Complex-mode Galerkin approach in transverse vibration of an axially accelerating viscoelastic string

Under the consideration of harmonic fluctuations of initial tension and axially velocity, a nonlinear governing equation for transverse vibration of an axially accelerating string is set up by using the equation of motion for a 3-dimensional deformable body with initial stresses. The Kelvin model is used to describe viscoelastic behaviors of the material. The basis function of the complex-mode Galerkin method for axially accelerating nonlinear strings is constructed by using the modal function of linear moving strings with constant axially transport velocity. By the constructed basis functions, the application of the complex-mode Galerkin method in nonlinear vibration analysis of an axially accelerating viscoelastic string is investigated. Numerical results show that the convergence velocity of the complex-mode Galerkin method is higher than that of the real-mode Galerkin method for a variable coefficient gyroscopic system.     

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.     

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.     

Energetics and conserved quantity of an axially moving string undergoing three-dimensional nonlinear vibration

Nonlinear three-dimensional vibration of axially moving strings is investigated in the view of energetics. The governing equation is derived from the Eulerian equation of motion of a continuum for axially accelerating strings. The time-rate of the total mechanical energy associated with the vibration is calculated for the string with its ends moving in a prescribed way. For a string moving in a constant axial speed and constrained by two fixed ends, a conserved quantity is proved to remain unchanged during three-dimensional vibration, while the string energy is not conserved. An approximate conserved quantity is derived from the conserved quantity in the neighborhood of the straight equilibrium configuration. The approximate conserved quantity is applied to verify the Lyapunov stability of the straight equilibrium configuration. Numerical simulations are performed for a rubber string and a steel string. The results demonstrate the variation of the total mechanical energy and the invariance of the conserved quantity.     

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。     

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.     

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.     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

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。     

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.     

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。     

轮带系统横向振动的行波消去法

考虑作动器中张紧轮质量的影响,研究轴向运动弦线和作动器所组成的耦合系统的横向振动控制。此系统被作动器分成受控和未控两部分,在频域内利用Green函数法求解出系统的响应,采用行波消去法设计出控制律。在初始条件和激励作用下,利用Durbin拉氏变换数值反演法将受控系统的振动响应转化到时域内,并利用Matlab进行数值仿真。算例结果表明:在脉冲激励和正弦激励作用下,系统振动在3秒内分别减小到0和未受控制时的1/5,验证了控制律的有效性。     

DYNAMIC STABILITY OF AXIALLY MOVING VISCOELASTIC BEAMS WITH PULSATING SPEED

IntroductionThe class of systems with axially moving materials involves power transmission chains,band saw blades and paper sheets during processing. Vibration of such systems is generallyundesirable. The traveling tensioned Euler-Bernoulli beam is the pr…     

Nonlocal and strain gradient effects on nonlinear forced vibration of axially moving nanobeams under internal resonance conditions

Based on the nonlocal strain gradient theory(NSGT), a model is proposed for an axially moving nanobeam with two kinds of scale effects. The internal resonanceaccompanied fundamental harmonic response of the external excitation frequency in the vicinities of the first and second natural frequencies is studied by adopting the multivariate Lindstedt-Poincaré(L-P) method. Based on the root discriminant of the frequencyamplitude equation under internal resonance conditions, theoretical analyses are p...     

简谐激励作用下轴向运动梁横向振动特性分析

为分析简谐激励作用下轴向运动梁的横向振动问题,采用单元数目及长度固定不变、节点参数在不同时间步下无缝传递的节点生死方法,建立了时变系统的动力学有限元模型,通过已有实例验证了模型的准确性和有效性。在此基础上,分析了架设速度、激励力频率和幅值对某型平推式军用桥梁架设过程横向动力响应的影响规律。结果表明,在架设过程中,当桥梁的时变固有频率与激励力频率接近时,桥梁位移动态响应呈共振的特点,据此提出了减小某型平推式军用桥梁架设过程动力响应的措施。     

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.     

Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime

Summary Nonlinear dynamics of one-mode approximation of an axially moving continuum such as a moving magnetic tape is studied. The system is modeled as a beam moving with varying speed, and the transverse vibration of the beam is considered. The cubic stiffness term, arising out of finite stretching of the neutral axis during vibration, is included in the analysis while deriving the equations of motion by Hamilton's principle. One-mode approximation of the governing equation is obtained by the Galerkin's method, as the objective in this work is to examine the low-dimensional chaotic response. The velocity of the beam is assumed to have sinusoidal fluctuations superposed on a mean value. This approximation leads to a parametrically excited Duffing's oscillator. It exhibits a symmetric pitchfork bifurcation as the axial velocity of the beam is varied beyond a critical value. In the supercritical regime, the system is described by a parametrically excited double-well potential oscillator. It is shown by numerical simulation that the oscillator has both period-doubling and intermittent routes to chaos. Melnikov's criterion is employed to find out the parameter regime in which chaos occurs. Further, it is shown that in the linear case, when the operating speed is supercritical, the oscillator considered is isomorphic to the case of an inverted pendulum with an oscillating support. It is also shown that supercritical motion can be stabilised by imposing a suitable velocity variation. Received 13 February 1997; accepted for publication 29 July 1997     

Nonlinear three-dimension analysis for axially symmetrical circular plates and multilayered plates

Analytic nonlinear three-dimension solutions are presented for axially symmetrical homogeneous isotropic circular plates and multilayered plates with rigidly clamped boundary conditions and under transverse load.The geometric nonlinearily from a moderately large deflection is considered.A developmental perturbation method is used to solve the complicated nonlinear three-dimension differential equations of equilibrium.The basic idea of this perturbation method is using the two-dimension solutions as a basic form of the corresponding three-dimension solutions,and then processing the perturbation procedure to obtain the three-dimension perturbation solutions.The nonlinear three-dimension results in analytic expressions and in numerical forms for ordinary plates and multilayered plates are presented.All of the plate stresses are shown in figures.The results show that this perturbation method used to analyse nonlinear three-dimension problems of plates is effective.     

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.     

ON GALERKIN DISCRETIZATION OF AXIALLY MOVING NONLINEAR STRINGS

A computational technique is proposed for the Galerkin discretization of axially moving strings with geometric nonlinearity. The Galerkin discretization is based on the eigenfunctions of stationary strings. The discretized equations are simplified by regrouping nonlinear terms to reduce the computation work. The scheme can be easily implemented in the practical programming. Numerical results show the effectiveness of the technique. The results also highlight the feature of Galerkin＇s discretization of gyroscopic continua that the term number in Galerkin＇s discretization should be even. The technique is generalized from elastic strings to viscoelastic strings.