### ASA 129th Meeting - Washington, DC - 1995 May 30 .. Jun 06

## 3aSA1. The equation of diffusion applied to energy density of vibrating
beams.

**Myriam Djimadoum
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*L.V.A., I.N.S.A. Bat 303, Villeurbanne, France
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**Jean Louis Guyader
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*Institut National des Sciences Appliquees, 69621 Villeurbanne Cedex,
France
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Inspired by precedent studies on the possibility of writing simple
differential equations governing the evolution of vibratory energy density [D.
J. Nefske and S. H. Sung, Trans. ASME 111, 94--100 (1989)], and equation of
diffusion and its related conditions at discontinuities have been developed for
an energetic quantity (Omega) in the case of flexural waves propagating in
beams, where (Omega) represents the space-averaged far-field part of the
displacement autospectrum, is proportional to energy density, but does not need
extra time averaging. Several assumptions based on frequency and space
averaging allow writing energetic conditions at discontinuities and defining a
complete formalism for monodimensional problems. The procedure is applied to
two coupled Euler Bernoulli beams: (Omega) is numerically in very good
agreement with ``exact'' space and frequency-averaged far-field results and so
the conditions at discontinuities are validated. Knowing that (Omega) allows
for the ability to obtain an approximation of the energy flow in the beams and
so gives an idea of how the energy propagates in the structure. These good
results are encouraging, but the generalization of the procedure to plates is
not trivial and requires further assumptions.