Optimal Design of Vibrating Structures

The problem of optimal design of vibrating elastic systems with damping is discussed. Linear constitutive relations in differential form for viscoelastic materials are cited with remarks concerning some experimental results of optimal identification of material damping. These relations are used in the derivation of selected variational principles in complex formulation, together with analysis of forced vibrations and eigenproblems. It is demonstrated that material damping, described in commonly used linear models of viscoelastic materials, significantly changes mechanical properties of a system with even an arbitrarily small amount of dissipated energy. It is shown that a continuous viscoelastic system, in contrast to a perfectly elastic one, may have a finite number of eigenfrequencies and resonant frequencies. Moreover, some of the eigenfrequencies associated with high order eigenmodes may be shifted down, due to dissipation of energy. Therefore the eigenvalue optimization problem is directly related to material characteristics and frequently cannot be formulated in such a manner as for perfectly elastic systems. The formulation of sensitivity analysis and optimization of structures for two important classes of problems, namely forced steady state harmonic vibration and eigenproblems, are presented and discussed.