FREE VIBRATION ANALYSIS OF A CANTILEVER BEAM CARRYING ANY NUMBER OF ELASTICALLY MOUNTED POINT MASSES WITH THE ANALYTICAL-AND-NUMERICAL-COMBINED METHOD

The natural frequencies and the corresponding mode shapes of a uniform cantilever beam carrying “any number of” elastically mounted point masses are determined by means of the analytical-and-numerical-combined method (ANCM). One of the key points for the present method is to replace each spring–mass system (with spring constantkmxand mass magnitudemmx) by a massless “effective” spring with spring constantkeffx−kmx]/(1−γr2). Where γris the frequency ratio defined by γr=ωmx/ω, in which ωmx=√kmx]/mmxis the natural frequency of thevth spring–mass system with respect to the attached beam and ω is the natural frequency of the “constrained” beam. The present method is much better than the conventional finite element method (FEM), since it consumes less than 30% of the CPU time required by the conventional FEM to achieve approximately the same accuracy of the lowest five natural frequencies of the “constrained” beam. It is also superior to the existing analytical (or semi-analytical) approaches, since the latter is available only for the eigenvalue problems with “one or two” elastically mounted point masses but the former (the ANCM) easily solves the eigenvalue problems with “any number of” spring–mass attachments. To confirm the reliability of the present method, all the results obtained from the ANCM were checked by those calculated with the conventional FEM. For this purpose two kinds of techniques were presented to derive the stiffness matrix and mass matrix of the associated finite “constrained” beam element: (i) increasing one degree of freedom for each spring–mass attachment and (ii) replacing each spring–mass attachment by a massless effective spring.