modeling14 - conditions ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1...

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Modeling Q UESTION 14 For the system shown in Figure 14, 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 . 1 N/m 11 N/m 7 Ns/m 1 Ns/m 7 kg 2 kg 8 kg x 1 (t) x 2 (t) x 3 (t) f (t) 2 Ns/m 2 Ns/m 2 Ns/m Figure 14 The Free Body Diagrams are given in Figure 14.1 below. Applying Newton’s Second Law to the 7 kg mass and the constitutive laws for springs and dampers ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 2 3 1 1 2 1 7 11 2 1 x t f t f t f t x t x t x t x t = - + = - - + - (1) Applying Newton’s Second Law to the 2 kg mass and the constitutive laws for springs and dampers ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 3 4 5 2 1 2 3 2 2 1 2 1 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 8 kg mass and the constitutive laws for springs and dampers ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 5 6 7 3 2 3 3 8 1 2 7 x t f t f t f t x t x t x t x t = - - + = - - - - (3)
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Modeling Taking the Laplace Transform of equations (1)–(3), respectively, and assuming zero initial
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Unformatted text preview: conditions ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1 1 2 1 7 11 2 s X s X s sX s sX s sX s = --+-(4) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 1 2 3 2 2 2 s X s sX s sX s sX s X s X s F s = ---+-+ (5) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 3 3 2 3 3 8 2 7 s X s X s X s sX s sX s = ----(6) Putting equations (4)(6) in matrix form ( 29 ( 29 ( 29 ( 29 2 1 2 2 2 3 7 3 11 2 3 1 1 1 1 8 9 1 s s s X s s s s X s F s s s X s + +- -+ +-= -+ + (7) 7 kg f 3 f 1 2 kg f 3 f 5 8 kg f 7 f 5 f f 4 f 2 f 6 Figure 14.1 2...
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modeling14 - conditions ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1...

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