Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

In the conventional finite element method (FEM), the dynamic characteristics of a longitudinally vibrating rod with mass density p(r), Young's modulus E-r, cross-sectional area A(r) and total length l(r) are considered to be the same as those of a helical spring with stiffness constant k(r) = A(r)E(r)/t(r) and total mass m(r) = p(r)A(r)l(r). For a lumped-mass model, the mass matrix of a rod element is a 2 x 2 diagonal one with each of its non-zero coefficients to be equal to one half of the total rod mass (i.e., 0.5m(r)). Furthermore, the dynamic characteristics of a rod on the basis of last lumped-mass model have been found to be very close to those on the basis of consistent-mass model. Thus, one can easily take into account of the inertial effect of a helical spring using a massless one with one half of its total mass, respectively, concentrated at its two ends (in Method 2) instead of modeling it by an elastic rod with uniform mass per unit length (in Method 1). When one more spring-mass system is attached to the beam, the total number of unknown constants increases one in Method 2 and two in Method 1, thus, Method 2 will reduce more effort than Method 1 for studying the dynamic behaviors of a beam carrying a number of spring-mass systems with mass of each helical spring considered. In this paper, the formulations of Methods 1 and 2 are presented first and then the numerical examples are illustrated to confirm the reliability of the presented theory and the developed computer programs. Finally, the effect concerning mass of each helical spring of the spring-mass systems is studied. (c) 2006 Elsevier Ltd. All rights reserved.