单自由度体系有阻尼自由振动振幅包络线的性质研究与数学证明  

Characteristics of the Envelope of Damped Free Vibrations of SDOF Systems and Its Proof

作　　者：李镇明[1,2] 张瑀[1,2] 周广春[1,2] 

机构地区：[1]哈尔滨工业大学结构工程灾变与控制教育部重点实验室,黑龙江哈尔滨150090 [2]哈尔滨工业大学土木工程学院,黑龙江哈尔滨150090

出　　处：《数学的实践与认识》2015年第21期227-232,共6页Mathematics in Practice and Theory

基　　金：哈尔滨工业大学教育科研项目(HIT20140127)

摘　　要：针对单自由度体系有阻尼自由振动,探究了振幅包络线与振动位移一时间曲线(以下简称振动曲线)的交点个数、振幅包络线与振动曲线交点位置以及振幅包络线值与结构阻尼比的关系三方面的问题.通过理论推导,给出了一条振幅包络线在单周期内与振动曲线只有一个切点且切点位于振动曲线峰值点稍靠右侧处的证明过程.通过改变相关参数的取值,发现了在某些初始条件下在振动初始阶段振幅包络线的绝对值不与结构的阻尼比呈负相关的有趣现象,并给出了直观的图像说明与理论证明.Aiming at deep analyse of the envelopes of damped free vibrations of SDOF systems, this paper discusses three issues about it. They are： the number of points of the intersection of the envelope curves and the displacement-time curve, the positions of those points and the relationship between the envelope curves and the damping ratio. This paper presents the process proving that there is only one point of the intersection of the envelope and the displacement-time curve in one cycle, and that the envelope curves touch the displacement- time curve at points slightly to the right of its peak values. Also, the paper presents a new discovery that the absolute values of the envelope curve doesn＇t decrease with the damping ratio in a very short time after the beginning of vibration under some initial conditions and then gives proofs and explanations of this phenomena.

关 键 词：振幅包络线 振动曲线 交点 阻尼比 数学证明 

分 类 号：O342[理学—固体力学] O328[理学—力学]