Abstract

In this paper, we discuss the problem of searching for equilibrium states of Atwood’s machine in which a pulley of a finite radius is replaced by two separate small pulleys and both masses can oscillate in a vertical plane. Differential equations of motion for the system are derived, and their solutions are computed in the form of power series in a small parameter. It is shown that, in the case of equal masses, the equilibrium position \(r = {\text{const}}\) of the system exists only if the oscillation amplitudes and frequencies of the masses are the same and the phase shift is α = 0 or \(\alpha = \pi \). In addition, there is a dynamic equilibrium state when both masses oscillate with the same amplitudes and frequencies and the phase shift is \(\alpha = \pm \pi {\text{/}}2\). In this case, the lengths of the pendulums also oscillate around a certain equilibrium value. Comparison of the results with the corresponding numerical solutions of the motion equations confirms their correctness. All necessary computations are carried out using the Wolfram Mathematica computer algebra system.

中文翻译：

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用计算机代数搜索具有两个振荡体的阿特伍德机的平衡态

摘要

在本文中，我们讨论了寻找阿特伍德机器的平衡状态的问题，在该机器中，有限半径的皮带轮被两个单独的小皮带轮代替，并且两个质量都可以在垂直平面内振荡。推导了系统的运动微分方程，并以小参数的幂级数形式计算了它们的解。结果表明，在质量相等的情况下，仅当质量的振荡幅度和频率相同且相移为时，系统的平衡位置\（r = {\ text {const}} \\）才存在。 α= 0或\（\ alpha = \ pi \）。此外，当两个质量块以相同的幅度和频率振荡并且相移为\（\ alpha = \ pm \ pi {\ text {/}} 2 \）。在这种情况下，摆的长度也围绕某个平衡值振荡。将结果与运动方程式的相应数值解进行比较，可以证实其正确性。使用Wolfram Mathematica计算机代数系统执行所有必要的计算。