Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 4, pp 413–420

# Energy Dissipation during Vibrations of Heterogeneous Composite Structures: 2. Method of Solution

• L. V. Parshina
• V. M. Ryabov
• B. A. Yartsev
Mathematics

## Abstract

This paper describes the method of numerical solution of decaying vibration equations for heterogeneous composite structures. The system of algebraic equations is generated by applying the Ritz method with Legendre polynomials as coordinate functions. First, real solutions are found. To find complex natural frequencies of the system, the obtained real natural frequencies are taken as initial values, and then, by means of the third-order iteration method, complex natural frequencies are calculated. The paper discusses the convergence of numerical solution of the differential equations describing the motion of layered heterogeneous structures, obtained for an unsupported rectangular two-layered plate. The bearing layer of the plate is made of unidirectional CRP, its elastic and dissipation properties within the investigated band of frequencies and temperatures are independent of vibration frequency. The bearing layer has one of its outer surfaces covered with a layer of “stiff” isotropic viscoelastic polymer characterized by a temperature-frequency relationship for the real part of complex Young’s modulus and loss factor. Validation of the mathematical model and numerical solution performed through comparison of calculation results for natural frequencies and loss factor versus test data (for two composition variants of a two-layered unsupported beam) has shown good correlation.

## Keywords

solution method Legendre polynomials linear algebraic equations damping convergence of numerical solution validation natural frequency loss factor

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## Authors and Affiliations

• L. V. Parshina
• 1
Email author
• V. M. Ryabov
• 2
• B. A. Yartsev
• 1
• 2
1. 1.Krylov State Research CenterSt. PetersburgRussia
2. 2.St. Petersburg State UniversitySt. PetersburgRussia