Modal Analysis of Elastic Vibrations of Incompressible Materials Based on a Variational Multiscale Finite Element Method

Date

2021-01-01

Author

Codina, Ramon
Türk, Önder

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

139
views

0
downloads

© 2021, Springer Nature Switzerland AG.In this study, we extend the standard modal analysis technique that is used to approximate vibration problems of elastic materials to incompressible elasticity. The second order time derivative of the displacements in the inertia term is utilized, and the problem is transformed into an eigenvalue problem in which the eigenfunctions are precisely the amplitudes, and the eigenvalues are the squares of the frequencies. The finite element formulation that is based on the variational multiscale concept preserves the linearity of the eigenproblem, and accommodates arbitrary interpolations. Several eigenvalues and eigenfunctions are computed, and then the time approximation to the continuous solution is obtained taking a few modes of the whole set, those with higher energy. We present an example of the vibration of a linear incompressible elastic material showing how our approach is able to approximate the problem. It is shown how the energy of the modes associated to higher frequencies rapidly decreases, allowing one to get good approximate solution with only a few modes.

URI

https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85106441510&origin=inward
https://hdl.handle.net/11511/98814

DOI

https://doi.org/10.1007/978-3-030-55874-1_101

Conference Name

European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2019

Collections

Graduate School of Applied Mathematics, Conference / Seminar

Suggestions

OpenMETU
Core

Modal analysis of elastic vibrations of incompressible materials based on a variational multiscale finite element method Codina, Ramon; Türk, Önder (null; 2019-10-04) In this study, we extend the standard modal analysis technique that is used to approximate vibration problems of elastic materials to incompressible elasticity. In modal analysis, the second order time derivative of the displacements in the inertia term is utilized, and the problem is transformed into an eigenvalue problem in which the eigenfunctions are precisely the amplitudes, and the eigenvalues are the squares of the frequencies. While this approach is applied directly to compressible materials in dif...
Modal analysis of elastic vibrations of incompressible materials using a pressure-stabilized finite element method Codina, Ramon; Türk, Önder (2022-09-01) This paper describes a modal analysis technique to approximate the vibrations of incompressible elastic solids using a stabilized finite element method to approximate the associated eigenvalue problem. It is explained why residual based formulations are not appropriate in this case, and a formulation involving only the pressure gradient is employed. The effect of the stabilization term compared to a Galerkin approach is detailed, both in the derivation of the approximate formulation and in the error estimat...
Modal identification of nonlinear substructures and implementation in structural coupling analysis Arslan, Özge; Özgüven, Hasan Nevzat; Department of Mechanical Engineering (2008) In this work, a new method is suggested for the modal identification of nonlinear structures and for the use of the modal data in calculating response of the nonlinear system to harmonic excitation. Nonlinearity in mechanical structures is usually encountered in connection regions. In this study, the nonlinear part of such a structure is modeled as a single nonlinear element and modal parameters of the structure are obtained as a function of displacement amplitude. Identification and modeling of nonlinear e...
Static and dynamic analysis of shear deformable composite shells of revolution by semi-analytical approach Kayran, Altan (2013-10-18) In the present study, multi-segment numerical integration technique is applied for the static and dynamic analysis of macroscopically anisotropic shells of revolution including transverse shear deformation. Application of the multi-segment numerical integration technique is achieved through the use of finite exponential Fourier transform of the fundamental shell of revolution equations governing the static loading and free vibration of the shell of revolution. For the non-axisymmetrically loaded shells of r...
Static and free vibration analyses of small - scale functionally graded beams possessing a variable length scale parameter using different beam theories Aghazadeh, Reza; Dağ, Serkan; Ciğeroğlu, Ender; Department of Mechanical Engineering (2013) This study presents static and free vibration analyses of functionally graded (FG) micro - beams on the basis of higher order continuum mechanics used in conjunction with classical and higher order shear deformation beam theories. Unlike conventional ones, higher order elastic theories consider the size effect for the beam. Strain gradient theory (SGT) and modified couple stress theory (MCST) are the two common non-classical continuum approaches capable of capturing the size effect. Shear deformation beam t...

Citation Formats

R. Codina and Ö. Türk, “Modal Analysis of Elastic Vibrations of Incompressible Materials Based on a Variational Multiscale Finite Element Method,” Egmond aan Zee, Hollanda, 2021, vol. 139, Accessed: 00, 2022. [Online]. Available: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85106441510&origin=inward.