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

L. V. Parshina, V. M. Ryabov, B. A. Yartsev

Research output

### 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.

Original language English 413-420 8 Vestnik St. Petersburg University: Mathematics 51 4 https://doi.org/10.3103/S106345411804012X Published - 1 Oct 2018

### Fingerprint

Composite Structures
Energy Dissipation
Natural Frequency
Vibration
Numerical Solution
Third-order Method
Ritz Method
Legendre polynomial
Young's Modulus
Iteration Method
Algebraic Equation
Dissipation
Polymers
Mathematical Model
Differential equation
Motion

### Scopus subject areas

• Mathematics(all)

### Cite this

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title = "Energy Dissipation during Vibrations of Heterogeneous Composite Structures: 2. Method of Solution",
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 = "convergence of numerical solution, damping, Legendre polynomials, linear algebraic equations, loss factor, natural frequency, solution method, validation",
author = "Parshina, {L. V.} and Ryabov, {V. M.} and Yartsev, {B. A.}",
year = "2018",
month = "10",
day = "1",
doi = "10.3103/S106345411804012X",
language = "English",
volume = "51",
pages = "413--420",
journal = "Vestnik St. Petersburg University: Mathematics",
issn = "1063-4541",
number = "4",

}

In: Vestnik St. Petersburg University: Mathematics, Vol. 51, No. 4, 01.10.2018, p. 413-420.

Research output

TY - JOUR

T1 - Energy Dissipation during Vibrations of Heterogeneous Composite Structures

T2 - 2. Method of Solution

AU - Parshina, L. V.

AU - Ryabov, V. M.

AU - Yartsev, B. A.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

KW - convergence of numerical solution

KW - damping

KW - Legendre polynomials

KW - linear algebraic equations

KW - loss factor

KW - natural frequency

KW - solution method

KW - validation

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