Horizontal Nonlinear Vibration Of Visco-elastic Belt

The belt drive system is widely used in various industries to transfer power nowadays. Despite many advantages of this device, as the belt system is a flexible mechanism;it will bring noises and vibrations (especially transverse vibration) when the machine runs. The vibrations will cause the imprecision and jiggles of the transmission and abrasions of the apparatus, these defects have limited its applications. In order to perform the vibration control system, the transverse vibration of the transmission belt should be has an intensive analysis. In this paper, some nonlinear dynamic problems of the visco-elastic transmission belt are investigated.Firstly, the paper systematically introduces the research background and research progress of the transverse vibration of transmission belt at present, in order to reflect the true dynamic characters, some of nonlinear factors (such as material's nonlinear constitutive relation, geometry nonlinear of deformation) cannot be ignored in practice, based on the predecessor's achievement, the visco-elastic of the belt's material, the geometric nonlinear of deformation and flexural rigidity for the visco-elastic moving belt are considered simultaneously in the paper, the nonlinear equation of planar transverse oscillation for visco-elastic moving belt is established using elastic mechanics method. The Kelvin visco-elastic model is adopted to describe the relation between the stress and strain for visco-elastic material. The motion partial equation contains more nonlinear polynomial expressions compared with other's work, the forced vibration equation of the belt subjected to periodic oscillation of the axially speed is also obtained.Secondly, It is continued to investigate the dynamic stability of the nonlinear transmission belt subjected to small harmonic variations of the belt's mean transmission velocity, the multiple scale method is applied directly to the nonlinear partial differential governing equation of the Kelvin visco-elastic model into a set of ordinary differential equations respect to small perturbation parameter, the modal functions invented by Kong are used in the roots of zero-order linear equation. When these roots are substituted into one-order equation the solvability conditions of the nonlinear equation are obtained--the permanent polynomials of the equationshould be canceled. The analysis shows that the principal parametric resonance will occur when the frequency of velocity fluctuations is close to two times the natural frequencies. Dynamic stability problems of parametric vibration are also considered. Numerical examples indicate that the velocity and bending stiffness of the belt both influence the resonance stability regions;the dynamic stability of the trivial and nontrivial solution is studied respectively by the Lyapunov stability theory. The stable and the instable regions for the belt's nonlinear transverse vibration are numerically presented.Finally, The second order truncation Galerkin approach is applied to the partial equation obtaining a group of second-order generalized coordinates nonlinear equations that decoupled in time and space coordinates. The numerically simulation is applied directly to the nonlinear partial differential equations of the belt to analyze the bifurcation and chaotic phenomena. Numerical simulation method is used to yield the response of the system via mathematic software MATLAB;the chaotic phenomenon such as sensibility of initial value and continues power spectrum are observed, the effect of the velocity perturbation (such as the mean velocity varies or the amplitude of perturbation varies) is discussed emphatically. The bifurcation figures and Poincare maps of all parameters (the mean transport speed, the amplitude and frequency of transport speed fluctuation, the belt's stiffness and the belt's dynamic viscosity) are studied by using step-length varied Runge-Kutta algorithm, from these figures it seems that the belt may undergo periodic and chaotic motions around different equilibrium in certain parameter regions, and the chaotic motion will occur through period double bifurcation. The Poincare map of continuous varied frequencies indicated that the amplitude of the chaos increases when the frequency of the external excitation increases...