Abstract

We consider a simplified model of a synchronous electric motor, which is described by a second-order differential equation, not containing electrical currents. F. Tricomi found that the phase portrait of this equation refers to one of three types, depending on whether the damping coefficient it contains is greater than, less than or equal to some critical value. Since an explicit expression for the critical value is not available, the efforts of many mathematicians were focused on deriving explicit upper and lower analytical estimates for this value. We use a computer to obtain phase portraits of this equation; the features of its phase trajectories are revealed, which are difficult to notice in known phase portraits obtained by analytical methods. Computer calculations are used to plot the critical value of the damping factor in this equation as a function of the main steady-state value of the angular variable. Linear and sinusoidal approximations of this curve are proposed, and the absolute and relative errors of such approximations are computed.

中文翻译：

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同步无电流电动机模型的计算机分析

摘要

我们考虑同步电动机的简化模型，它由二阶微分方程描述，不包含电流。F. Tricomi 发现该方程的相图指的是三种类型之一，具体取决于它包含的阻尼系数是否大于、小于或等于某个临界值。由于无法获得临界值的明确表达式，因此许多数学家的工作重点是导出该值的明确的上限和下限分析估计。我们使用计算机获得该方程的相图；揭示了其相轨迹的特征，这些特征在通过分析方法获得的已知相图中很难注意到。使用计算机计算来绘制该方程中阻尼因子的临界值作为角度变量的主稳态值的函数。提出了该曲线的线性和正弦近似，并计算了这种近似的绝对误差和相对误差。