Enter length, mass, damping, and starting conditions below. Choose units and evaluate at any time. Compare under, critical, or over damping with clarity today.
Enter values, select a damping definition, then calculate.
| Case | L (m) | m (kg) | ζ | ωn (rad/s) | ωd (rad/s) | Regime |
|---|---|---|---|---|---|---|
| Light damping | 1.00 | 1.00 | 0.05 | 3.132 | 3.128 | Underdamped |
| Near critical | 0.50 | 0.80 | 1.00 | 4.429 | — | Critically damped |
| Heavy damping | 1.20 | 1.50 | 1.40 | 2.858 | — | Overdamped |
For small angles, the damped pendulum is modeled by:
θ″ + 2β θ′ + ωn² θ = 0
If you provide a viscous torque coefficient b (N·m·s), the model uses: β = b / (2 m L²).
A damped pendulum is a swinging mass that gradually loses energy to air drag, bearing friction, or fluid resistance. The motion can still be modeled accurately for small angles using a linear equation with constant damping. This calculator estimates angle, velocity, and key time constants from your inputs. It is common in labs, clocks, vibration tests, and suspended instruments today.
Length and gravity set the restoring torque, while mass affects some damping models. The initial angle θ₀ and angular velocity ω₀ define the starting state. Choose a target time t to evaluate θ(t) and ω(t). You can describe damping as ζ, β, Q, or a viscous torque coefficient b.
For small oscillations the undamped natural frequency is ω₀ = √(g/L). Longer lengths reduce ω₀ and increase the period. If angles become large, the true period grows slightly; use this calculator as a first estimate or keep θ₀ modest for best accuracy.
The damping ratio ζ compares damping strength to the critical level. Underdamped (ζ < 1) oscillates with a decaying envelope. Critically damped (ζ = 1) returns fastest without oscillation. Overdamped (ζ > 1) returns slowly and never crosses through repeated swings.
When the system is underdamped, the oscillation frequency becomes ωd = ω₀√(1−ζ²). The damped period Td = 2π/ωd is slightly longer than the undamped period. As ζ increases toward 1, ωd drops and oscillations stretch out before disappearing.
The envelope of the angle decays approximately as e^(−βt), where β = ζω₀ for the standard second‑order form. From θ(t) you can compute how long it takes for the amplitude to halve or fall to any chosen fraction. For underdamped motion, the logarithmic decrement δ relates to ζ through δ = 2πζ/√(1−ζ²).
Use θ(t) to predict clearance or sensor range at a given time, and use ω(t) to estimate peak velocities. Compare the energy fraction to see how quickly the motion dies out. If you have measured successive peak angles, you can back‑fit ζ or Q and then rerun the calculator to validate your model.
ζ is a dimensionless measure of damping compared with the critical value. If ζ < 1 the pendulum oscillates while decaying, ζ = 1 is the fastest no‑oscillation return, and ζ > 1 returns slowly without oscillations.
β is an exponential decay rate in 1/s for the amplitude envelope, while ζ is dimensionless. In the standard model they are linked by β = ζ·ω₀, where ω₀ is the undamped natural frequency.
Use Q when your data comes from resonance or bandwidth measurements. For underdamped systems, Q relates directly to ζ by ζ = 1/(2Q), making it convenient to translate measured “sharpness” into time‑domain decay.
For small angles, ω₀ = √(g/L) so the undamped period depends on length and gravity, not mass. Mass matters only when you model damping through a torque coefficient b, because b is converted using m and L.
The linear equation uses sin(θ) ≈ θ, which is accurate when θ is modest. At large angles the period increases and the waveform is no longer perfectly sinusoidal. Keep θ₀ small for best agreement, or treat results as an approximation.
Measure two successive peak angles A1 and A2 and compute the logarithmic decrement δ = ln(A1/A2). For underdamped motion, ζ = δ / √( (2π)² + δ² ). Enter ζ (or Q) and compare predicted peaks with your data.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.