The transverse vibration of an axially moving string supported by a viscoelastic foundation is analysed using the complex modal method. The equation of motion is developed using the generalized Hamilton principle. The...The transverse vibration of an axially moving string supported by a viscoelastic foundation is analysed using the complex modal method. The equation of motion is developed using the generalized Hamilton principle. The exact closed-form solution of eigenvalues and eigen- functions are obtained. The governing equation is represented in a canonical state space form defined by two matrix differential operators, and the eigenfunctions and adjoint eigenfunctions are proved to be orthogonal with respect to each operator. This orthogonality is applied so that the response to arbitrary external excitations and initial conditions can be expressed in modal expansion. Numerical examples are presented to validate the proposed approach.展开更多
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. ...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.展开更多
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, t...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（△t^2 ＋ △x^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.展开更多
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 discre...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.展开更多
A numerical method is proposed to simulate the transverse vibrations of a viscoelastic moving string constituted by an integral law. In the numerical computation, the Galerkin method based on the Hermite functions is ...A numerical method is proposed to simulate the transverse vibrations of a viscoelastic moving string constituted by an integral law. In the numerical computation, the Galerkin method based on the Hermite functions is applied to discretize the state variables, and the Runge- Kutta method is applied to solve the resulting differential-integral equation system. A linear iterative process is designed to compute the integral terms at each time step, which makes the numerical method more efficient and accurate. As examples, nonlinear parametric vibrations of an axially moving viscoelastic string are analyzed.展开更多
The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear...The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear partial-differential equation that governs the transverse vibration of the string was derived by use of the Hamilton principle. The method of multiple scales was applied directly to the equation. The solvability condition of eliminating the secular terms was established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one parametric resonance were obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation were presented. The Liapunov linearized stability theory was employed to derive the instability conditions of the trivial solution and the nontrivial solutions for the two-to-one parametric resonance. Some numerical examples highlighting influences of the related parameters on the instability conditions were presented.展开更多
When we learn about Chinese traditional culture,we can find many interesting things.One of them is a special musical instrument—Erhu.The Erhu is a two-stringed instrument played with a bow.People often call it the Ch...When we learn about Chinese traditional culture,we can find many interesting things.One of them is a special musical instrument—Erhu.The Erhu is a two-stringed instrument played with a bow.People often call it the Chinese violin or Chinese fiddle.It is one of the most famous and loved instruments in Chinese music.The Erhu has a small wooden body and a long neck.It is held on the lap,and the bow moves between the two strings to make music.展开更多
The modal method is applied to analyze coupled vibration of belt drive systems. A belt drive system is a hybrid system consisting of continuous belts modeled as strings as well as discrete pulleys and a tensioner arm....The modal method is applied to analyze coupled vibration of belt drive systems. A belt drive system is a hybrid system consisting of continuous belts modeled as strings as well as discrete pulleys and a tensioner arm. The characteristic equation of the system is derived from the governing equation. Numerical results demenstrate the effects of the transport speed and the initial tension on natural frequencies.展开更多
In this paper,a fractional differential equation is introduced to describe the transverse vibrations of an axially moving viscoelastic string.An iterative algorithm is constructed to analyze the dynamical behavior.By ...In this paper,a fractional differential equation is introduced to describe the transverse vibrations of an axially moving viscoelastic string.An iterative algorithm is constructed to analyze the dynamical behavior.By conveying the memory effect of the fractional differential terms step by step,the computation cost can be greatly reduced.As a numerical example,the effects of the viscoelastic parameters on a moving string are investigated.展开更多
Nonlinear dynamic analysis is performed on moving belts subjected to geometric nonlinearity and initial tension fluctuation. To incorporate more accurately the damping mechanism of belt material, linear viscoelastic m...Nonlinear dynamic analysis is performed on moving belts subjected to geometric nonlinearity and initial tension fluctuation. To incorporate more accurately the damping mechanism of belt material, linear viscoelastic models are adopted in a unified form of differential operators. To circumvent high-order differential vibration equation of time-varying coefficients and with gyroscopic and nonlinear terms, where analytical solution is almost impossible, a systematic approach is presented by reforming the motion equation and directly using the method of multiple scales. To exemplify the procedure, the solutions at principal resonance are obtained and their stability conditions are derived for employing a Kelvin-Voigt model to reflect the property of the belt material. The solutions and stability conditions successfully reduce to those for using Kelvin model and elastic model, which validate the present approaches. Numerical simulations highlight the effects of tension fluctuations and translating speeds on the stability of the belt vibration.展开更多
基金Project supported by the State Key Program of National Natural Science of China(No.11232009)the National Natural Science Foundation of China(Nos.11372171 and 11422214)
文摘The transverse vibration of an axially moving string supported by a viscoelastic foundation is analysed using the complex modal method. The equation of motion is developed using the generalized Hamilton principle. The exact closed-form solution of eigenvalues and eigen- functions are obtained. The governing equation is represented in a canonical state space form defined by two matrix differential operators, and the eigenfunctions and adjoint eigenfunctions are proved to be orthogonal with respect to each operator. This orthogonality is applied so that the response to arbitrary external excitations and initial conditions can be expressed in modal expansion. Numerical examples are presented to validate the proposed approach.
