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timeresponse8

timeresponse8 - &(3 Time Response Taking the Laplace...

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Time Response Q UESTION 8 For the system shown in Figure 8.1, find the gear ratio 1 2 r r such that the settling time for a constant input torque τ ( t ) is 20 s. 4 kg · m 2 7 N · m/rad τ (t) 2 N · m/(rad/s) r 2 r 1 3 kg · m 2 Figure 8.1 Denote the angular position of the mass on the left as θ 1 and the angular position of the mass on the right as θ 2 . The torque and angular position relationships for the gear system is 1 1 2 2 2 1 r r τ θ τ θ = = . Summing the torques acting on the mass on the left ( 29 ( 29 ( 29 1 1 3 t t t θ τ τ = - && (1) Summing the torques acting on the mass on the right ( 29 ( 29 ( 29 ( 29 2 2 2 2 4 2 7 t t t t θ θ θ τ = - - + && & (2) Utilizing the gear relationships and combining equations (1) and (2) ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 1 1 4 3 2 7 r r t t t t r r θ θ θ τ +

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Unformatted text preview: & (3) Time Response Taking the Laplace Transform of equation (3), assuming zero initial conditions, and rearranging, the transfer function ( 29 ( 29 2 s s Θ Τ is ( 29 ( 29 2 2 1 2 2 2 1 4 3 2 7 r s r s r s s r Θ = Τ + + + ÷ (4) The settling time is 4 s n T ζϖ = and, from equation (4), 2 2 1 2 2 4 3 n r r = + ÷ . Therefore 2 2 1 20 4 4 3 r r = + ÷ and 1 2 3 r r = 2...
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timeresponse8 - &(3 Time Response Taking the Laplace...

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