David J. Scholl
Ford Motor Company
Providing a smooth, quiet environment for drivers and passengers is one of the many challenges faced by automobile manufacturers, and mathematical tools are required to meet this challenge. These tools are used both for modeling, to predict the performance of a specific design, and for data analysis, to assess the performance of real vehicles throughout their service lives. Techniques based on Fourier transforms, such as modal analysis and statistical energy analysis, have proven indispensable for both performance prediction and assessment. However, it is becoming increasingly clear, as these Fourier-based techniques mature, that there is an important class of sounds and vibrations for which the Fourier transform is simply not very useful. The archetype of this class is the rattle, which is a repetitive series of mechanical impacts between two or more objects. Rattle sounds are characterized by a low duty cycle (the time duration of the individual event is short relative to the period of silence between successive events) and the lack of a perceptible musical pitch (the resonant vibrations that result from the impacts are heavily damped). These two characteristics of rattles make them especially suited for analysis with wavelet transforms, since the individual wavelet basis vectors share these same properties. This is in contrast to humming or whistling sounds, which have a high duty cycle and a definite musical pitch, much like the sines and cosines that make up the Fourier basis. To illustrate the utility of this approach, a new technique for wavelet-based transient waveform visualization developed at Ford Research Laboratories will be discussed and applied to examples of realistic automotive rattle sounds. The "Industrial Problems" to be discussed arise in two different areas. First, there are some specific problems prompted by the waveform visualization algorithm mentioned earlier. Second, and more generally, there is the question of how far the analogy between the engineering applications of the Fourier Transform and the Wavelet Transform can be extended. The wavelet-based data visualization algorithm mentioned earlier is analogous to the calculation of Fourier-based energy spectral density. Do any of the many other Fourier-based engineering algorithms have wavelet-based analogs, still waiting to be discovered?