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Analysis on the Second Frequency Spectrum of Timoshenko Beam

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    Incorporating the boundary conditions, initial parameter solutions of vibration differential equations for Timoshenko beam are used to derive the frequency equation of a simply-supported beam. When the natural frequency is less than the critical frequency, the frequency equation can be factorized into the hyperbolic sine function and the trigonometric sine function, while, when the natural frequency is greater than the critical frequency, the frequency equation can be factorized into double trigonometric sine functions, which is the crucial reason for the existence of the second frequency spectrum. Frequency equations for two-span and three-span continuous Timoshenko beams with uniform cross sections and equal spans are derived. Other structures with the second frequency spectrum are forecasted theoretically. The formulas for the first and second frequencies are deduced for simply-supported Timoshenko beam. The existence of the second frequency spectrum is confirmed through the examples. Through solving the differential equation of motion, the critical frequency is proven to be an efficient part of the natural frequencies for the framed structures. The corresponding mode shape of the critical frequency contributes to the displacement mode shape with zero amplitude and rotation mode shape with constant amplitude. Due to the truncation error of the computer, the critical frequency predicted by the finite element method shows error, and the mode shape of the displacement is very irregular.

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  • Online: November 24,2021
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