We deal with the exact determination of eigenfrequencies of a beam with intermediate elastic constraints and generally restrained ends. It is the purpose of this paper to use the calculus of variations to obtain the equations of motion and the natural boundary conditions, and particularly those at the intermediate constraints. Numerical values for the first five natural frequencies are presented in a tabular form for a wide range of values of the restraint parameters. Several particular cases are presented and some of these cases have been compared with those available in the literature.
Ricardo Oscar Grossi. Carlos Marcelo Albarracín. "Eigenfrequencies of generally restrained beams." J. Appl. Math. 2003 (10) 503 - 516, 24 September 2003. https://doi.org/10.1155/S1110757X03203065