Free vibration of beams with finite mass rigid tip load and flexural–torsional coupling

The free vibration of an Euler–Bernoulli beam with a rigid tip mass whose centre of gravity need not be coincident with its point of attachment to the beam is examined. The beam is assumed to experience torsional deformation in conjunction with a planar elastic bending deformation. Explicit expressions are presented for the characteristic (or frequency) equation, mode shapes and orthogonality relation. The system fundamental frequency is observed to be dependent on several parameters such as the magnitude of the tip mass, the offset of the tip mass centre of gravity from the point of attachment, the moments of inertia of the tip mass about the centre of gravity, the length of the beam, the slenderness ratio, and the bending stiffness and torsional rigidity of the beam. The effects of these parameters are examined using equivalent nondimensional terms.