Frequency analysis of multi-span shifts

Some basic properties of receptance functions are employed to determine the stationary values of the critical frequencies of a multi-span shaft in which the position of one of the bearings can be varied. Details are given of the determination of the maximum and minimum values of the first two critical frequencies. The theory is then extended to obtain the spacing of the intermediate supports for the first critical to be a maximum. The simple result thus obtained generalizes Darnley's solution for the special case of a uniform shaft with simply-supported ends. The method is also employed to determine the maximum fundamental frequency of a beam or shaft having a freely articulating internal joint of variable position.