Closed form solutions for frequency equation and mode shapes of elastically supported Euler-Bernoulli beams

Abstract

This paper is concerned with the problem of free vibration of an Euler-Bernoulli beam supported with an arbitrary number of translational springs of different constant stiffnesses. The natural frequencies and mode shapes are calculated by means of the Green's function method. Seven different boundary conditions are considered at both ends of the beam: fixed, pinned, sliding, free, translational spring, rotational spring and combined translational-rotational spring. Each boundary condition on one end is combined with all other boundary conditions on the other, yielding a total of 49 possible cases. It has been shown in this paper that a Green's function for any set of boundary conditions can be represented as a linear combination of function coefficients calculated for homogeneous boundary conditions, and also a new form of the general solution is formulated using this principle. Green's functions are calculated and tabulated for each of the 49 possible cases, which in essence closes the question of analytical solutions for this particular kind of problem using Green's functions. The tabulated solutions were used as the starting point in the derivation of the general formulae and to provide a new insight into the underlying mathematical structure of the solutions formulated with Green's functions. This can be of practical interest to the end user who wishes to implement them in computer code. The validity of the obtained solutions is demonstrated with examples, whose solutions are compared to the results in the published literature and to the results obtained with FEM.