Free vibration analysis of variable cross-section beam with internal discontinuities

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Journal of Vibration and Shock ›› 2022, Vol. 41 ›› Issue (19) : 164-171.

GU Yeqing, CHANG Tingting, BAO Siyuan, SHEN Feng

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Abstract

Aiming at solving the free vibration of an Euler-Bernoulli beam with variable cross-section and internal spring subjected to axial compression load under arbitrary boundary conditions, a strategy based on the improved Fourier series and Garlerkin method is proposed to study the vibrational characteristics of beams. In order to simulate arbitrary boundaries, the transverse displacement spring and rotational spring are set at both ends of the beam. By assigning different values to the stiffness coefficient of the spring, the internal discontinuity condition is simulated. First, the displacement admissible function of the vibration displacement is established, in which the auxiliary polynomial is derived from the value of the end function or derivative. Then, the Garlerkin method is used to deal with the governing differential equation for free vibration of beams, thus the vibration problem is transformed into matrix eigenvalue problem. Finally, the numerical simulation is carried out by using the numerical calculation software. When the boundary conditions and mid-span discontinuity are changed, the mass matrix and stiffness matrix of the system can be determined by changing the spring stiffness coefficient, and the corresponding vibrational characteristics can be obtained. Numerical results show that the proposed method is simple, reasonable and of good practical significance.
Keywords: Euler-Bernoulli beam; internal discontinuity; the improved Fourier series; free vibration

Key words

Euler-Bernoulli beam / internal discontinuity / the improved Fourier series / free vibration

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GU Yeqing, CHANG Tingting, BAO Siyuan, SHEN Feng. Free vibration analysis of variable cross-section beam with internal discontinuities[J]. Journal of Vibration and Shock, 2022, 41(19): 164-171

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