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.

[1]  David G. Jones,et al.  Vibration and Shock in Damped Mechanical Systems , 1968 .

[2]  H. A. Luther,et al.  Applied numerical methods , 1969 .

[3]  P.A.A. Laura,et al.  On the dynamic behaviour of structural elements carrying elastically mounted, concentrated masses , 1977 .

[4]  R. G. Jacquot Optimal dynamic vibration absorbers for general beam systems , 1978 .

[5]  Earl H. Dowell,et al.  On Some General Properties of Combined Dynamical Systems , 1978 .

[6]  Toshio Yoshimura,et al.  A finite element method prediction of the vibration of a bridge subjected to a moving vehicle load , 1984 .

[7]  Lawrence A. Bergman,et al.  FREE VIBRATION OF COMBINED DYNAMICAL SYSTEMS , 1984 .

[8]  Toshio Yoshimura,et al.  Vibration analysis of non-linear beams subjected to a moving load using the finite element method , 1985 .

[9]  Toshio Yoshimura,et al.  Vibration analysis of a non-linear beam subjected to moving loads by using the galerkin method , 1986 .

[10]  H. N. Özgüven,et al.  Suppressing the first and second resonances of beams by dynamic vibration absorbers , 1986 .

[11]  P.A.A. Laura,et al.  Analytical and experimental investigation on continuous beams carrying elastically mounted masses , 1987 .

[12]  Toshio Yoshimura,et al.  An active suspension for a vehicle travelling on flexible beams with an irregular surface , 1990 .

[13]  Yih-Hwang Lin,et al.  Dynamic Modeling and Analysis of a High Speed Precision Drilling Machine , 1990 .

[14]  Yih-Hwang Lin,et al.  Finite element analysis of elastic beams subjected to moving dynamic loads , 1990 .

[15]  M. Crocker,et al.  Vibration Absorbers for Hysterically Damped Mass-Loaded Beams , 1991 .

[16]  P.A.A. Laura,et al.  Natural frequencies of a Bernoulli beam carrying an elastically mounted concentrated mass , 1992 .

[17]  Ming Une Jen,et al.  Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System , 1993 .

[18]  Metin Gurgoze,et al.  ON THE EIGENFREQUENCIES OF A CANTILEVER BEAM WITH ATTACHED TIP MASS AND A SPRING-MASS SYSTEM , 1996 .

[19]  Metin Gurgoze,et al.  ON THE ALTERNATIVE FORMULATIONS OF THE FREQUENCY EQUATION OF A BERNOULLI–EULER BEAM TO WHICH SEVERAL SPRING-MASS SYSTEMS ARE ATTACHED IN-SPAN , 1998 .

[20]  C.-Y. Chang,et al.  Vibration analysis of beams with a two degree-of-freedom spring-mass system , 1998 .

[21]  Jong-Shyong Wu,et al.  FREE VIBRATION ANALYSIS OF A CANTILEVER BEAM CARRYING ANY NUMBER OF ELASTICALLY MOUNTED POINT MASSES WITH THE ANALYTICAL-AND-NUMERICAL-COMBINED METHOD , 1998 .

[22]  L Fryba,et al.  VIBRATION OF SOLIDS AND STRUCTURES UNDER MOVING LOADS (3RD EDITION) , 1999 .