Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system)

The analysis of the energy transfer between subsystems coupled in a hybrid system is an urgent problem for various applications. We present an analytic investigation of the energy transfer between linear and nonlinear oscillators for the case of free vibrations when the oscillators are statically or dynamically connected into a double-oscillator system and regarded as two new hybrid systems, each with two degrees of freedom. The analytic analysis shows that the elastic connection between the oscillators leads to the appearance of a two-frequency-like mode of the time function and that the energy transfer between the subsystems indeed exists. In addition, the dynamical linear constraint between the oscillators, each with one degree of freedom, coupled into the hybrid system changes the dynamics from single-frequency modes into two-frequency-like modes. The dynamical constraint, as a connection between the subsystems, is realized by a rolling element with inertial properties. In this case, the analytic analysis of the energy transfer between linear and nonlinear oscillators for free vibrations is also performed. The two Lyapunov exponents corresponding to each of the two eigenmodes are expressed via the energy of the corresponding eigentime components. Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 796–814, June, 2008.