ME 375 FINAL EXAM Friday, May 6, 2005
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1 ME 375 FINAL EXAM Friay, May 6, 005 Divisio: Kig 11:30 / Cuigham :30 (circle oe) Name: Istructios (1) This is a close book examiatio, but you are allowe three crib sheets. () You have two hours to work all the problems o the exam. (3) Use the solutio proceure we have iscusse: what are you give, what are you aske to fi, what are your assumptios, what is your solutio, oes your solutio make sese. You must show all of your work to receive ay creit. (4) You must write eatly a shoul use a logical format to solve the problems. You are ecourage to really thik about the problems before you start to solve them. Please write your ame i the top right-ha corer of each page. (5) A table of Laplace trasform pairs a properties of Laplace trasforms is attache at the e of this exam set. Sectio A Sectio B Problems Score Problems Score 1. (10) 1. (5). (6). (0) 3. (6) 3. (30) 4. (6) 4. (5) 5. (8) 6. (15) 7. (15) 8. (6) 9. (6) 10. (8) 11. (6) 1. (8) Total (A) /100 Total (B) /100 Total (A+B) /00
2 December 7, 005 Name A.1 (10%) Usig Laplace trasforms, fi the total respose ( t) of motio give by: whe the iput () t y + 8 y + 16y = 3u 0 = y 0 =. u is a uit step, y ( ) 0, a ( ) 0 y of a system with the equatio
3 December 7, 005 Name 3 A. (6%) A two-egree-of-freeom mechaical system is escribe by the followig two ifferetial equatios: x1+ 3x 1 x + 5x1 5x = 0 4 x x + 3x 5x + 5x = F 1 1 Write the equatios i Laplace omai a explai how to etermie the trasfer fuctio X 1 (s)/f(s) A.3 (6%) The motio of a system is govere by y + 4 y + 9y = 7u, where y is the output of the system a u is the iput of the system. Give the trasfer fuctio from the iput to the output. Next, fi the poles of the trasfer fuctio a etermie the stability of this system, givig a brief explaatio.
4 December 7, 005 Name 4 A.4 (6%) A seco orer system is escribe by y + y + 5y = 0u( t), where y is the output of the system, a u is the iput of the system. Fi the atural frequecy, ampig ratio, a static gai of this system. A.5 (8%) The steay-state torque-spee relatioship for a c motor is give by the followig formula: KT RAB + KT K b TL = ei ωss RA RA where e i is the omial iput voltage. Give the torque costat, K T, the back emf costat, K b, a the spee loa map below, calculate the armature resistace, R, a the bearig ampig, B. A K T K b N m = amp V sec = ra T L N m e i = 6 V ra 37 sec ω ss
5 December 7, 005 Name 5 A.6 (15%) Draw a iagram of a aalogous mass-sprig-amper mechaical system for the electrical system show i the figure below. Assume force is the effort variable a velocity is the flow variable.
6 December 7, 005 Name 6 A.7 (15%) As a egieer for a appliace maufacturer, you ee to moel a turkey roastig i a covectio ove. The covective air flow i the ove is kept at the set temperature, T ove, the covective heat trasfer coefficiet is hove bir, a the turkey has a heat trasfer area, A bir. Assume that as the turkey cooks its temperature is uiform throughout, T bir. hove bir, Abir T bir T ove (a) Derive the ifferetial equatio that escribes the evolutio of the temperature of the roastig turkey, T bir, where the iput is T ove. (Assume the turkey has heat capacity c p a mass m bir.) (b) I terms of the moel parameters you efie i (a), how log oes it take for the turkey temperature, T bir, to rise to withi 1% of its fial temperature?
