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Unformatted text preview: Division Meckl 10:30 / King 1:30 (circle one) ME 375 FINAL EXAM
Monday, May 3, 2004 Name_§_DjA£9_r\_ M (1) This is a closed book examination, but you are allowed three 8.5x11 crib sheets. (2) You have two hours to work all the problems on the exam. (3) Use the solution procedure we have discussed: what are you given, what are you asked to
find, what are your assumptions, what is your solution, does your solution make sense.
You must show all of your work to receive any credit. (4) You must write neatly and should use a logical format to solve the problems. You are
encouraged to really “think” about the problems before you start to solve them. Please
write your name in the top righthand corner of each page. (5) A table of Laplace transform pairs and properties of Laplace transforms is attached at the end of this exam set. Section A Section B Problems Score Problems Score l.(lO)
2.(8)
3.(5)
4.(5)
5.(6)
6.(10)
7.(12)
8.(6)
9.(10)
10.(8)
ll.(6)
12.(14) Total (A) / 100 Total (B) / 100 May 3, 2004 Name ._____... 2
ALL (10%) Using Laplace transforms, find the total response y(t) of a system with equation of
motion given by: ji+6y+13y=13u when the input u(t) is a unit step, y(0) = 0, and j) (0) : 0. (51+ és+l3>V($) = /3 (Am = 13 S
BSi—Q
Yo) = ___’_§___~_ : A + I
. 3<St+es+‘3) s s «f—(gS'f‘lS 2 ,Aszl— 6245+/3/4 + 352+ C3 s (s'LJ 65413) May 3, 2004 Name 3 A._2 (8%) A two—degreeof—freedom mechanical system is described by the following two
differential equations: Assuming that the state variables are given by q1 = x1, q2 = 221, q3 = x2 , q4 2 X2 , the input is F, and
the output is x2, write down the state and output equations for this system. ét :67? p
7"?» ’2 “5.71 “372 +§73+ 2’74 73: 7+
_ 3; 5“ 3 1 s‘ ._, _,_ __
7+ iqt+4ll 4?? +7+++F 31:73 A; (5%) The motion of a system is governed by 33 + 4y = 3_u , where y is the output of the system, and u is the input of the system. Get the transfer function from input to output. Find the poles of
the transfer function. Determine the stability of this system with a brief explanation. (5+ 4) rcs) : 2 (Ms) May 3, 2004 Name 4 M (5%) A second order system is described by j} + 3y + 36y = 3611, where y is the output of the system, and u is the input of the system. Find the natural frequency, damping ratio, and static gain
of this system. M (6%) You have been given the following parameters for a dc motor: RA 2 0.38 ohm
KT = 0.05 Nm/amp
K b = 0.05 V/(rad/sec) B = 2.9x104 Nm/(rad/sec) The steadystate torquespeed relationship is given by the following formula: where e, represents the nominal input voltage. (a) Compute the stall torque (where speed is zero) when the nominal input voltage is 15 volts. T :: NM. L. (b) Compute the noload speed when the nominal input voltage is 15 volts. ec‘ : 1F. 3 . May 3, 2004 Name 5 t M» sow>126
: Kl (9001’ K! CI Qour+ 0°01. :‘ Q IA) _A_._6_ (10%) Consider the flow control system shown below: _—>‘
Qm :a 1 im The tank serves to smooth out the outlet ﬂow rate and provides backup flow in case the inlet flow
is interrupted. However, because of the tank dynamics, the inlet ﬂow must be properly controlled
to maintain desired outlet flow. Thus, a flow sensor is used to measure the outlet flow rate QOUT.
This value is compared with the desired flow rate QDES and the difference is used in a proportional
controller to set the proper valve opening to regulate the inlet flow QIN. Generate a block diagram
that describes this flow control system. @ouT‘ A.7 (12%) Consider the ﬂuid system shown below. Derive the differential equation relating the
gage pressure at the bottom of the tank pmnk to the pump gage pressure [95. F‘l‘hk = lac—P" =Pcr May 3, 2004 Name 7 Y
A35 (6%) Obtain the transfer function GCL (S) = (S) for the system described by the following block diagram. Express it as function of K. = p , 5K (5+2) 6;" (S) 1&2)— : 0'5 CS+~OCS+§
e“) /+ o i r/c (3+2)
. (S+I)(5+§) M (10%) The performance requirements 0 a c ennc process require that its output have a two—
percent settling time less than 1 second and an overshoot less than 5 percent. Sketch the region on
the real—imagi ary plane that satisfies these performance requirements. ‘3 1.
