Natural Frequency Maximization of Beam with an Internal Sliding Joint of Translational Restraint

Abstract

The natural frequency variation of a uniform Euler–Bernoulli beam with an internal sliding joint of the translational restraint is studied intensively such that the optimal design of the connecting point is pursued for maximization of a natural frequency of the system. First, the compatibility conditions at the linkage are presented in detail, and the dynamic characteristic equation of the beam system is accordingly derived. Next, the influences of both the translational restraint stiffness and the position of the sliding joint upon the natural frequency and mode shape are described comprehensively. Then, the optimal position and the corresponding minimum translational stiffness of the internal joint are explored exclusively for maximizing a natural frequency of the beam structure. In this context, the essential requirements for the optimization of the internal sliding joint are presented, respectively, in two different ways to reveal their mechanical implications. It turns out that the optimal joint location can be determined readily from the corresponding mode shape of the equivalent continuous beam. Afterward, numerical results are provided with two classical beams of a single internal sliding joint to demonstrate the variations of their dynamic characteristics as well as to offer significant physical insight into the vibratory behaviors of the beam system with maximization of a natural frequency of interest.