Transverse Vibration And Stability Of Moving Viscoelastic Plate

Axially moving systems have extensive application in many fields such as machinery, textile, electron and spaceflight. Therefore, the study of the tranverse vibration and stability of the axially moving systems is of great significance. However, the existent studies have been mostly confined to unidimension problem, that is, the research objects are viscoelastic column and viscoelastic beam, few papers have been presented on the axially moving viscoelastic plate. The tranverse vibration and stability of the axially moving uniform plate and plate with varying thickness, the stability of axially moving non-conservative plate and the parametric vibration of the axially accelerating viscoelastic plate are analyzed in this paper, respectively. The main research work is as follows.(1) The transverse vibration and stability of the axially moving uniform viscoelastic plate are investigated. On the basis of the thin plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equation of the axially moving viscoelastic rectangular plate in the Laplace domain is deduced, the equation is suitable for various viscoelastic differential models. Then, the differential equation of motion of the viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt model for distortion in time domain is derived. The complex eigen-equation of axially moving viscoelastic plate is established by the differential quadrature method. The curves of real parts and imaginary parts of the first three order dimensionless complex frequencies versus dimensionless axially moving speed, the critical speed and the instability type are obtained by solving the complex eigen-equation.(2) The transverse vibration and stability of the axially moving viscoelastic rectangular plate with linearly and parabolically varying thickness in the direction orthogonal to the direction of motion are analyzed. The differential equation of motion of the axially moving viscoelastic plate with varying thickness in time domain is established, by introducing dimensionless variables, the mode of vibration equation of axially moving viscoelastic plate with varying thickness is obtained. The complex eigen-equation is derived by the discretization of the mode of vibration equation and boundary conditions using the differential quadrature method. The first three modes of the system is solved by Matlab program and effects of the aspect ratio, thickness ratio, the dimensionless moving speed and the dimensionless delay time on the transverse vibration and stability of the axially moving viscoelastic plate are analyzed.(3) The instability type and the critical load of viscoelastic rectangular plate subjected to uniformly distributed tangential follower force are studied. The curves of the dimensionless complex frequencies versus dimensionless follower force under three different boundary conditions are given, the factors influencing the instability type and critical load of the non-conservative visoelastic rectangular plate are discussed.(4) The differential equation of motion of axially moving non-conservative viscoelastic plate is a four order partial differential equation with variable coefficients, and the variable coefficients issue from tangential follower force. By the Levy method and power series method, the complex eigenvalue problem of the differential equation is derived. The complex frequency and the instability type of the axially moving non-conservative is obtained, thereby effects of the dimensionless delay time, non-conservative force and dimensionless axially moving speed on the stability of axially moving non-conservative viscoelastic plate is analyzed.(5) The parametric vibration of the axially accelerating viscoelastic plate is studied. The axially moving speed is a simple harmonic fluctuation about a constant mean speed, the differential equation of motion of axially accelerating viscoelastic plate is established. By the discretization of spatial variable in the equation, a third-order differential system of equations containing periodic time-varying coefficient is derived. Introducing the state vector, the first-order state equation is obtained, and it is solved by implicit expression of Runge-Kutta method. Based on the Floquet theory, the dynamic stability regions of the axially moving viscoelastic plate is determined, effects of the constant mean speed and the amplited of the simple harmonic fluctuation on the dynamic stability of the axially accelerating viscoelastic plate are discussed.