Finite element synthesis of structural or acoustic receptances in view of practical design applications

The present study addresses the modal synthesis of dynamic responses or receptances, which can be considered as a major issue in vibratory or acoustic design. A new method termed “method of orthocomplement” offers to increase the accuracy of modal superposition. Two apparently independent domains are considered: first, the numerical control of round-off errors in finite-dimensional models; second, the regularization of modal boundary discontinuities in infinite-dimensional formulations. Concerning the finite-dimensional systems, using the proposed method in conjunction with pseudo-inversion theorems, proves that “inertia relief corrections” remedy modal truncation problems in singular floating systems, just as static corrections do for simply supported systems. The proof is based on hitherto unpublished expressions of inertial contributions in terms of orthogonal projectors. It is worth noting that such expressions present by themselves some technical interest. Modal formulae being delivered in closed form, a thorough analysis of errors becomes possible. Special “spectrograms” are introduced to appraise the convergence of ordinary or modified modal sums. When applied to infinite-dimensional modal boundary discontinuities, the method of orthocomplement eliminates the Gibbs oscillations that affect the ordinary spectral sums, and produces the same corrected formulae as it did for finite-dimensional examples. Effective solutions for the computation of vibratory or acoustic boundary receptances, are developed in that way. The paper once again illustrates how orthogonal projectors can contribute to the determination of static corrections. Future applications to coupling problems, model reduction and modal synthesis are briefly presented at the end of the text.