Spring Pendulum (Elastic Pendulum) Simulator

The motion is described in polar coordinates (r, θ). The first equation governs the radial (bouncing) acceleration. It has three parts: centrifugal force from swinging, gravity's radial component, and the spring's restoring force.

Here, $\ddot{r}$ is radial acceleration (m/s²), $r\dot{\theta}^2$ is the centrifugal term, $+g\cos\theta$ is gravity's pull along the spring, and $\frac{k}{m}(L_0 - r)$ is the spring acceleration towards its natural length $L_0$.

The second equation governs the angular (swinging) acceleration. It's similar to a pendulum but modified by the fact the "rod" length r is changing.

Here, $\ddot{\theta}$ is angular acceleration (rad/s²). The term $-g\sin\theta/r$ is the pendulum-like swing. The crucial coupling term $- 2\dot{r}\dot{\theta}/r$ appears because the changing length r affects angular momentum conservation. This term is the source of the energy transfer between bouncing and swinging.

Flexible Robot Arms & Cranes: Long, slender robotic arms or construction crane booms can flex and sway. Their dynamics are modeled as a series of coupled spring-pendulum segments. Engineers use these simulations to prevent destructive resonant oscillations during operation.

Atomic Force Microscopy (AFM): The AFM's microscopic cantilever tip scans surfaces. It's essentially a tiny, damped spring pendulum. Understanding its nonlinear resonance is critical for improving image resolution and measuring nanoscale forces.

Cable Structures & Tethers: The dynamics of suspended cables, mooring lines for ships, or tethers for space elevators involve coupled stretch and swing. Parametric resonance from wave or wind forces can lead to unexpected, large-amplitude motions that must be designed against.

Energy Harvesting Devices: Some prototypes use a spring pendulum mechanism to capture energy from ambient vibrations (like bridges or machinery). The chaotic, wide-frequency response of the system can make it efficient at harvesting energy from irregular real-world vibrations.

First, you might tend to think that "a softer spring makes the pendulum's motion slower," but in reality, it's not that simple. While it's true that reducing the spring constant k makes the spring's extension and contraction slower, the period of the pendulum's swing is primarily determined by its initial length. If you make k extremely small, creating a "floppy" state, the extension/contraction and the swing become strongly coupled, sometimes resulting in motion that appears irregular. This isn't about the motion being slower; it's about it becoming more complex.

Next, a pitfall when setting parameters: do not confuse the "natural length L0" with the "initial length r0". The length of the spring at the start of the simulation is r0. For example, if you set L0=1.0m and r0=1.5m, the spring is initially at rest while stretched by 0.5m. If you gently release it from this state (zero initial velocity), the weight will begin pendulum motion while falling straight down—an unexpected movement. To create your intended initial state, be mindful of the relationship between r0 and L0.

Finally, regarding the interpretation of simulation results: it's crucial to understand that while the law of conservation of energy always holds in an ideal system without damping, the spring energy and the pendulum energy are not individually conserved. The essence of the system is that the two vigorously exchange energy over time. Even if the breakdown of energy fluctuates wildly on the graph, if the total sum remains constant, your simulation is working correctly. In reality, motion almost never starts from a state where energy is completely biased to one side.