The exact solution for free vibration of uniform beams carrying multiple two-degree-of-freedom spring–mass systems

The literature regarding the “exact” solutions of natural frequencies and mode shapes of a uniform beam carrying multiple two-degree-of-freedom (2-dof) spring–mass systems is rare, thus, this paper aims at studying this problem using the numerical assembly method (NAM). First of all, the equivalent springs for replacing the effect of a 2-dof spring–mass system are determined. Next, the coefficient matrix for a 2-dof spring–mass system attached to the uniform beam is derived based on the compatibility of deformations and equilibrium of forces (including moments). The coefficient matrices for the left end and right end of the beam are also derived based on the various boundary conditions of the beam. Combining the coefficient matrices for all the 2-dof spring–mass systems attached to the beam and the coefficient matrices for the boundary conditions of the beam, one obtains the overall coefficient matrix of the constrained beam (i.e., the beam carrying any number of 2-dof spring–mass systems). The product of the overall coefficient matrix and the vector for all the integration constants yields a set of simultaneous equations. Let the coefficient determinant of the last simultaneous equations equal to zero, one obtains the frequency equation. The roots of the last frequency equation denote the natural frequencies of the constrained beam. Substituting the roots of the frequency equation into the set of simultaneous equations one may determine the associated mode shapes of the constrained beam. In this paper, the “exact” solution refers to the one obtained from the “continuous” model instead of the “discrete” mode, besides, the accuracy of the analytical-and-numerical combined method (ANCM) given by the existing literature is dependent on the total number of vibration modes considered, but this is not true for the accuracy of the NAM adopted here. To confirm the reliability of the presented theory, all the numerical results obtained from NAM are compared with the corresponding ones obtained from the conventional finite element method (FEM) and good agreement is achieved.