Combination resonance of parametrically excited coupled second order systems with non-linear damping

Abstract

The parametrically excited oscillation of two coupled second order systems with nonlinear damping is investigated. The coupling is through the small parametric excitation terms in which the off-diagonal terms dominate, indicating combination resonance type motion with primary instability regions near the forcing frequencies ω F = Ω i + Ω j , with each mode oscillating near its own natural frequency, Ω i . Through an asymptotic analysis simple formulae are obtained for evaluating the steady state amplitudes in the first instability region and the non-dimensional solution surfaces are plotted. These are found to be analogous to those for a single second order system and represent master plots. The type of damping considered here includes the important case of velocity squared, fluid dynamic type damping. The theoretical results are compared with those from an experiment in which this type of damping was observed and measured. The measured lateral bending-torsion amplitudes are well predicted by the theory, the results of which are also in agreement with the numerical integration results.