Energy Sharing in the Random Vibration of Non-linearly Coupled Modes

If small non-linear terms couple the (linear) modes of a multidegree-of-freedom system, energy can flow from one mode to another and vice versa. In this paper a perturbation method is used to calculate the average rate of energy transfer between nonlinearly coupled modes. Each mode is assumed to be subjected to an independent source of stationary, Gaussian, random excitation, and the coupling terms, which are conservative, are of the form x_rx_s, $xs2$ and x_r$xs2$, where x_r and x_s are the normal co-ordinates of the linearized system. The principal result is that, if the excitation is white noise, no flow of energy occurs when each mode has the same energy. Thus, as predicted by the results of statistical mechanics, the equilibrium state for white noise excitation is that of equipartition of energy between the modes. When the modal energies are not equal, energy flows from modes of higher energy to modes of lower energy, and the rate of energy transfer is greatest for those modes whose natural frequencies satisfy internal resonance combinations. An application of the results of the paper is in calculating noise transmission between connected structures, where vibrational modes may be coupled by small nonlinear terms. If the natural frequencies of the coupled modes are suitably related, it is shown that very small coupling may cause considerable excitation of otherwise dormant parts of a structure. The paper may also be of interest in the study of molecular reactions and biological rhythms, and of non-linear wave interactions in such areas as plasma dynamics, turbulence, and the theory of water waves.