Non-linear vibrations of transversely isotropic rectangular'' plates

The large amplitude free flexural vibration of transversely isotropic rectangular plate, incorporating the effects of transverse shear and rotatory inertia, is studied using the von Karman field equations. A mode shape, consisting of three generalised-coordinates together with the Galerkin technique, results in a system of three non-linear simultaneous ordinary differential equations which govern the motion of the plate. These equations are integrated using a fourth-order Runge-Kutta method to obtain the period for each amplitude of vibration. The non-linear period vs amplitude behaviour is of the hardening type and it is also found that transverse shear and rotary inertia effects increase the period and that this increase is quite significant even for thin transversely isotropic plates. The results are compared with earlier results which were based on a one-term or one generalised coordinate solution and using the Berger approximation or the von Karman field equations.     

计算圆板大振幅非线性振动频率的平均刚度法

本文用平均刚度法研究圆板大振幅非线性振动的频率问题，导出了相应的非线性广义特征值方程，构造了一种避免发散并能加速收敛的加权平均迭代法，计算结果与Ｋａｎｔｏｒｏｖｉｃｈ时间平均法的解十分吻合。     

Accurate analytical perturbation approach for large amplitude vibration of functionally graded beams

The present work derives the accurate analytical solutions for large amplitude vibration of thin functionally graded beams. In accordance with the Euler–Bernoulli beam theory and the von Kármán type geometric non-linearity, the second-order ordinary differential equation having odd and even non-linearities can be formulated through Hamilton's principle and Galerkin's procedure. This ordinary differential equation governs the non-linear vibration of functionally graded beams with different boundary constraints. Building on the original non-linear equation, two new non-linear equations with odd non-linearity are to be constructed. Employing a generalised Senator–Bapat perturbation technique as an ingenious tool, two newly formulated non-linear equations can be solved analytically. By selecting the appropriate piecewise approximate solutions from such two new non-linear equations, the analytical approximate solutions of the original non-linear problem are established. The present solutions are directly compared to the exact solutions and the available results in the open literature. Besides, some examples are selected to confirm the accuracy and correctness of the current approach. The effects of boundary conditions and vibration amplitudes on the non-linear frequencies are also discussed.     

Large amplitude free flexural vibrations of laminated composite skew plates

Here, the large amplitude free flexural vibration behaviors of thin laminated composite skew plates are investigated using finite element approach. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman's assumptions is introduced. The non-linear governing equations obtained employing Lagrange's equations of motion are solved using the direct iteration technique. The variation of non-linear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, number of layers, fiber orientation, boundary condition and aspect ratio. The influence of higher vibration modes on the non-linear dynamic behavior of laminated skew plates is also highlighted. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and lamination parameters of the plate. Also, the degree of hardening behavior increases with the skew angle and its rate of change depends on the level of amplitude of vibration.     

Double plate system with a discontinuity in the elastic bonding layer

The double plate system with a discontinuity in the elastic bonding layer of Winker type is studied in this paper. When the discontinuity is small, it can be taken as an interface crack between the bi-materials or two bodies (plates or beams). By comparison between the number of multifrequencies of analytical solutions of the double plate system free transversal vibrations for the case when the system is with and without discontinuity in elastic layer we obtain a theory for experimental vibration method for identification of the presence of an interface crack in the double plate system. The analytical analysis of free transversal vibrations of an elastically connected double plate systems with discontinuity in the elastic layer of Winkler type is presented. The analytical solutions of the coupled partial differential equations for dynamical free and forced vibration processes are obtained by using method of Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one mode vibration corresponds an infinite or finite multi-frequency regime for free and forced vibrations induced by initial conditions and one-frequency or corresponding number of multi-frequency regime depending on external excitations. It is shown for every shape of vibrations. The analytical solutions show that the discontinuity affects the appearance of multi-frequency regime of time function corresponding to one eigen amplitude function of one mode, and also that time functions of different vibration basic modes are coupled. From final expression we can separate the new generalized eigen amplitude functions with corresponding time eigen functions of one frequency and multi-frequency regime of vibrations. The English text was polished by Keren Wang.     

