Alternative approach for free vibration of beams carrying a number of two-degree of freedom spring-mass systems

The purpose of this paper is to determine the natural frequencies and mode shapes of beams carrying any number of two-degree of freedom (DOF) spring-mass systems by means of two finite element methods FEM1 and FEM2. For convenience, a beam without attachment is called the unconstrained (or bare) beam and that carrying attachment(s) is called the constrained (or loading) beam. FEM1 is the conventional finite element method (FEM), in which each two-DOF spring-mass system is considered as a finite element and then the assembly technique is used to establish the overall property matrices of the constrained beam. FEM2 is an alternative approach, in which each two-DOF spring-mass system is replaced by four effective springs with spring constants keff,ij(v) (i,j= 1,2) and then the overall property matrices of the constrained beam are obtained by considering the whole structural system as the unconstrained beam elastically supported by the effective springs. Based on the above-mentioned two approaches, the eigenvalue equations for the constrained beams are obtained and free vibration analyses are performed. The theoretical and numerical analyses in this paper reveal that the so-called exact solution reported in the existing literature is available only for the case of neglecting the effects of the coupling effective spring constants (i.e., the coupling terms), keff,12(v) and keff,21(v). In other words, the existing exact solution is only a special case of FEM2 presented in this paper.