Visualise the time response of a standard second-order system to a step input. Adjust the natural frequency ωn and the damping ratio ζ to see the percent overshoot, peak time, rise time and settling time update in real time, and build an intuition for time-domain control specifications.
Width of the tolerance band around the final value
Results
—
Damped freq. ωd (rad/s)
—
Overshoot (%)
—
Peak time (s)
—
Rise time (s)
—
Settling time (s)
—
Response type
—
Step response animation
The unit-step response rises toward the target, overshoots and rings before settling. The green band is the ±tolerance band, the dot marks the peak (overshoot) and the vertical line marks the settling time.
Percent overshoot Mp, settling time ts and damped natural frequency ωd. tol is the tolerance band (0.02 for ±2 %, 0.05 for ±5 %). The damping ratio ζ shapes the response (overshoot), while the natural frequency ωn sets its speed.
Peak time tp and rise time tr (0–100 %). Both are defined only for the underdamped case (ζ<1); a critically or over-damped system shows no overshoot or oscillation, so they do not apply.
What is the Second-Order Step Response?
🙋
The "second-order step response" is always the first thing in a control textbook. Why is it so important?
🎓
In a nutshell, because a huge number of real systems behave, near their operating point, like a standard second-order system. A motor position loop, the needle of an analogue meter, a car suspension, an RLC circuit, a building swaying in the wind — they can all be approximated as second-order. And a second-order system is fixed by just two numbers — the natural frequency ωn and the damping ratio ζ. So once you can read this one picture, you can read the behaviour of countless systems at a glance.
🙋
Just two numbers. When I raise ωn on the left, the graph squeezes sideways and gets faster. So ωn is the "speed" knob?
🎓
Exactly. ωn sets the overall speed of the response — double ωn and the rise time and settling time both roughly halve. The damping ratio ζ, on the other hand, is the "shape" knob. Set ζ to a small value like 0.4 and the response overshoots the target and bounces a few times before settling. The smaller ζ is, the bigger the overshoot and the longer the ringing. Try moving ζ around and watch.
🙋
When I set ζ to 1, the overshoot dropped to zero and the peak time and rise time turned into "—". What's happening there?
🎓
ζ = 1 is "critical damping": the system reaches the target as fast as possible with no overshoot at all. It does not oscillate, so there is no "time of the peak" — hence the "—" for peak time. The rise time also loses its usual 0–100 % definition for an oscillatory system, so the tool shows "—" too. Push ζ above 1 and you get an "overdamped" system: no overshoot, but the response becomes sluggish.
🙋
The "settling criterion" lets me pick ±2 % or ±5 %. What's the difference?
🎓
The settling time is "how long until the response stays close to the final value and never leaves again". But "close" is a matter of definition. You can say it has settled once it stays inside a ±2 % band around the final value, or use ±5 %. The ±5 % band is wider, so it is judged "settled" earlier and the settling time comes out shorter. In formulas, ts ≈ 4/(ζωn) for ±2 % and ts ≈ 3/(ζωn) for ±5 %. You pick whichever the specification is written against.
🙋
So to reduce overshoot I just raise ζ. Is there no downside to that?
🎓
Good question. Raising ζ does shrink the overshoot Mp = exp(−πζ/√(1−ζ²)) steadily. But it is not free: more damping makes the rise sluggish and the whole response slower. Overshoot and speed are a tug-of-war. In practice, where overshoot is dangerous — an aircraft control surface, a chemical-reactor temperature — engineers keep ζ around 0.7 or higher for safety and recover the speed they need through ωn instead.
Frequently Asked Questions
An enormous range of real systems — a motor position loop, a moving-coil instrument, a vehicle suspension, an RLC circuit, a building swaying in the wind — behave near their operating point like a standard second-order system. The response is governed by just two numbers, the natural frequency and the damping ratio, so once you can read this single picture you can read the behaviour of countless systems at a glance. That is why the second-order step response is the single most-studied figure in control engineering.
The natural frequency ωn sets the speed of the response and the damping ratio ζ sets its shape. Raising ωn compresses the time axis, so rise time and settling time both shorten. Lowering ζ increases the overshoot and lengthens the ringing before the response settles. At ζ = 1 the system is critically damped and reaches the target as fast as possible with no overshoot; for ζ > 1 it is overdamped, with no overshoot but a sluggish response. Designing a controller is largely placing ωn and ζ to meet the target specifications.
The settling time ts is the time until the response has entered, and stays within, a tolerance band (±2 % or ±5 %) around its final value. The common envelope estimate is ts = −ln(tol)/(ζωn). With the ±2 % criterion, tol = 0.02 and ts ≈ 4/(ζωn); with the ±5 % criterion, tol = 0.05 and ts ≈ 3/(ζωn). The wider ±5 % band gives a shorter settling time. Which one you use depends on the specification — how close the response must be before it counts as "finished".
