Frequency Variation of Beam with an Internal Hinge of Rotational Restraint

The frequency variation problem of a classical beam with a rotationally restrained internal hinge arbitrarily located is studied intensively. The characteristic equations of the natural vibration are derived rigorously upon the corresponding boundary conditions and the compatibility conditions at the interconnecting point such that the effects of both the rotational restraint stiffness and the hinge position on the natural frequencies and mode shapes are described comprehensively. In the context, the critical locations of an internal hinge are investigated extensively to obtain the extreme values of a natural frequency of the beam system, and the associated requirements are explored in depth. It appears that the essential conditions for obtaining the critical points with regard to the maximum or minimum frequency come, respectively, from the same compatibility relation of the rotational flexibility in view of the opposite criteria. Numerical results are provided with two representative uniform beams to show the variations of their dynamic characteristics and to gain new insights into the transition conditions in the two distinct cases.