THE EXACT SOLUTIONS FOR THE NATURAL FREQUENCIES AND MODE SHAPES OF NON-UNIFORM BEAMS WITH MULTIPLE SPRING–MASS SYSTEMS

Abstract

For a beam carrying n spring–mass systems, if the left side and right side of each attaching point and each end of the beam are regarded as nodes, then considering the compatibility of deformations and the equilibrium of forces between the two adjacent beam segments at each attaching point and incorporating with the equation of motion for each spring–mass system, simultaneous equations may be obtained for the ν th attaching point, where the unknowns for the simultaneous equations are composed of the integration constants for the eigenfunctions of the νth and ( ν+ 1)th beam segments and the associated modal displacements of the ν th sprung mass. It is evident that if these unknowns are considered as the nodal displacements, then the coefficient matrix of the simultaneous equations will be equivalent to the element stiffness matrix for the ν th attaching point (associated with the ν th and ( ν+ 1)th beam segments). In view of the last fact, one may use the numerical assembly method (NAM) for the conventional finite element method to obtain the overall simultaneous equations for the overall ( n) attaching points (associated with the overall ( n+ 1) beam segments) by taking into account the boundary conditions of the whole beam. The solutions for the coefficient determinant of the overall simultaneous equations to be equal to zero will give the “exact” natural frequencies of the constrained beam (carrying multiple ( n) spring–mass systems) and the substitution of each corresponding values of the integration constants into the associated eigenfunctions for each attaching point will determine the corresponding mode shapes. Since no discretization on the continuous beam was made in the present approach (NAM), the natural frequencies and the corresponding mode shapes obtained are the exact ones.