Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

Abstract

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam

“…The increase in number of Hopf bifurcation points and an additional closed-loop-type solution as illustrated in Fig. 4 represents significant qualitative changes in the present study in contrast with [23].…”

“…Besides in the present study, both the modes have no upper bounds in the maximum nontrivial amplitude, but in the single-frequency parametric excitation case, the second mode response (a 2 ) is bounded, while there is no upper bound in nontrivial amplitude of first mode (a 1 ). Further, the number of Hopf bifurcation points and the range of instability associated with the Hopf bifurcations in the nontrivial equilibrium solution branches is substantially reduced in the present case compared to the case of [23]. In addition, the Hopf bifurcation points in the trivial state totally vanish in the present case compared to [23]; however, the saddle-type instability in trivial solution increases to some extent.…”

“…Further, the number of Hopf bifurcation points and the range of instability associated with the Hopf bifurcations in the nontrivial equilibrium solution branches is substantially reduced in the present case compared to the case of [23]. In addition, the Hopf bifurcation points in the trivial state totally vanish in the present case compared to [23]; however, the saddle-type instability in trivial solution increases to some extent.…”

“…4]. Unlike the case of [23], there are no saddle node (SN) bifurcations in the present case involving external damping. Besides in the present study, both the modes have no upper bounds in the maximum nontrivial amplitude, but in the single-frequency parametric excitation case, the second mode response (a 2 ) is bounded, while there is no upper bound in nontrivial amplitude of first mode (a 1 ).…”

“…In this study, the effects of mass numbers and mass ratios on the nonlinear natural frequency and amplitude of the system have been investigated. The authors of the paper [23,24] have performed the stability and bifurcation analysis for a traveling beam under single-frequency parametric excitation in presence of 3:1 internal resonance using direct method of multiple scales.…”