System Dynamics (4th Edition) by Katsuhiko Ogata

By Katsuhiko Ogata

</B> this article offers the elemental thought and perform of process dynamics. It introduces the modeling of dynamic platforms and reaction research of those structures, with an advent to the research and layout of keep an eye on structures. <B>KEY TOPICS particular bankruptcy themes comprise The Laplace remodel, mechanical structures, transfer-function method of modeling dynamic platforms, state-space method of modeling dynamic structures, electric structures and electro-mechanical structures, fluid platforms and thermal platforms, time area analyses of dynamic structures, frequency area analyses of dynamic platforms, time area analyses of keep an eye on structures, and frequency area analyses and layout of keep an eye on platforms. <B></B> For mechanical and aerospace engineers.

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Equation (2-3) is called the differentiation theorem. f(t), the values of /(0+) and 1(0-) may be the same or different, as illustrated in Figure 2-8. f(t) has a discontinuity at t = 0, because, in such a case, dJtt)/dt will Sec. 2-3 Laplace Transformation 25 /(/) /(/) /(0 +) Flgure 2-8 Step function and sine function indicating initial values at 1 = 0- and 1 = 0+. involve an impulse function at t modified to = O. If 1(0+) ::F 1(0-), Equation (2-3) must be ~+[:,t(t)] = sF(s) ~-[:,t(t) ] = [(0+) sF(s) - /(0-) To prove the differentiation theorem, we proceed as follows: Integrating the Laplace integral by parts gives roo -st 10 I(t)e- dt = I(t) ~s st I (:JO[ d 00 0 - 10 ] dtl(t) ~s dt -sf Hence, [d 1 -/(t) F(s) = -1(0) + -;£ s s dt ] It follows that ~[:t[(t) ] = sF(s) - [(0) Similarly, for the second derivative of I(t), we obtain the relationship ~[;:[(t) ] = h(s) - s/(O) - j(O) where 1(0) is the value of dl(t)ldt evaluated at t = O.

FW) • (n-l) 1 where f(O), f(O), ... , f(O) represent the values of f(t), dl(t)ldt, ... , d n- f(t)1 dt n- 1, respectively, evaluated at t = O. If the distinction between :£+ and ;£_ is necessary, we substitute t = 0+ or t = 0- into I(t), df(t)ldt, ... , dn-1f(t)ldtn-1, depending on whether we take ;£+ or ;E_. Note that, for Laplace transforms of derivatives of f(t) to exist, dnf(t)ldrn (n = 1, 2, 3, ... ) must be Laplace transformable. Note also that, if all the initial values of f(t) and its derivatives are equal to zero, then the Laplace transform of the nth derivative off(t) is given by snF(s).

Thus, l OO f (T)l(T)e- S(T+a> dT °O -a = f f (7) 1(T)e-S(T+a) dT 10 = ["'j(T)e-STe-as dT 1 00 = e-as j( T)e-ST dT = e-asp(s) where P(s) = ~(f(t)] = 1°Oj (tv" dl Hence, ;Eff(t - a)l(t - a)] = e-aSF(s) a~O This last equation states that the translation of the time functionf(t)l(t) by a (where a ~ 0) corresponds to the multiplication of the transform F(s) bye-as. Pulse function. Consider the pulse function shown in Figure 2-6, namely, A f(t) = to =0 for 0 < t < to for t < 0, to < t where A and to are constants.

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