Frequency Analysis of Beams by Flexibility Method

The general matrix formulation of the vibrating beam problem, starting with the string polygon equation developed for static elastic systems, is presented. The distributed mass system has been replaced by an equivalent lumped mass system, and dynamic equilibrium is established by equating to zero the beam shears and inertia forces. A linear stiffness matrix is derived directly from the angular flexibility matrix, and a linear flexibility matrix is defined. Iteration techniques are used to solve for the natural frequencies and the corresponding mode shapes of the system. The general method is directly applicable to continuous beams with intermediate constraints. Extension to continuous systems with elastic restraints is direct. Numerical examples illustrating the procedure of application to single span and continuous beams are included.