Strongly Nonlinear Subharmonic Resonance and Chaotic Motion of Axially Moving Thin Plate in Magnetic Field

Abstract

In this paper, the nonlinear vibration and chaotic motion of the axially moving currentconducting thin plate under external harmonic force in magnetic field is studied. Improved multiple-scale method is employed to derive the strongly nonlinear subharmonic resonance bifurcation-response equation of the strip thin plate in transverse magnetic field. By using the singularity theory, the corresponding transition variety and bifurcation, which contain two parameters of the universal unfolding for this nonlinear system, are obtained. Numerical simulations are carried out to plot the bifurcation diagrams, corresponding maximum Lyapunov exponent diagrams, and dynamical response diagrams with respect to the bifurcation parameters such as magnetic induction intensity, axial tension, external load, external excited frequency, and axial speed. The influences of different bifurcation parameters on period motion, period times motion, and chaotic motion behaviors of subharmonic resonance system are analyzed. The results show that the complex dynamic behaviors of resonance system can be controlled by changing the corresponding parameters.

Vibrations of axially traveling plates partially coupled with a viscous, orthogonally flowing liquid

“…where N is the shape function matrix of order (1,12), and δ e is the displacement vector of the element of order (12, 1), which is written as…”

“…Dynamics of structures is important issue in engineering structures [3][4][5][6][7]. Many dynamics studies have been conducted on axially moving structures in vacuum [8][9][10][11][12][13][14]. Higher-dimensional periodic and chaotic oscillations for a parametrically excited viscoelastic moving belt with multiple internal resonances were investigated by Zhang and Song [15].…”

Dynamical Behavior of Fractional-Order Delayed Feedback Control on the Mathieu Equation by Incremental Harmonic Balance Method

“…For the numerical integration method, because of its slow convergence speed, it is very difficult to analyze the effects of those parameters in complex dynamical systems. However, some efficient analytical methods, such as the generalized multiple-scale method with its improved version, the perturbation-incremental method and the homotopy analysis method [22][23][24][25] could provide satisfactory results for those strongly nonlinear oscillators in some complex engineering. IHBM is a classical, effective, and semianalytical method with many advantages.…”