Study oscillator decay with precise physical inputs. Compare regimes through clear metrics and responsive plots. Turn raw inputs into readable insights for faster analysis.
This static table shows a typical setup and sample outcome values for a lightly damped oscillator.
| Mass | Damping | Stiffness | x₀ | v₀ | Duration | Regime | Approx. ζ |
|---|---|---|---|---|---|---|---|
| 1.20 kg | 0.80 N·s/m | 20.00 N/m | 0.05 m | 0.00 m/s | 12.0 s | Underdamped | 0.0816 |
| 2.00 kg | 12.65 N·s/m | 20.00 N/m | 0.03 m | -0.10 m/s | 10.0 s | Critically damped | 1.0000 |
| 1.00 kg | 12.00 N·s/m | 16.00 N/m | 0.04 m | 0.20 m/s | 8.0 s | Overdamped | 1.5000 |
Governing equation: m x″ + c x′ + k x = 0
Natural frequency: ωn = √(k / m), critical damping: cc = 2√(km), damping ratio: ζ = c / cc
Underdamped case (ζ < 1):
x(t) = e-αt[A cos(ωdt) + B sin(ωdt)]
Here α = c / (2m), ωd = ωn√(1 - ζ²), A = x₀, B = (v₀ + αx₀) / ωd
Critically damped case (ζ = 1):
x(t) = (C₁ + C₂t)e-αt, where C₁ = x₀ and C₂ = v₀ + αx₀
Overdamped case (ζ > 1):
x(t) = C₁er₁t + C₂er₂t, with r₁, r₂ = (-c ± √(c² - 4mk)) / (2m)
Acceleration and energy:
a(t) = -(c x′ + kx) / m
E(t) = ½mv² + ½kx²
Enter mass, damping coefficient, and stiffness first. These define the oscillator and determine whether the response is underdamped, critically damped, or overdamped.
Provide initial displacement and initial velocity next. These set the starting state and control the full transient motion shown in the results and chart.
Choose the total simulation duration and number of sample points. Longer durations reveal decay behavior, while more samples create smoother curves and denser export data.
Press the calculate button. The page will show the result block below the header and above the form, including summary metrics, a Plotly graph, and a data preview.
Use the CSV button when you need the full time-series data. Use the PDF button when you want a portable report of the visible results section.
The damping ratio compares actual damping to critical damping. Values below one oscillate while decaying, exactly one returns fastest without oscillation, and above one returns without oscillation more slowly.
A viscous damper removes mechanical energy from the system over time. The calculator models that loss through the damping term, so total energy falls as motion decays.
The damped frequency exists only for underdamped motion. Critically damped and overdamped systems do not oscillate, so a repeating period and oscillation frequency are not reported.
Yes, but all inputs must stay consistent. For example, if stiffness and damping use custom units, mass, displacement, velocity, and time must match that same unit system.
Critical damping gives the fastest non-oscillatory return to equilibrium. Designers often target it in suspensions, instruments, and control systems that must settle quickly without overshoot.
The reported value is an approximate 5% settling measure based on decay behavior. It helps estimate how long the motion needs before becoming practically small.
Initial displacement alone creates a spring restoring force. That force immediately produces acceleration, even if the object starts from rest at the first instant.
For most studies, 200 to 500 points works well. Use more points for smoother graphs or longer simulations, but remember larger datasets increase export size.
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.