Midterm_1_solutions

# Midterm_1_solutions - Name ME 670 Midterm 1 1 March 2007...

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Unformatted text preview: Name ME 670 Midterm 1 1 March 2007 Problem 1 32 pts Problem 2 36 pts Problem 3 32 pts Total 100 pts Egg“ 1 7; wa Problem 1. Answer each separate question. a) For the given electrical circuit, write the differential equation(s) of the system using graph theory and KVL, and ﬁnd the transfer ﬁinction Eo(s)/Ii(s) in standard form. Make sure you deﬁne standard parameters. 62.. ' it x “ Adv Ami. ,OL helm/’52 ‘A/X‘x 80 LC. 1» Q» a K 4‘ "ﬁg/{Lao 4 ZlZCA/W/Um: LZQ/LL-t RCA, + 06%: £2. «2/ K 2. 50 (mi mﬁwmmmfmmmmm K A. t ‘ x , /7 : wﬁﬂﬂ‘ ft: kiwi" i” + E Q“ m K5 A41 M "JV/{16¢} 0-”: t Mw/‘A l .‘ I,” if“ L34 LEhﬁfg K: wk :1 wq“£‘g L’a' 44;" a A f K b) For the given mechanical system, write the equations of motion using Newton’s law. Assume small movement. _ X2 X; \C X; k5, (x “’ng “m” “'5’ ' Z a”? \z is”: Wt?” gig ' m ‘ \ r a g .. “a / ._ \quluq (x\$~x&}+e‘{x-2<&\:NZ>\¢ j a! , / ﬂ 3” :ij ,, v _ Cantilever spring MA “MI ; 4°" "'2 “‘7! 7‘ , M m ern N Problem 2. Answer the following questions for the given mechanical system. ( roblem continues on next a e) a) Using a free body diagram of the system, write the equations of motion with Ti as the input and 60 as the output. Assume that R>>t . 3/1 ——El at 2M : m£<xamg§ﬁ +T-*(§; : 3C9 +M§R R>>“hl a \ W O 0 {:0 mi Q; 2 u ﬁve/"ET M31“ '3; X SJL.‘ *- K .7369; tags} ~ a (35 + a a —.. i 4;: JQD “Ft/“W; Goa b) Find a single equation relating 60 to Ti and deﬁne the standard parameters. 6 ii a a. Each «:3 {3? ﬁrm“? 690 4 Cglﬂgﬂe i- KR @o 3 7:; k (1 5mm Ea a} it“ *2} k a: it .9 w 2 ‘° I’HLJ ~F PM @o + K: £9 Jr (99: (qu why. w“ 4, K KL 337 - k : “L t Lon: Tfﬁtﬂ’” “a Kat “ ha Ami a 555 mmnm 0) Deﬁne state variables and put the system into state—space form. L >8: 90 X1: QO w k 2, >452 m {(09.4 xf Lu“ x‘ + king,“ 74,4 fa // G d) Solve the state-space equations for 90 (t) given no input (Ti = 0) and initial conditions 90(0) = 1 anil 90(0) = . 0 1 0 \ Assume that the vector-matrlces are: A =[ 4 0 4] 32L] C =[1 O] D:0 \ ~~~~~~~~~~~ I ’77 O No unfair q .\ n ' mi , km t 1%)»:le 22103 4r “‘4 g “WM —1 4 m ~( Awe» w?» T" Ami. \ {AI‘AZ‘3 ‘i Atmy 5 t .g’@ vi A4 A“ _ n \ Cmtwrtmi ‘i‘\ ‘1‘ A + 69 A— Z > “t V! \$343 [Aw-3:“ A} [( 1 : AWLEQZ A} % Liwhage (4%} O A1 to 4%,“; CA3“: 2 ~ 7‘ AL+¢V+A+C¥ W‘Xt[ Awiwmav I _ o I WMWWWWW .— gGDaisyaﬁion «\$33 ; Ck v7 33 3*“ “Wig/u c, ‘9’ ~‘ «i g “‘ A t 27"“ J [W " , WK z 3;? @oCﬂe‘ at A” + L? \‘f ‘ A A2460 Tm“? Aﬂoﬂjmw _ W. - w t 2 __ —fbuht u M \t _, L‘ 63—13— 2 f “‘ Wx “‘ Q £no(whﬁ~~sh i; “1b.” i “1 ‘t‘ ) a Qigéwwggw My ‘\ 05x H Itwa Problem 3. All questions refer to the given electrical system. a) Write the differential equation(s) of the system using Kirchoﬁ" s voltage loop law. A: 2 ~ 9/; + if 3»: a“; ‘5‘ 2mg€gfvbgjtﬂ£ 36> {mg VT? ,TWW I t k E 'g + i is; swim +ALR2‘3 < ,4} r2: .: «2.0 b) Using Cramer’s method, ﬁnd the transfer ﬁmction Eo(s)/Ei(s). 1 ‘ .L ‘ u i t ‘ k i s. w ‘ :AH/N/K “CAAL - «u “‘11? Zk‘ QE" Cacao A 2" W9 —1 ‘ at w 2’ g“ M; +(~L+c»1€~o Qwé +i~£@% R3&’2L‘:Q ‘ a 0 2 «Age;- 60 2° ‘i a) CA8“: (A e”: “ Ea(4.al N RC A, m t 3 g 9 E‘ Z ~yR a i “A”; All 2mm,“ c) Solve for the steady-state output of eo(t) if ei(t) = -cos(10t). All initial conditions (IC’s) are zero. Assume that the differential equation is 4de" +ea 2233 .3, 2:25 : i 0* 3A) gait dt dt M ( )EJMQ aegfﬂ I i» w “3 in Lt0'\Cto) iktafmwﬁi w» M £9; Z ' lit-E Mata ' 5? mi “mt! aﬁﬁasmﬂ i; MW at,“ % Am lemma aha)? J??? i 2 i -l W 2 \ — M '7 w g ((4213 4‘14"” H may». “M ‘6 Ce} .3 4M"? (to) Sis(’is-£~m1w() r: ’ wa’wm (“w ...
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