Transverse vibrations of an Euler–Bernoulli uniform beam carrying several particles

The Euler–Bernoulli uniform beam considered in this paper consist of n portions and carry (n+1) particles, two of which are at the beam ends. For the classical beam eigen-value technique developed here, n co-ordinate systems are chosen with origins at the particle locations. The mode shape of the jth portion of the beam is expressed in the form Y_j(X_j)=AU_j(X_j)+BV_j(X_j) in which U_j(X_j) and V_j(X_j) are ‘modified’ mode shape functions applicable to that portion but the constants A and B are common to all the portions. From the boundary conditions at the right end, the frequency equation was expressed in closed form as a second-order determinant equated to zero. Schemes are presented to compute the four elements of the determinant (from a recurrence relationship) and to evaluate the roots of the frequency equation. Computational difficulties were not encountered in the implementation of the schemes. The first three natural frequency parameters of 16 combinations of the classical boundary conditions are tabulated for beams with three and up to nine portions for selected particle location and mass parameters. Frequency parameters of beams with one and up to 500 equi-spaced, equi-mass systems are also tabulated. The approaches in previous publications include those based on various approximate methods like finite element, Rayleigh–Ritz, Galerkin, transfer matrix, etc. The results in the present paper may be used to judge the accuracy of values obtained by approximate methods.