考虑切向摩擦力时弹性地基梁的振动

The Fourier series solutions for the vibration of elastic foundation beams by considering horizontal frictional resistances

假定切向摩擦力与梁底面的纵向位移成正比,通过引入广义剪力,得到了梁的位移型平衡方程。将位移及荷载展开为带附加项的Fourier级数,利用平衡方程和边界条件研究了弹性地基梁的自由振动和简谐振动。通过算例结果分析表明:纵向摩擦力对梁的固有频率、位移和内力均有影响。梁的最大挠度、转角、弯矩及剪力随着地基纵向反力系数的增大而减小;梁的固有频率、轴向位移和轴力则随着地基纵向反力系数的增大而增大;同时轴力引起的轴向位移和转角引起的梁底面纵向位移具有同一数量级。

By supposing the longitudinal friction is proportional to the beam’s base longitudinal displacement and introducing a general shear force,equations of equilibrium in the form of displacements are obtained.By expanding the displacements and loads to Fourier series with additional terms,the free and harmonic vibrations of elastic foundation beams are analytically discussed by using the equilibrium equations and boundary conditions.Results obtained indicate that the inherent frequencies and displacements and internal forces of beams are related to the longitudinal friction.The maximum values of deflection,slope,bending moment and shear force are decreased with the increase of the frictional force,but the inherent frequencies and the maximum values of axial displacements and axial forces are increased with the increase of the frictional force.Results obtained also indicate that the axial displacements originated by the axial forces and the longitudinal displacements of beam’s bottom originated by the slope are to be the same order.