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán’s theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.     

A finite element study on the large amplitude flexural vibration characteristics of FGM plates under aerodynamic load

The large amplitude flexural vibration characteristics of functionally graded material (FGM) plates are investigated here using a shear flexible finite element approach. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of the constituents. The effective material properties are then evaluated based on the rule of mixture. The FGM plate is modeled using the first-order shear deformation theory based on exact neutral surface position and von Kármán’s assumptions for large displacement. The third-order piston theory is employed to evaluate the aerodynamic pressure. The governing equations of motion are solved by harmonic balance method to study the vibration amplitude of FGM plates under supersonic air flow. Thereafter, the non-linear equations of motion are solved using Newmark’s time integration technique to understand the flexural vibration behavior of FGM plates in time domain (simple harmonic or periodic or quasi-periodic). This work is new in the sense that it deals with the non-linear flutter characteristics of FGM plates under high supersonic airflow accounting for both the geometric and aerodynamic non-linearities. Some parametric study is conducted to understand the influence of these non-linearities on the flutter characteristics of FGM plates.     

Non-linear analysis of functionally graded fiber reinforced composite laminated plates,Part I: Theory and solutions

This paper deals with the large amplitude vibration, non-linear bending and postbuckling of fiber reinforced composite laminated plates resting on an elastic foundation in hygrothermal environments. Two kinds of fiber reinforced laminated plates, namely, uniformly distributed and functionally graded reinforcements, are considered. The material properties of fiber reinforced laminated plates are estimated through a micromechanical model and are assumed to be temperature-dependent and moisture-dependent. The motion equations are based on a higher order shear deformation plate theory that includes plate-foundation interaction and the hygrothermal effect. A two-step perturbation technique is employed to determine the non-linear to linear frequency ratios of plate vibration, the load-deflection and load-bending moment curves of plate bending, and postbuckling equilibrium paths of laminated plates.     

Thermoelastically coupled axisymmetric nonlinear vibration of shallow spherical and conical shells

The problem of axisymmetric nonlinear vibration for shallow thin spherical and conical shells when temperature and strain fields are coupled is studied. Based on the large deflection theories of yon Ktirrntin and the theory of thermoelusticity, the whole governing equations and their simplified type are derived. The time-spatial variables are separated by Galerkin ‘ s technique, thus reducing the governing equations to a system of time-dependent nonlinear ordinary differential equation. By means of regular perturbation method and multiple-scales method, the first-order approximate analytical solution for characteristic relation of frequency vs amplitude parameters along with the decay rate of amplitude are obtained, and the effects of different geometric parameters and coupling factors us well us boundary conditions on thermoelustically coupled nonlinear vibration behaviors are discussed.     

功能梯度材料明德林矩形微板的热弹性阻尼

功能梯度材料微板谐振器热弹性阻尼的建模和预测是此类新型谐振器热?弹耦合振动响应的新课题. 本文采用数学分析方法研究了四边简支功能梯度材料中厚度矩形微板的热弹性阻尼. 基于明德林中厚板理论和单向耦合热传导理论建立了材料性质沿着厚度连续变化的功能梯度微板热弹性自由振动控制微分方程. 在上下表面绝热边界条件下采用分层均匀化方法求解变系数热传导方程, 获得了用变形几何量表示的变温场的解析解. 从而将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的偏微分方程. 然后,利用特征值问题在数学上的相似性,求得了四边简支条件下功能梯度材料明德林矩形微板的复频率解析解, 进而利用复频率法获得了反映谐振器热弹性阻尼水平的逆品质因子. 最后, 给出了材料性质沿板厚按幂函数变化的陶瓷?金属组分功能梯度矩形微板的热弹性阻尼数值结果. 定量地分析了横向剪切变形、材料梯度变化以及几何参数对热弹性阻尼的影响规律. 结果表明, 采用明德林板理论预测的热弹性阻尼值小于基尔霍夫板理论的预测结果, 而且两者的差别随着相对厚度的增大而变得显著.      

