An application of Mueller's method for determining eigenfrequencies of vibrations of viscoelastic bodies with frequency-dependent characteristics of a material

A search for optimal damping properties of structures using methods of numerical modelling is as a rule associated with a large number of computations. Alongside this an application of mechanical problem of natural vibrations of structures for this purpose allows estimating damping properties of structures regardless external force and kinematic impacts. This fact leads to sufficient decrease in computational costs. The results of the solution to the problem of natural vibrations of piecewise-homogeneous viscoelastic bodies are complex natural vibration frequencies, the real part of which is a frequency of vibrations and imaginary part is damping index (rate of vibration damping). A mechanical behavior of a viscoelastic material is described by the linear theory of Boltzman–Volterra. Within the frameworks of this theory mechanical properties of a viscoelastic material can be represented as complex dynamic moduli (shear modulus and bulk modulus). As a rule, these properties depend on frequency of external excitation. In current paper an algorithm which allows obtaining solution to the problem on natural vibrations, in case when components of complex dynamic moduli are frequency-dependent, is represented. The algorithm is based on using capabilities of the ANSYS software package and also the Mueller's method which allows solving partial problem of complex eigenvalues. An efficiency and productivity of the algorithm is demonstrated on the example of a two-layered cantilever plate. One layer of the plate is made of an elastic material and the second one is made of a viscoelastic material. Reliability of the obtained results is proved by comparison natural vibration frequencies obtained as a result of solution to the problem of natural vibrations and resonant frequencies at frequency response plots of the displacements obtained as a result of solution to the problem of forced steady-state vibrations using the ANSYS software package.