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.     

Dynamic stability of a viscoelastic plate with concentrated masses

The paper addresses the geometrically nonlinear problem of dynamic stability of a viscoelastic plate with concentrated masses. The Bubnov-Galerkin method based on polynomial approximation is used to reduce the problem to a system of nonlinear Volterra-type integro-differential equations with singular relaxation kernels. This system is solved by numerical method based on quadrature formulas. The critical loads are found and their dependence on the arrangement and number of concentrated masses is studied for a wide range of mechanical and geometrical parameters of the plate. The choice of a relaxation kernel for dynamic problems for viscoelastic thin-walled plate-like structures is justified. Results produced by different theories are compared __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 109–118, February 2008.     

Nonlinear damped vibrations of simply-supported rectangular sandwich plates

The nonlinear, forced, damped vibrations of simply-supported rectangular sandwich plates with a viscoelastic core are studied. The general, nonlinear dynamic equations of asymmetrical sandwich plates are derived using the virtual work principle. Damping is taken into account by modelling the viscoelastic core as a Voigt-Kelvin solid. The harmonic balance method is employed for solving the equations of motion. The influence of the thickness of the layers and material properties on the nonlinear response of the plates is studied.     

Vibrations of rectangular plates supported at corner points

This paper discusses by energy Theorem the method of approximate compulation for the lowest eigenfrequcncies of rectangular plates, on which there are symmetrical concentrated masses, supported at corner points. In the case of several concentrated masses, by using the principle of superposition we may find the reduced coefficients of masses conveniently. Hence we can obtain the lowest eigenfrequencies of thin plates. In the paper a good many numerical calculating examples are illustrated.     

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.     

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.     

Exact solutions for the stability of viscoelastic rectangular plate subjected to tangential follower force

An exact solution procedure is formulated for the stability analysis of viscoelastic rectangular plate with two opposite edges simply supported and other two edges clamped as well as the viscoelastic rectangular plate with one edge clamped and other three edges simply supported under the action of tangential follower force. Firstly, by assuming the transverse displacement (W) as independent functions which automatically satisfies the simply supported boundary conditions, the governing partial differential equation is reduced to an ordinary differential equation with variable coefficients. Then, by the normalized power series method and applying the boundary conditions yield the eigenvalue problem of finding the roots of a fourth-order characteristic determinant. The results show that the aspect ratio λ and the dimensionless delay time H have great effects on the types of instability and the critical loads of the viscoelastic plates.     

Nonlinear Random Response of Cylindrical Panels to Acoustic Excitations Using Finite Element Modal Method

This paper investigates large amplitude multi-mode free vibration andrandom response of thin cylindrical panels of rectangular planform usinga finite element modal formulation. A thin laminated composite doublycurved element is developed. The system equation in structural nodal DOFis transformed into the modal coordinates by the using the modes of theunderlying linear system. The nonlinear stiffness matrices are alsotransformed into nonlinear modal stiffness matrices. Numericalintegration is employed to determine free vibration and random response.Single-mode free vibration results are compared with existing classicalanalytical solutions to validate the nonlinear modal formulation.Nonlinear random analysis results for cylindrical panels have shown thatthe root mean square of panel deflections could be larger than thoseobtained using the linear structure theory. Time histories, probabilitydistribution functions, power spectral densities, and phase plane plotsare also presented.     

EFFECT OF DAMAGE ON NONLINEAR DYNAMIC PROPERTIES OF VISCOELASTIC RECTANGULAR PLATES

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

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.     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

Nonlinear analysis of chatter vibration in a cylindrical transverse grinding process with two time delays using?a?nonlinear time transformation method

A nonlinear dynamic system of cylindrical transverse grinding process is studied in this paper. The system consists of a grinding wheel and a workpiece, which are connected to the base by spring-damper elements, interacting with nonlinear normal forces. This two DOF model includes two time delays originated from the regenerative effects of the workpiece and the grinding wheel. Bifurcation points are located using a numerical algorithm by which we can find all the eigenvalues in a given rectangular region on the complex plane for the delayed differential equations. Supercritical bifurcation has been found for some sets of system parameter values. The amplitudes of the limit cycles are predicted using a nonlinear time transformation method, which is similar to the harmonic balance approach in that a periodic solution is approximated by a Fourier series. However, the main difference is that a nonlinear time ? is introduced in the Fourier series rather than the physical time t. The analytical solutions of stable limit cycles up to the third harmonics are compared with numerical simulations for the retarded system. It is shown that the proposed method gives accurate approximate solutions.     

