Energy Flows between Arbitrary Configurations of Conservatively Coupled Multi-Modal Elastic Subsystems

In several recent publications exact energy flow results (sometimes referred to as `power flows') have been derived for pairs of vibrating multi-modal, elastic subsystems coupled by conservative springs at single points. These results were used to study the assumptions inherent in traditional statistical energy analysis (SEA) techniques but, although giving many insights, the limitations of the model adopted restricted the number of situations where they could be applied. More recently this analysis has been extended to lay down the theoretical groundwork enabling the most severe of these restrictions to be relaxed. Specifically, the theory developed allows for arbitrary numbers of subsystems and these may be coupled in any fashion desired, including having more than one coupling between pairs of subsystems. The current work draws on and extends this theory to enable deterministic and ensemble average energy flow calculations to be done for a number of configurations of interest. The examples discussed here range from a problem with one coupling and two subsystems, through three couplings and three subsystems up to 45 couplings and ten subsystems. All are one dimensional with low modal overlap factors and have no more than one coupling between any pair of subsystems, although the theory developed is not restricted to such problems. The results from these examples agree well with traditional SEA calculations when the subsystems are lightly coupled or the geometric configurations are of a random nature but differ markedly at high coupling strengths or for systems with marked patterns in their connections (such as chains of subsystems). One feature hitherto not widely discussed in the literature concerns the reversals of energy flows through couplings that can occur as driving frequencies or coupling strengths are varied. These various studies represent the first stage of a programme of research primarily aimed at quantifying deviations from the mean energy flows predicted by SEA. They are of particular relevance to engineering structures which commonly have small numbers of subsystems, few interacting modes, strong couplings or repetitive geometries.