Engineering systems with noise and vibration problems are investigated in a model of source, path and receiver where the source can be quantifi ed or modifi ed by changing its frequency content, levels, etc.; the path can be
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quantifi ed or modifi ed by changing mass, stiffness and damping or changing geometry, adding isolators, foams, fi bres, etc.; and the receiver can be quantifi ed or optimized by design for subjective response, sound quality, reduction in levels, etc.
In order to predict noise and vibration, classic low-frequency approaches are applied to establish system dynamic properties or system dynamic equa- tions in a large number of degrees of freedom by use of fi nite element dynamic analysis or an experimental modal analysis method, then to calcu- late the system responses from excitation and the dynamic properties or equations. For a typical saloon car, there are 3 × 106 structural modes and 1 × 106 acoustic modes at frequencies less than 10kHz. Higher-order modes are extremely sensitive to uncertainties in boundary conditions, material properties and physical properties. It takes a long computational time to solve the dynamic equations in the large number of degrees of freedom in the high frequencies, and the results of the solutions are uncertain due to the uncertainties caused by the material, manufacturing and assembly process variations as shown in Fig. 7.1. This is because subsystem properties and boundary conditions are not known precisely; short-wavelength responses in higher-order modes are very sensitive to small uncertainties.
Therefore, traditional deterministic analysis methods are not appropriate for problem solutions at high frequencies due to the expense and amount of detail required. Alternative approaches are needed for high-frequency noise and vibration prediction.
Statistical energy analysis (SEA) was originally developed in the 1960s as a method of predicting the high-frequency response of dynamic struc- tures [1]. Statistical energy analysis is a fi eld of study in which subsystems are statistically described in order to simplify the analysis of complicated structural–acoustic problems; the ‘S’ indicates systems drawn from a popu- lation or ensemble, ‘E’ represents the fact that energy is the primary response quantity of interest, and ‘A’ illustrates that SEA is an analysis frame instead of one method. The power injection method and experimen- tal techniques in comparison with analytical calculations such as imped- ance/modal density calculations were developed in the 1970s. Commercial codes such as VAPEPs, SEAM, AutoSEA, etc., appeared in the 1980s.
Leppington et al. [2] proposed radiation effi ciency formulations and k-space approaches; more generic coupling loss factor calculations based on line wave impedances were developed in the 1990s. Langley and co-workers [3–6] generalized the wave approach and developed the hybrid FEA–SEA method; based on the wave approach, variance estimation of energy vari- ables was extended to generic subsystems in the 2000s [7].
In SEA, the whole structure and acoustic cavity system are considered as a network of subsystems coupled through joints. A subsystem is defi ned as a fi nite region with a resonant behaviour, involving a number of modes
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of the same type. Each subsystem’s response is represented by a space and frequency band averaged energy level. Based on the principle of conserva- tion of energy, a band limited power balance matrix equation for the con- nected subsystems can be derived and easily resolved. Because of the reduced number of degrees of freedom of the models, the SEA calculations are usually very fast (from few minutes to few hours), allowing a fast analysis process in the design phase. The method predicts the ‘mean’ energy level in each of the subsystems, in the sense that the system is considered to have random properties. The method would, for example, predict the mean interior noise level for a fl eet of cars manufactured on the same pro- duction line. A variance prediction method for built-up structures was developed by Langley and Cotoni [7] in which prediction of the statistics of the response of a highly random system without knowledge of the nature of the underlying physical uncertainties became possible. The method
Frequency (Hz) (a)
(b)
0 50
60 40 20 0 –20 –40 –60
100 150 200 300 350 400 450 500
FRF (dB re 1 Pa/N)
60 40 20 0 –20 –40 –60
FRF (dB re 1 Pa/N)
250
Frequency (Hz)
0 50 100 150 200 250 300 350 400 450 500
7.1 Uncertainty of measured frequency response functions (FRF);
sampled data range: (a) one vehicle, repeated 12 times; (b) 98 nominally identical vehicles. The excitation and response points are identical for the FRF measurement.
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allows for the long-established SEA method to be extended to variance prediction. Both the mean energy level and its variance give estimations of subsystem responses. However, this method is only applicable in mid- to high-frequency ranges.
The objectives of this chapter are to provide an overview of the modern predictive SEA method, to highlight the problems associated with model- ling noise and vibration at mid- and high frequencies, to summarize the derivation of the SEA equations and list the main assumptions, and to illustrate the concepts and methods used to derive the parameters of an SEA model.