Chaotic Analysis of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow

In this paper chaotic behavior of nonlinear viscoelastic panels in asupersonic flow is investigated. The governing equations, based on vonKàarmàn's large deflection theory of isotropic flat plates, areconsidered with viscoelastic structural damping of Kelvin's modelincluded. Quasi-steady aerodynamic panel loadings are determined usingpiston theory. The effect of constant axial loading in the panel middlesurface and static pressure differential have also been included in thegoverning equation. The panel nonlinear partial differential equation istransformed into a set of nonlinear ordinary differential equationsthrough a Galerkin approach. The resulting system of equations is solvedthrough the fourth and fifth-order Runge–Kutta–Fehlberg (RKF-45)integration method. Static (divergence) and Hopf (flutter) bifurcationboundaries are presented for various levels of viscoelastic structuraldamping. Despite the deterministic nature of the system of equations,the dynamic panel response can become random-like. Chaotic analysis isperformed using several conventional criteria. Results are indicative ofthe important influence of structural damping on the domain of chaoticregion.     

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.     

黏弹性板的大挠度蠕变屈曲

研究了具有初始小挠度受轴向压载黏弹性板的蠕变屈曲问题，在建立控制方程时，利用了von Karman非线性应变-位移关系，并考虑了初始挠度，用标准线性固体模型描述材料的黏弹性特性，在求解非线性积分方程时，利用梯形公式计算记忆积分式，将非线性积分方程化为非线性代数方程进行数值求解，得到了结构的蠕变变形过程，又将问题退化到小挠度情况进行研究，得到了挠度随时间扩展的解析解，分析了瞬时失稳临界载荷、持久临界载荷的物理意义，讨论了考虑几何非线性对黏弹性板蠕变屈曲的影响。     

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。     

Nonlinear Vibrations of a Simply Supported Rectangular Metallic Plate Subjected to Transverse Harmonic Excitation in the Presence of a One-to-One Internal Resonance

The nonlinear response of rectangular and square metallic plates subjectto transverse harmonic excitations is studied. The nonlinearitiesoriginate from the use of Von Kármán strains. The method of multiplescales is used to solve the system of differential equationsapproximately. Frequency response curves are presented for both squareand rectangular plates for primary resonance of either mode in thepresence of a one-to-one internal resonance. Stability of steady statesolutions is investigated. Bifurcation points and their types arediscussed.     

Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach

A semi-analytical approach to the elastic nonlinear stability analysis of rectangular plates is developed. Arbitrary boundary conditions and general out-of-plane and in-plane loads are considered. The geometrically nonlinear formulation for the elastic rectangular plate is derived using the thin plate theory with the nonlinear von Kármán strains and the variational multi-term extended Kantorovich method. Emphasis is placed on the effect of destabilizing loads and on the derivation of the solution methodologies required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The performance of the continuation procedures and the convergence of the multi-term extended Kantorovich method are examined through the solution of the two-dimensional Bratu–Gelfand benchmark problem. The applicability of the proposed approach to the tracking of the nonlinear equilibrium path in the post-buckling range is demonstrated through numerical examples of rectangular plates with various boundary conditions.     

The effect of support locations on the steady-state response of viscoelastically point-supported rectangular orthotropic plates

The vibration of orthotropic rectangular plates having viscoelastic point supports symmetrically located on its diagonals is analyzed. The plate is under the effect of a sinusoidally varying force at its center. The Lagrange equations are used in the solution process. To apply the Lagrange equations, the trial function denoting the deflection of the plate is expressed in polynomial form. By using the Lagrange equations, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, the damping of the supports and the locations of point supports on the steady-state response of the viscoelastically point-supported rectangular plates is investigated numerically for the concentrated load at center for various values of the mechanical properties characterizing the anisotropy of the plate material and for various values of damping and location of the supports for a certain stiffness value of the supports. The results are given for the considered frequency range of the external force. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with other solutions based on the Kirchhoff–Love plate theory.     

含基体横向损伤的黏弹性板的蠕变后屈曲分析

基于Schapery三维黏弹性损伤本构关系,引入沈为和Kachanov损伤演化方程,建立了基体横向损伤的纤维单一方向铺设黏弹性板的损伤模型;应用von Karman板理论,导出了考虑损伤效应的具初始挠度的纤维单一方向铺设黏弹性矩形板的非线性压屈平衡方程. 对未知变量在空间上采用差分法离散,时间上采用增量算法和Newton-Cotes积分法离散,控制方程被迭代求解. 算例中讨论了损伤以及有关参数对黏弹性复合材料板后屈曲行为的影响,且与已有文献的结果进行了比较. 数值结果表明：随着外载荷或者初始挠度的增大,板后屈曲趋于稳定时的挠度就愈大,损伤的影响愈明显;而随着长宽比的增大,板后屈曲趋于稳定时的挠度愈小,损伤的影响却随之增大.