ME 375 FINAL EXAM SOLUTIONS Friday December 7, 004 Diviion Adam 0:30 / Yao :30 (circle one) Name Intruction () Thi i a cloed book eamination, but you are allowed three 8.5 crib heet. () You have two hour to work all i problem on the eam. (3) You mut how all of your work to receive any credit. (4) You mut write neatly and hould ue a logical format to olve the problem. You are encouraged to really think about the problem before you tart to olve them. Pleae write your name in the top right-hand corner of each page. Problem No. (0%) Problem No. (5%) Problem No. 3 (0%) Problem No. 4 (30%) Problem No. 5 (5%) Problem No. 6 (0%) TOTAL (***/00%)
December 7, 004 POBLEM (0%): Conider the feedback ytem below. (a) Show that the cloed loop characteritic equation i given by: a = 0 00 The cloed loop tranfer function between () and Y() i given by: Y( ) 00 = a ( ) 00 Therefore the characteritic equation i: a 00 = 0 which can be rearranged to the following: a = 0 00
December 7, 004 (b) Draw the root locu for thi cloed loop ytem a the parameter, a, varie from 0 to infinity and calculate and provide all of the root locu information required to do thi (open loop pole, open loop zero, break-in point, aymptote, etc.). From the characteritic equation, the open loop pole are at ±j0 rad/ and the open loop zero i at =0 rad/. Therefore, there are two root loci and they tart at the two pole and end at the zero and infinity. In order to end at the origin and infinity, the two root loci mut break-in to the real ai at the following location: a da d 00 uch that = 0 d d =, ( ) 00 0 = 00 0 = 0 = 00 =± 0 The root at -0 i the break-in point. At thi tate, the only other root locu rule needed i that there mut be root locu on the real ai to the left of an odd number of real pole/zero; therefore, loci mut be drawn on the real ai to the left of the origin. The final reult i hown below. 0 8 6 4 Imag Ai 0 - -4-6 -8-0 - -0-8 -6-4 - 0 eal Ai
December 7, 004 POBLEM (5%) Conider the control ytem hown below where Y d () i the Laplace tranform of the deired output, E() i the control error, Y() i the Laplace tranform of the actual output, U() i the Laplace tranform of the control input, D() i the diturbance, and G c () i the compenator. Anwer the following erie of quetion. Y d (), Deired Output - Error G c () U() D(), Diturbance - Y(), Output (a) You are aked to tabilize the teady tate repone of the cloed loop ytem to a tep input. Firt, ue a proportional control law, G c ()=K p, and plot the root locu a K p varie from 0 to on the grid provided below. Then comment on whether or not you can tabilize the ytem uing proportional control and, if o, give the tabilizing range of K p (5 pt). We plot the root of the cloed loop characteritic equation, K p = 0, by firt finding the open loop pole and zero and then drawing root locu to the left of an odd number of real pole and/or zero. Imag - - - eal We cannot tabilize the ytem uing proportional control becaue the root never pa into the left half plane. The bet we can do i to place the pole at the origin, which will provide an infinite teady tate repone to a tep input. -
December 7, 004 (b) Aume a control law having the form of G c ()=K/(-0.5). Plot the root locu a K varie from 0 to on the grid provided below. Then comment on whether or not you can tabilize the ytem uing uch a control law. If not, why. If ye, what i the range of K that tabilize the cloed-loop ytem. We plot the root of the cloed loop characteritic equation, K 0 0.5 =, by firt finding the open loop pole and zero and then drawing root locu to the left of an odd number of real pole and/or zero. Then we find the break away and break in point a follow:, ( )( ) K = = dk d 0.5.5 0.5 (.5) (.5 0.5) = 0.5 = = 0 =± 0.5 =± 0.707 The imaginary ai croing can be found by ubtituting = jω into the cloed-loop characteritic equation: ( jω0.5)( jω ) Kjω = 0 ω.5jω 0.5 jkω = 0 which lead to the following two equation to olve for K and ω : ω 0.5 = 0 ω =± 0.5 =± 0.707.5ω Kω = 0 K =.5 Imag - - - eal - Thu the cloed-loop ytem with the above control law will be table if K >.5.