7 December 7, 005 Name 7 A.8 (6%) Use block iagram methos to fi the correspoig G(s) for a uity feeback system give the o-uity feeback system show i the figure. A.9 (6%) The performace requiremets of a automotive cruise cotrol are less tha percet overshoot a lest tha secos two-percet settlig time. Sketch the regio that satisfies these performace criteria o the complex plae below Imagiary Axis Real Axis
8 December 7, 005 Name 8 A.10 (8%) Fi the steay-state respose y ss (t) of a system with trasfer fuctio 0 G ( s) = whe the iput is u ( t) = si(5t). Make sure you compute all require terms. s + s + 5 A.11 (6%) Give the followig trasfer fuctio: y(t) is the output a u(t) is the iput. s + 6 G ( s) =, etermie the ODE if s + s + 5
9 December 7, 005 Name 9 A.1 (8%) The step respose of a staar seco-orer system (assume ζ < 1) with iitial coitios is give by K ω 1 s y( 0) + y ( 0) + ζω y( 0) Y () s = + s + ζω + ω s s + ζω + ω where ω = ω 1 ζ. ( ) The Partial Fractio Expasio for this system is give below. ( ) Y () s = A + s s + ( 0) + ζω y( 0) B ω C ( s + ζω ) y( 0) ( s + ζω ) ω ζω s + ω ( s + ζω ) ( ) s + ω s + ζω ζω + ω + s + ω y ω (a) Above the Partial Fractio Expasio, ietify the free a force portios of the respose. (b) Below the Partial Fractio Expasio, ietify the trasiet a steay-state portios of the respose.
10 December 7, 005 Name 10 B.1 (5%) Sketch the root locus for the followig close-loop characteristic equatio: ( s + 5) 1+ K = 0 s + 4 s + 16 Be sure to calculate (a clearly label) ay asympototes, break-i/break-away poits, arrival/eparture agles, a imagiary axis crossigs (if ay) Imagiary Axis Real Axis State the rage of gai K for which this system is stable.
11 December 7, 005 Name 11
12 December 7, 005 Name 1 B. (0%) Give the followig Boe iagram etermie the trasfer fuctio:
13 December 7, 005 Name 13 B.3 (30%) Give the followig c motor feeback system with cotroller C () s, R(s) + E(s) - C() s 1 s s ( + ) Y(s) esig a PD cotroller, C() s K K s = +, usig the followig steps. p (a) Determie the esire performace regio o the complex plae below usig the followig performace criteria: t s % 1sec % OS 5% Imagiary Axis Real Axis
14 December 7, 005 Name 14 (b) Use the itersectio poits of the performace costrait lies as the esire close-loop pole locatios, p 1 a p, a use these to write the esire close-loop characteristic equatio. (Hit: ( s + p1 ) ( s + p ) = 0 where p 1 a p may be complex cojugates.) Multiply the equatio out to get a sigle polyomial. (c) Write out the actual close-loop characteristic equatio where you let C() s K + K s Leave K p a K as variables i your polyomial. =. p () Compare coefficiets betwee the esire close-loop characteristic equatio a the actual close-loop characteristic equatio to calculate the values for K a K. This metho esures that the root locus will pass through the esire close-loop pole locatios. p
15 December 7, 005 Name 15 (f) Fially, usig the values for K p a system. Remember that C() s K s + p = K ( s a) efie by K you calculate i () raw the root locus for this K = + K such that the cotroller gai is K a = p. K K = K a the cotroller zero locatio is efie by
16 December 7, 005 Name 16 B.4 (5%) Give the followig mechaical system: (a) Draw the free boy iagrams: (b) Write the elemetal equatios (c) Determie the equatios of motios
17 December 7, 005 Name 17 Laplace Trasform Pairs f F (s) Commet () t δ () t 1 Uit Impulse 1 for t > 0 t for > 0 1 s 1 s t at e si ( ωt) cos ( ωt) () t s s 1 s + a ω + ω s + ω Uit Step Uit Ramp Expoetial Sie Cosie f F (s) Fuctio f () t t f t () t s ( s) f ( 0 ) sf First Derivative 1 f 0 th F () ( ) ( ) s s f 0 s! s t + 1 f () t F ( s + a ) e at at te ( ) t e at e at si ωt 1 s + a! + ( s + a) 1 ( ) ( s + a) + ω ( s + a) cos( ) ( s + a) + ω e at ωt + Iitial Value Theorem f ( 0 ) = lim sf( s) ω s Fial Value Theorem lim f ( t) = lim sf( s) t s 0 t Derivative
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