---
title: "Servo differential equation: ζ, frequency and overshoot"
subject: "Control Systems"
chapter: 3
key: "semester_3/control_systems:servo-de-64z"
source: https://www.bctengineer.com/q/semester_3/control_systems/servo-de-64z
site: BCT Engineering
---

# Servo differential equation: ζ, frequency and overshoot

Control Systems · Chapter 3: System Transfer Function and Responses

- Asked in: 2073 Shrawan, 4 marks
- Video: [Problem to calculate damping ratio,natural frequency and output response | Control system |](https://www.youtube.com/watch?v=xqYiYB1yYRk) — Mathspedia
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/servo-de-64z
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3

## Question

A closed loop servo is represented by the differential equation d²y/dt² + 8 dy/dt = 64z, where 'y' is the displacement of the output shaft and 'u' is the displacement of the input shaft and z = u − y. Determine frequency of sustained oscillation, damping ratio and percentage maximum overshoot for unit step input.

## Short answer

The transfer function is 64/(s²+8s+64), yielding ωn=8 rad/s, ζ=0.5. Sustained oscillation frequency is ωn=8 rad/s (damped ωd=6.93 rad/s). Maximum percent overshoot = 16.3%.

## Answer

Take the Laplace transform with zero initial conditions and compare with the standard second order form $\omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2)$.

#### Transfer function
$$
\frac{d^2y}{dt^2} + 8\frac{dy}{dt} = 64(u - y)
\;\Rightarrow\;
(s^2 + 8s + 64)Y(s) = 64U(s)
$$
$$
\frac{Y(s)}{U(s)} = \frac{64}{s^2 + 8s + 64}
$$

#### Parameters
$$
\begin{aligned}
\omega_n^2 &= 64 \;\Rightarrow\; \omega_n = 8\ \text{rad/s} \\
2\zeta\omega_n &= 8 \;\Rightarrow\; \zeta = \frac{8}{16} = 0.5
\end{aligned}
$$

#### Frequency of sustained oscillation
Sustained (non-decaying) oscillation would occur with zero damping, at the undamped natural frequency:
$$
\omega_n = 8\ \text{rad/s} \quad (f_n = \omega_n/2\pi = 1.27\ \text{Hz})
$$
With the actual damping ($\zeta = 0.5$), the step response oscillates and decays at the damped frequency:
$$
\omega_d = \omega_n\sqrt{1 - \zeta^2} = 8\sqrt{0.75} = 6.928\ \text{rad/s}
$$

#### Maximum overshoot
$$
M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}} = e^{-\pi(0.5)/0.866} = e^{-1.814} = 0.1630 = 16.30\%
$$

**Answer:** frequency of sustained oscillation $\omega_n = 8$ rad/s (damped frequency $\omega_d = 6.93$ rad/s), $\zeta = 0.5$, $M_p = 16.3\%$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With z = u - y, what is Y/U?
   A. 64/(s^2 + 8s + 64)
   B. 64/(s^2 + 8s)
   C. 8/(s^2 + 64)
   D. 64/(s^2 + 64s + 8)
   Answer: A. (s^2 + 8s)Y = 64(U - Y) gives (s^2 + 8s + 64)Y = 64U. (Key point: Transfer function)

2. What is the damping ratio?
   A. 1
   B. 0.5
   C. 0.125
   D. 8
   Answer: B. wn = 8 and 2 zeta wn = 8 gives zeta = 0.5. (Key point: zeta)

3. What frequency does the step response oscillate at?
   A. wd = 8 sqrt(0.75) = 6.93 rad/s
   B. wn = 8 rad/s
   C. wd = 8 x 0.5 = 4 rad/s
   D. 64 rad/s
   Answer: A. A damped system oscillates at the damped frequency wd. (Key point: Frequency of oscillation)

4. What is Mp?
   A. 16.3%
   B. 50%
   C. 0.5%
   D. 32.6%
   Answer: A. exp(-pi x 0.5/0.866) = exp(-1.814) = 0.163. (Key point: Max overshoot)

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