Vibration of Functionally Graded Rotating Beams Including the Effects of Nonlocal Elasticity

Some exact solutions for rotating inhomogeneous beams including the effects of nonlocal elasticity was studied. Both clamped–clamped and clamped–free boundary conditions are studied. Mode shapes are postulated as fourth-order polynomials satisfying the geometric boundary conditions, whereas the flexural rigidity is expressed in the form of a fourth-order polynomial with unknown coefficients. In the case of clamped–clamped boundary conditions, the semi-inverse method yields closed-form solutions for the entire problem, whereas for the case of clamped–free boundary conditions, a system of nonlinear polynomial equations is derived for the unknown coefficients.