Complex Modal Analysis of Transverse Vibration of Modified Timoshenko Beam on Viscoelastic Winkler Foundation

WANG Jia-le

doi:10.11988/ckyyb.20190822

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Journal of Changjiang River Scientific Research Institute ›› 2020, Vol. 37 ›› Issue (9) : 87-95. DOI: 10.11988/ckyyb.20190822

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Complex Modal Analysis of Transverse Vibration of Modified Timoshenko Beam on Viscoelastic Winkler Foundation

WANG Jia-le

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Abstract

In the light of the modified Timoshenko beam theory, a vibration-control differential equation of the modified Timoshenko beam on viscoelastic Winkler foundation is established in consideration of the viscous damping of foundation. The present equation overcomes the shortcoming of the classical Timoshenko beam on viscoelastic foundation that “one mode has two frequencies”. The frequency solution to the transcendental equation and its modal function under various boundary conditions are derived by complex modal analysis. When the influence of shear inertia caused by shear deformation of beams is not taken into account, the equation degrades to the classical Timoshenko beam model of viscoelastic Winkler foundation; when the shear deformation of beams and the viscous damping of foundation are not taken into account, the equation decays to the common Euler beam vibration model of elastic foundation. The computational model in this paper is a more general one. The calculation differences of viscoelastic foundation beams under different theories are analyzed. The results show that Euler beams on viscoelastic foundation have great errors in high frequency band as well as in low frequency band with short beams. The error of classical Timoshenko beams on viscoelastic Winkler foundation is smaller than that of Euler beams; but the relative error increases gradually with the increase of order, and the influence cannot be ignored in high frequency band. The research finding proves that the viscous damping of foundation has great influence on the low frequency vibration of foundation beam.

Key words

viscoelastic foundation beam / modified Timoshenko beam / complex mode / modal function / dual-spectrum

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WANG Jia-le. Complex Modal Analysis of Transverse Vibration of Modified Timoshenko Beam on Viscoelastic Winkler Foundation[J]. Journal of Changjiang River Scientific Research Institute. 2020, 37(9): 87-95 https://doi.org/10.11988/ckyyb.20190822

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