Vibrational Power Flow Analysis of Rods and Beams

A new method to model vibrational power flow and predict the resulting energy density levels in uniform rods and beams is investigated. This method models the flow of vibrational power in a manner analogous to the flow of thermal power in a heat conduction problem. The classical displacement solutions for harmonically excited, hysteretically damped rods and beams are used to derive expressions for the vibrational power flow and energy density in the rod and beam. Under certain conditions, the power flow in these two structural elements will be shown to be proportional to the energy density gradient. Using the relationship between power flow and energy density, an energy balance on differential control volumes in the rod and beam leads to a Poisson's equation which models the energy density distribution in the rod and beam. Coupling the energy density and power flow solutions for rods and beams is also discussed. It is shown that the resonant behavior of finite structures complicates the coupling of solutions, especially when the excitations are single frequency inputs. Two coupling formulations are discussed, the first based on the receptance method, and the second on the travelling wave approach used in Statistical Energy Analysis. The receptance method is the more computationally intensive but is capable of analyzing single frequency excitation cases. The traveling wave approach gives a good approximation of the frequency average of energy density and power flow in coupled systems, and thus, is an efficient technique for use with broadband frequency excitation.