Double Pendulum Simulator Calculator

Enter Simulation Inputs

Use degrees for initial angles, radians per second for angular velocities, and consistent units for masses and lengths.

Mass 1

Mass 2

Length 1

Length 2

Initial Angle 1 (deg)

Initial Angle 2 (deg)

Initial Angular Velocity 1

Initial Angular Velocity 2

Gravity

Damping Coefficient

Time Step

Total Time

Example Data Table

This sample illustrates a typical starting configuration for exploring highly sensitive motion.

Mass 1	Mass 2	Length 1	Length 2	Angle 1	Angle 2	Omega 1	Omega 2	Gravity	Step	Total Time
1.00	1.00	1.00	1.00	120°	-10°	0.00	0.00	9.81	0.01	20.00
1.20	0.80	1.10	0.90	95°	15°	0.20	-0.10	9.81	0.02	15.00
1.00	1.00	1.30	0.70	135°	25°	-0.30	0.40	9.81	0.01	25.00

Formula Used

The simulator uses the standard double pendulum equations with a fourth-order Runge-Kutta method. This improves numerical accuracy compared with a simple Euler step.

State variables: θ₁, ω₁, θ₂, ω₂

Position equations:

x₁ = l₁ sin(θ₁)
y₁ = -l₁ cos(θ₁)
x₂ = x₁ + l₂ sin(θ₂)
y₂ = y₁ - l₂ cos(θ₂)

Angular rate equations:

dθ₁/dt = ω₁
dθ₂/dt = ω₂

Angular acceleration equations:

dω₁/dt = [ -g(2m₁ + m₂)sinθ₁ - m₂g sin(θ₁ - 2θ₂) - 2m₂sin(θ₁ - θ₂)(ω₂²l₂ + ω₁²l₁cos(θ₁ - θ₂)) ] / [ l₁(2m₁ + m₂ - m₂cos(2θ₁ - 2θ₂)) ]

dω₂/dt = [ 2sin(θ₁ - θ₂) (ω₁²l₁(m₁ + m₂) + g(m₁ + m₂)cosθ₁ + ω₂²l₂m₂cos(θ₁ - θ₂)) ] / [ l₂(2m₁ + m₂ - m₂cos(2θ₁ - 2θ₂)) ]

Optional damping is added as -cω₁ and -cω₂ to the angular acceleration terms.

Total energy:

E = K + U
K = 0.5m₁v₁² + 0.5m₂v₂²
U = -(m₁ + m₂)gl₁cosθ₁ - m₂gl₂cosθ₂

How to Use This Calculator

Enter both masses and both rod lengths using consistent units.
Provide initial angles in degrees.
Set starting angular velocities in radians per second.
Enter gravity, damping, time step, and total simulation time.
Click Run Simulation to calculate the motion.
Review the summary cards, tables, and Plotly graphs.
Use the CSV button for raw data export.
Use the PDF button to save the visible results block.

Frequently Asked Questions

1) Why is the double pendulum considered chaotic?

Tiny changes in starting angles or velocities can produce very different trajectories over time. That sensitive dependence is a classic sign of chaotic behavior.

2) Are these results exact?

No. They are numerical approximations. The page uses a fourth-order Runge-Kutta method, which is accurate for many cases, but still depends on time-step size.

3) What units should I use?

Use one consistent unit system. For example, kilograms, meters, seconds, and meters per second squared work well together.

4) Why can energy drift slightly during simulation?

Numerical integration introduces small rounding and truncation effects. A smaller time step usually reduces drift, though it also increases the number of calculations.

5) What does the damping input do?

Damping gradually removes energy from the system. Higher damping values reduce oscillation intensity and can suppress extremely chaotic motion.

6) Why should I reduce the time step sometimes?

If the motion becomes very fast or the energy plot looks unstable, a smaller step often improves accuracy and smoothness.

7) What does the trajectory graph represent?

It traces the second bob's x-y path over time. Dense, looping, or tangled shapes often reveal strong nonlinear behavior.

8) Can this page help compare scenarios?

Yes. Change one parameter at a time, rerun the model, and compare summaries, tables, and plots to study sensitivity clearly.