Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification
Transverse non-linear vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about a constant mean speed. The material time derivative is used in the viscoelastic constitutive relation....
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Shock and Vibration |
| Online Access: | http://dx.doi.org/10.3233/SAV-2011-0648 |
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| _version_ | 1832548435207651328 |
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| author | Li-Qun Chen Hu Ding C.W. Lim |
| author_facet | Li-Qun Chen Hu Ding C.W. Lim |
| author_sort | Li-Qun Chen |
| collection | DOAJ |
| description | Transverse non-linear vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about a constant mean speed. The material time derivative is used in the viscoelastic constitutive relation. The transverse motion can be governed by a non-linear partial-differential equation or a non-linear integro-partial-differential equation. The method of multiple scales is applied to the governing equations to determine steady-state responses. It is confirmed that the mode uninvolved in the resonance has no effect on the steady-state response. The differential quadrature schemes are developed to verify results via the method of multiple scales. It is demonstrated that the straight equilibrium configuration becomes unstable and a stable steady-state emerges when the axial speed variation frequency is close to twice any linear natural frequency. The results derived for two governing equations are qualitatively the same, but quantitatively different. Numerical simulations are presented to examine the effects of the mean speed and the variation of the amplitude of the axial speed, the dynamic viscosity, the non-linear coefficients, and the boundary constraint stiffness on the instability interval and the steady-state response amplitude. |
| format | Article |
| id | doaj-art-e6d806a905c74f0aa1f025c5bfe31aea |
| institution | Kabale University |
| issn | 1070-9622 1875-9203 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Shock and Vibration |
| spelling | doaj-art-e6d806a905c74f0aa1f025c5bfe31aea2025-02-03T06:14:05ZengWileyShock and Vibration1070-96221875-92032012-01-0119452754310.3233/SAV-2011-0648Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature VerificationLi-Qun Chen0Hu Ding1C.W. Lim2Department of Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, ChinaDepartment of Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, ChinaDepartment of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong, ChinaTransverse non-linear vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about a constant mean speed. The material time derivative is used in the viscoelastic constitutive relation. The transverse motion can be governed by a non-linear partial-differential equation or a non-linear integro-partial-differential equation. The method of multiple scales is applied to the governing equations to determine steady-state responses. It is confirmed that the mode uninvolved in the resonance has no effect on the steady-state response. The differential quadrature schemes are developed to verify results via the method of multiple scales. It is demonstrated that the straight equilibrium configuration becomes unstable and a stable steady-state emerges when the axial speed variation frequency is close to twice any linear natural frequency. The results derived for two governing equations are qualitatively the same, but quantitatively different. Numerical simulations are presented to examine the effects of the mean speed and the variation of the amplitude of the axial speed, the dynamic viscosity, the non-linear coefficients, and the boundary constraint stiffness on the instability interval and the steady-state response amplitude.http://dx.doi.org/10.3233/SAV-2011-0648 |
| spellingShingle | Li-Qun Chen Hu Ding C.W. Lim Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification Shock and Vibration |
| title | Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification |
| title_full | Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification |
| title_fullStr | Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification |
| title_full_unstemmed | Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification |
| title_short | Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification |
| title_sort | principal parametric resonance of axially accelerating viscoelastic beams multi scale analysis and differential quadrature verification |
| url | http://dx.doi.org/10.3233/SAV-2011-0648 |
| work_keys_str_mv | AT liqunchen principalparametricresonanceofaxiallyacceleratingviscoelasticbeamsmultiscaleanalysisanddifferentialquadratureverification AT huding principalparametricresonanceofaxiallyacceleratingviscoelasticbeamsmultiscaleanalysisanddifferentialquadratureverification AT cwlim principalparametricresonanceofaxiallyacceleratingviscoelasticbeamsmultiscaleanalysisanddifferentialquadratureverification |