A low-cost experimental averaging method for fast–slow mechanical systems via long-exposure video: Application to the Kapitza pendulum

Nonlinear mechanical systems are typically challenging to analyze experimentally, particularly when high-frequency motions require high temporal resolution, often making data collection difficult and expensive. This research explores an alternative approach by utilizing inexpensive and readily available long-exposure cameras, such as those found in billions of mobile devices. The method is suited to fast–slow systems in which a periodic, high-frequency motion rides on a much slower drift—examples include vibratory energy harvesters, tuned-mass dampers, and the Kapitza pendulum studied here. When the camera’s exposure covers at least one fast period, pixel brightness becomes proportional to the classical probability density (CPD) of the fast motion. Substituting this CPD into the standard averaging integral yields the governing equations of the slow subsystem without resolving the high-frequency motion in time. We demonstrate the experimental feasibility of the idea on a Kapitza pendulum whose pivot vibrates at
48 Hz. The blurred motion of three LEDs is captured on a 30fps HD video. For each frame, the spatial average is evaluated in MATLAB
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, allowing for the recovery of the slow trajectory. Consequently, a system parameter identification is performed. Furthermore, with some uncertainty, it is even possible to recover the slow system dynamics of the unexcited, physical pendulum without directly measuring it.
Long-exposure imaging thus extracts relevant slow-dynamics information with pocket-sized hardware. It eliminates the need for costly high-speed cameras, providing a practical and low-budget tool for experimental averaging and system identification.