Dynamics of axially moving beams: A finite difference approach

Abstract

In this paper, a finite difference approach is being presented as the most reliable technique to investigate the non-linear transverse vibration of an axially moving beam. Firstly, Hamilton’s principle is used to derive the governing hyperbolic partial differential equation (HPDE) of free transverse vibration. Then finite difference along with state space method is used to convert the partial differential equation into a system of coupled first-order ordinary differential equations (ODE’s). Comparison of the proposed finite difference model is made with both the numerical and analytical models. It is noted that the finite difference model is in excellent agreement with the modal analysis approach (analytical solutions) as compared to the generalized integral transform technique GITT (numerical method). A parametric study was also presented to examine the impact of system parameters, such as axial translation speed and flexure rigidities of the beam on the transverse response amplitude, frequencies and instabilities. The transverse vibration of the beam at selected points along its length demonstrates higher-frequency oscillations that corresponds to a lower axial speed. With the increase of beam flexural rigidity, the divergence critical speed increases. Coupled effect of axially moving beam and rotating rolls is also considered.

Keywords

• Finite difference
• Partial differential equations
• Axially moving beam
• Roll-to-roll systems