Page 374 - The Mechatronics Handbook

0066_Frame_C18 Page 10 Wednesday, January 9, 2002 5:14 PM

FIGURE 18.14 Spring–mass–damper system.

FIGURE 18.15 Second-order system—step response.
The response of an underdamped 0 ≤ ζ < 1) second-order system to a unit step input can be deter-
(
mined as:
ζ
2 
yt() = 1 e – ζω t  cos ω n 1 ζ t + ------------------sin ω n 1 ζ t 
2
–
–
–
n
1 ζ– 2
This second-order system step response is often characterized by a set of time response parameters
illustrated in Fig. 18.15.
These time response parameters are functions of the damping ratio ζ and the natural frequency ω n :
• peak time, T P : the time required to reach the first (or maximum) peak
π
T P = -------------------------
2
ω n 1 ζ–
• percent overshoot, %OS: amount the response exceeds or overshoots the steady-state value

2
–
%OS = 100e ( – ζπ/1 ζ )
• settling time, T S : the time when the system response remains within ±2% of the steady-state value

4
T S = ---------
ζω n
• rise time, T R : time required for the response to go from 10% to 90% of the steady-state value.
Figure 18.16 shows the nondimensional rise time (ω n T R ) as a function of damping ratio, z.
A frequently used approximation relating these two parameters is
ω n T R ≈ 2.16ζ + 0.6 0.3 ≤ ζ 0.8

Figures 18.17 and 18.18 show the unit step response of a second-order system as a function of damping
ratio ζ.

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