A Swing of Beauty: Pendulums, Fluids, Forces, and Computers

Abstract

While pendulums have been around for millennia and have even managed to swing their way into undergraduate curricula, they still offer a breadth of complex dynamics to which some has still yet to have been untapped. To probe into the dynamics, we developed a computational fluid dynamics (CFD) model of a pendulum using the open-source fluid-structure interaction (FSI) software, IB2d. Beyond analyzing the angular displacements, speeds, and forces attained from the FSI model alone, we compared its dynamics to the canonical damped pendulum ordinary differential equation (ODE) model that is familiar to students. We only observed qualitative agreement after a few oscillation cycles, suggesting that there is enhanced fluid drag during our setup’s initial swing, not captured by the ODE’s linearly-proportional-velocity damping term, which arises from the Stokes Drag Law. Moreover, we were also able to investigate what otherwise could not have been explored using the ODE model, that is, the fluid’s response to a swinging pendulum—the system’s underlying fluid dynamics.

Contributions to the Teaching and Learning of Fluid Mechanics

“…In the article by Mongelli and Battista [17], the authors undertake a systematic study of pendulum dynamics by properly and fully accounting for the flow around a moving body which is not captured through the classical mechanical pendulum equations (see, also, another recent study that examines this issue through the lens of the least action principle [18]). The authors develop a computational fluid dynamics (CFD) model of a pendulum using the open-source fluid-structure interaction (FSI) software, IB2d.…”

Lagrangian particle tracking velocimetry investigation of vortex shedding topology for oscillating heavy spherical pendulums underwater

“…The first known study of pendulum motion was carried out by Galileo Galilei in 1605 as he discovered that the period of the swings remained constant. Since then, pendulums have been subject to much research and utilised in many technical applications, such as, among others, the pendulum clock, ballistic pendulums, seismometers, metronomes, viscosimeters and mass dampers in high-rise buildings (Mongelli & Battista 2020;Worf et al 2022). Also, the pendulum is an 'educational classic' and a standard device to study the concept of oscillating motions, starting with the undamped case and simple harmonic motion (Mongelli & Battista 2020).…”

“…Since then, pendulums have been subject to much research and utilised in many technical applications, such as, among others, the pendulum clock, ballistic pendulums, seismometers, metronomes, viscosimeters and mass dampers in high-rise buildings (Mongelli & Battista 2020;Worf et al 2022). Also, the pendulum is an 'educational classic' and a standard device to study the concept of oscillating motions, starting with the undamped case and simple harmonic motion (Mongelli & Battista 2020). The physical pendulum also shares a long tradition in fluid flow and fluid-structure interaction studies, including concepts such as added mass and fluid friction.…”

“…Their findings show that even with the cylinder, which could be interpreted as a 2-D flow, only the 3-D analysis can explain adequately the measured vortex shedding phenomena. Mongelli & Battista (2020) performed numerical fluid-structure interaction simulations of pendulums with a spherical bob. However, their simulations were 2-D, which resembled a disk or cylinder slice rather than a sphere.…”

Damped harmonic oscillator revisited: the fastest route to equilibrium

“…Under which conditions does the damped harmonic oscillator return to the equilibrium state the fastest? The standard textbook answer to that question is that the oscillator returns to the equilibrium state the fastest in the critically damped regime, regardless of the initial conditions [1][2][3][4][5] . Since none of the solutions to this model ever exactly reach the equilibrium state, but only asymptotically approach it, the authors usually justify this answer by considering the speed of convergence of the general underdamped, critically damped and overdamped solutions towards the equilibrium state in the infinite time limit 1 .…”