Percent overshoot Mp depends on the damping ratio ζ alone: Mp = exp(−πζ/√(1−ζ²)). Increasing ζ monotonically reduces the overshoot, reaching exactly zero at ζ = 1. But more damping slows the rise time and makes the whole response sluggish — overshoot and speed pull against each other. Where overshoot is dangerous, as on an aircraft control surface or a chemical-reactor temperature, engineers keep ζ around 0.7 or higher and gain speed through ωn instead.
Real-World Applications
Servo and motion control: The feed axis of a machine tool, the joint of an industrial robot, the stage of a semiconductor lithography machine — every positioning mechanism is judged by its second-order step response. How fast it reaches the commanded position (rise time), how far it overshoots, and how quickly it falls inside the tolerance band (settling time) translate directly into cycle time and machining accuracy. Shortening the settling time is at the heart of higher productivity.
Mechanical vibration and suspension: A car's suspension is a textbook second-order system of spring, mass and damper. The motion of the body after hitting a bump is exactly a step response: with a small damping ratio ζ the ride is soft but keeps oscillating, while with a large ζ the motion stops quickly but the ride turns harsh. Tuning ζ to balance ride comfort against road holding is what chassis design is about.
Electrical circuits and instruments: The voltage response of a series RLC circuit and the needle movement of an analogue voltmeter or galvanometer can both be described as second-order. In an instrument, ζ is designed slightly below 1 (roughly 0.6–0.7) so the needle does not overshoot the scale and cause a misreading, settling quickly on the indicated value with only a small overshoot.
Process control and controller tuning: Temperature, pressure and flow PID loops can often be approximated, as a closed loop, by a second-order system. Tuning is fundamentally about adjusting the gains while watching the overshoot and settling time for a step change in setpoint, and a time-response map like this tool shows intuitively what happens when you turn the gains up or down.
Common Misconceptions and Pitfalls
A common misconception is to treat ts = −ln(tol)/(ζωn) as an exact value. This formula is a convenient estimate based on the approximation that the response amplitude decays inside an exponential envelope e−ζωnt; the real response grazes the tolerance band once before it truly stays inside, so the exact settling time has to be found by tracking the band crossings. With a small damping ratio, or a wide band such as ±5 %, the estimate and the exact value can differ by 10–20 %. It is fine for early design work, but confirm the final value by tracking the response waveform in simulation.
Next, the belief that a real system is genuinely a second-order system. This tool handles a standard second-order system with no zeros, but a real transfer function may carry a zero in the numerator or have third- or higher-order poles. A transfer-function zero can enlarge the overshoot or cause an inverse response (the output first moving the wrong way), and non-negligible higher-order poles push the response away from "nearly second-order". The second-order approximation is a first-order picture that looks only at the two dominant poles.
Finally, the misconception that more damping is always better. Raising ζ does reduce the overshoot, but it also slows the response. An overdamped system with ζ above 1 has zero overshoot, yet it dawdles toward the target and recovers sluggishly from disturbances. In practice, the "just-right damping" of ζ ≈ 0.7 is often chosen because it gives the fastest response that still keeps the overshoot within an acceptable range (often around 5–10 %). Chasing zero overshoot by pushing too far into overdamping costs you the very responsiveness you wanted.
How to Use
Set the natural frequency ωn (rad/s) using wnNum or wnRange slider; typical values range 1–100 rad/s for mechanical and electrical systems.
Adjust damping ratio ζ (zeta) via zetaNum or zetaRange; ζ < 1 produces underdamped oscillation, ζ = 1 is critical damping, ζ > 1 is overdamped.
Enter step input amplitude (ampNum or ampRange) in appropriate engineering units (volts, newtons, pressure).
Observe real-time plots of displacement/voltage response and damped frequency ωd, overshoot percentage, peak time, rise time, and settling time (2% band).
Worked Example
A precision servo motor with ωn = 25 rad/s and ζ = 0.65 receives a 5 V step input. The simulator calculates: damped frequency ωd = 18.4 rad/s, overshoot = 8.2%, peak time = 0.171 s, rise time = 0.089 s, and settling time = 0.24 s. The response exhibits moderate oscillation typical of position control loops in industrial robots, converging to 5 V steady-state within 240 ms.
Practical Notes
For hydraulic actuators with ζ = 0.4 and ωn = 12 rad/s, expect 25% overshoot; increase ζ to 0.7 to reduce overshoot to 4.6% if stability margin permits.
Critical damping (ζ = 1.0) eliminates oscillation but slows response; use for precision positioning where settling speed trumps rise time.
Verify ωd remains real (ωd = ωn√(1−ζ²)); the simulator flags complex roots for overdamped cases.
Compare settling time across ζ values: typical LC circuits at ζ = 0.5 settle in 8/ωn seconds; tune ζ to balance transient behavior against measurement bandwidth.