ME 375 FINAL EXAM Wedneday, May 6, 9 Diviion Meckl :3 / Adam :3 (circle one) Name_ Intruction () Thi i a cloed book examination, but you are allowed three ingle-ided 8.5 crib heet. A calculator i NOT allowed. () You have two hour to work all five problem on the exam. (3) Ue the olution procedure we have dicued: what are you given, what are you aked to find, what are your aumption, what i your olution, doe your olution make ene. You mut how all of your work to receive any credit. (4) Circle or box-in your anwer. (5) 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. (6) A table of Laplace tranform pair and propertie of Laplace tranform i attached at the end of thi exam et. Problem No. () Problem No. (5) Problem No. 3 (5) Problem No. 4 (3) Problem No. 5 () TOTAL (***/)
PROBLEM : (%) For the following tranfer function: (i) determine the pole (ii) compute time contant, natural frequency and/or damping ratio (iii) indicate which time repone (,, 3, or 4) below repreent the unit tep repone of each tranfer function. 5 3 (a) (b) + 6+ 5 + 3 (c) + + (d) 5 + 3+ 5 () () Amplitude.9.8.7.6.5.4.3.. Step Repone..4.6.8..4.6.8 Time (ec) Amplitude.4..8.6.4. Step Repone.5.5.5 3 3.5 4 Time (ec) (3) (4) Amplitude.4..8.6.4. Step Repone..4.6.8..4.6 Time (ec) Amplitude.9.8.7.6.5.4.3.. Step Repone 3 4 5 6 Time (ec)
3 PROBLEM : (5%) The frequency repone for a flexible mechanical ytem i given in the Bode plot below: Magnitude (db) Phae (deg) - - -3-4 -5-45 -9-35 Bode Diagram -8 Frequency (rad/ec) (a) Determine the natural frequency, damping ratio, and tatic gain for thi ytem. (b) Generate an expreion for the tranfer function G() for thi ytem. (c) Find the magnitude and phae of G(jω) at ω = and 65 rad/ec. (d) Find the teady-tate output y (t) of the ytem to an input of ut ( ) = 3 + in( t) + in(65 t).
4 PROBLEM : (cont)
5 PROBLEM 3: (5%) A hock aborber (with attached ma M) can be repreented a in the chematic diagram below: x A M p r p L q R p r Aume there i no leakage of fluid between the piton and cylinder, and alo no friction damping between piton and cylinder. Alo aume that the hydraulic fluid i incompreible. (a) Derive a differential equation that decribe the free repone of the piton diplacement x. (b) What i the equivalent damping contant b for thi hock aborber in term of the parameter given in the chematic above?
6 PROBLEM 3: (cont)
7 PROBLEM 4: (3%) Conider the feedback control ytem repreented below in block diagram form. The reference input R(), ytem output Y(), and diturbance D() are denoted in the figure along with the error E() and control effort F(). You will deign the control law G c () to achieve certain performance criteria. Anwer the following quetion (aume D()= in all part except part (i)): (a) Show that the tranfer function relating the reference R() to the output Y() i given by: ( ) Y R () = G c () +.+ G c () (b) Auming a proportional control law, G c () = K p, chooe K p to limit the teady tate error between the reference and the output to 5% for a unit tep reference. (c) For the proportional control law, G c () = K p, chooe K p o that 63% of the teady tate repone for a unit tep input i reached in exactly econd. (d) Explain why you cannot achieve both criteria lited in (b) and (c) uing a proportional control law. What control law would you ue (and why) to achieve both of thee criteria? (e) For an integral control law, G c () = K I /, find the teady tate error between the reference and the output for a unit tep reference. (f) Chooe K I in the integral control law from part (e) to limit the peak overhoot to %. (g) Auming a proportional-integral control law, G c () = K P + K I /, chooe K I and K P to imultaneouly limit the peak overhoot to % and ettling time (for repone to be within % of the final value) to econd. (h) Show that the tranfer function relating the diturbance D() to the error E() i given by, ( ) E D () = +.+ G c () (i) For a unit tep diturbance, calculate the total teady tate error in the output Y() auming you ue the proportional control deign from part (b).
8 PROBLEM 4: (cont)
9 PROBLEM 5: (%) Conider the feedback control ytem from PROBLEM 4 to anwer the following quetion about root locu. (a) For a proportional control law, G c () = K p - draw the root locu for the cloed loop characteritic equation a K p increae; - calculate where the root locu tart for K p = and indicate thi on the diagram with an X ; - decribe what i happening to the tranient repone of the feedback control ytem a K p increae.
(b) For an integral control law, G c () = K I / - draw the root locu for the cloed loop characteritic equation a K I increae; - calculate where the root locu tart for K I = and indicate thi on the diagram with an X ; - decribe what i happening to the tranient repone of the feedback control ytem a K I increae.
Laplace Tranform Pair f(t) F() Comment. δ(t) Unit impule. for t> 3. t for t> Unit tep Unit ramp 4. e -at + a 5. in ωt ω + ω Exponential Sine 6. co ωt + ω Coine 7. f(t) F() Function 8. df() t dt F() f( ) Firt Derivative 9. n d f() t n dt n n n F () f( ). t n n n! + df( ) dt n th Derivative at. e f() t F( + a). te -at + ( a) n! 3. t n e -at ( + a) n + 4. e -at in ωt 5. e -at co ωt ω ( + a) + ω ( + a) ( + a) + ω 6. Initial Value Theorem: f( ) = lim F( ) + 7. Final Value Theorem: lim f() t = lim F() t