基金supported by the National Natural Science Foundation of China(10472021,10421002 and 10032030)the NSFC-RFBR Collaboration Project(1031120166/10411120494)the Scientific Research Foundation for the Retumed 0verseas Chinese Scholars,State Education Ministry.
文摘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.
基金Project supported by the National Natural Science Foundation of China(No.10472060)the Natural Science Foundation of Shanghai Municipality(No.04ZR14058)the Shanghai Leading Academic Discipline Project(No.Y0103)
文摘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（△t^2 ＋ △x^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.
基金supported by the National Outstanding Young Scientists Fund of China(No.10725209)the National Natural Science Foundation of China(No.10672092)+3 种基金Shanghai Subject Chief Scientist Project(No.09XD1401700)Shanghai Municipal Education Commission Scientific Research Project(No.07ZZ07)Shanghai Leading Academic Discipline Project (No.S30106)Changjiang Scholars and Innovative Research Team in University Program(No.IRT0844).
文摘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.
基金supported by the National Outstanding Young Scientists Fund of China (No. 10725209)the National ScienceFoundation of China (No. 10672092)+1 种基金Shanghai Municipal Education Commission Scientific Research Project (No. 07ZZ07)Shanghai Leading Academic Discipline Project (No. Y0103).
文摘A numerical method is proposed to simulate the transverse vibrations of a viscoelastic moving string constituted by an integral law. In the numerical computation, the Galerkin method based on the Hermite functions is applied to discretize the state variables, and the Runge- Kutta method is applied to solve the resulting differential-integral equation system. A linear iterative process is designed to compute the integral terms at each time step, which makes the numerical method more efficient and accurate. As examples, nonlinear parametric vibrations of an axially moving viscoelastic string are analyzed.
文摘The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear partial-differential equation that governs the transverse vibration of the string was derived by use of the Hamilton principle. The method of multiple scales was applied directly to the equation. The solvability condition of eliminating the secular terms was established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one parametric resonance were obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation were presented. The Liapunov linearized stability theory was employed to derive the instability conditions of the trivial solution and the nontrivial solutions for the two-to-one parametric resonance. Some numerical examples highlighting influences of the related parameters on the instability conditions were presented.
文摘When we learn about Chinese traditional culture,we can find many interesting things.One of them is a special musical instrument—Erhu.The Erhu is a two-stringed instrument played with a bow.People often call it the Chinese violin or Chinese fiddle.It is one of the most famous and loved instruments in Chinese music.The Erhu has a small wooden body and a long neck.It is held on the lap,and the bow moves between the two strings to make music.
基金Project supported by the National Natural Science Foundation of China(Nos.10672092 and 10725209)Scientific Research Project of Shanghai Municipal Education Commission(No.07ZZ07)Shanghai Leading Academic Discipline Project(No.Y0103)
文摘The modal method is applied to analyze coupled vibration of belt drive systems. A belt drive system is a hybrid system consisting of continuous belts modeled as strings as well as discrete pulleys and a tensioner arm. The characteristic equation of the system is derived from the governing equation. Numerical results demenstrate the effects of the transport speed and the initial tension on natural frequencies.
基金supported by the National Natural Science Foundation of China(Nos.11072120 and 11002075)supported partially by Brazilian National Council for Scientific and Technological Development(CNPq).
文摘In this paper,a fractional differential equation is introduced to describe the transverse vibrations of an axially moving viscoelastic string.An iterative algorithm is constructed to analyze the dynamical behavior.By conveying the memory effect of the fractional differential terms step by step,the computation cost can be greatly reduced.As a numerical example,the effects of the viscoelastic parameters on a moving string are investigated.
基金Supported by the Basic Research Fund of Tsinghua University
文摘Nonlinear dynamic analysis is performed on moving belts subjected to geometric nonlinearity and initial tension fluctuation. To incorporate more accurately the damping mechanism of belt material, linear viscoelastic models are adopted in a unified form of differential operators. To circumvent high-order differential vibration equation of time-varying coefficients and with gyroscopic and nonlinear terms, where analytical solution is almost impossible, a systematic approach is presented by reforming the motion equation and directly using the method of multiple scales. To exemplify the procedure, the solutions at principal resonance are obtained and their stability conditions are derived for employing a Kelvin-Voigt model to reflect the property of the belt material. The solutions and stability conditions successfully reduce to those for using Kelvin model and elastic model, which validate the present approaches. Numerical simulations highlight the effects of tension fluctuations and translating speeds on the stability of the belt vibration.
