Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System

Abstract

An exact solution for the natural frequencies and mode shapes for a beam elastically constrained at its ends and to which a rigid mass is elastically mounted is obtained. The attached mass can both translate and rotate. The general solution is obtained using the Laplace transform with respect to the spatial variable and yields the exact solutions to several previously published simpler configurations that were obtained using approximate methods. Numerous numerical results are presented for the natural frequency coefficients that extend previously reported results and that show the transition between various limiting cases. In addition, values are presented for the lowest two natural frequency coefficients for a beam that is clamped at both ends and is carrying a two dof spring-mass system. Representative mode shapes at selected values of the system’s parameters are also given.

The Natural Frequencies and Mode Shapes of a Uniform Cantilever Beam With Multiple Two-Dof Spring–mass Systems

“…spring}mass system with the beam, x T G and x T I respectively. In reference [22], it has been found that the exact solution presented in reference [18] is correct only if the e!ects of the coupling e!ective spring constants k T C ff and k T C ff de"ned by equation (24) are negligible.…”

“…Besides, FryH ba [11], Hino et al [12,13], Yoshimura et al [14,15] and Lin et al [16,17] have investigated the vibration problem of single and multiple spring}mass systems moving along a beam by considering the interactions between the suspension systems and the beam. In 1993, Jen and Magrab [18] presented an&&exact'' solution for the natural frequencies and mode shapes for beams carrying a&&single'' two-d.o.f. spring}mass system.…”

“…spring}mass system) were derived based on the "nite element formulation and in references [18,19] the same work was done by using the Laplace transform with respect to the spatial variable. Instead of the &&entire'' system, the two-d.o.f.…”

Dynamic Responses of Two Beams Connected by a Spring-Mass Device

“…A lot of researches on the system dynamics of a beam carrying spring-mass devices have been studied extensively [1][2][3][4][5][6][7][8][9]. The natural frequencies of this system may deviate considerably from those of the beam itself.…”

“…The natural frequencies of this system may deviate considerably from those of the beam itself. The solutions of this problem had been obtained by means of the analytical methods [1][2][3], or the numerical methods [4,5]. Gurgoze [6] used the Lagrange multiplier formalism to obtain the natural frequencies of a beam to which several spring-mass devices are attached.…”