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<p>In this work the free vibration response of the <i>anisotropic</i> pendulum is analyzed. The <i>spherical</i> pendulum has three primary periodic motions of free vibration, but two are degenerate and have the same frequency, and therefore all three are stable. In contrast, the anisotropic pendulum has three <i>distinct</i> primary periodic motions of free vibration, but only if the level of energy is above a critical threshold. In this case, the primary periodic motions have distinct frequencies. It is shown that the primary periodic motion with the intermediate frequency is inherently unstable and sensitive to initial conditions. When the system is released to freely vibrate in this primary periodic motion, the dynamic response will exhibit sequentially and repeatedly, Dzhanibekov-like transitions. If the energy level of the anisotropic pendulum is below the critical threshold, then it has only two primary periodic motions of free vibration and both are stable. It is shown that the critical threshold decreases with decreasing anisotropy. Even for a slight level of anisotropy the instability of the primary periodic motion with the intermediate frequency is apparent. Léon Foucault identified in 1851 (as did others after him) that his seemingly spherical pendulum is sensitive to a slight anisotropy of its suspension. Our analysis provides a rational explanation for this very old question. A simple and straightforward experimental setup is used to qualitatively corroborate the theoretical predictions.</p>

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The intermediate frequency effect and Dzhanibekov-like transitions in the response of the anisotropic pendulum

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月27日

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  • Eli Benvenisty,
  • David Elata

摘要

In this work the free vibration response of the anisotropic pendulum is analyzed. The spherical pendulum has three primary periodic motions of free vibration, but two are degenerate and have the same frequency, and therefore all three are stable. In contrast, the anisotropic pendulum has three distinct primary periodic motions of free vibration, but only if the level of energy is above a critical threshold. In this case, the primary periodic motions have distinct frequencies. It is shown that the primary periodic motion with the intermediate frequency is inherently unstable and sensitive to initial conditions. When the system is released to freely vibrate in this primary periodic motion, the dynamic response will exhibit sequentially and repeatedly, Dzhanibekov-like transitions. If the energy level of the anisotropic pendulum is below the critical threshold, then it has only two primary periodic motions of free vibration and both are stable. It is shown that the critical threshold decreases with decreasing anisotropy. Even for a slight level of anisotropy the instability of the primary periodic motion with the intermediate frequency is apparent. Léon Foucault identified in 1851 (as did others after him) that his seemingly spherical pendulum is sensitive to a slight anisotropy of its suspension. Our analysis provides a rational explanation for this very old question. A simple and straightforward experimental setup is used to qualitatively corroborate the theoretical predictions.