ABSTRACT
This chapter summarizes the efforts and results in the search for closed-form solutions in vibration and buckling of inhomogeneous rods, beams, columns and plates. For specific columns with variable cross-section the mode shape is represented by polynomial functions, while for those with uniform cross-sections one needs to solve a transcendental equation. The following task was posed: find polynomial expressions for the stiffness of a column or a beam, such that the associated buckling modes of vibration mode shapes will also constitute polynomial expressions. The statement was added that a manuscript on the vibration aspect was under way, which was true — albeit “in embryo.” Even in the latter with the exact solution available, the vibration frequency and the buckling load are given in terms of the number p and the accuracy of the result depends on that of the latter.
