Exact Solutions for the Natural Frequencies and Mode Shapes of the Uniform Beams Carrying Any Number of Spring-Mass Systems and in-span support

• Main Author: 江鑑原
• Other Authors: 王紀瑞
• Format: Others
• Language: zh-TW
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/77094006511023701126

碩士 === 建國科技大學 === 自動化工程系暨機電光系統研究所 === 99 === Abstract From the motion equation of the “bare” uniform beam (without any number of spring-mass systems and in-span support), an eigenfunction consisting of five integration constants is obtained. Where the last eigenfunction is substituted into the four compatible equations, one force-equilibrium equation and incorporating with the equation of motion for each attaching point of the spring-mass system, and the boundary equations for the two ends of the beam, a matrix equation of the form is got. The solutions of = 0 (where denotes a determinant) give the “exact” natural frequencies of the “constrained” beam (carrying any number of spring-mass systems and in-span support) and the substitution of each corresponding values of into the associated eigenfunction for each attaching points will determine the corresponding mode shapes. Since the order of is 5n+4, where n is the total number of spring-mass systems and in-span support, the “explicit” mathematical expressions for the existing approach becomes lengthy intractable if n > 2. The “numerical assembly method” introduced in this paper aims at improving the last drawback of the existing approach. The “exact” solutions in this paper refer to the numerical results obtained from the “continuum” models for the classical analytical approaches rather than from the “discretized” ones for the conventional finite element methods. Keyword：spring-mass system, in-span support, eigenfunction, numerical assembly method