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Chap2+Sec3A,4A+-+Second+Order+Systems__Sakai_Derivations_for_Students_ONLY_

Chap2+Sec3A,4A+-+Second+Order+Systems__Sakai_Derivations_for_Students_ONLY_

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Chapter 2 Lecture Notes for Students *************************** Section 1 ************************************ Notes on Conversion of Two First Order Equations to a Second Order Model A linear second order system is sometimes represented as a system of two first order differential equations like those in Eqs and below. ( ) dx ax by f t dt = + + ( ) dy cx dy g t dt = + + Suppose a single equation relating the dependent variable ( ) x x t = and the inputs ( ) f f t = and ( ) g g t = is required. We can use the Laplace operator, s, to convert the two differential equations (Eqs 3.22, 3.23) into algebraic equations. F bY aX sX + + = (3.23a) G dY cX sY + + = (3.23b) Solving for Y in Eq (3.23b) gives G cX Y d s + = - ) ( (3.23c) or d s G cX Y - + = (3.23d) Substitute Eq (3.23d) into Eq (3.23a) gives F d s G cX b aX sX + - + + = (3.23e) Multiplying both sides of the equation gives F d s G cX b X d s a X d s s ) ( ) ( ) ( ) ( - + + + - = - (3.23f) or 2.3 - 1

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dF Fs bG bcX adX asX dsX X s - + + + - = - 2 (3.23g) Collecting like terms gives dF bG sF X bc ad sX d a X s - + = - + + - ) ( ) ( 2 (3.23h) Convert back to time domain give f d g b dt df x bc ad dt dx d a dt x d - + = - + + - ) ( ) ( 2 2 (3.23i) *** ************************** End Section 1 ****************************** *** *************************** Section 2 ********************************* Example 3.2 The well-mixed tanks shown in Figure 3.4 contain uniform salt concentrations of 1 1 ( ) c c t = and 2 2 ( ), c c t = respectively. Concentration of salt in the input to the first tank
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Chap2+Sec3A,4A+-+Second+Order+Systems__Sakai_Derivations_for_Students_ONLY_

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