Orthogonality condition for a multi-span beam, and its application to transient vibration of a two-span beam

This paper treats the orthogonality condition for a multi-span beam, and its application to forced (transient) vibration of a two-span beam. The beam is modeled as a Bernoulli–Euler beam. The boundary conditions for the particular case of two-span beam are clamped–pinned–pinned. An exact closed-form solution is obtained for this problem. Even though there has been an enormous amount of work on beam vibration, most of the studies are conducted on a single-span beam. There are some studies on the multi-span beam vibration. However, their treatment is rather specialized in terms of the applied loading and the initial conditions. None of the studies in the past treats an exact solution for a forced (transient) vibration of a general two-span beam with arbitrary initial conditions and arbitrary forcing functions. Therefore, the solution obtained in this paper is new. The key development in the solution is the orthogonality condition for a multi-span beam. The method of solution developed in this paper establishes a general methodology for the forced (transient) vibration of a multi-span beam. The closed-form solution obtained in this paper can be used as a benchmark solution for the transient vibration of a two-span beam.