70$:(00ar (3‘45‘2 f5 giw < "Tl c. ”‘
,— 3 o. as
Jzo‘ 713‘ = (Azo'°f(lgfz) I Ja___ _I____ I
I
I
I
I

I
I
I
I
I
I I
I
 May 3, 2004 Name 8 A.10 (8%) Find the steady—state response yss(t) of a system with transfer function G(s)= 11(3) zTL when the input is u(t)=28in(6t). Make sure you compute all
U(s) s +3s+36 ‘ 3Q
required terms. g “a :7 ﬂ—t—f—
ZQ‘ou +J3°°
L15: Cf) = 2 / ago] 5,; (Cf 4— LGCJ'GD
G _ _____.?‘ :
/C(J°°)’ = 3 z =3 [Cgai‘ xe 2‘ A.11 (6%) Given the following two transfer functions: (1) —3—3——— (2) 46 s2+6s+13 : 32+15s+50 (a) Determine damping ratio for each system. Which system has more overshoot for a step
res onse?
P w, : J7; 2 3.4, .4, col =JT—cT = 1.07 H: 2551‘“ ‘ (D a j! 2 0'33 / 252‘“ z [5. =7 ‘___—_.. 3754*». U) Ans hawa OVfr'Slxbﬁ'é'. (b) Determine static gain for each system. Which system has the higher gain? 6,0) = 1/ 63(0) =_—;E—§ = 0.92. 37344:.» C A00 tub LL; j‘au‘n . May 3, 2004 Name 9 Q (14%) The cooling system in your automobile engine serves to remove excess heat from the
engine block. We will model this system as follows. The combustion process is assumed to
provide a heat flow rate Qi(t) to the engine block, which has total mass m, specific heat cp, and
temperature Tb. Coolant ﬂows through the block and removes heat via convection, with
convection coefficient hi and total interface area Ai. Coolant at temperature Tc flows to the
radiator, where it exchanges heat with ambient air at temperature T ,1 via convection, with
convection coefficient ho and total interface area A0. Neglect thermal capacitance of the coolant. Derive a differential equation that describes how the engine block temperature Tb reacts to the heat
flow rate Qi(t). p. i
élock Cééié—A May 3, 2004 Name 10 a; (20%) The following schematic shows a rack and pinion system: In the above schematic, F is the force acting at the center of the disk, which has moment of inertia l J = Emsz and mass mg. The disk has radius R and makes an angle 6with the vertical. Note that :9: 0 when x: = 0; 31131 (10%) Draw free body diagrams of all the elements and write down the elemental equations. Use the positive directions given in the above schematic. Be sure that the motions and
forces are indicated as necessary. x. x MI W" r”! V5)“ F3” lax \.
F v—EEwee—j, ‘ W5 1:
we {a IC May 3, 2004 Name 11 B.1§b[ (10%) With F being the input and x1 being the output, derive the inputoutput model. You may use the Laplace transform in the process of derivation. However, be sure that the final result is
a differential equation. May 3, 2004 Name 12 Q (20%) Draw a straight—line approximation bode magnitude plot for the system with transfer
20003 function: G(s) = 2—————
S +1001s+1000 . Clearly show the break frequencies. GO) 2 23
(5+ 0(5/m0 i 1) am = 2 6,0) = s
can): —i— go): d?..PlOF%Mag9éF94¢ {. l ’1
It “i
w r Magnitude (dB) 0.01 0.1 1 10 100 110
Frequency (rad/s) May 3, 2004 13 Name 3.; (30%) Given the following system: (a) Design PD controller gains Kay and KP such that the closed—loop system has poles at Sz—7ij7.
€16): Vb) : 5Q: +519
[6(5) (desa—S“) +§Qs+Sl<P
WW 51+ C(+S"<4)s + (Si—SK?) .2. (j
chick.41 Jar. 2,7: CS+¥+J ?)(3+1 j?) = 52}. [45+ 98 eﬁu—ait W‘C’RC‘aﬂ—J‘s:
(«pkg/<4 = (4 e 5“+5I§0'= C98 ==> [KP—ewe (b) What is the steadystate error for a unit step reference? J.
QLC“): __5:_EE_ :1 S % €552 /“ A : "5“
 ‘j 5 {+519 stay/go
§+ C)  A—
C” “ /+ Isa
(c) If the error is nonzero, suggest a different controller that will eliminate the error.
FIZZ or F3: :9 14 Name May 3, 2004 5+6
(s+2)(s—2) 1+K B.4 (30%) Sketch the root locus for the following characteristic equation: Be sure to calculate (and clearly label) any asymptotes, breakinfbreakaway points, and imaginary axis crossings (if any). _____l_____ ____ ___————.._.L._____J__—__ J—__. m r. m m 0.0 Am A.» m.
0
\/.. V»
a m .
1
3
G? .,.T o m c a.
4. \./% . a Q .5
+ [w .1.» s 1‘ L2 ( u
2 .r. G C H ¢1 41 H O
85 c. s + {a . .
~ & 2. S _ r2
: (k 4.. R 2 a
2 z + S S
K K’s h ..
lac“ . s 3/ 14 16 Provide the range of gain K for which this system is stable. 2. s+~l<
H—A LC (SAL—C.) 7 4 : O ‘
i 320 3:.
3 =>x<=~ 6K=4 $47M 42.. lz<>1§— ...
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This note was uploaded on 12/22/2011 for the course ME 375 taught by Professor Meckle during the Fall '10 term at Purdue.
 Fall '10
 Meckle

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