⚡ electrical engineering

Dynamics of the Thermomagnetic Pendulum

This paper introduces and models a thermomagnetic pendulum that achieves sustained oscillations through the coupling of gravity, magnetic forces, and heat transfer, where heating and cooling cycles drive the ferromagnetic bob across its Curie point to modulate magnetic attraction.

Original authors: Ryan Thompson, Ethan Wang, Nilay Kant

Published 2026-06-03

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CC BY 4.0

Original authors: Ryan Thompson, Ethan Wang, Nilay Kant

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a playground swing, but instead of a child pushing it, the swing is powered by a clever game of "hot and cold" played with magnets. This is the story of the Thermomagnetic Pendulum, a device described in the paper by Ryan Thompson, Ethan Wang, and Nilay Kant.

Here is how it works, broken down into simple concepts:

1. The Cast of Characters

The Swing (The Pendulum): A metal ball (the "bob") hangs from a string. It's made of Nickel, a special metal that loves magnets when it's cool but hates them when it's hot.
The Magnet: A permanent magnet sits nearby, but it's slightly off to the side, not directly under the swing.
The Heater: A invisible "sun" shines on the swing, but only when the ball swings into a specific spot.
The Air: The air around the ball acts as a cooling fan, but only when the ball swings away from the heater.

2. The Magic Trick: The "Curie Point"

The secret sauce of this experiment is a property called the Curie Point. Think of this as a "magnetic melting point."

When the Nickel ball is cool: It acts like a magnet. It gets pulled strongly toward the permanent magnet.
When the Nickel ball gets hot: It loses its "magnetic personality." It stops caring about the magnet and becomes just a regular metal ball.

3. The Dance: How It Moves

The pendulum doesn't need a battery or a person to push it. It moves on its own through a repeating cycle of heating and cooling:

The Pull: The ball swings toward the magnet. Because it's cool, the magnet grabs it, pulling it in.
The Heat: As the ball swings into the heated zone, it gets hot. Once it gets hot enough (crossing the Curie point), it suddenly loses its magnetic attraction. The magnet lets go.
The Release: Without the magnet holding it, gravity takes over. The ball swings away from the magnet, out of the hot zone.
The Cool Down: As it swings away, the air cools the ball down. Once it's cool again, it "wakes up" magnetically.
The Return: Now that it's magnetic again, the magnet grabs it and pulls it back toward the start, and the cycle repeats.

4. The Computer Model: A Digital Twin

The authors didn't just build this; they built a virtual version on a computer to understand exactly how it works. They had to solve a tricky puzzle because three things are happening at once:

Heat: How fast does the ball heat up or cool down? (Like tracking how fast a cup of coffee cools).
Magnetism: How strong is the pull? (This changes instantly as the ball's temperature changes).
Motion: How does the ball swing? (Physics of a pendulum).

Usually, scientists might just guess how strong the magnet pulls. But here, the authors built a system where the computer calculates the temperature of every tiny part of the ball, figures out how magnetic that specific part is right now, and then calculates the pull based on that. It's like a video game where the physics engine updates the character's strength every single frame based on their body temperature.

5. What They Found

When they ran the simulation, they saw a few interesting things:

The "Snap" Effect: The magnetic pull doesn't fade slowly. It stays strong until the ball gets very close to the Curie point, and then it drops like a stone. It's an "on/off" switch rather than a dimmer switch.
The Wobble: Because the magnet is placed slightly off-center, the pull isn't perfectly symmetrical. The swing behaves slightly differently depending on which way it's moving, creating a unique, lopsided rhythm.
The Endless Loop: Despite friction and air resistance, the system finds a steady rhythm. It heats up, loses its grip, cools down, gets pulled back, and keeps swinging forever (in the simulation).

Summary

In short, the paper describes a self-swinging pendulum that uses heat to turn off a magnet and cooling to turn it back on. The authors created a complex computer model to prove that this "hot-and-cold" dance creates a stable, repeating motion without any external power source other than the heat applied to it.

