Nonlinear flutter of viscoelastic rectangular plates and cylindrical panels of a composite with a concentrated masses

The problem of flutter of viscoelastic rectangular plates and cylindrical panels with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate and panel, the effect of concentrated masses is accounted for using the δ-Dirac function. The problem is reduced to a system of nonlinear ordinary integrodifferential equations by using the Bubnov-Galerkin method. The resulting system with a weakly singular Koltunov-Rzhanitsyn kernel is solved by employing a numerical method based on quadrature formulas. The behavior of viscoelastic rectangular plates and cylindrical panels is studied and the critical flow velocities are determined for real composite materials over wide ranges of physicomechanical and geometrical parameters.     

Nonlinear vibrations of a viscoelastic plate with concentrated masses

The problem of vibrations of a viscoelastic plate with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate, the action of the concentrated masses is taken into account using Dirac δ-functions. The problem is reduced to solving a system of Volterra type ordinary nonlinear integrodifferential equations using the Bubnov-Galerkin method. The resulting system with a singular Koltunov-Rzhanitsyn kernel is solved using a numerical method based on quadrature formulas. The effect of the viscoelastic properties of the plate material and the location and amount of concentrated masses on the vibration amplitude and frequency characteristics is studied. A comparison is made of numerical calculation results obtained using various theories. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 158–169, November–December, 2007.     

Nonlinear vibrations of viscoelastic cylindrical shells taking into account shear deformation and rotatory inertia

The vibration problem of a viscoelastic cylindrical shell is studied in a geometrically nonlinear formulation using the refined Timoshenko theory. The problem is solved by the Bubnov–Galerkin procedure combined with a numerical method based on quadrature formulas. The choice of relaxation kernels is substantiated for solving dynamic problems of viscoelastic systems. The numerical convergence of the Bubnov–Galerkin procedure is examined. The effect of viscoelastic properties of the material on the response of the cylindrical shell is discussed. The results obtained by various theories are compared.     

Dynamic stability of viscoelastic circular cylindrical shells taking into account shear deformation and rotatory inertia

The present work discusses the problem of dynamic stability of a viscoelas- tic circular cylindrical shell,according to revised Timoshenko theory,with an account of shear deformation and rotatory inertia in the geometrically nonlinear statement.Pro- ceeding by Bubnov-Galerkin method in combination with a numerical method based on the quadrature formula the problem is reduced to a solution of a system of nonlinear integro-differential equations with singular kernel of relaxation.For a wide range of vari- ation of physical mechanical and geometrical parameters,the dynamic behavior of the shell is studied.The influence of viscoelastic properties of the material on the dynamical stability of the circular cylindrical shell is shown.Results obtained using different theories are compared.     

Dynamic Stability of Viscoelastic Plates under Increasing Compressing Loads

The dynamic stability problem of viscoelastic orthotropic and isotropic plates is considered in a geometrically nonlinear formulation using the generalized Timoshenko theory. The problem is solved by the Bubnov-Galerkin procedure combined with a numerical method based on quadrature formulas. The effect of viscoelastic and inhomogeneous properties of the material on the dynamic stability of a plate is discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 165–175, March–April, 2006.     

Nonlinear vibrations and dynamic stability of a viscoelastic circular cylindrical shell with shear strain and inertia of rotation taken into account

We consider nonlinear vibration and dynamic stability problems for a viscoelastic circular cylindrical shell according to the refined Timoshenko theory, which takes into account the shear strain and the inertia of rotation, in a geometrically nonlinear setting. The problem data are reduced to systems of nonlinear integro-differential equations with singular relaxation kernels, which can be solved by the Bubnov-Galerkin method combined with a numerical method based on quadrature formulas. We study the numerical convergence of the Bubnov-Galerkin method. We analyze the shell dynamic behavior in a wide range of physical-mechanical and geometric parameters. We demonstrate the influence of the viscoelastic properties of the material on the nonlinear vibrations and dynamic stability of a circular cylindrical shell. We also compare the results obtained according to different theories.     

变温场中具损伤粘弹性矩形板的非线性动力响应分析

基于热粘弹性理论、Von Karman板理论和连续损伤力学,导出了二维状态下各向同性材料的变温粘弹性本构方程,建立了含损伤效应的各向同性粘弹性矩形板在变温场中的非线性运动控制方程,且应用有限差分法对问题进行求解.算例中,讨论了损伤演化及温度场等因素对粘弹性矩形板非线性动力学行为的影响,得出一些有意义的结论.     

