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Unformatted text preview: M Time domain analysis 49 3.6 Time domain response of secondorder systems
3.6.1 Standard form Consider a secondorder differential equation
a d2 xo
dxo
cxo exi (t)
b
2
dt
dt (3:40) Take Laplace transforms, zero initial conditions
as2 Xo (s) bsXo (s) cXo (s) eXi (s)
(as2 bs c)Xo (s) eXi (s) (3:41) The transfer function is
G(s) Xo
e
( s) 2
as bs c
Xi To obtain the standard form, divide by c
G ( s) a
c s2 e
c
bs
c 1 which is written as
G(s) K
12
s
!2
n 2
! s 1
n (3:42) This can also be normalized to make the s2 coefficient unity, i.e.
G(s) K !2
n
s2 2!n s !2
n (3:43) Equations (3.42) and (3.43) are the standard forms of transfer functions for a secondorder system, where K steadystate gain constant, !n undamped natural
frequency (rad/s) and damping ratio. The meaning of the parameters !n and
are explained in sections 3.6.4 and 3.6.3. 3.6.2 Roots of the characteristic equation and their
relationship to damping in secondorder systems As discussed in Section 3.1, the transient response of a system is independent of the
input. Thus for transient response analysis, the system input can be considered to be
zero, and equation (3.41) can be written as
(as2 bs c)Xo (s) 0
If Xo (s) T 0, then
as2 bs c 0 (3:44) //SYS21/D:/B&H3B2/ACE/REVISES(080801)/ACEC03.3D ± 50 ± [35±62/28] 9.8.2001 2:26PM 50 Advanced Control Engineering
Table 3.4 Transient behaviour of a secondorder system
Discriminant Roots Transient response type b > 4ac s1 and s2 real
and unequal
(Àve) Overdamped
Transient
Response b2 4ac s1 and s2 real
and equal
(Àve) Critically
Damped Transient
Response b2 < 4ac s1 and s2 complex
conjugate of the
form: s1 , s2 À Æ j! Underdamped
Transient
Response 2 This polynomial in s is called the Characteristic Equation and its roots will determine
the system transient response. Their values are
s1 , s2 Àb Æ p
b2 À 4ac
2a (3:45) The term (b2 À 4ac), called the discriminant, may be positive, z...
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This note was uploaded on 01/18/2014 for the course MECH 107 taught by Professor Sali mon during the Winter '12 term at Bingham University.
 Winter '12
 Sali mon

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