Technical Summary: Dynamics of the Thermomagnetic Pendulum

Problem Statement
This paper addresses the modeling and simulation of a passive thermomagnetic pendulum, a coupled thermo-magnetic-mechanical system designed for low-grade thermal energy harvesting. The system consists of a ferromagnetic spherical bob suspended as a rigid pendulum near an offset permanent magnet. The core challenge lies in the fact that the magnetic torque driving the system cannot be prescribed by mechanical equations alone. Instead, the torque emerges from a complex interaction between the transient thermal state of the bob, the temperature-dependent magnetic response of the material (specifically near the Curie point), and the geometric configuration of the pendulum relative to the external magnetic field. Previous studies have highlighted the need for coupled thermal and mechanical treatments in such devices, as the magnetic force vanishes when the material exceeds its Curie temperature ( $T_c$ ) and recovers upon cooling, creating a cyclic motion that is difficult to predict without a multiphysics approach.

Methodology
The authors develop a comprehensive computational framework that couples three interacting subsystems:

Thermal Subsystem: The transient three-dimensional heat conduction within the spherical bob is modeled using the heat equation in spherical coordinates. The model accounts for uniform external heating over a specific angular region, convective cooling (dependent on instantaneous surface velocity), and radiative cooling. The convective coefficient is calculated using forced-convection correlations for flow over a sphere, thereby coupling the thermal field to the mechanical velocity.
Magnetic Subsystem: The interaction between the bob and the permanent magnet is modeled under magnetostatic conditions. The external field is computed analytically using a uniformly magnetized cuboid model. Inside the bob, the effective magnetic field is determined by subtracting the demagnetizing field (approximated using a spherical demagnetization factor of $N=1/3$ ) from the external field. The magnetization is treated as a nonlinear, temperature-dependent constitutive response. Below the Curie point, the saturation magnetization follows a critical scaling law, and the local magnetization aligns with the effective field via a hyperbolic tangent approximation (Langevin-type response). Above $T_c$ , magnetization vanishes.
Mechanical Subsystem: The pendulum dynamics are governed by a second-order differential equation including inertia, viscous damping, gravity, and the magnetic torque. The magnetic torque is derived by integrating the magnetic body-force density (calculated from the gradient of the external field acting on the spatially varying magnetization) over the bob's volume.

Numerical Implementation
The coupled model is implemented using a node-centered finite-volume formulation on a structured spherical grid. The temperature and magnetization fields are co-located on the same mesh to ensure local evaluation of thermo-magnetic coupling without interpolation. The thermal field is solved implicitly, and the magnetization is updated iteratively at each time step to satisfy the nonlinear constitutive relation and the effective field equation. The resulting torque is then used to advance the pendulum's mechanical state.

Key Results
Simulations of the system reveal several critical dynamic behaviors:

Temperature Dependence: The magnetic force decreases nonlinearly with temperature, exhibiting a rapid collapse as the temperature approaches the Curie point, eventually reaching zero for $T \ge T_c$ .
Torque Asymmetry: The magnetic torque is asymmetric with respect to the pendulum angle due to the offset orientation of the permanent magnet. The torque magnitude is also sensitive to the gap distance between the bob and the magnet, decreasing as the gap increases.
Sustained Oscillations: The time-domain simulations demonstrate that the cyclic heating and cooling process induces sustained oscillatory motion. As the bob enters the heated zone, magnetization drops, reducing magnetic attraction and allowing gravity to pull the bob away. As it exits the heated zone, the bob cools, magnetization recovers, and the magnetic attraction re-establishes, driving the return motion. The system converges to a steady-state periodic motion.

Significance and Claims
The paper claims to provide a necessary multiphysics modeling framework that resolves the transient 3D thermal evolution of the bob and links it directly to the magnetic torque in the mechanical equations. The authors emphasize that this problem cannot be reduced to a simple pendulum with an externally prescribed forcing law; rather, the forcing is an emergent property of the coupled system. The study demonstrates that the system's dynamics are highly sensitive to the pendulum angle, bob temperature, and the bob-magnet gap. The work establishes a foundation for understanding passive thermomagnetic devices, with future work proposed to include more general geometries, refined heat-transfer models, improved magnetic constitutive laws, and experimental validation.