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For a double pendulum system with equal masses, equal string lengths, and under the small-angle approximation, analyze the equations of motion for the two bobs and solve for the characteristic frequencies (angular frequencies) of the system. Assume both bobs have same mass $m$ and the massless string length $l$. All pivots are assumed to be frictionless. Under the small angle approximation, $sin \theta \approx \theta$ and $cos\theta =1-\frac{\theta^2}{2}$. ...

Introduction In the article of Small amplitude oscillations of a double pendulum , the author experimentally investigate a double pendulum model under small angles. By using Tracker to analyze the motion of two pendulum bobs and performing Fourier transformation on their movements, the results can be showed that the motion of bobs can be described as a superposition of two sinusoidal functions with different oscillating modes. Notably, the author mentioned that when the mass of the lower bob is much smaller than that of the upper bob, an exception can occur: the oscillation frequencies of the two modes become approximately equal, leading to the observation of beat phenomena and the energy of one pendulum gradually transferred to the other. ...