Power Flow between Linearly Coupled Oscillators

Abstract

The power flow between two independently and randomly excited harmonic oscillators is calculated assuming small linear coupling. It is found that for conservative coupling the power flow is proportional to the temperature (average modal energy) difference of the two oscillators. The constant of proportionality is symmetric in the parameters of the two modes and is positive definite although its magnitude depends on the relative sign of the inertial and stiffness coupling. An equivalent circuit for the energy flow between the modes is developed. Finally, we apply the formalism to obtain the motion transmitted through a two-stage vibration isolator. The method is then extended to the problem where two multimodal systems interact. A particular application is the power flow between several structural modes and a reverberant acoustic field. The reverberant sound field is considered as a temperature bath in which the structural modes are immersed. Using this model the steady-state partition of energy between the two systems and the parameters which govern this partition are computed. Analog circuits for this problem are also constructed. The radiation resistance for some of the normal modes of a beam are calculated and results of the formalism are illustrated experimentally.