Closed-form solutions for dynamic analysis of extensional circular Timoshenko beams with general elastic boundary conditions

Closed-form solutions for dynamic analysis of extensional circular Timoshenko beams with general elastic boundary conditions are derived. Taking the Laplace transform and some procedures, the system composed of three coupled governing differential equations and six coupled boundary conditions is uncoupled and reduced to a single equation in terms of the angle of rotation due to bending. The explicit relations between the inward radial displacement, the tangential displacement and the angle of rotation due to bending are revealed. Six exact normalized fundamental solutions of the uncoupled governing differential equation are obtained by the Frobenius method. The exact transformed general solution of the uncoupled system is expressed in terms of the six fundamental solutions, using the generalized Green function given by Lin. The systems based on the Rayleigh and Bernoulli-Euler beam theories can be obtained by taking the corresponding limiting procedures. Without the Laplace transform, the exact solutions for the steady and free vibrations of the general system are obtained. The effects of the spring constants, the opening angle, the rotary inertia and the shear deformation on the natural frequencies are investigated.