10.1119/10.0001389.1The azimuthal-radial pendulum is a coupled pendulum-beam system where a pendulum and a supporting string are attached to the free end of a cantilever beam with its other end clamped. Conversion between radial (parallel to cantilever) and azimuthal (perpendicular to cantilever) modes of oscillations is observed when the radial frequency is twice the azimuthal frequency. A theoretical Lagrangian formulation of the system is presented and numerical predictions are compared to experimental data, showing good agreement. A non-linear coupling term arising from the inextensibility of the cantilever beam is shown, via simplifying assumptions that reduce the equations of motion to the Mathieu equation for parametric oscillators, which is responsible for the parametric resonance observed in the system. Experimental work involved measurements of the dynamic trajectory of the pendulum bob in the lab frame, and systematic investigation into the maximum conversion amplitude, beat frequency, and parametric resonance regime boundaries as a function of bob mass, string length, cantilever length, initial release amplitude, and other relevant parameters, shows good agreement with numerical results.

中文翻译：

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摆梁耦合系统中的动态轨迹与参数共振

10.1119/10.0001389.1 方位-径向摆锤是一种耦合摆梁系统，其中摆锤和支撑绳连接到悬臂梁的自由端，而悬臂梁的另一端被夹紧。当径向频率是方位频率的两倍时，观察到振荡的径向（平行于悬臂）和方位（垂直于悬臂）振荡模式之间的转换。提出了系统的理论拉格朗日公式，并将数值预测与实验数据进行了比较，显示出良好的一致性。通过将运动方程简化为参数振荡器的 Mathieu 方程的简化假设，显示了由悬臂梁的不可延伸性引起的非线性耦合项，该方程负责在系统中观察到的参数共振。