The Effect of Uncertainty in the Excitation on the Vibration Input Power to a Structure

In structural dynamic systems, there is inevitable uncertainty in the input power from a source to a receiver. Apart from the nondeterministic properties of the source and receiver, there is also uncertainty in the excitation. This comes from the uncertainty of the forcing location on the receiver and, for multiple contact points, the relative phases, the force amplitude distribution at those points, and also their spatial separation. This paper investigates quantification of the uncertainty using possibilistic or probabilistic approaches. These provide the maximum and minimum bounds and the statistics of the input power, respectively. Expressions for the bounds, mean, and variance are presented. First the input power from multiple point forces acting on an infinite plate is examined. The problem is then extended to the input power to a finite plate described in terms of its modes. The uncertainty due to the force amplitude is also discussed. Finally, the contribution of moment excitation to the input power, which is often ignored in the calculation, is investigated. For all cases, frequency band-averaged results are presented. 1. Introduction The treatment of structure-borne sound sources remains a challenging problem. Structural excitation to a building floor, for example, by active components like pumps, compressors, fans, and motors, is an important mechanism of sound generation. To obtain an accurate prediction of the injected input power from such sources, both the source and the receiver must firstly be characterised. However in practical application, the variability of source and receiver properties including the lack of knowledge in the excitation force creates uncertainty in the input power. The problem is exacerbated because in practice there will usually be multiple contact points (typically four) and 6 degrees of freedom (3 for translation and 3 for rotation) at each, and that force and moment components at each contact point will contribute to the total input power. Therefore to assess the uncertainty, some quantification of the bounds, mean, and variance of the input power is of interest. The uncertainty in vibrational energy due to random properties, for example, dimensions, shapes, boundary conditions, and so on in a simple receiver structure, such as a plate, has been described by Langley and Brown [1, 2], where expressions for the mean vibrational energy and its variance were developed. A closed form solution was presented for the relative variance as a function of modal overlap factor and the nature of the excitation [1]. In [2], the