A contribution to the exact modal solution of in-plane beam structures

The exact vibration modes and natural frequencies of planar structures and mechanisms, comprised Euler–Bernoulli beams, are obtained by solving a transcendental, non-linear, eigenvalue problem stated by the dynamic stiffness matrix (DSM). To solve this kind of problem, the most employed technique is the Wittrick–Williams algorithm, developed in the early seventies. By formulating a new type of eigenvalue problem, which preserves the internal degrees-of-freedom for all members in the model, the present study offers an alternative to the use of this algorithm. The new proposed eigenvalue problem presents no poles, so the roots of the problem can be found by any suitable iterative numerical method. By avoiding a standard formulation for the DSM, the local mode shapes are directly calculated and any extension to the beam theory can be easily incorporated. It is shown that the method here adopted leads to exact solutions, as confirmed by various examples. Extensions of the formulation are also given, where rotary inertia, end release, skewed edges and rigid offsets are all included.