midterm_1_solutions_2008.03.27

# midterm_1_solutions_2008.03.27 - é 2. ME 670 Midterm 1 6...

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Unformatted text preview: é 2. ME 670 Midterm 1 6 March 2008 Problem 1 33 pts Problem 2 34 pts Problem 3 33 pts Total 100 pts Name 3 f HMS) Ava €97»? "a 2’ 52+; Problem 1. Answer each separate question. ‘ . . it + x * y = u a) Use Cramer’s method to ﬁnd the transfer function X(s)/U(s) from the coupled differential equations: Axa‘tM'Yz'wz @Mlx’éﬂryﬁw 2X” it} wil “x <3 3 30 A,» A‘s/U x+y20 “Eu 1%,»! 13¢" all? i l ti A.» XW “admitqu ” vim “" "irritate 9: p g, b) Given )5: + 4x = 214 , draw a simulation (simulink) diagram of the system equation. k:~‘l”X+Z’Z« @ c) For the given electrical circuit, use graph theory to deﬁne current loops for KVL analysis. Make sure you show the current loops in the given circuit, but do not write the loop equations. \ whim” " a» z , 3‘1” Problem 2. Answer the following questions for the given mechanical system. (problem continues on next page) Note that parts c), d) and e) do not depend on a) and b). l 2 a) Using free body diagrams of the system, write the equations of motion. Note that fi is the input force, and x and 9 are the dependent variables. The wire rolls on the disk with no slippage. ; i J E R , /i ‘ it)”: t: '2 E u I \‘QX WWW k 0 X} mmwmw-Mz at / "‘“Wyyﬁmw “v: 5' f; ‘ ‘ M ' )‘m M i J a pm It, MELT: a x‘” a M iv; : i wee/42s “H “N” ,g‘» ?N " ‘Wm‘m‘mwml‘ ““Zhgmwmb if B “Ix x Z 316% 2- M x 445 ~ g3; +l<214152©~x§=ﬁ~f1 ‘x f” (<3 E? I w «m s :0 = 0’3 W 6.,» \(A C E6)“ X» l ' T-Qyé.z IéDQ mmmwmmmww «mm jg g — KlCae—xjﬁ - {<39 \$3673; (:9 b) Assume J = 0 and K3 = 0. Find the transfer function X(s)/Fi(s) and deﬁne the standard parameters. . a} g gx KACmewx ) -= H x ‘ Q??? —l<L(rzc9»><3K : CD =~' KLCEQWJ Sagd‘iiyi“ 5 WT ﬁnd x(t) for a step input of force, given the initial (s + l O i 5. c) Assuming that the transfer function reduces to X(s)/Fi(s) = conditions x(0) = 0 , 2(0) .= 2 . ,r L 9 M g p LA 4: laymen mpg} X054 : .3 am; 3.4 i 20 3:. “is %@@“ﬁ yak: 'Lﬁ « ﬂaw/s26??? +20wa; «~20 xii; + awiw‘xarS“ E; A.» ,0 / {A} +2mxsiwi~ tars l 74 i/EJX «1“ Z"- )? AVC e. 2. L 5 " -mwmmmimWWWWM Jr WWWMWWWWW WV“ Lilon Abutmf‘ _-t V L , _ .40: ~ mist “tag: i Vibes]: XCJES‘Y: zte 1r 56:; “when” state; ‘ «mt , _. at ~i0t «XCtlsztﬂ. + {lg<\\[email protected] ) 3' d) What is the ﬁnal value of x(t) for part c)? 2 e) If J at 0 for the given system, deﬁne the states that you would use to write state space equations. Do not write the state space equations. x : \ x1: swig “:9 ﬁt ,a‘ m”? . . . Problem 3. All questions refer to the glven electrlcal system. (problem continues on next page) Note that parts 0) and d) do not depend on a) and 1)). ~ oi a) Write the diﬁ‘erential equation(s) of the system using Kirchoff‘s current node law. The current MD is the input and the output is eo (t) a E‘ ' ‘ > E ‘ l ’ a A e ‘ m C A“ Re» {A at ' E E " “E ” C it“ + I: - w m M W k 4 {52% W” NAM” new; 1‘94“ E f A/K “A, r: (3;) @Q A” :M [w W V» a. \/ Q; Vb ‘ Wmme - A/ 3' C) R z. \ c? b) Put the equations from part a) in state space form. am-mmmmm wwmwmwmwxmwm 3‘7an .0 «A ‘x t; Va“, K)» 3 A: j “0‘92. Ow “a; __ 4 as g , OM .fmk am“ .M 23%,) [ 5" c) Find the eigenvalues of the system, assuming that the state space matlices are: [421:0 4] 32m C=[0- 3]D:O. 1—3 5 KNO 2 gi‘l’twﬁaifm : l «l {\kgk C) :1. ﬁx‘agifklallz 4% d) Solve the state-space equations for e0 (t) given Ii(t) = 6 (t) , an impulse, and all initial conditions equal to zero. . . O — 2 1 As m part c), assume that the vector—matnces are: A = [I 3] 13::[0] C = [0 3] 13:0. § § § a i g g 3 5 [#43 ~23: “a """" " \ m» [kl i W l A“ Em a: - l A ll {Aka/M i A4933 - Liw ewzé W m ,..,,.,.W. m\ M “i . i E A” TM ‘WA \ L] l .l ‘4 ME 4 r A. \‘3, m: :I M ‘3” ‘11 2 Ci WW C k A»: I ('9 .t WM“ \ W'“"W“W”‘““"‘"‘"’WW” As? it»? \ l muniﬁi i) ~~~ " Mi —.l ‘ 3 MW e 09,: [a 33% w?“ OZ [gym/ma g M W 3' wZ‘l? em 2W6. \wﬁ) ...
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## This note was uploaded on 04/17/2008 for the course ME 670 taught by Professor Fussel during the Spring '08 term at New Hampshire.

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midterm_1_solutions_2008.03.27 - é 2. ME 670 Midterm 1 6...

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