Optimization of dissipative characteristics of structures on the basis of problems on natural vibrations of viscoelastic solids

The damping ability of a material plays an important role in the dynamical behavior of structures. It is responsible for decay of free vibrations, drastic decrease in amplitudes of the displacements and stresses arising in structures subjected to dynamical actions. To date, a lot of approaches have been developed which describe the mechanism of internal friction of materials, which causes the energy dissipation under vibrations [

1

]. The Boltzmann-Volterra theory [

2

] is the most general linear one, reflecting practically all peculiarities of the quasistatic and dynamic behavior of viscoelastic materials.

The damping for structures can be positive or negative factor. A quantitative assessment of the dissipative properties of structures is generally based on the results of solving free vibrations. In this case the dissipation of a system leads to the decay of vibrations, and the rate of decay estimates quantitatively the dissipative properties of a system. The higher is the decay rate of vibrations, the greater are the dissipative properties.

The problem of natural damped vibrations is formulated using a complex analog to the Boltzman-Volterra equations. The transition to the complex analog is made under some assumptions. To confirm the validity of the algorithm for optimization of the dissipative properties of a construction, the optimization search by solving the problem of forced steady-state vibrations is also performed. The application of the finite element method to the stated problem reduces it to the algebraic problem of eigenvalues for complex matrices.

To have assurance that the lowest vibration modes defining the damping properties of the system will be determined and to decrease significantly the volume of calculations, the method is proposed which is based on expansion of the viscoelastic problem solution into finite series with respect to eigenforms of vibrations of the corresponding elastic structure. The optimization problem is solved within the framework of nonlinear programming. Mechanical and geometrical parameters of the system are used as optimization parameters.

The performed numerical experiments showed that the optimal structure could be achieved even in the case when the real viscous properties of the material are given in rather rough manner.