Dynamic characteristics of a beam and distributed spring-mass system

This paper studies dynamic characteristics of a beam with continuously distributed spring-mass which may represent a structure occupied by a crowd of people. Dividing the coupled system into several segments and considering the distributed spring-mass and the beam in each segment being uniform, the equations of motion of the segment are established. The transfer matrix method is applied to derive the eigenvalue equation of the coupled system. It is interesting to note from the governing equations that the vibration mode shape of the uniformly distributed spring-mass is proportional to that of the beam at the attached regions and can be discontinuous if the natural frequencies of the spring-masses in two adjacent segments are different. Parametric studies demonstrate that the natural frequencies of the coupled system appear in groups. In a group of frequencies, all related modes have similar shapes. The number of natural frequencies in each group depends on the number of segments having different natural frequencies. With the increase of group order, the largest natural frequency in a group monotonically approaches the natural frequency of corresponding order of the bare beam from the upper side, whereas the others monotonically move towards those of the independent spring-mass systems from the lower side. Numerical results show that the frequency coupling between the beam and the distributed spring-mass mainly occurs in the low order of frequency groups, especially in the first group. In addition, vibratory characteristics of the coupled system can be approximately represented by a series of discrete multi-degrees-of-freedom system. It also demonstrates that a beam on Winkler elastic foundation and a beam with distributed solid mass are special cases of the proposed solution.