ABSTRACT
This chapter presents a selective review (with emphasis on the word “selective”) of the direct, semi-inverse and inverse eigenvalue problem for structures described by a differential equation with variable coefficients. It gives only a taste of the extensive research that has been conducted since 1759, when Leonhard Euler posed, apparently for the first time, a boundary value problem. Since then numerous studies have been conducted for rods, Bernoulli–Euler beams, Bresse–Timoshenko beams, Kirchhoff–Love and Mindlin–Reissner plates and shells, and structures analyzed via finer, high-order theories. This selective review classifies the solutions as belonging to one of three main classes: (1) direct problems, (2) semi-inverse problems, (3) inverse problems. In addition, some new closed-form solutions are reported that have been obtained via posing an inverse vibration problem. Due to the huge body of literature, the author limits himself to discussing classic theories of structures.
