modeling6 - conditions Modeling 29 29 29 29 29 2 1 1 2 1 4...

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Modeling Q UESTION 6 For the system shown in Figure 6, set up the equations in matrix form that would allow you to determine the transfer functions ( 29 ( 29 1 X s F s , ( 29 ( 29 2 X s F s , and ( 29 ( 29 3 X s F s . 12 N/m 4 N/m 3 Ns/m 5 Ns/m 8 Ns/m 4 kg 5 kg 1 kg x 1 (t) x 2 (t) x 3 (t) f (t) Figure 6 The Free Body Diagrams are given in Figure 6.1 below. Applying Newton’s Second Law to the 4 kg mass and the constitutive laws for springs and dampers ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 2 1 2 1 4 4 5 x t f t f t x t x t x t = + = - + - (1) Applying Newton’s Second Law to the 5 kg mass and the constitutive laws for springs and dampers ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 3 2 4 2 2 1 3 2 5 8 5 12 x t f t f t f t f t x t x t x t x t x t f t = - + + = - - - + - + (2) Applying Newton’s Second Law to the 1 kg mass and the constitutive laws for springs and dampers ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 4 5 3 2 3 12 3 x t f t f t x t x t x t = - + = - - - (3) Taking the Laplace Transform of equations (1)–(3), respectively, and assuming zero initial
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Unformatted text preview: conditions Modeling ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1 2 1 4 4 5 s X s X s s X s X s = -+-(4) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 1 3 2 5 8 5 12 s X s sX s s X s X s X s X s F s = ---+-+ (5) ( 29 ( 29 ( 29 ( 29 ( 29 2 3 3 2 3 12 3 s X s X s X s sX s = ---(6) Putting equations (4)–(6) in matrix form ( 29 ( 29 ( 29 ( 29 2 1 2 2 2 3 4 5 4 5 5 5 13 12 12 1 12 3 12 s s s X s s s s X s F s s s X s + +- -+ +-= -+ + (7) 4 kg f 2 f 1 5 kg f 2 f 3 f 4 1 kg f 5 f 4 f Figure 6.1 2...
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modeling6 - conditions Modeling 29 29 29 29 29 2 1 1 2 1 4...

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