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首页> 外文期刊>Applied Mathematical Modelling >Deterministic and stochastic aspects of the stability in an inverted pendulum under a generalized parametric excitation
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Deterministic and stochastic aspects of the stability in an inverted pendulum under a generalized parametric excitation

机译：广义参数激励下倒立摆的稳定性的确定性和随机性

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In this paper, we explore the stability of an inverted pendulum under a generalized parametric excitation described by a superposition of N cosines with different amplitudes and frequencies, based on a simple stability condition that does not require any use of Lya-punov exponent, for example. Our analysis is separated in 3 different cases: N = 1, N = 2. and N very large. Our results were obtained via numerical simulations by fourth-order Runge-Kutta integration of the non-linear equations. We also calculate the effective potential also for N ＞ 2. We show then that numerical integrations recover a wider region of stability that are not captured by the (approximated) analytical method of the effective potential. We also analyze stochastic stabilization here: firstly, we look the effects of external noise in the stability diagram by enlarging the variance, and secondly, when N is large, we rescale the amplitude by showing that the diagrams for survival time of the inverted pendulum resembles the exact case for N= 1. Finally, we find numerically the optimal number of cosines corresponding to the maximal survival probability of the pendulum.
机译：在本文中，我们基于不需要任何Lya-punov指数的简单稳定性条件，研究了由不同振幅和频率的N个余弦的叠加描述的广义参数激励下倒立摆的稳定性。 。我们的分析分为3种不同的情况：N = 1，N = 2，N非常大。我们的结果是通过非线性方程的四阶Runge-Kutta积分通过数值模拟获得的。我们还计算了N＞ 2时的有效电势。然后，我们证明了数值积分可以恢复较宽的稳定性区域，而有效电势的（近似）分析方法无法捕获该范围。在这里，我们还分析了随机稳定：首先，通过扩大方差，在稳定性图中查看外部噪声的影响，其次，当N大时，我们通过显示倒立摆的生存时间图类似于图来重新调整幅度。 N = 1的确切情况。最后，我们在数字上找到对应于摆的最大生存概率的最佳余弦数。

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