TY - JOUR

T1 - Free vibration frequencies of a circular thin plate with nonlinearly perturbed parameters

AU - Vasiliev, G. P.

AU - Smirnov, A. L.

N1 - Publisher Copyright: © 2021 Journal of Turkish Sleep Medicine. All rights reserved. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/5/25

Y1 - 2021/5/25

N2 - Transverse vibrations of an inhomogeneous circular thin plate are studied. The plates, which geometric and physical parameters slightly differ from constant ones and depend only on the radial coordinate, are analyzed. After separation of variables the obtained homogeneous ordinary differential equations together with homogeneous boundary conditions form a regularly perturbed boundary eigenvalue problem. For frequencies of free vibrations of a plate, which thickness and/or Young's modulus nonlinearly depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. As examples, free vibrations of a plate with parameters quadratically or exponentially depending on the radial coordinate, are examined. The effect of the small perturbation parameter on the behavior of frequencies is also analyzed under special conditions: I) for a plate, the mass of which is fixed, if the thickness is variable and ii) for a plate with the fixed average stiffness, if Young's modulus is variable. Finally, effects of the boundary conditions and values of the wave numbers on the corrections to frequencies are studied. For a wide range of small parameter values, the asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4 and the numerical results of other authors.

AB - Transverse vibrations of an inhomogeneous circular thin plate are studied. The plates, which geometric and physical parameters slightly differ from constant ones and depend only on the radial coordinate, are analyzed. After separation of variables the obtained homogeneous ordinary differential equations together with homogeneous boundary conditions form a regularly perturbed boundary eigenvalue problem. For frequencies of free vibrations of a plate, which thickness and/or Young's modulus nonlinearly depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. As examples, free vibrations of a plate with parameters quadratically or exponentially depending on the radial coordinate, are examined. The effect of the small perturbation parameter on the behavior of frequencies is also analyzed under special conditions: I) for a plate, the mass of which is fixed, if the thickness is variable and ii) for a plate with the fixed average stiffness, if Young's modulus is variable. Finally, effects of the boundary conditions and values of the wave numbers on the corrections to frequencies are studied. For a wide range of small parameter values, the asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4 and the numerical results of other authors.

KW - Free vibrations of plates

KW - Inhomogeneous circular plate

KW - Perturbation method

UR - http://www.scopus.com/inward/record.url?scp=85108102294&partnerID=8YFLogxK

U2 - 10.18500/1816-9791-2021-21-2-227-237

DO - 10.18500/1816-9791-2021-21-2-227-237

M3 - Article

AN - SCOPUS:85108102294

VL - 21

SP - 227

EP - 237

JO - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics

JF - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics

SN - 1816-9791

IS - 2

ER -