轴向变速运动正交各向异性板的动态稳定性

研究面内载荷作用下轴向变速运动正交各向异性薄板的横向振动及其稳定性。利用Galerkin法与平均法,在激励频率为2倍固有频率或为两阶固有频率之和附近时得到自治的常微分方程组;在参数激励频率和激励振幅平面上,分析由共振引发的失稳区域。数值算例验证了面内载荷、轴向速度、加速度参数对失稳区域的影响。     

Dynamic response of axially moving Timoshenko beams: integral transform solution

The generalized integral transform technique (GITT) is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations (ODEs) in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.     

Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams

The dynamic stability of axially moving viscoelastic Rayleigh beams is presented. The governing equation and simple support boundary condition are derived with the extended Hamilton’s principle. The viscoelastic material of the beams is described as the Kelvin constitutive relationship involving the total time derivative. The axial tension is considered to vary longitudinally. The natural frequencies and solvability condition are obtained in the multi-scale process. It is of interest to investigate the summation parametric resonance and principal parametric resonance by using the Routh-Hurwitz criterion to obtain the stability condition. Numerical examples show the effects of viscosity coefficients, mean speed, beam stiffness, and rotary inertia factor on the summation parametric resonance and principle parametric resonance. The differential quadrature method (DQM) is used to validate the value of the stability boundary in the principle parametric resonance for the first two modes.     

Transverse vibration characteristics of axially moving viscoelastic plate

The dynamic characteristics and stability of axially moving viscoelastic rect- angular thin plate are investigated.Based on the two dimensional viscoelastic differential constitutive relation,the differential equations of motion of the axially moving viscoelastic plate are established.Dimensionless complex frequencies of an axially moving viscoelastic plate with four edges simply supported,two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method.The effects of the aspect ratio,moving speed and dimensionless delay time of the material on the trans- verse vibration and stability of the axially moving viscoelastic plate are analyzed.     

数值仿真轴向运动黏弹性梁非线性参激振动

分别通过两种直接数值方法研究速度变化的经典边界条件下轴向运动黏弹性梁参数振动的稳定性。在控制方程的推导中,采用物质导数黏弹性本构关系和只对时间取偏导数的黏弹性本构关系;分别运用有限差分法和微分求积法对两种经典边界下轴向变速运动黏弹性梁的非线性控制方程求数值解,计算得到梁中点非线性参数振动的稳定稳态响应。数值结果表明,两种黏弹性本构关系对应的稳态响应存在明显差别,同时发现两种直接数值方法的仿真结果基本吻合,证明数值仿真具有较高精度。     

轴向运动梁非线性振动内共振研究

采用多元L-P方法分析轴向运动梁横向非线性振动的内共振，首先根据哈密顿原理建立轴向运动梁的横向振动微分方程，然后利用Galerkin方法分离时间和空间变量，再采用多元L-P方法进行求解，推导了内共振条件下频率-振幅方程的求根判别式，理论分析发现内共振与强迫力的振幅有关，而且可以从理论上决定这一界乎不同内共振的强迫力振幅的临界值，典型算例获得了轴向运动梁横向非线性振动内共振复杂的频率一振幅响应曲线，揭示了很多复杂而有趣的非线性振动特有的现象，多元L-P方法的数值结果，在小振幅时与IHB法的结果一致。     

Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions

The dynamic stability of axially accelerating plates is investigated. Longitudinally varying tensions due to the acceleration and nonhomogeneous boundary conditions are highlighted. A model of the plate combined with viscoelasticity is applied. In the viscoelastic constitutive relationship, the material derivative is used to take the place of the partial time derivative. Analytical and numerical methods are used to investigate summation and principal parametric resonances, respectively. By use of linear models for the transverse behavior in the small displacement regime, the plate is confined by a viscous damping force. The generalized Hamilton principle is used to derive the governing equations, the initial conditions, and the boundary conditions of the coupled planar vibration. The solvability conditions are established by directly using the method of multiple scales. The Routh-Hurwitz criterion is used to obtain the necessary and sufficient condition of the stability. Numerical examples are given to show the effects of related parameters on the stability boundaries. The validity of longitudinally varying tensions and nonhomogeneous boundary conditions is highlighted by comparing the results of the method of multiple scales with those of a differential quadrature scheme.     

