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

Abstract

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.

Mechanics of Moving Materials

“…It was shown that the natural frequency of each mode decreases when the transport speed increases, and that the travelling string and beam both experience divergence instability at a suciently high speed. However, in the case of the string, this result was recently contrasted by Wang et al (2005), who showed using…”

“…the buckling analysis of travelling panels and plates is based on this idea. However, in their analysis of the travelling ideal string, Wang et al (2005) caution that steady-state solutions may exist without indicating an instability, if the eigenfunctions remain linearly independent at the critical parameter value. Thus, we may conclude that a static instability can only arise from a steady state, but in a rigorous analysis, the existence of a steady state should be taken only as a necessary condition for static instability, not a sucient one.…”

Theoretical study on travelling web dynamics and instability under non-homogeneous tension

“…Studies by Mote include dynamic stability analysis of axially moving strings under periodic tension variations [9] or under axial acceleration [10]. Recently, Wang et al [11] showed using Hamiltonian mechanics that there is no instability at the critical velocity in the 2 case of a travelling string.…”

Estimates for Divergence Velocities of Axially Moving Orthotropic Thin Plates

“…They presented the expressions for the critical transport velocities analytically. However, recently Wang et al (2005) showed analytically that no static instability occurs for the transverse motion of an ideal string at the critical velocity. For axially moving beams with small flexural stiffness, Kong and Parker (2004) found closed-form expressions for the approximate frequency spectrum by a perturbation analysis.…”