Vibrational energy distributions in large built-up structures - a wave chaos approach

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Predicting the response of a large, complex mechanical system such as a car or an aeroplane to high frequency vibrations is a remarkably difficult task. Still, obtaining good estimates for the distribution of vibrational energy in such structures, including coupling between sub-components, damping and energy loss in form of acoustic radiation, is of great importance to engineers. An increasing demand for low vibration, low noise products to meet performance specifications and to reduce noise pollution makes any improvement in predicting vibrations response characteristics of immediate interest for industrial applications. Demand for improved virtual prototyping, as opposed to the use of expensive and time-consuming physical prototypes, is another area of application in reducing development costs and time scales. Numerical tools are often based on 'Finite Element Analysis' (FEM). While these methods work well in the low frequency regime, that is, tackling wavelengths of the order of the size of the system, they become too expensive computationally in the mid-to-high frequency regime. In particular, FEM fails to describe accurately so-called mid-frequency problems where sub-components are characterized by a wide variation of wave-lengths. While FEM is suitable for handling 'stiff' elements such as the body frame in a car, it cannot routinely capture energy transport through 'soft' components such as thin, flexible plates coupled to stiff components. A common numerical tool for predicting the vibrational contribution of short wave length components is Statistical Energy Analysis (SEA); it is, however, based on a set of restrictive assumptions which, so far, are often hard to control and generally only fulfilled in the high frequency limit and for low damping. Thus, SEA can not deliver the degree of reliability necessary to make it attractive for a wider end user community in industrial R & D departments. It is suggested here that mathematical tools from wave or quantum chaos can considerably improve the situation sketched above. Recent results by the PI Tanner show that by combining methods ranging from operator theory, dynamical systems theory and small wavelength asymptotics, SEA can be embedded into a more general theory. The new approach is based on semiclassical expansions of the full Green function in terms of rays and describing the nonlinear ray-dynamics in terms of linear operators. The resulting method captures the full correlations in the ray dynamics and has such a much improved range of validity compared to SEA. The method could revolutionise the treatment of vibrations in complex mechanical systems. Not only does it allow (i) to give quantitative bounds for the applicability of SEA (of interest to SEA users); it will also (ii) improve predictive capability in situation where SEA does not apply at a moderate computational overhead; in addition, (iii) it can be easily combined with FEM methods thus making it an ideal candidate for tackling mid-frequency problems. The approximations made are well controlled by starting from a semiclassical approach which makes it possible (iv) to systematically incorporate wave interference effects (absent in standard SEA treatments) into the method.By tackling the issues addressed above we will be able to provide improved and conceptually completely new solution methods to the engineering community based on advanced mathematical methods. The proposed research evolved out of pump-prime EPSRC funding in terms of a Springboard Fellowship. The project is thus by default of interdisciplinary nature and will be tackled jointly by the PI Tanner (Nottingham, Mathematics) and PI Mace (Southampton, ISVR, Engineering) with industrial partners from the FEM/software side (inuTech) and an engineering consulting firm (DS2L) providing input about end-user demands.