Vibration and stability of an axially moving viscoelastic beam with hybrid supports

Vibration and stability are investigated for an axially moving beam constrained by simple supports with torsion springs. A scheme is proposed to derive natural frequencies and modal functions from given boundary conditions of an elastic beam moving at a constant speed. For a beam constituted by the Kelvin model, effects of viscoelasticity on the free vibration are analyzed via the method of multiple scales and demonstrated via numerical simulations. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams in parametric resonance. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity.     

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

The weakly forced vibration of an axially moving viscoelastic beam is investigated.The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved.The nonlinear equations governing the transverse vibration are derived from the dynamical,constitutive,and geometrical relations.The method of multiple scales is used to determine the steady-state response.The modulation equation is derived from the solvability condition of eliminating secular terms.Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation.The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.     

随从力作用下热弹耦合轴向运动梁的稳定性

研究了切向均布随从力作用下热弹耦合轴向运动梁的稳定性问题。建立了热弹耦合轴向运动梁 在随从力作用下的运动微分方程,采用归一化幂级数法,推导出了2种边界条件下热弹耦合轴向运动梁在随 从力作用下的特征方程。计算了系统的前3阶量纲一复频率,分析了量纲一运动速度、量纲一热弹耦合系数 和量纲一随从力等参数对梁的稳定性的影响。     

Dynamic stability of axially accelerating Timoshenko beam: Averaging method

This study investigates dynamic stability in transverse parametric vibrations of an axially accelerating tensioned beam of Timoshenko model on simple supports. The axial speed is assumed as a harmonic fluctuation about the constant mean speed. The Galerkin method is applied to discretize the governing equation into a finite set of ordinary differential equations. The method of averaging is applied to analyze the instability phenomena caused by subharmonic and combination resonance. Numerical examples demonstrate the effects of the mean axial speed, bending stiffness, rotary inertia and shear modulus on the instability boundaries.     

Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams

Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially uniform and temporally harmonic. The transverse motion of axially moving beams is governed by a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The material time derivative is used in the viscoelastic constitutive relation. The method of multiple scales is applied to the governing equations to investigate primary resonances under general boundary conditions. It is demonstrated that the mode uninvolved in the resonance has no effect on the steady-state response. Numerical examples are presented to demonstrate the effects of the boundary constraint stiffness on the amplitude and the stability of the steady-state response. The results derived for two governing equations are qualitatively the same,but quantitatively different. The differential quadrature schemes are developed to verify those results via the method of multiple scales.     

Approximate solutions of axially moving viscoelastic beams subject to multi-frequency excitations

Approximate solutions of axially moving viscoelastic beams subject to multi-frequency excitations are presented. A non-linear partial-differential equation governing the transverse vibration of the beams is derived from Newton's second law, the Kelvin constitutive relationship, and the Lagrangian strain. Based on 1-term Galerkin's truncation, the governing equation is reduced to an ordinary differential equation. Three cases, including superharmonic resonance case, subharmonic resonance, and combination resonance are studied. The approximate solutions of the transverse vibration of the beams are obtained. Numerical results show that the approximate solutions are in good agreement with numerical results.     

Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models

Principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed. A nonlinear partial-differential equation governing the transverse vibration is derived from the dynamical, constitutive, and geometrical relations. Under certain assumption, the partial-differential reduces to an integro-partial-differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams. The method of multiple scales is applied to two equations to calculate the steady-state response. Closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelascity, and nonlinearity and to compare results obtained from two equations.     

