弹性边界条件下带有任意分布弹簧质量系统的梁自由振动的解析解

基于正弦展开方法,对弹性边界条件下带有任意分布弹簧质量系统的梁的振动微分方程进行了求解,获得了一种近似解析解.运用该方法分析了带有均匀分布弹簧质量系统的梁的自由振动,模态频率的计算结果与参考文献中的数值结果一致,验证了该文算法的正确性.以此为基础,进一步研究了弹簧质量系统五种不同的分布形式对梁归一化模态频率的影响,结合不带弹簧质量系统的梁的振型图可得：弹簧质量系统分布形式在梁某阶模态振型幅值最大处的分布范围越广、分布密度越大,对该阶模态频率影响越大.

AN ANALYTIC SOLUTION FOR A BEAM WITH ARBITRARILY DISTRIBUTED SPRING-MASS SYSTEMS UNDER ELASTIC BOUNDARY CONDITION

An analytic solution for a beam with arbitrarily distributed spring-mass systems under elastic boundary condition is obtained by using sine expansion method. It is applied to solving free vibration of a beam carrying uniformly distributed sprung masses, and results compared with those in reference show good agreement, which validates the methodology. Additional, effects of five different distributions of spring-mass system on dimensionless natural frequencies of beam are studied. Considering modes of beam without spring-mass, it is concluded that the higher the density and the wider the distribution of spring-mass system locating at the largest magnitude of a mode, the greater influences the spring-mass system has on the dimensionless natural frequencies of that mode.