December 7, 004 POBLEM 3 (0%) Derive the differential equation of motion in term of coordinate θ for the camera auto-focu mechanim hown below. The gear rotate about it center of ma and the two rack, M and M, roll without lipping on the gear a it rotate. The coordinate you hould ue are defined with repect to the deformed poition of the pring. M K Fied point of rotation J θ τ(t) M C K (a) Draw the free body diagram for the ytem. The free body diagram for thi ytem are given below. K M C F J θ F τ K M
December 7, 004 (b) Write down Newton and Euler law for thee free bodie. Newton and Euler equation for the free body diagram in part (a) are given below: ) ( K F M C K F M F F t J = = = τ θ (c) Derive the differential equation of motion for an input torque τ(t) and output motion θ(t). After eliminating all coordinate ecept for the rotational coordinate by ubtituting the econd two equation into the firt and applying the two rolling contraint, =θ and =θ, the following ingle equation of motion i obtained: ( ) ( ) ( ) ( ) ) ( ) ( t K K C M M J K M C K M t J τ θ θ θ τ θ = =
December 7, 004 POBLEM 4 (30%) You are deigning a temperature control ytem for the automobile paenger cabin hown in Figure. The differential equation of thi ytem i given by: where the thermal capacitance of the cabin i C = 00 J/ o F; the thermal reitance to conduction through to the outide i = 0.6 W/ o F; T c i the temperature in the cabin; T a i the outide temperature; and q i i the heat/ac (in Watt) upplied to the cabin. The block diagram of the control ytem you need to deign i alo hown below. Anwer the following quetion.
December 7, 004 (a) Show that the differential equation i a given above. If it aumed that the temperature of the outer urface of the car i the ame a the ambient temperature, then convection to the outide i zero and the following equation can be derived: (b) For a proportional controller, G c () = K P, find the cloed loop tranfer function between T r and T c and the cloed loop tranfer function between T a and T c. (c) Aume the input T r i a unit tep (T r = o F) and T a = 0 o F. If you want T c to reach % of it teady-tate value in 80 ec, what value of K P hould you chooe?
December 7, 004 (d) What i the teady-tate error between T r and T c for a unit tep input (T r = o F) for the cloed loop ytem with proportional control and the K P value you obtained in part (c)? (e) For a proportional integral controller, G c () = K α, find the cloed loop tranfer function between T r and T c. (f) Find the teady-tate error between T r and T c for a unit tep input (T r = o F) in the cloed loop ytem with proportional-integral control a in part (e).
December 7, 004 POBLEM 5 (5%) The hydraulic lift below i ued to ervice automobile by poitioning them vertically with the hydraulic upply preure, P, and oil flow rate, Q. Important parameter are hown in the figure including the gravitational contant g, vicou damping coefficient b, piton area A and hydraulic capacitance C. C Ue thi chematic to find the differential equation that relate the vertical poition of the vehicle y to the fluid flow Q. (a) Draw the free body diagram of the ram-vehicle ytem The free body diagram for thi ytem i given below y F f mg where F f = (P P)A r by
December 7, 004 (b) Write down Newton econd law for the ram-vehicle ytem F = F mg by = my y f my by = (P P )A mg r (c) Write down the element law for the piton that relate P to Q and y. P = QAy C [ ] (d) Derive the differential equation relating output y to input Q. mg m b Y() = P() P () A ( ) ( r ) P() = Q() AY() C [ ] mg m b Y() = Q() AY() P r () A C ( ) [ ] A A my by y = Q(t) AP r (t) mg δ(t) C C
December 7, 004 POBLEM 6 (0%) Ue the frequency repone data given below to anwer the following quetion: (a) If the ytem teady-tate repone i given by, y (t) = 0 co(0.6t-0.6 rad) cm, what i the input u(t)?
December 7, 004 (b) If you know that the tranfer function correponding to the frequency repone data i, find A G() = A/ (.6 4)