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چکیده
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In this paper, by considering the moment of inertia caused by shear deformation, and using the Fourier transform, Laplace transform and their inverse transforms, the analytical solutions of the vertical and torsional-flexural natural frequencies and vibration responses of the modified Timoshenko beams under moving harmonic loads are derived. This method is simple, fast and precise. On the basis of this analytical solution, the analytical or semi-analytical solutions of the vertical and torsional-flexural natural frequencies and vibration responses of the other four beam theories (i.e., Euler-Bernoulli beam, Shear beam, Rayleigh beam, and Tymoshenko beam) can be obtained without considering the shear deformation or bending rotary inertia or shear rotary inertia, and the natural frequencies and vibration responses of five beams of different lengths are further numerically analyzed. It is found that the shear deformation has the greatest influence on the vertical and torsional-flexural natural frequency of beam. The influence of shear rotary inertia on the vertical and bending-torsional frequency of beams is greater than that of bending rotary inertia, which improves the accuracy by about 6.00%. Additionally, for vertical vibration response of short beams, Shear beam theory, Timoshenko beam theory or modified Timoshenko beam theory should be used; for bending-torsion vibration response of long beams, Rayleigh beam theory, Timoshenko beam theory or modified Timoshenko beam theory should be used. This study shows that the proposed modified Timoshenko beam is necessary to calculate the natural frequency and vibration response of the beam under moving harmonic load.
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