Bifurcations and Chaotic Dynamics of an Axially Accelerating Hinged–Clamped Viscoelastic Beam

Abstract

This paper investigates stability, bifurcation and dynamic behavior with chaotic response of an axially accelerating hinged–clamped viscoelastic beam under the combined effect of parametric and internal resonant conditions. The beam material obeys the Kelvin–Voigt model in which the material time derivative is used. The axial time-dependent velocity is characterized as a simple harmonic variation about the constant mean speed. The direct method of multiple scales is applied to the nonlinear integro-partial differential equation of motion to reduce it into a set of first-order equations. Here, in particular, aim is fixed to examine the system response in the close proximity of a 3:1 internal resonance between first two transverse modes. Specifically, the fixed point responses are obtained using continuation technique and the dynamic solution is obtained through direct time integration. The steady-state solution has both trivial and two-mode nontrivial solutions with hardening type of nonlinear characteristics. The system displays various bifurcations including super and subcritical pitchfork, Hopf and saddle node bifurcations. The multiple equilibrium solutions get transition to instability of both a saddle node type and a Hopf bifurcation. The damping has different effects on different modes. With decrease in external damping, the maximum amplitude of directly excited first mode is nearly unaffected, but the maximum amplitude of the indirectly excited second mode grows. This shows the energy exchange between the involved modes is encouraged due the presence of internal resonance. Internal damping has stabilizing effect on the transverse modes in the Hopf bifurcation region of the frequency and amplitude response equilibrium curves. It suppresses the nonlinear characteristics, viz. the number of Pitchfork, saddle node and Hopf bifurcation points even to zero for certain cases. Most importantly, multiple nontrivial equilibrium solution curves reduce to two in case of frequency response analysis and a single isolated closed-loop nontrivial solution in amplitude response analysis. Dynamic solution reveals the existence of not only stable periodic, mixed mode and quasiperiodic solutions, but also unstable chaotic oscillations depending on the system parameters and initial conditions in the neighborhoods of Hopf bifurcation regions. The system displays a wide array of interesting dynamic results equipped with various nonlinear features which are not present in the available literature on nonlinear traveling systems.