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ME3514_Partial_Fraction_Expansion

# ME3514_Partial_Fraction_Expansion - ME3514 Partial Fraction...

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ME3514 Partial Fraction Expansion Solving ODE using Time Domain Laplace Domain Step # 1: Take Differential equation with initial conditions • We want to solve for x(t) Algebraic equation Step # 2: Solve for the of the response X ( s ) Many times we can not find the -1 [ X(s) ] in the aplace table Step # 3: Partial fraction expansion to simplified X ( s ) Laplace table Then, we need to use partial fraction expansion tep # 4: Solve for sing - esponse (t) M. Remillieux Step # 4: Solve for x ( t ) using 1 Response x(t) 1

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ME3514 Partial Fraction Expansion Example 2 () 21 0 s Fs s s Find inverse Laplace transform of: 2 2 22 1 0.3 1 10 10 n n   We can use the table! 10 . 3 3 . 1 2 1 ) s in311 03 t s e t        2 1 2 in3. 11 0.3 . 3 1 s in3 0 3 t e t  M. Remillieux 2 .3 . 3 tan 3.17 0.3  2
ME3514 2 0 xx  Partial Fraction Expansion Example 32 xxx  EOM I.C.   01 x   00 x and    2 xs X ss    0 X sx We know that and       2 03 0 s X s X s X s    22 2 33 2 Xs s  Step # 3: Solve for x ( t ) using -1 [ X ( s )]= x ( t ) 23 n  3 1 2  Can we use lines 22 and 23 of the table? NO 2 2 n 2 n  M. Remillieux 3 So, how do we find x(t) or -1 ? Using PARTIAL FRACTION EXPANSION

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ME3514 Partial Fraction Expansion What is it used for? 1. We have a complicated ratio of polynomials he inverse Laplace transform of this complicated ratio is not in the table 2. The inverse Laplace transform of this complicated ratio is not in the table Need to break this ratio into forms that are in the table of Laplace transforms 1 () Po lynomialof order m ) Pol nomialof order n m om bb s b s Bs Xs   1 ( ) Polynomialof order n n on A sa a s a s Partial Fraction Expansion 2 ) ... , terms X s F s F s F sn  M. Remillieux 4       12 ( ) e s n sss s n With F(s) , very simple functions of s , 4
ME3514 Partial Fraction Expansion ) m b s b s s Possible Cases 1 1 () om n on bb Bs Xs A sa a s a s  istinct Poles ultiple Poles 2 cases: Distinct Poles Multiple Poles n m By-hand/MATLAB By-hand/MATLAB This lecture m>n MATLAB M. Remillieux 5

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ME3514 Partial Fraction Expansion ) m b s b s s Before you use partial fractions, you want to rewrite the polynomial such that a n =1 . 1 1 () om n on bb Bs Xs A sa a s a s   r 1 m o m ab b s bs or 1 1 1 no n n m X s aa s a s a b s b s 1 1 1 n o n nn a a a s s M. Remillieux 6
ME3514 ) b b Partial Fraction Expansion Poles and Zeros 1 1 () m om n o bb s b s B s Xs A sa a s s   oles: • Values of s that leads to or   Poles: You need to know the poles to do the partial fraction expansion • Roots of A ( s ):      2 01 2 1 2 ( ) 0 ... ...

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ME3514_Partial_Fraction_Expansion - ME3514 Partial Fraction...

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