# Buckling Induced Strongly Nonlinear Vibration of Supercritical Axially Moving Beam
## 1 Introduction  
Axially moving systems, such as threads, textiles, wires, magnetic tapes, belts and cables, are present in a wide class of engineering structures. The vibration of such devices has been studied extensively [1, 2, 3, 4, 5, 6]. In general, one-dimensional string or beam theory can be used to model axially moving elements, depending on flexural stiffness [7, 8, 9, 10, 11, 12, 13, 14].  
It is known that the linear frequency of a moving beam decreases as its axial speed increases. At certain critical speeds, the linear frequency vanishes and buckling occurs [3, 7]. In previous studies, the linear and nonlinear vibration of axially moving beams were studied in the subcritical regime, and referred to as subcritical axially moving beams. Herein, we refer to beams traveling in the supercritical regime as supercritical axially moving beams. [PERSON] [3] studied the linear vibration of an axially moving beam with time-dependent velocity below critical value. The multi-scales method was used to investigate the resulting parametric resonance. [PERSON] and [PERSON] [15, 16] described three-dimensional parametric vibrations of a simply-supported pipe conveying pulsating fluid, and found that the jump phenomenon for the parametric resonance in the post-buckling state could be clearly observed. [PERSON] and [PERSON] [17, 18] derived a new governing equation for the traveling beam using geometric nonlinearity, which was used to account for finite stretching of the beam; the nonlinear vibration problem was studied using the perturbation method. [PERSON] et al. [19, 20, 21] contributed to the field of forced dynamics of a subcritical axially moving viscoelastic beam and investigated the beam's bifurcation and chaotic motions.  
The literature regarding supercritical axially moving beams is rather limited. [PERSON] and [PERSON] [22, 23] studied the influence of the initial curvature caused by the support wheel and pulley on the bifurcation and stability in the supercritical regime. [PERSON] [24, 25] included the nonlinear stretching effects for a moving beam and analyzed both the subcritical and supercritical speed range. [PERSON] et al. [26, 27, 28, 29] utilized the Galerkin method and other numerical methods to calculate the supercritical natural frequency and equilibrium configurations for different boundary conditions. [PERSON] et al. [30]studied the stability of an axially moving beam subjected to harmonic excitation. They employed the multi-scales method to calculate the steady-state response amplitude for the first single mode; the strongly nonlinear state was shown in some situations.  
In this paper, we focus on the dynamics of supercritical axially moving beams. In this high-speed region, the translating beam undergoes large overall buckling deformations leading to strongly nonlinear vibration. This situation is common; however, the solution is not available in the literature. Using Galerkin truncation, we found that the partial differential equation can be simplified and that the dynamic process can be described by an ordinary differential equation, which differs from the previous literature. Without using transformation, we directly used the homotopy analysis method to address the strongly nonlinear problem.