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exam_1_w06_soln

# exam_1_w06_soln - I—tH—EE—Eldldb U'Kidti P'llzL'H&...

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Unformatted text preview: I—tH—EE—Eldldb U'Kidti P'llzL'H & HtHLl IzNLillelelNLi b'i’d titi4 DUHU I—‘.|£11 3600 — System Dynamics, Winter 2006 Exam I — Closed Book (40 pts, 40% of the total points) Short Answers: l. (6 pts) An impulse is a standard input function used in analyzing dynamic systems. Give’a' “real-world" example of physical system that couldhe‘modeled by an impulse response. 2. (4 pts) Is the following differential equation a) linear time-invariant (LTD, 13) linear time-varying, or c) nonlinear? y+4y+3yeosﬁt=u 3. (4 pts) True False: when deriving a transfer function (3(5), the speciﬁc nature of the input function 11(1) must be known in order to completely detemiine (3(5). Short Problemstr; '. 4. (12 pts) A unifonn disk with moment of inertia J is rotating on an supported by bearings. Noinput torque from an external source is appliedwto the :disk. The rotating disk :is immersed in a stationary cylinder that is ﬁlled with. hydraulic ﬂuid. Using the definitions for work and energy, derive an equation for the rate of energy loss’(i.e., power dissipated) due to viscous friction. ' 5. A system’s model is given below: 33+3j3+3j2+20y =16u a) (3 pts) Derive the transfer function for this system, and show all steps. b) (6 pts) If‘ilie input is u = 7.5 = constant (step input), then compute the steady— state response y(co). Explain your answer. EBSHEETE E2442 1DUSHEE?S 2 @- {LE-Bag FEB—EE—EUUb UT=JH thH K HtHU tNUthtHlNU byj 554 bUHU H-UE ._..__ ._. W WW-__‘L':um ,---- _; a. Wm.-,_—J~.h—.-J_‘_-. yaw—m" v .- ,———.. ._. .__ {GEE—'45P Earn-k I l . i {A a - ; ‘£*£§:_C # f /— Tﬂf‘w‘Pii-M SfanSe '. 1) ALL—{b SHSFEHSMH‘ .kl'gfﬁﬂ‘. P94" liar/e. ' H 2) (Quévﬁ'é'r éHZ/¢+ M4,.” fMﬂq’JE... Hammer SHE/‘43.: ’Srysmn +5 CW' Cr'+"3- Vflépra-Jf‘l'iqnaal' Made: __—-- "2.. ‘ ﬁhau vary-’3 005- 3_ J’i‘ﬁgﬁﬁ' 22-144 200% is 4‘. A, fit—E : (Eque)(‘lﬂﬁh(ar L i ‘_ Pnﬁfilbnaf uasms “fa/rim r. T =- +6. rm: Hzlzl—EE—Eldldb U'K: dbl P‘llzL'H E: HtHLI IzNLillelelNLi {Zr-Yd BEE-4 51115111 l-‘ . [£15 3600 — System Dynamics, Winter 2006 I — Open Book (60 pts, 60% of the total points) Problem 1. g (Zilfpts) A two—mass translational mechanical system is’ shown below. The applied: force LU) acts on mass m[. Displacements 21 and z; are absolute positions of masses 'm; and mg, respectively, measured relative to fixed coordinates (the static equilibrium positions with 120) = 0). An oil ﬁlm withwi-s’oous friction coefﬁcient b separates masses m1 and M2. Derive the system model and show your steps. ‘ ,Foil ﬁlm, friction coefﬁcient b Problem 2.“ (Ellipts) Obtain a complete state-space representationtSSR) for the system model below, with input :1 and output y = i. ’ 42" + ZOE + 842: = 0.122; Problem 1 Eating) Solve the following ODE using Laplace transform methods. Show all steps. " ‘ ' 2j3+12j2+68y=0 with initial conditions y(0)=0 and y(«03_,=’2.5 I-tH-EE-Elﬂlﬂb HTin MtL'H 3: HtHLI tNLithtHlNL-j _ b'ﬂjjtm blﬂ'ﬂlﬂ |-'.li14 W '5'" ‘5'" FHA—ifdﬁﬂ’J-ﬂ H'“ __.__....,- .---—‘-——J- _' ' ' ' ...w.m awn v" u-- \ \ ‘ V - .. _._____.____..¢_uw..m - 1 ,, * ‘ Way I .‘ FEE ¢ £1 M . +b( kl. E: ("E a.) # ) ‘CI 5 CLC I! pp. JI‘ X N ‘._. El SHEET ~ 22-142 ma SH EETS 597' . xgérlrfpﬁﬂ 22—1111: 2am SHEETS 5 224111 I-tH—EE—Elﬂlﬁb UN 55 MtL'H E: leHLl IzNLillelelNLi b'l’j HES-4 DUHU I-‘.l£1'::- , ‘ . H“ K . ' ~ -'~ Frauen 2 I '12:; + x2? + (033 = 0 L Iiiu/ ﬂ 1:: hivﬁJJ; #1; ’_ :“ .‘J‘WJ; 1: " 0W 1 0.812“ 5.‘n(a) = (a “w H‘i‘)‘: J; ‘ I” L_ ‘ , A J.H-......,3" Fmﬂm . {j f;_ '—‘/.5¢=_- 3t 51:115-1- + Erg‘e‘ fen-551': r l :p "" mid; EoSr'ch?) + a 3.5 MI TEITFIL F' . IZIS J 4 Wm. _4i_,4_3.._ ...
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