Combination Parametric and Internal Resonances of an Axially Moving Beam

This paper deals with nonlinear planar vibration of a travelling beam subjected to combination parametric resonance in the presence of internal resonance. The beam is simply supported at both ends and the travelling velocity is assumed to be comprised of a harmonically varying component superimposed over a mean velocity. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the beam. The natural frequency of the second mode is approximately three times that of the natural frequency of the first mode for a range of mean velocity of the beam, resulting in a three-to-one internal resonance. The analysis is carried out using the Method of Multiple Scales (MMS) by directly attacking the governing nonlinear integral-partial-differential equations and the associated boundary conditions. The resulting set of first-order ordinary differential equations governing the modulation of amplitude and phase of the first two modes is analyzed numerically. The stability, bifurcation and response behavior of the beam is investigated for combination parametric resonance in presence of internal resonance. The system exhibits trivial and two mode closed loop and isolated solutions with saddle-node and Hopf bifurcations. The effects of higher magnitude of fluctuating velocity component and lower magnitude of internal frequency detuning parameter on the nonlinear interaction are investigated numerically. The dynamic response of the system is illustrated by periodic, mixed mode, quasiperiodic and chaotic behavior in terms of two dimensional phase portraits, Poincare maps, time traces and FFT power spectra. This wide array of dynamic response of the system shows the influence of internal resonance.