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.     

Eigenvalue and stability analysis for transverse vibrations of axially moving strings based on Hamiltonian dynamics

The Hamiltonian dynamics is adopted to solve the eigenvalue problem for transverse vibrations of axially moving strings. With the explicit Hamiltonian function the canonical equation of the free vibration is derived. Non-singular modal functions are obtained through a linear, symplectic eigenvalue analysis, and the symplectic-type orthogonality conditions of modes are derived. Stability of the transverse motion is examined by means of analyzing the eigenvalues and their bifurcation, especially for strings transporting with the critical speed. It is pointed out that the motion of the string does not possess divergence instability at the critical speed due to the weak interaction between eigenvalue pairs. The expansion theorem is applied with the non-singular modal functions to solve the displacement response to free and forced vibrations. It is demonstrated that the modal functions can be used as the base functions for solving linear and nonlinear vibration problems. The project supported by the National Natural Science Foundation of China (10472021, 10421002 and 10032030), the NSFC-RFBR Collaboration Project (1031120166/10411120494) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry The English text was polished by Keren Wang     

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.     

轴向运动弦线横向振动的频域分析

轴向运动弦线是多种工程系统的模型。为明确轴向运动横向振动的频域特性，及探索频域方法的应用特点．本文用频域方法分析轴向运动弦线的横向振动。基于轴向运动弦线横向振动方程和边界条件．通过Laplace变换导出频率域中的控制方程，并将该控制方程和边界条件用状态变量表示。由状态空间中的控制方程导出特征方程，从而求出固有频率。由轴向运动弦线的矩阵函数计算得到系统的传递函数，然后用留数定理计算传递函数的Laplace逆变换．这样就可以得到时域响应。最后分析了轴向运动弦线的横向共振，若简谐外激励的频率与系统固有频率相同，系统响应将随时间无限增加。     

Effects of rotary inertia on sub- and super-critical free vibration of an axially moving beam

The most important issue in the vibration study of an engineering system is dynamics modeling. Axially moving continua is often discussed without the inertia produced by the rotation of the continua section. The main goal of this paper is to discover the effects of rotary inertia on the free vibration characteristics of an axially moving beam in the sub-critical and super-critical regime. Specifically, an integro-partial-differential nonlinear equation is modeled for the transverse vibration of the moving beam based on the generalized Hamilton principle. Then the effects of rotary inertia on the natural frequencies, the critical speed, post-buckling vibration frequencies are presented. Two kinds of boundary conditions are also compared. In super-critical speed range, the straight configuration of the axially moving beam loses its stability. The buckling configurations are derived from the corresponding nonlinear static equilibrium equation. Then the natural frequencies of the post-buckling vibration of the super-critical moving beam are calculated by using local linearization theory. By comparing the critical speed and the vibration frequencies in the sub-critical and super-critical regime, the effects of the inertia moment due to beam section rotation are investigated. Several interesting phenomena are disclosed. For examples, without rotary inertia, the study overestimates the stability of the axially moving beam. Moreover, the relative differences between the super-critical fundamental frequencies of the two theories may increase with an increasing beam length.     

Supercritical vibration of nonlinear coupled moving beams based on discrete Fourier transform

Natural frequencies of nonlinear coupled planar vibration are investigated for axially moving beams in the supercritical transport speed ranges. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. The finite difference scheme is developed to calculate the non-trivial static equilibrium. The equations are cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. Under fixed boundary conditions, time series are calculated via the finite difference method. Based on the time series, the natural frequencies of nonlinear planar vibration, which are determined via discrete Fourier transform (DFT), are compared with the results of the Galerkin method for the corresponding governing equations without nonlinear parts. The effects of material parameters and vibration amplitude on the natural frequencies are investigated through parametric studies. The model of coupled planar vibration can reduce to two nonlinear models of transverse vibration. For the transverse integro-partial-differential equation, the equilibrium solutions are performed analytically under the fixed boundary conditions. Numerical examples indicate that the integro-partial-differential equation yields natural frequencies closer to those of the coupled planar equation.     

Equilibria of axially moving beams in the supercritical regime

Equilibria of axially moving beams are computationally investigated in the supercritical transport speed ranges. In the supercritical regime, the pattern of equilibria consists of the straight configuration and of non-trivial solutions that bifurcate with transport speed. The governing equations of coupled planar is reduced to a partial-differential equation and an integro-partial-differential equation of transverse vibration. The numerical schemes are respectively presented for the governing equations and the corresponding static equilibrium equation of coupled planar and the two governing equations of transverse motion for non-trivial equilibrium solutions via the finite difference method and differential quadrature method under the simple support boundary. A steel beam is treated as example to demonstrate the non-trivial equilibrium solutions of three nonlinear equations. Numerical results indicate that the three models predict qualitatively the same tendencies of the equilibrium with the changing parameters and the integro-partial-differential equation yields results quantitatively closer to those of the coupled equations.     

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.     

超临界轴向运动梁横向非线性振动频域分析

数值方法研究超临界速度下轴向运动梁横向非线性振动前两阶固有频率.通过对非平凡平衡位形做坐标变换,建立超临界轴向运动梁的标准控制方程,一个积分-偏微分非线性方程.利用有限差分法数值离散梁两端简支边界下控制方程,计算轴向运动梁中点的横向振动位移,并将计算结果作为时间序列,运用离散傅立叶变换得到超临界轴向运动梁横向振动的频率...     

