are derived. Then control of the plate/shells deflections and natural frequencies using high control voltages are studied in order to optimize the structural response. The present formulation of optimal design introduces boundaries of piezoelectric patches as new class of design variables. In addition, classical design variables in the form of ply orientation angles of orthotropic layers are also taken into account. For the actuator/actuator configuration, it was shown that the piezoelectric actuators can significantly reduce deformations/eigenfrequencies of the composite plate. Those effects were dependent on the value of the applied voltage. It was demonstrated that the proper choice of the actuator area is more efficient in reducing deflections/eigenfrequencies. The accuracy of optimal design are verified both with the aid of the FE package ABAQUS and using the standard Rayleigh-Ritz method. The results concerning active vibration control for axisymmetric cylindrical shells are also discussed.     

Nonlinear dynamic analysis of tapered sandwich plates with multi-layered faces subjected to air blast loading

The nonlinear dynamic behavior of simply supported tapered sandwich plates subjected to air blast loading is investigated theoretically and numerically. The plate is supposed to have both tapered core and tapered laminated face sheets and be subjected to uniform air blast load. The theory is based on a sandwich plate theory, which includes von Kármán large deformation effects, in-plane stiffnesses, inertias and shear deformations. The sandwich plate theory for plates with constant thickness which have one-layered face sheets found in the literature is developed to analyze the tapered sandwich plates with multi-layered face sheets. The equations of motion are derived by the use of the virtual work principle. Approximate solution functions are assumed for the space domain and substituted into the equations. The Galerkin method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The tapered sandwich plate subjected to air blast load is also modelled by using the finite element method. The displacement–time and strain–time histories are obtained. The theoretical results are compared with finite element results and are found to be in an agreement.     

Analytical geometrical responses in large deflection of simply supported piezoelectric layered plate under initial tension

Abstract

The analytical geometrical responses in large deflection of a simply supported and layered piezoelectric circular plate under initial tension due to lateral pressure are presented. The approach follows von Karman plate theory for large deflection with a consideration of a symmetrically laminated case including a piezoelectric layer. The related nonlinear governing equations are derived in a non‐dimensional form and are simplified by neglecting the arising nonlinear terms, yielding a modified Bessel equation or a standard Bessel equation for the lateral slope. The associated analytical solutions are developed by imposing the simply supported edge conditions of the problem. For a 3‐layered nearly monolithic plate under a low pretension and a low applied voltage upon the piezoelectric layer, the results agree well with those obtained by using the classical plate theory for a single‐layered plate under pure mechanical loading, and thus the developed approach is validated. Typical 3‐layered piezoelectric plates are then implemented and the results show that, no apparent edge effect was found for the present problem. In additions, a piezoelectric effect appears to be present only up to a moderate initial tension. For a relatively high pretension, the tension effect tends to be dominant, resulting in nearly the same results for the geometrical responses, regardless of the magnitude of the applied voltage.     

Buckling analysis of a porous circular plate with piezoelectric sensor–actuator layers under uniform radial compression

This study presents the buckling analysis of a solid circular plate made of porous material bounded with piezoelectric sensor–actuator patches. The porous material properties vary through the thickness direction of the plate following a given function. The general mechanical nonlinear equilibrium and linear stability equations are derived using the variational formulations to obtain the governing equations of the piezoelectric porous plate. The buckling load is derived for solid circular plates under uniform radial compressive loading for the clamped edge condition. The effects of piezoelectric layers on the buckling load of the plate, piezoelectric layer-to-porous plate thickness ratio, feedback gain, and variation of porosity are investigated. The results are verified with the known results in the literature.     

Duffing振子激励下平板声振特性研究

针对非线性振动激励下结构声辐射问题,由变分原理导出Duffing振子激励下平板声振耦合动力学方程,由模态展开法及增量谐波平衡法导出轻流体中耦合动力学方程的近似解析解,给出多频激励下平板表面平均振速及辐射声功率表达式,研究激励力频率、非线性项对系统振动及声辐射特性影响。结果表明,Duffing振子激励下平板的声振耦合问题为含离散与连续系统的复杂动力学问题;耦合运动下Duffing振子出现二次跳跃现象与新的共振特性;平板声振特性主要由三次谐波决定。研究结果可为隔振结构的声振设计提供理论依据。