### ASA 128th Meeting - Austin, Texas - 1994 Nov 28 .. Dec 02

## 5aSA9. Wavelet frame expansions and wavelet response functions.

**Taner Onsay
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*Dept. of Mech. Eng., Michigan State Univ., East Lansing, MI 48824
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Wavelet frame expansions are utilized to introduce ``wavelet response
functions'' (WRF). The convolution integral is replaced by an efficient
synthesis scheme by incorporating multiresolution and wavelet frame expansion
concepts. Wavelet frame expansion is used to partition a wide spectral range
into multiresolution frequency bands. The response from each resolution level
is obtained by using a sampling interval that is matched with the bandwidth.
The contributions coming from different resolution levels are combined to
construct the overall system response. In the applications, the attention is
focused on the synthesis of the transient time-domain vibration response of
proportionally damped (B[sub n]=(eta), f[sub n], n=modes) wideband linear
systems. Naturally, such systems, having impulse response durations that are
inversely proportional to the modal bandwidths (T[inf max[inf n] ]= 2.2/B[inf
n]), require a multiresolution synthesis scheme during the construction of
their transient response, which involves fast-decaying high-frequency
components that are superimposed on top of slowly decaying low-frequency modal
responses. The proposed construction technique averts the time-domain aliasing
problem associated with the DFT schemes. The effectiveness of the wavelet frame
synthesis scheme is demonstrated by considering the construction of the
transient vibration response of a finite dispersive system.