Vibration of beams with arbitrary discontinuities and boundary conditions

A general solution of the vibration of an Euler–Bernoulli beam with arbitrary type of discontinuity at arbitrary number of locations is presented in this paper. To account for the discontinuity term induced by various additional elements on the beam, Heaviside's function is used to express the modal displacement of the whole beam by a single function. This general modal displacement function is then solved by using Laplace transformation. This general solution consists of four types of basic modal shapes induced by four corresponding types of discontinuity terms at the discontinuity points. Various discontinuity terms are obtained and expressed by the boundary values of the modal displacement in a recursive way. Consequently, the modal displacement can be determined by examining only the conditions on the boundary. In such a way, the present solution reduces the vibration of beams with arbitrary discontinuities to the same order of the case without discontinuity point. To demonstrate the efficiency and applicability of the present method, three application examples are presented. Calculation example shows that the lead–zirconate–titanate (PZT) actuator should be placed as close to the fixed end as possible to achieve the best excitation effect on a cantilever beam. A new method to calculate the driving-point anti-resonance frequency is also proposed. Numerical results suggest that the variation of driving-point anti-resonance frequency can be used to determine the location and size of crack in beams. Due to the generic nature of the solution and the problem, the present method can be utilized in smart structures modeling and structural health monitoring of beam-type structures.