轴向运动弦线的纵向振动及其控制

综述轴向运动弦线纵向振动及其控制问题的研究进展.多种工程 系统如动力传送带、磁带、纸带、纺织纤维、带锯、空中缆车索道等均 涉及轴向运动弦线的纵向振动.对线性模型而言,除早期结果外,总结了 运动弦线的模态分析、具有复杂约束和耦合的运动弦线振动和运动弦线 参数振动的近期研究.对非线性模型而言,提出了轴向运动弦线大幅纵向 振动的运动微分方程,概述了离散化和直接近似解析分析、用黏弹性材 料模型化阻尼机制和动力传输系统的耦合振动研究的新进展.讨论了轴 向运动弦线振动主动控制的研究现状,包括能控性和能观性,控制分析的 频域方法和能量方法,振动的自适应控制和非线性振动的控制.最后指出 该研究方向今后需要研究的若干重要问题,包括运动弦线的非线性动力学 行为、黏弹性运动弦线的振动、含运动弦线的混杂系统的控制和轴向运 动弦线非线性振动的控制.     

燃料大幅晃动等几何分析仿真及航天器耦合动力学研究

现代航天器肩负许多周期长且复杂的航天任务,通常需要携带大量的液体燃料.贮箱中液体燃料大幅晃动会严重影响航天器的姿态稳定性和控制精度,是现代航天器耦合动力学建模和精确控制研究的重要问题.本文提出了一种新的液体大幅晃动数值仿真方法,采用等几何分析方法对贮箱内气体和液体整体进行建模和空间离散,采用压力修正的分步法对控制方程进行时间离散,结合水平集方法划分气体和液体区域并且实时追踪液体晃动自由面.提出了一种质量修正方法以消除水平集函数演化产生的液体质量误差.基于燃料大幅晃动等几何分析仿真方法,对携带太阳能帆板的充液航天器进行动力学建模和耦合运动数值仿真.对于太阳能帆板的振动问题则采用Kirchhoff-Love板理论建模和模态分析法数值求解.通过将数值仿真结果与解析解对比,证明了本文给出方法的正确性.本文还对燃料大幅晃动下的航天器刚-液-柔耦合运动进行了数值仿真,发现液体晃动对航天器的姿态变化和结构振动的幅值和频率具有不可忽视的影响.     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

Non-linear vibrations of suspension bridges with external excitation

Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency-response equation is derived and the jump phenomenon in the frequency-response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.     

Asymptotic solutions of coupled equations of supercritically axially moving beam

In supercritical regime, the coupled model equations for the axially moving beam with simple support boundary conditions are considered. The critical speed is determined by linear bifurcation analysis, which is in agreement with the results in the literature. For the corresponding static equilibrium state, the second-order asymptotic nontrivial solutions are obtained through the multiple scales method. Meantime, the numerical solutions are also obtained based on the finite difference method. Comparisons among the analytical solutions, numerical solutions and solutions of integro-partial-differential equation of transverse which is deduced from coupled model equations are made. We find that the second-order asymptotic analytical solutions can well capture the nontrivial equilibrium state regardless of the amplitude of transverse displacement. However, the integro-partial-differential equation is only valid for the weak small-amplitude vibration axially moving slender beams.     

Finite amplitude vibrations of an orthotropic circular plate with an isotropic core

The free and forced non-linear vibrations of a fixed orthotropic circular plate, with a concentric core of isotropic material, are studied. Existence of harmonic vibrations is assumed and thus the time variable is eliminated by a Ritz-Kantorovich method. Hence, the governing non-linear partial equations for the axisymmetric vibration of the composite circular plate are reduced to a set of ordinary differential equations which form a non-linear eigen-value problem. Solutions are obtained by utilizing the related initial-value problems in conjunction with Newton's integration method. The results reveal the effects of finite amplitude and anisotropy of materials upon the dynamic responses. Further, the method developed in this paper, which is used to solve the title problem, is one of some generality. It can be applied to many differential eigenvalue problems with piecewise continuous functions.     

