THE NATURAL FREQUENCIES AND MODE SHAPES OF A UNIFORM CANTILEVER BEAM WITH MULTIPLE TWO-DOF SPRING–MASS SYSTEMS

Abstract

Because of the complexity of the mathematical expressions, the literature concerning the free vibration analysis of a uniform beam carrying a “single” two degrees-of-freedom (d.o.f.) spring–mass system is rare and the publications relating to that carrying “multiple” two-d.o.f. spring–mass systems have not yet appeared. Hence the purpose of this paper is to present some information in this area. First of all, the closed form solution for the natural frequencies and the corresponding normal mode shapes of the uniform beam alone (or the “bare” beam) with the prescribed boundary conditions are determined analytically. Next, a method is presented to replace each two-d.o.f. spring–mass system by two massless equivalent springs with spring constants k(v)eq,iandk(v)eq,k , and then the foregoing natural frequencies and normal mode shapes for the“bare” beam are in turn used to derive the equation of motion of the “loading” beam (i.e., the bare beam carrying any number of two-d.o.f. spring–mass systems) by using the expansion theorem. Finally, the natural frequencies and the associated mode shapes of the“loading” beam are obtained from the last equation by using the numerical method. To confirm the reliability of the present method, all the numerical results obtained in this paper are compared with the corresponding ones obtained from the conventional finite element method (FEM) and good agreement is achieved. Because the order of the property matrices for the equation of motion of the“loading” beam derived from the present method is much lower than that derived from the FEM, the computer time required by the former is much less than that required by the latter. Besides, the equation of motion derived from the present method may always run on the cheaper personal computers, but that from the FEM may run only on the more expensive larger computers if the degree of freedom of the loading beam exceeds a certain limit.