粘弹性阻尼修改后结构系统动特性的预测

无阻尼或比例阻尼结构系统经粘弹阻尼修改后,可变为粘弹性阻尼系统。要获得其动特性,需求解复特征值问题。但是,随之带来了计算量大、费用高等问题。尤其是粘弹性材料特性随频率而变化,需求解高阶非线性复特征值问题,这对于一个自由度较大的结构,计算量太大,通常较难实现。本文在特征值修改方法的基础上,提出粘弹性阻尼局部动力修改方法,即仅需已知原结构系统的实模态参数,就可求出粘弹阻尼修改后系统的复模态参数。还发展了特征值和修改量同时迭代方法,有效地解决了粘弹材料复模量随频率变化引起的非线性复特征值问题。     

粘弹性圆薄板的动力学行为

基于线性粘弹性力学的Boltzmann叠加原理,给出粘弹性圆薄板动力学分析的初边值问题。通过一定的简化后得到描述薄板力学行为的四维非线性非自治动力系统。综合使用非线性动力学中的数值分析方法,研究了参数对粘弹性圆薄板动力学行为的影响。同时计算了吸引子的Lyapunov维、相关维和点形维。     

Identification of the dynamic properties of nonlinear vicoelastic materials and the associated wave propagation problem

A method is developed for the identification of the dynamic properties of nonlinear viscoelastic materials using transient response information arising from impact tests. The solutions of the identification problem and that of the associated nonlinear wave propagation problem are shown to be coupled. They are accomplished via application of the method of lines, the Runge-Kutta-Pouzet integration scheme with automatic step size control and Powell's method of unconstrained optimization. Numerical experiments are performed to demonstrate the feasibility, accuracy and stability of the solution procedure established, and wave propagation experiments are conducted to investigate the applicability of the method to a real physical system. The results are of particular interest in the modeling of nonlinear viscoelastic materials and the identification of systems governed by nonlinear hyperbolic partial-integro-differential equations.     

Dynamic response of an infinite Timoshenko beam on a nonlinear viscoelastic foundation to a moving load

The present paper investigates the dynamic response of infinite Timoshenko beams supported by nonlinear viscoelastic foundations subjected to a moving concentrated force. Nonlinear foundation is assumed to be cubic. The nonlinear governing equations of motion are developed by considering the effects of the shear deformable beams and the shear modulus of foundations at the same time. The differential equations are, respectively, solved using the Adomian decomposition method and a perturbation method in conjunction with complex Fourier transformation. An approximate closed form solution is derived in an integral form based on the presented Green function and the theorem of residues, which is used for the calculation of the integral. The dynamic response distribution along the length of the beam is obtained from the closed form solution. The derivation process demonstrates that two methods for the dynamic response of infinite beams on nonlinear foundations with a moving force give the consistent result. The numerical results investigate the influences of the shear deformable beam and the shear modulus of foundations on dynamic responses. Moreover, the influences on the dynamic response are numerically studied for nonlinearity, viscoelasticity and other system parameters.     

EFFECT OF DAMAGE ON NONLINEAR DYNAMIC PROPERTIES OF VISCOELASTIC RECTANGULAR PLATES

IntroductionRecently,compositestructureshavereceivedwideapplicationsinmodernindustriesincludingastronavigation ,aviation ,petroleumandchemicalindustry ,etc.However,thiskindofmaterialsgenerallyhavethepropertiesofviscoelasticitywiththeapparentcreepphenomenonandrelaxationbehaviors.Moreover,variousdamagewillemergeinthestructuresundertheactionofloading ,temperatureandenvironment.Damagemakesthemechanicalbehaviorsdeteriorategraduallybeforethefailure .Damageeffectswillsoftenthestiffnessofthestructure…     

Free vibration analysis of the axially moving Levy-type viscoelastic plate

In this paper, the viscoelastic theory is applied to the axially moving Levy-type plate with two simply supported and two free edges. On the basis of the elastic – viscoelastic equivalence, a linear mathematical model in the form of the equilibrium state equation of the moving plate is derived in the complex frequency domain. Numerical calculations of dynamic stability were conducted for a steel plate. The effects of transport speed and relaxation times modeled with two-parameter Kelvin–Voigt and three-parameter Zener rheological models on the dynamic behavior of the axially moving viscoelastic plate are analyzed.     

黏弹性复合材料梁动力学实验建模的研究

以ABS树脂为基材, 填充1%~10%的金红石纳米二氧化钛制成纳米复合材料样本系列,搭建了参数激励非线性振动实验系统. 采用实验建模的方法, 基于非线性增量谐波平衡识别理论,建立了黏弹性复合材料屈曲梁的动力学控制方程. 通过数值模拟与实验结果的比较, 验证了理论模型和实验系统在定性定量分析上的一致性, 并且对一类不同配比成分的纳米复合材料也有很好的适用性.      

Dynamical behaviors of nonlinear viscoelastic thick plates with damage

Based on the deformation hypothesis of Timoshenko's plates and the Boltzmann's superposition principles for linear viscoelastic materials, the nonlinear equations governing the dynamical behavior of Timoshenko's viscoelastic thick plates with damage are presented. The Galerkin method is applied to simplify the set of equations. The numerical methods in nonlinear dynamics are used to solve the simplified systems. It could be seen that there are plenty of dynamical properties for dynamical systems formed by this kind of viscoelastic thick plate with damage under a transverse harmonic load. The influences of load, geometry and material parameters on the dynamical behavior of the nonlinear system are investigated in detail. At the same time, the effect of damage on the dynamical behavior of plate is also discussed.     