Modeling and analysis of an axially acceleration beam based on a higher order beam theory

In this paper, a higher order model equation is presented for an axially accelerating beam. Based on a new kinematic frame of the beam and continuum mechanics theory, the coupled governing equations of nonlinear vibration for axially accelerating beam are obtained with the aid of the generalized Hamilton principle. The governing equations take into account the characteristic of the material, the shear strain, the rotation strain and the effect of longitudinally varying tension due to the axial acceleration. The equations are decoupled into a nonlinear partial-integro-differential equations when the transverse nonlinear vibration is small. For the principal parametric resonances, the steady-state frequency responses are obtained by the multiple scales method. The stable and unstable interval are analyzed for the trivial and nontrivial steady-state response. Effects of the system parameters on the amplitude have been investigated. The results show that the material parameter (i.e, in-plane Poisson ratio) has a significant effect on the amplitude and the nonlinear vibration behavior type. The amplitude decrease with the growth of the in-plane Poisson ratio. The total potential energy has play a very important role in determining the amplitude of frequency response according to model analysis. Lastly, comparisons among the analytical solutions and numerical solutions are made and good agreements for the amplitude are found.     

Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells

This paper discusses the derivation of discrete low-dimensional models for the non-linear vibration analysis of thin shells. In order to understand the peculiarities inherent to this class of structural problems, the non-linear vibrations and dynamic stability of a circular cylindrical shell subjected to dynamic axial loads are analyzed. This choice is based on the fact that cylindrical shells exhibit a highly non-linear behavior under both static and dynamic axial loads. Geometric non-linearities due to finite-amplitude shell motions are considered by using Donnell’s nonlinear shallow shell theory. A perturbation procedure, validated in previous studies, is used to derive a general expression for the non-linear vibration modes and the discretized equations of motion are obtained by the Galerkin method. The responses of several low-dimensional models are compared. These are used to study the influence of the modelling on the convergence of critical loads, bifurcation diagrams, attractors and large amplitude responses of the shell. It is shown that rather low-dimensional and properly selected models can describe with good accuracy the response of the shell up to very large vibration amplitudes.     

Dynamic analysis of a dielectric elastomer-based microbeam resonator with large vibration amplitude

A non-linear vibration equation with the consideration of large amplitude, gas damping and excitation is developed to investigate the dynamic performance of a dielectric elastomer (DE)-based microbeam resonator. Approximate analytical solution for the vibration equation is obtained by applying parameterized perturbation method (PPM) and introducing a detuning variable. The analysis exhibits that active tuning of the resonant frequency of the resonator can be achieved through changing an applied electrical voltage. It is observed that increasing amplitude will increase the natural frequency while it will decrease the quality factor of the resonator. In addition, it is found that the initial pre-stretching stress and the ambient pressure can significantly alter the resonant frequency of the resonator. The analysis is envisaged to provide qualitative predictions and guidelines for design and application of DE-based micro resonators with large vibration amplitude.     

Non-linear flexural oscillations of a partially tapered annular plate

The free finite amplitude axisymmetric oscillations of an isotropic annular plate with partially tapered thickness are investigated. The time variable is eliminated by a Ritz-Kantorovich averaging method. The von Karman plate equations are then reduced to two non-linear ordinary differential equations, which form a non-linear eigenvalue problem. Solutions to the problem are obtained by utilizing a direct computational method. The results reveal the effects of large amplitude upon the dynamic responses. Also, an annulus of constant thickness, which has the same boundary conditions and the same volume as the partially tapered one, is investigated. Their results, which may shed light on the optimal design of annular plates, are compared.