A NEW APPROACH FOR DETERMINING THE NATURAL FREQUENCIES AND MODE SHAPES OF A UNIFORM BEAM CARRYING ANY NUMBER OF SPRUNG MASSES

Abstract In theory, one may obtain five equations from each attaching point of a spring–mass system and two boundary–equations from each end of the uniform beam. Hence, for a uniform beam carrying n spring–mass systems, simultaneous equations of the form [ B ]{ C }= 0 ] will be obtained. The solutions of ∣ B ∣= 0 (where ∣ · ∣ represents a determinant) give the natural frequencies of the “constrained” beam and the substitution of each corresponding values of C j ( j =1∽4) into the eigenfunction will define the associated mode shapes. While the foregoing theory is simple, the lengthy explicit mathematical expressions become impractical if the total number of spring–mass systems is larger than “two”. For this reason, it was applied to do the free vibration analysis of a uniform beam carrying “one” spring–mass system only in the existing literature. The purpose of this paper is to present a numerical technique to apply the foregoing theory to obtain the exact solutions for the lowest several natural frequencies and mode shapes of a uniform beam carrying “any number of” spring–mass systems with various boundary conditions. To this end, each integration constant C vi and each mode displacement Z v ( v =1∽ n , i =1∽4) at the attaching point and the two ends of the beam are considered as nodal displacements of a finite beam element and are assigned an appropriate degree of freedom (dof). Hence, each associated coefficient matrix will be equivalent to the stiffness matrix of a beam element, and the conventional numerical assembly technique for the finite element method (FEM) may be used to determine the “overall” coefficient matrix [ B ]. Therefore, the eigenvalue equation [ B ]{ C }= 0 is easily obtained.