Abstract

The problem of finding a periodic motion of generalized Atwood’s machine in which the pulley of a finite radius is replaced by two small pulleys and one of the objects may oscillate in the vertical plane is discussed. Using symbolic computations, the equations of motion are derived and their periodic solutions in the form of power series in a small parameter in the case of small oscillations are obtained. It is shown that if the difference of the objects' masses is small, then the system has a dynamic equilibrium state in which the oscillating object behaves like a pendulum the length of which performs small oscillations. In this case, the frequency resonance \(2:1\) is observed; i.e., the pendulum length oscillation frequency is twice as large as the oscillation frequency of the angular variable. The comparison of the analytical results with the numerical solutions to the equations of motion confirms the validity of the analytical computations. All computations are performed using the computer algebra system Wolfram Mathematica.

中文翻译：

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利用计算机代数构造广义阿特伍德机运动方程的周期解

摘要

讨论了寻找通用Atwood机器的周期性运动的问题，在该机器中，有限半径的滑轮被两个小滑轮代替，并且其中一个物体可能在垂直平面内振荡。使用符号计算，导出运动方程，并在小振荡的情况下以小参数获得幂级数形式的周期解。结果表明，如果物体质量的差异很小，则系统具有动态平衡状态，在该状态下，振动物体的行为像摆锤一样，其长度会产生较小的振荡。在这种情况下，频率共振\（2：1 \）被观察；也就是说，摆长振荡频率是角度变量的振荡频率的两倍。分析结果与运动方程数值解的比较证实了分析计算的有效性。所有计算均使用计算机代数系统Wolfram Mathematica执行。