轴向运动弦线横向振动的线性反馈控制

研究轴向运动弦线和作动器组成的耦合系统的横向振动控制。此系统被作动器分成未控和受控两部分,通过作用在作动器上的控制力对受控部分的横向振动进行控制。采用能量方法获得反馈控制规律,得到两种控制力,并用半群理论证实受控弦线横向振动的渐近稳定性和指数稳定性。在初始扰动和激励力作用下,通过数值仿真证实控制规律的有效性。     

横向载荷作用下轴向运动梁的屈曲失稳以及非线性振动特性

梁的轴向运动会诱发其产生横向振动并可能导致屈曲失稳,对结构的安全性和可靠性产生重大的影响。本文重点研究了横向载荷作用下轴向运动梁的屈曲失稳及横向非线性振动特性。基于Hamilton变分原理,建立了横向载荷作用下轴向运动梁的动力学方程,获得了梁的后屈曲构型。使用截断Galerkin法,将控制方程改写成Duffing方程的形式。用同伦分析方法确定载荷作用下轴向运动梁的非线性受迫振动的封闭形式的表达式。结果表明,后屈曲构型对轴向速度和初始轴向应力有明显的依赖性。通过同伦分析法得出非线性基频的显式表达式,获得了初始轴向力会影响非线性频率随初始振幅和轴向速度的线性关系。另外,轴向外激励的方向也会改变系统固有频率。     

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.     

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.     

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.     

Principal parametric resonance of axially accelerating rectangular thin plate in magnetic field

Nonlinear parametric vibration and stability is investigated for an axially accelerating rectangular thin plate subjected to parametric excitations resulting from the axial time-varying tension and axial time-varying speed in the magnetic field. Consid- ering geometric nonlinearity, based on the expressions of total kinetic energy, potential energy, and electromagnetic force, the nonlinear magneto-elastic vibration equations of axially moving rectangular thin plate are derived by using the Hamilton principle. Based on displacement mode hypothesis, by using the Galerkin method, the nonlinear para- metric oscillation equation of the axially moving rectangular thin plate with four simply supported edges in the transverse magnetic field is obtained. The nonlinear principal parametric resonance amplitude-frequency equation is further derived by means of the multiple-scale method. The stability of the steady-state solution is also discussed, and the critical condition of stability is determined. As numerical examples for an axially moving rectangular thin plate, the influences of the detuning parameter, axial speed, axial tension, and magnetic induction intensity on the principal parametric resonance behavior are investigated.     

Dynamical analysis of axially moving plate by finite difference method

The complex natural frequencies for linear free vibrations and bifurcation and chaos for forced nonlinear vibration of axially moving viscoelastic plate are investigated in this paper. The governing partial differential equation of out-of-plane motion of the plate is derived by Newton’s second law. The finite difference method in spatial field is applied to the differential equation to study the instability due to flutter and divergence. The finite difference method in both spatial and temporal field is used in the analysis of a nonlinear partial differential equation to detect bifurcations and chaos of a nonlinear forced vibration of the system. Numerical results show that, with the increasing axially moving speed, the increasing excitation amplitude, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into periodic motion, and then the periodic motion becomes chaotic motion by period-doubling bifurcation.     

Nonlinear vibrations of axially moving multi-supported strings having non-ideal support conditions

In this study, nonlinear vibrations of an axially moving multi-supported string have been investigated. The main difference of this study from the others is in that there are non-ideal supports allowing minimal deflections between ideal supports at both ends of the string. Nonlinear equations of the motion and boundary conditions have been obtained using Hamilton’s Principle. Dependence of the equations of motion and boundary conditions on geometry and material of the string have been eliminated by non-dimensionalizing. Method of multiple scales, a perturbation technique, has been employed for solving the equations of motion. Axial velocity has been assumed a harmonically varying function about a constant value. Axially moving string has been investigated in three regions. Vibrations have been examined for three different cases of the velocity variation frequency. Stability has been analyzed and stability boundaries have been established for the principal parametric resonance case. Effects of the non-ideal support conditions on stability boundaries and vibration amplitudes have been investigated.

     

Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

Axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries.Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli(EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.     

Frequency analysis of nonlinear free vibration of a cross string

This work concerns nonlinear free vibration of a cross string under large amplitude. The equations governing the nonlinear vibration of the cross string are derived at first from the Hamilton principle, and they take the form of Duffing equation. Then the perturbation method is used to solve the nonlinear coupled natural frequency of the cross string. The nonlinear natural frequency not only has the characteristic of nonlinearity, but also reflects the coupled characteristic, i.e., the natural frequency of the cross string varying with that of its constituent strings. The results show that the overall effect on the cross string is somehow averaged due to the nonlinearity of each constituent string, i.e., the natural frequencies of the cross string contain both the linear natural frequencies of the constituent strings and the nonlinear parts that depend upon the vibration amplitude, the diameter of one constituent string, the length ratio of the two strings, etc., but the contribution of each constituent string to the natural frequency is in different proportions.     

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.