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Test2-S08

Course: ME 3015, Fall 2008
School: Georgia Tech
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Word Count: 982

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Print) Name:_____Solutions_____________________________(Please Georgia Institute of Technology Woodruff School of Mechanical Engineering ME3015 System Dynamics and Control Second mid-term: April 2, 2008 You need to show clearly all the work. Only answers or partial steps are not acceptable. Transfer functions must be written in standard form: G(s) =N(s)/D(s) where N(s) and D(s) are polynomials of...

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Print) Name:_____Solutions_____________________________(Please Georgia Institute of Technology Woodruff School of Mechanical Engineering ME3015 System Dynamics and Control Second mid-term: April 2, 2008 You need to show clearly all the work. Only answers or partial steps are not acceptable. Transfer functions must be written in standard form: G(s) =N(s)/D(s) where N(s) and D(s) are polynomials of "s" in descending order. All problems are equally weighed. Answer three of the four problems. Closed book/ closed notes, one-page formula sheet (with no examples) must be turned in with the test. Check problem to be graded: Problem 1:_____ Problem 2:_____ Problem 3:_____ Problem 4:_____. [1] (33.33%) Figure 1 shows a mechanical system, where x is measured from equilibrium position. To determine the viscous friction coefficient b and the spring stiffness k, a sinusoidal force p(t ) = P sin t is applied to the massless platform with P=1N. The steady state amplitude of the displacement x is 0.05m when =2rad/s, and is 0.02m when =10rad/s. Determine the values of b and k. G(s) = X ( s) 1 = P ( s ) bs + k G ( j ) = 1 b j + k 1 (b ) 2 + k 2 G ( j ) = Figure 1 With =2rad/s, G ( j 2) P = 0.05 1 (2b) 2 + k 2 = 0.05 (1) when =10rad/s, G ( j10) P = 0.02 1 (10b) 2 + k 2 = 0.02 (2) Solve (1) and (2) simultaneously for b and k: b = 4.68 Ns/m and k = 17.67 N/m [2] (33.33%) Figure 1a shows a milling machine supported by an isolation table (with spring stiffness k1 and damping coefficient b1) and has a combined machine/table mass M. The isolation table is designed to reduce vibration effects of the milling machine that exerts a force of p (t ) = P cos t as shown in diagram, where x is measured from equilibrium position when the milling machine is turned off. (a) Find the sinusoidal transfer function G ( j ) = X ( j ) / P( j ) . Obtain the steady state expression of x(t) in terms of k1, b1, ,P , M and . What is the value of such that the isolation table vibrates at maximum amplitude? X ( s ) / P ( s ) = 1/ ( Ms 2 + b1s + k1 ) G ( j ) = X ( j ) / P( j ) = 1/((k1 - M 2 ) + b1 j ) G ( j ) = 1/ (k1 - M 2 ) 2 + (b1 ) 2 G ( j ) = - tan -1 (b1 /(k1 - M 2 )) x(t ) = G ( j ) P cos (t + G ( j ) ) Figure 1a Find that maximizes G ( j ) = 1/ (k1 - M 2 )2 + (b1 )2 d G ( ) d =0 = k1 b - M 2M 2 2 1 Figure 1b (b) The force transmitted ft(t) to the base of the isolation table is the sum of the spring and damper forces. F ( ) 1 Show that in the absence of the damper, t = where r = . 2 P ( ) 1 - r n ft (t ) = kx Ft ( s ) = kX ( s ) Ft ( s ) k1 X ( s ) k1 = = P( s) P( s) Ms 2 + k1 Ft ( ) k1 1 1 = = = 2 2 P ( ) k1 - M 1 - M / k1 1 - r 2 (c) In order to reduce the vibration amplitude of the isolation table, an addition mass m1 is bolted to the table as shown in Figure 1b. If k1=20N/m2, b1=1N-s/m, m1=20kg, M=100kg, P=25N and is1200rpm, find x(t). G '( j ) = 1/ (k1 - (( M + m1 ) 2 ) 2 + (b1 ) 2 = 5.28 10-7 G '( j ) = - tan -1 (b1 /(k1 - ( M + m1 ) 2 )) = -6.63 10-5 x(t ) = G '( j ) P cos (t + G '( j ) ) = 1.32 10-5 cos ( 40 t - 6.63 10-5 ) [3] (33.33%) Consider the system shown below, where R(s) is a reference input; D(s) is step K disturbance; and Gc ( s ) = as 2 + bs + 1 . Using the following steps, you are asked to find the values of s K, a and b such that the response to any step disturbance will be damped out quickly (in 2 seconds in terms the 2% settling time). D(s) R(s)=0 + E(s) _ Gc(s) + + 1 s + 3.6s + 9 2 C(s) (a) Show that the transfer function C(s)/D(s) can be written in the following form: C (s) s s = = 3 2 2 2 2 2 D( s ) ( s + p)( s + 2n s + n ) s + ( p + 2n ) s + (2n p + n ) s + pn Express p, and n in terms of K, a and b C (s) G(s) = D( s ) 1 + G ( s )Gc ( s ) 1 1 Kas 2 + Kbs + K 1+ 2 s 2 + 3.6 s + 9 s + 3.6 s + 9 s s = 3 2 s + 3.6 s + 9s + Kas 2 + Kbs + K s = 3 2 s + (3.6 + Ka) s + (9 + Kb) s + K = where G ( s) = Hence, pn 2 = K 2n p + n 2 = 9 + Kb p + 2n = 3.6 + Ka 1 . s + 3.6 s + 9 2 (b) In order to reduce the order of the system, we choose p=5n and the pair of dominant closed loop poles to be a damping ratio of 0.5. Find the values of K, a and b that will meet the settling time specification. ts = 4 / n n = 4 a = ( p + 2n - 3.6) / K = 0.065 2 K = pn 2 = 5n 3 = 160 b = (2n p + n - 9) / K = 0.2938 (c) With the design in (b), n2 C (s) s . Estimate and sketch neatly the response to the 2 2 2 D( s ) pn s + 2n s + n unit-step disturbance input. c(t ) 1.4 1.2 1 Amplitude n2 1 d -1 1 dcs (t ) L 2 = ...

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