轴向变速粘弹性Rayleigh梁参数共振稳定性

运用近似解析方法和数值方法研究轴向变速运动黏弹性Rayleigh梁的次谐波共振和组合共振的稳定性区域。基于变分原理,考虑梁断面旋转惯性的影响,推导轴向速度有周期波动的微变形梁横向振动的数学模型;采用多尺度方法建立前两阶次谐波共振和组合共振范围内的参数振动的可解性条件;进而确定梁两端简支边界条件下,因共振而产生的失稳区域;通过微分求积方法求解表征细长Rayleigh梁横向振动的运动微分方程。数值算例分析了黏弹性系数和扭转系数对梁振动失稳区域的影响,将数值仿真结果与近似解析方法的结论进行比较。算例表明:近似解析解的精度较高,第一、第二阶主共振的最大误差分别为3.206%、4.213%。     

Complex modes and traveling waves in axially moving Timoshenko beams

Complex modes and traveling waves in axially moving Timoshenko beams are studied. Due to the axially moving velocity, complex modes emerge instead of real value modes. Correspondingly, traveling waves are present for the axially moving material while standing waves dominate in the traditional static structures. The analytical results obtained in this study are verified with a numerical differential quadrature method.     

DYNAMIC RESPONSES OF VISCOELASTIC AXIALLY MOVING BELT

Based on the Kelvin viscoelastic differential constitutive law and the motion equation of the axially moving belt, the nonlinear dynamic model of the viscoelastic axial moving belt was established. And then it was reduced to be a linear differential system which the analytical solutions with a constant transport velocity and with a harmonically varying transport velocity were obtained by applying Lie group transformations. According to the nonlinear dynamic model, the effects of material parameters and the steady-state velocity and the perturbed axial velocity of the belt on the dynamic responses of the belts were investigated by the research of digital simulation . The result shows:1) The nonlinear vibration frequency of the belt will become small when the relocity of the belt increases . 2) Increasing the value of viscosity or decreasing the value of elasticity leads to a deceasing in vibration frequencies. 3) The most effects of the transverse amplitudes come from the frequency of the perturbed veloc     

Supercritical equilibrium solutions of axially moving beams with hybrid boundary conditions

In this paper supercritical equilibria and critical speeds of axially moving beams constrained by sleeves with torsion springs are deduced. Transverse vibration of the beams is governed by a nonlinear integro-partial-differential equation. In the supercritical regime, the corresponding static equilibrium equation for the hybrid boundary conditions is analytically solved for the equilibria and the critical speeds. In the view of the non-trivial equilibrium, comparisons are made among the integro-partial-differential equation, a nonlinear partial-differential equation for transverse vibration, and coupled equations for planar motion under the hybrid boundary conditions.     

A FINITE DIFFERENCE METHOD FOR SIMULATING TRANSVERSE VIBRATIONS OF AN AXIALLY MOVING VISCOELASTIC STRING

A finite difference method is presented to simulate transverse vibrations of an axially moving string.By discretizing the governing equation and the equation of stress- strain relation at different frictional knots,two linear sparse finite difference equation systems are obtained.The two explicit difference schemes can be calculated alternatively, which make the computation much more efficient.The numerical method makes the nonlinear model easier to deal with and of truncation errors,O(Δt~2 Δx~2).It also shows quite good stability for small initial values.Numerical examples are presented to demonstrate the efficiency and the stability of the algorithm,and dynamic analysis of a viscoelastic string is given by using the numerical results.     

Principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string

To investigate the principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string, the method of multiple scales is applied directly to the nonlinear partial differential equation that governs the transverse vibration of the string. To derive the governing equation, Newton‘s second law, Lagrangean strain, and Kelvin‘s model are respectively used to account the dynamical relation, geometric nonlinearity and the viscoelasticity of the string material. Based on the solvability condition of eliminating the secular terms, closed form solutions are obtained for the amplitude and the existence conditions of nontrivial steady-state response of the principal parametric resonance. The Lyapunov linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions in the principal parametric resonance. Some numerical examples are presented to show the effects of the mean transport speed, the amplitude and the frequency of speed variation.