Forced vibration analysis of a spinning Timoshenko beam under axial loads by means of the three-dimensional Green's functions

Vibration of spinning structure is a very important and common problem in rotation machines and oil drilling. This paper studies dynamic responses of a spinning Timoshenko beam (STB) under the combined action of axial force and external excitation, and the three-dimentional (3D) steady-state Green's function of forced vibration of the spinning beam is derived, in a systematic manner, which are actually constructed by two components i.e. two Green's functions in two vertical directions. The Hamilton's principle is used to establish forced vibration equations of the STB. By employing the separation of variables method and Laplace transform method, the 3D Green's functions of STBs are obtained for different boundary conditions. By setting the shear correction factor k to infinity and the moment of inertia gamma is set to zero, the present 3D Green's functions can be reduced to the spinning Rayleigh beam and Euler-Bernoulli beam cases. In numerical section, the present analytical solution is verified by finite element method results, experimental results, and results in references in this work. Influences of the cross-section shape, such as circle, square, and ring, to the present solutions are discussed, and influences of some important geometric and physical parameters, such as the spinning speed and axial force, to the present solutions are also discussed.