Aeroelastic dynamics stability of rotating sandwich annular plate with viscoelastic core layer

A dynamic model for a rotating sandwich annular plate with a viscoelastic core layer is developed. All fundamental equations and boundary conditions are established based on Hamilton’s principle, and the rotation effect and viscoelastic properties of the sandwich structure are taken into account. The aerodynamics force acting on the plate is described by a rotating damping model, and the constitutive behavior of the viscoelastic core layer is formulated by the frequency-dependent complex modulus. The effects of geometrical and material parameters on frequencies and damping of forward and backward traveling waves and the dynamic stability for the rotating sandwich plate are numerically analyzed by means of Galerkin’s method. The results show that the critical and flutter speeds of the rotating plate can be increased at some certain parameters of the viscoelastic core layer.     

Optimal bounded control of stochastically excited MDOF nonlinear viscoelastic systems

A new procedure for designing optimal bounded control of stochastically excited multi-degree-of-freedom (MDOF) nonlinear viscoelastic systems is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the system is formulated as a quasi-integrable Hamiltonian system with viscoelastic terms and each viscoelastic term is replaced approximately by an elastically restoring force and a visco-damping force based on the randomly periodic behavior of the motion of quasi-integrable Hamiltonian system. Thus, a stochastically excited MDOF nonlinear viscoelastic system is converted to an equivalent quasi-integrable Hamiltonian system without viscoelastic terms. Then, by applying stochastic averaging, the system is further reduced to a partially averaged system of less dimension. The adjoint equation and maximum condition for the optimal control problem of the partially averaged system are derived by using the stochastic maximum principle, and the optimal bounded control force is determined from the maximum condition. Finally, the probability and statistics of the stationary response of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation (FPK) associated with the fully averaged Itô equation of the controlled system. An example is worked out to illustrate the proposed procedure and its effectiveness.     

黏弹性环形板的临界载荷及动力稳定性

利用线性黏弹性力学的Boltzmann叠加原理,在考察位移单值性条件的基础上,给出黏弹性环形板非线性动力学分析的初边值问题。通过Galerkin方法和引进新的状态变量,将其化归为四维非线性非自治常微分方程组,从而得到黏弹性环形板的四种临界载荷,同时考察了几何缺陷对黏弹性薄板临界载荷的影响。根据Floquet理论,得出黏弹性形板在周期激励下的线性动力稳定性判据。综合使用非线性动力学中的数值分析方法,研究了参数对黏弹性环形板非线性动力稳定性的影响。     

黏弹夹芯板的有限元建模及实验研究

从有限元分析和数值模拟及实验验证的角度研究了黏弹夹芯板的频率依赖振动特性。夹芯板中间层为黏弹性材料，其刚度和阻尼的频率依赖性行为直接影响系统的模态频率和阻尼，并导致振动模式求解的复杂化。采用三阶七参数Biot模型描述黏弹性材料频率相关的黏弹性行为。开发了三层四节点28自由度的夹芯板单元，基于经典板理论和哈密顿原理建立了黏弹夹芯板的有限元动力学方程。通过引入辅助耗散坐标，将Biot模型和黏弹夹芯板的有限元动力学模型结合起来，并将其转化为常规二阶线性系统形式，极大简化了求解非线性振动特性的过程。对一边固定、另三边自由的黏弹夹芯板进行了前三阶固有频率和损耗因子的预测，并与实验结果对比。数值模拟结果和实验结果吻合良好，说明所提有限元方法是正确有效的。     

A dynamical model for nonlinear viscoelastic corrugated circular plates with shallow sinusoidal corrugations

An integrated mathematic model and an efficient algorithm on the dynamical behavior of homogeneous viscoelastic corrugated circular plates with shallow sinusoidal corrugations are suggested. Based on the nonlinear bending theory of thin shallow shells, a set of integro-partial differential equations governing the motion of the plates is established from extended Hamilton’s principle. The material behavior is given in terms of the Boltzmann superposition principle. The variational method is applied following an assumed spatial mode to simplify the governing equations to a nonlinear integro-differential variation of the Duffing equation in the temporal domain, which is further reduced to an autonomic system with four coupled first-order ordinary differential equation by introducing an auxiliary variable. These measurements make the numerical simulation performs easily. The classical tools of nonlinear dynamics, such as Poincaré map, phase portrait, Lyapunov exponent, and bifurcation diagrams, are illustrated. The influences of geometric and physical parameters of the plate on its dynamic characteristics are examined. The present mathematic model can easily be used to the similar problems related to other dynamical system for viscoelastic thin plates and shallow shells.