Analysis of rotating nonuniform pretwisted beams with an elastically restrained root and a tip mass

Using Hamiltonʹs principle derives the governing differential equations for the coupled bending–bending vibration of a rotating pretwisted beam with an elastically restrained root and a tip mass, subjected to the external transverse forces and rotating at a constant angular velocity. Using the mode expansion method derives the closed-form solutions of the dynamic and static systems. The orthogonal condition for the eigenfunctions of the system with elastic boundary conditions is discovered. The self-adjointness of the system is proved. Moreover, the Green functions of the system are obtained. The symmetric properties of the Green functions are revealed. The frequency response on the steady response of the beam is also investigated.