Optimisation of mechanical properties of viscoelastic structures

The spectral problem of determining the optimal distribution of mechanical properties of an arbitrary inhomogeneous body taking into account damping is investigated. The optimisation problem consists in finding the maximum first natural frequency. Damping is taken into account in the framework of the linear viscoelasticity model on the basis of the complex modules concept for the standard viscoelastic body model. The functions characterising the instantaneous and long-time modules are used as control functions. The problem formulation includes isoperimetric conditions that are imposed on the control functions and determine their average volume distribution. A Relye functional is constructed, and the optimality condition, which consists in the constancy of the potential energy, is found in a variational manner. As a model problem the problems of maximisation of the first natural frequency (bending and longitudinal vibrations) of a functionally graded cantilever beam with consideration of damping are considered. Analytical expressions for the laws of variation of the instantaneous and long-range modules are obtained. It is checked that the problem in the limiting case (when the relaxation time is equal to zero) is reduced to the elastic case. To determine the optimal value of the first natural frequency, a cubic equation is constructed and solved numerically. Asymptotic formulae for determining the optimal natural frequency at small values of relaxation time are obtained. Calculations have been carried out to evaluate the optimality of the obtained solution. For example, in comparison with the case of constant modules, the gain in the value of the first natural frequency is about 27% for the case of bending.