The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections

In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fo...

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Autores principales:	Guofang Li, Gang Wang, Junfang Ni, Liang Li
Formato:	article
Lenguaje:	EN
Publicado:	SAGE Publishing 2021
Materias:	Control engineering systems. Automatic machinery (General) TJ212-225 Acoustics. Sound QC221-246
Acceso en línea:	https://doaj.org/article/a9449e7421cc406c99164e4b8ab8e68b
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spelling	oai:doaj.org-article:a9449e7421cc406c99164e4b8ab8e68b2021-12-02T01:34:16ZThe vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections1461-34842048-404610.1177/14613484211019648https://doaj.org/article/a9449e7421cc406c99164e4b8ab8e68b2021-12-01T00:00:00Zhttps://doi.org/10.1177/14613484211019648https://doaj.org/toc/1461-3484https://doaj.org/toc/2048-4046In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fourier cosine series, which are mathematically capable of representing any variable cross section. The Young’s modulus, the mass density and the shear modulus varying along the lengthwise direction of the beam, are also expanded into Fourier cosine series. The translational displacement and rotation of cross section are expressed into the Fourier series by adding some polynomial functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. According to Hamilton’s principle, the eigenvalues and the coefficients of the Fourier series can be obtained. Some examples are presented to validate the accuracy of this method and study the influence of the parameters on the vibration of the beam. The results show that the first four natural frequencies gradually decrease as the coefficient of the radius β increases, and decreases as the gradient parameter n increases under clamped–clamped end supports. The stiffness of the functionally Timoshenko beam with arbitrary cross sections is variable compared with the uniform beam, which makes the vibration amplitude of the beam have different changes.Guofang LiGang WangJunfang NiLiang LiSAGE PublishingarticleControl engineering systems. Automatic machinery (General)TJ212-225Acoustics. SoundQC221-246ENJournal of Low Frequency Noise, Vibration and Active Control, Vol 40 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Control engineering systems. Automatic machinery (General) TJ212-225 Acoustics. Sound QC221-246
spellingShingle	Control engineering systems. Automatic machinery (General) TJ212-225 Acoustics. Sound QC221-246 Guofang Li Gang Wang Junfang Ni Liang Li The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections
description	In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fourier cosine series, which are mathematically capable of representing any variable cross section. The Young’s modulus, the mass density and the shear modulus varying along the lengthwise direction of the beam, are also expanded into Fourier cosine series. The translational displacement and rotation of cross section are expressed into the Fourier series by adding some polynomial functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. According to Hamilton’s principle, the eigenvalues and the coefficients of the Fourier series can be obtained. Some examples are presented to validate the accuracy of this method and study the influence of the parameters on the vibration of the beam. The results show that the first four natural frequencies gradually decrease as the coefficient of the radius β increases, and decreases as the gradient parameter n increases under clamped–clamped end supports. The stiffness of the functionally Timoshenko beam with arbitrary cross sections is variable compared with the uniform beam, which makes the vibration amplitude of the beam have different changes.
format	article
author	Guofang Li Gang Wang Junfang Ni Liang Li
author_facet	Guofang Li Gang Wang Junfang Ni Liang Li
author_sort	Guofang Li
title	The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections
title_short	The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections
title_full	The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections
title_fullStr	The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections
title_full_unstemmed	The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections
title_sort	vibration analysis of the elastically restrained functionally graded timoshenko beam with arbitrary cross sections
publisher	SAGE Publishing
publishDate	2021
url	https://doaj.org/article/a9449e7421cc406c99164e4b8ab8e68b
work_keys_str_mv	AT guofangli thevibrationanalysisoftheelasticallyrestrainedfunctionallygradedtimoshenkobeamwitharbitrarycrosssections AT gangwang thevibrationanalysisoftheelasticallyrestrainedfunctionallygradedtimoshenkobeamwitharbitrarycrosssections AT junfangni thevibrationanalysisoftheelasticallyrestrainedfunctionallygradedtimoshenkobeamwitharbitrarycrosssections AT liangli thevibrationanalysisoftheelasticallyrestrainedfunctionallygradedtimoshenkobeamwitharbitrarycrosssections AT guofangli vibrationanalysisoftheelasticallyrestrainedfunctionallygradedtimoshenkobeamwitharbitrarycrosssections AT gangwang vibrationanalysisoftheelasticallyrestrainedfunctionallygradedtimoshenkobeamwitharbitrarycrosssections AT junfangni vibrationanalysisoftheelasticallyrestrainedfunctionallygradedtimoshenkobeamwitharbitrarycrosssections AT liangli vibrationanalysisoftheelasticallyrestrainedfunctionallygradedtimoshenkobeamwitharbitrarycrosssections
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The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections

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