Adjust pendulum length, initial angle, damping coefficient, and gravitational acceleration for real-time RK4 numerical integration. Explore nonlinear oscillations across pendulum animation, angular displacement waveform, and phase space (θ-ω plane) tabs.
Parameters
Preset
Equation of motion (numerical integration with RK4)
$$\ddot{\theta} + \gamma\dot{\theta} + \frac{g}{L}\sin\theta = 0$$
Period for small-angle approximation:
$$T \approx 2\pi\sqrt{\frac{L}{g}}$$
Energy (when $\gamma=0$):
$$E = \frac{1}{2}L^2\dot{\theta}^2 + gL(1-\cos\theta)$$
Results
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Period T (linear approximation)
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Current angle θ
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Angular velocity ω = dθ/dt
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Total energy ratio
Pendulum Animation
Angular Displacement vs Time
Phase Space
Pend
Phase
💬 Conversation to Deepen Understanding
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I've heard the term "isochronism" — is it really true that the period stays the same even if the swing amplitude changes? It seems like a big swing would take more time...
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"For small amplitudes" it's the same — that's the key. Try θ₀=5° and 10°, you'll get almost the same period. But switch to the "Large amplitude" preset (θ₀=120°) and the period clearly becomes longer. That's because the approximation sinθ ≈ θ breaks down and nonlinearity kicks in. Galileo observed a "small-swing pendulum," which is why he discovered isochronism.
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When I turn on damping, the amplitude gradually decreases. Does isochronism hold even with damping?
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For small amplitudes and linear damping, the period stays almost the same. The period of a damped oscillation is T_d = 2π/√(ω₀² - γ²/4), so if γ is small, the difference from T₀ is tiny. The important thing is that "the amplitude decays exponentially." Look at the angular displacement tab — the envelope follows e^(-γt/2). Building vibration dampers and car shock absorbers use this damping force.
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I looked at the phase space tab — without damping it's a closed ellipse, and with damping it becomes a spiral. What does that mean?
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A closed ellipse is evidence of energy conservation. Going around the same ellipse forever = sustained oscillation. A spiral means the trajectory is sucked into the origin (rest) = energy dissipates and stops. Just by looking at the "flow pattern" in the phase plane, you can instantly see the system's stability and long-term behavior — this is a very important concept in CAE vibration analysis and control design.
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With the Moon (g=1.62) preset, the period became about 2.5 times longer. If I took a pendulum clock to the Moon, it would be off, right?
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Since T ∝ 1/√g, on the Moon T_moon/T_earth = √(9.81/1.62) ≈ 2.46 times. A pendulum that ticks once per second on Earth would tick only 0.41 times per second on the Moon — meaning it runs 2.46 times faster. Real precision pendulum clocks need altitude and latitude corrections (due to tiny g variations). And GPS satellite clocks actually compensate for general relativistic time dilation — clocks in high orbits where gravity is weaker run faster.
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How is RK4 different from the Euler method? Why do we use RK4?
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The Euler method (first-order accuracy) accumulates O(dt²) error per step, causing energy to increase or diverge in long simulations. For a pendulum, the amplitude will spontaneously grow after dozens of swings. RK4 (fourth-order accuracy) has O(dt⁵) error, giving much better precision for the same time step. In CAE structural dynamics, choosing between Newmark-β, Runge-Kutta, and other time integration methods is a key trade-off between "numerical dissipation and computational cost."
Frequently Asked Questions
Is the period of a simple pendulum determined only by its length?
For small amplitude approximation, T ≈ 2π√(L/g), so it depends only on length L and gravitational acceleration g. It is independent of mass and amplitude (isochronism). For large amplitudes, T ≈ 2π√(L/g)×(1 + θ₀²/16 + ...), adding an amplitude correction. Since this simulator uses RK4 retaining sinθ as is, nonlinearity at large amplitudes is accurately computed.
What happens if you set the pendulum angle to 170° (almost straight up)?
As θ₀ approaches 180° (straight up), the period diverges. This is due to energy conservation: the time to reach the 'unstable equilibrium point' at θ=180° becomes theoretically infinite. In phase space, this point becomes a 'saddle point', and trajectories split into closed oscillatory orbits and rotational orbits, forming a 'separatrix'. You can observe the period dramatically increasing when you try it.
How does damping occur in a real pendulum?
Mainly three sources: ① Air resistance (proportional to velocity, as modeled in this simulator) ② Friction at the pivot and acoustic radiation ③ Internal damping of the material. Real pendulum clocks use an escapement to replenish energy and compensate for damping. With a high Q (low damping) pendulum, the amplitude remains nearly constant even after thousands of swings.
Can you explain the relationship between a pendulum and a building's seismic response?
The first natural vibration mode of a building resembles a pendulum, with a natural period of about T ≈ 0.1×N (N: number of floors) seconds. When the dominant period of seismic waves matches this, resonance occurs and the building shakes violently (e.g., in the 1995 Kobe earthquake, 5-6 story buildings were severely damaged). A Tuned Mass Damper (TMD) places a pendulum on top of a building, swinging in opposite phase to absorb vibrations — essentially a system of two coupled pendulums like in this simulator.
What is the similarity between a simple pendulum and a spring-mass system?
The small-angle equation for a simple pendulum, θ'' + (g/L)θ = 0, and the spring-mass system equation, x'' + (k/m)x = 0, are mathematically identical. Here, g/L corresponds to k/m. Thus, the pendulum's natural angular frequency ω₀=√(g/L) perfectly matches the spring system's ω₀=√(k/m). In CAE vibration analysis, complex structures are often simplified into 'equivalent pendulums (or spring-mass systems)' to estimate natural periods quickly.
Can a nonlinear pendulum become chaotic?
A 'forced nonlinear pendulum' with an external force can become chaotic. Adding a periodic external force F₀cos(Ωt) to a simple pendulum results in a mix of periodic and chaotic motion depending on parameters (F₀ and Ω). A double pendulum (two links) is an extension of this simulator and is a classic example of chaos, where tiny differences in initial conditions grow exponentially. You can experience double pendulum chaos with NovaSolver's 'pendulum-chaos' tool.
Set pendulum length (L) in meters using slider vL; typical range 0.5–2.0 m for laboratory setups.
Adjust initial angle θ₀ in degrees (sT0) between 5–60° to observe nonlinear effects; small angles (<10°) approximate simple harmonic motion.
Input damping coefficient (drag) and gravitational acceleration (g, default 9.81 m/s²) to match experimental conditions.
Click simulate to run 4th-order Runge–Kutta integration with real-time θ and ω output.
Worked Example
For a 1.2 m steel pendulum with initial angle 25°, damping coefficient 0.05 s⁻¹, and g = 9.81 m/s²: linear period ≈ 2.20 s. RK4 integration shows actual period ≈ 2.24 s due to nonlinearity. After 10 oscillations (~22 s simulation), energy ratio drops to ~0.88 from 1.0, indicating 12% amplitude loss to air resistance and pivot friction.
Practical Notes
Use damping coefficients 0.01–0.1 s⁻¹ to model realistic laboratory pendulums; higher values simulate high-viscosity fluids or heavy damping.
RK4 step size automatically scales; for angles >45°, expect 10–15% deviation from small-angle approximation T = 2π√(L/g).
Energy ratio tracks conservation; ratios <0.95 indicate numerical drift or insufficient RK4 resolution—reduce step size if needed.
Compare Earth (g=9.81) versus lunar conditions (g=1.62 m/s²) to validate gravitational scaling on period.