Free Vibration of a Timoshenko Beam with Arbitrary Nonuniformities, Discontinuities and Constraints

Natural frequencies and mode shapes of a Timoshenko beam with arbitrary non-uniformities, discontinuities, discrete spring/mass constraints and boundary conditions are computed by developing a new method based on the state transition matrix for spatially varying state equations. Algorithms to treat discontinuities in material and geometrical properties and discontinuities due to discrete spring constraints are clearly presented. Equations for natural frequencies and mode shapes are derived for following boundary conditions: clamped-free, attached springs/masses at both ends and pinned–pinned. Numerical results are presented and compared to those in the existing literature.