Closed-form solutions for non-uniform Euler–Bernoulli free–free beams

• There exists a certain class of beams, having free–free boundary condition, which have a closed-form polynomial solution.
• The height and breadth variations of these physically feasible beams with rectangular cross-section are presented.
• The derived mass and stiffness properties are used as test functions to validate a p-version finite element code.

In this paper, the free vibration of a non-uniform free–free Euler–Bernoulli beam is studied using an inverse problem approach. It is found that the fourth-order governing differential equation for such beams possess a fundamental closed-form solution for certain polynomial variations of the mass and stiffness. An infinite number of non-uniform free–free beams exist, with different mass and stiffness variations, but sharing the same fundamental frequency. A detailed study is conducted for linear, quadratic and cubic variations of mass, and on how to pre-select the internal nodes such that the closed-form solutions exist for the three cases. A special case is also considered where, at the internal nodes, external elastic constraints are present. The derived results are provided as benchmark solutions for the validation of non-uniform free–free beam numerical codes.