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Unformatted text preview: M Time domain analysis 49 3.6 Time domain response of second-order systems 3.6.1 Standard form Consider a second-order differential equation a d2 xo dxo ‡ cxo ˆ exi (t) ‡b 2 dt dt (3:40) Take Laplace transforms, zero initial conditions as2 Xo (s) ‡ bsXo (s) ‡ cXo (s) ˆ eXi (s) (as2 ‡ bs ‡ c)Xo (s) ˆ eXi (s) (3:41) The transfer function is G(s) ˆ Xo e ( s) ˆ 2 as ‡ bs ‡ c Xi To obtain the standard form, divide by c G ( s) ˆ a c s2 e c ‡ bs c ‡1 which is written as G(s) ˆ 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) ˆ 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 ˆ steady-state gain constant, !n ˆ undamped natural frequency (rad/s) and  ˆ 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 second-order 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) ˆ 0 If Xo (s) Tˆ 0, then as2 ‡ bs ‡ c ˆ 0 (3:44) //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC03.3D ± 50 ± [35±62/28] 9.8.2001 2:26PM 50 Advanced Control Engineering Table 3.4 Transient behaviour of a second-order system Discriminant Roots Transient response type b > 4ac s1 and s2 real and unequal (Àve) Overdamped Transient Response b2 ˆ 4ac s1 and s2 real and equal (Àve) Critically Damped Transient Response b2 < 4ac s1 and s2 complex conjugate of the form: s1 , s2 ˆ 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 ˆ 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.

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