<abstract xmlns="http://www.w3.org/1999/xhtml"><p> Viscoelastic, or composite materials that are hereditary deformable, have been characterised by exponential and weakly singular kernels in a hereditary equation. An exponential kernel is easy to be numerically implemented, but does not well describe complex vibratory behaviour of a hereditary deformable system. On the other hand, a weakly singular kernel is known to describe the complex vibratory behaviour, but is nontrivial to be numerically implemented. This study presents a numerical formulation for solving a hereditary equation with a weakly singular kernel. Recursive algebraic equations, which are numerically solvable, are formulated by using the Galerkin method enhanced by a numerical integration and elimination of weak singularity. Numerical experiments showed that the present approach with a weakly singular kernel is well fitted into a realistic vibratory behaviour of a hereditary deformable system under dynamic loads, as compared to the same approach with an exponential kernel.</p></abstract>

Viscoelastic, or composite materials that are hereditary deformable, have been characterised by exponential and weakly singular kernels in a hereditary equation. An exponential kernel is easy to be numerically implemented, but does not well describe complex vibratory behaviour of a hereditary deformable system. On the other hand, a weakly singular kernel is known to describe the complex vibratory behaviour, but is nontrivial to be numerically implemented. This study presents a numerical formulation for solving a hereditary equation with a weakly singular kernel. Recursive algebraic equations, which are numerically solvable, are formulated by using the Galerkin method enhanced by a numerical integration and elimination of weak singularity. Numerical experiments showed that the present approach with a weakly singular kernel is well fitted into a realistic vibratory behaviour of a hereditary deformable system under dynamic loads, as compared to the same approach with an exponential kernel.

A Numerical Solution of Hereditary Equations with a Weakly Singular Kernel for Vibration Analysis of Viscoelastic Systems / Vienâdojumu Ar Vâjo Singulâro Kodolu Skaitliskais Risinâjums Iedzimto Viskoelastîgo Sistçmu Vibrâciju Analîzei

Faculty of Aviation, Tashkent State Technical University, Universitetskaya 2, 100095, Tashkent, UZBEKISTAN

Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences.

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

Viscoelastic, or composite materials that are hereditary deformable, have been characterised by exponential and weakly singular kernels in a hereditary...

A Numerical Solution of Hereditary Equations with a Weakly Singular Kernel for Vibration Analysis of Viscoelastic Systems / Vienâdojumu Ar Vâjo Singulâro Kodolu Skaitliskais Risinâjums Iedzimto Viskoelastîgo Sistçmu Vibrâciju Analîzei

Published Online: Feb 19, 2016

Page range: 326 - 330

Received: Apr 22, 2015

DOI: https://doi.org/10.1515/prolas-2015-0048

Keywords
viscoelastic system, vibration analysis, hereditary equation, weakly singular kernel, Galerkin method

© by Botir Usmonov

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

A Numerical Solution of Hereditary Equations with a Weakly Singular Kernel for Vibration Analysis of Viscoelastic Systems / Vienâdojumu Ar Vâjo Singulâro Kodolu Skaitliskais Risinâjums Iedzimto Viskoelastîgo Sistçmu Vibrâciju Analîzei

Published Online: Feb 19, 2016

Page range: 326 - 330

Received: Apr 22, 2015

DOI: https://doi.org/10.1515/prolas-2015-0048

Keywordsviscoelastic system, vibration analysis, hereditary equation, weakly singular kernel, Galerkin method

© by Botir Usmonov

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

Keywords
viscoelastic system, vibration analysis, hereditary equation, weakly singular kernel, Galerkin method