弹簧耦合摆在复杂情况下的小振动解

Small Vibration Solutions in Complex Situations of Spring Coupled Pendulum

求解弹簧耦合摆的简正频率是理论力学中一道非常经典的题目,用分析力学的角度研究该题是一种重要的方法.目前对弹簧摆的研究多针对两单摆间的距离恰好等于弹簧原长;而当距离略大于或小于弹簧原长时,平衡状态下单摆摆线不再处于竖直状态,由于涉及不同的小角度近似,系统动能和势能的表示都变得更为复杂.本文探究了弹簧耦合摆系统的简正频率与系统静止时摆线偏移竖直方向角度的关系,利用MATLAB计算对比了是否使用小角度近似的运动过程,得到了近似解的适用情况.最后,在不考虑小角度的情况下,计算了拍频与弹簧固有频率、单摆固有频率和单摆平衡状态下偏移角度的关系.

Solving the normal frequency of spring coupled pendulum is a very classical problem in theoretical mechanics.It is a valuable method to study this problem from the perspective of analytical mechanics.At present,the study on the spring pendulum mainly focuses on the circumstance where the distance between two simple pendulum is exactly equal to the original spring length;however,when the distance becomes slightly larger or smaller than the original spring length,the representation of the system′s kinetic energy and potential energy will be more complicated with the different small-angle approximations involved.In this paper,a general method of constructing normal coordinates is introduced.Starting from normal frequency,the relationship between a normal frequency of the system and the offset angle of the system at rest is explored by controlling the initial release angles of the two pendulum balls to be negative to each other.The motion process is simulated by MATLAB and compared.After that,the application of the small angle approximate solution is obtained and the relationship between the beat frequency and the natural frequency of spring,the natural frequency of pendulum and the offset angle of a pendulum at equilibrium is verified.