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Unformatted text preview: 110 lynholle Hath Symbol“: Math J Chapter 2 MUdElll'lg Ill the Frequency Domain Problems 1. Derive the Laplace transform for the following time functions:
a. uﬂ)
b. m0)
c. sin mm“)
11. cos wt 110‘} 2. Using the Laplace transform pairs of Table 2.1 and the Laplace transform
theorems of Table 22, derive the Laplace transforms for the following time
functions. a. 8“” sin (emf!)
h. 6“" cos altar!)
c. that) 3. Repeat Problem 15 in Chapter 1, using Laplace transforms. Assume that the
forcing functions are zero prior to t = 0—. 4. Repeat Problem [6 in Chapter I, using Laplace transforms. Assume that the
forcing functions are zero prior to r = 0— 5. Use MATLAB and the Symbolic Math Toolbox to ﬁnd the Laplace transform of the
iollowrng tlme functions: a. fit) = 5t2cosi3t + 45°)
b. lit} = Ste—2’srni4t + 60“) 6. Use MATLAB and the Symbolic Math Toolbox to ﬁnd the Inverse Laplace transform of
the lollowrng frequency functions: [52 + 35 + 7M5 + 2} a. GM = m
b Glsl= 53+452+65+5 (s + Site2 + 85 + Falls2 + 55 + 7]
7. A system is described by the following differential equation: d3); day dy d3x dzx dx
Er+3d—ﬂ+SE+}’ EE+4E+6E+SX Find the expression for the transfer function of the system, st) X(s). 8. For each of the following transfer functions, write the corresponding differ
ential equation .. w  __1
FLS) sz+25+7
{ﬂ = 10 ' F(s) (s+7){s+ 8)
XEﬁ 5+2 '3' H3)  s3+8s2+93+15 Figure 92.2 mm: MITLAI lymholle Hath Problems 111 9. Write the differential equation for the system shown in Figure P2. l. R”) 35 + 2.5""1 + 453 + 52 + 3 6(5)
36+ 79+ 334+ 253+52+3 10. Write the differential equation that is mathematically equivalent to the block
diagram shown in Figure P22. Assume that rt!) = 313. ﬁg} 54+ 2.13 + 5:2 + s + 1 CU}
55+ 354+ 253+ 451+ Ss+2 11. A sys1em is described by the following differential equation: d'zx dx
F + + 3x  1
with the initial conditlons x(0} = I, x(0) = —I. Show a block diagram of the system. giving its transfer function and all pertinent inputs and outputs.
(Hint: the initial conditions will show up as added inputs to an effective
system with zero initial conditions.) 12. Use MATLAB to generate the transfer function M
sls + 55llsz + 55 + 301ls + Stills? + 275 + 52) In the following ways: Glsl = a. the raho of factors;
I). the ratio of polynomials.
13. Repeat Problem 12 for the following transfer function: 5' + 2551' + 2le2 + 155 + 42
s5 + 135‘ + 953 + 31's2 + 355 + 50 14. Use MATLAB to generate the parhalfraction expansion of the following lunetlon:
lofts + lﬂlls + 60)
sls + 40lls + EDJlS2 + 75 + l‘ilIISlls2 + 65 + 901 15. Use MATLAB and the Symbolic Math Toolbox to input and form L11 objects In
polynomial and factored form for the following frequency tunetlons: 45ls2 + 31's + "Milsf + 2852 + 325 + 16}
is + 39lls + ﬁlls? + 25 + 100ll53 + 2752 + 185 + 15) 56ls + I‘ll{53 + 4952 + 625 + 53]
{53 + 8152 + 765 + stills? + 385 + 33llsz + 565 + Til Gls) = Flsl = a. Glsl = b. Gls) = 112 Chapter 2 Modeling In the Frequency Bertram 16. Find the transfer function, 6(5) = V43) 14(5). for each network shown in Figure P23 Flgure P23.
1 Q I Q l H
+ I I I I +
'
urn " is") turn 1 F Iain
[M (a) IT. Find the transfer functions, 6(3) 2 ﬁts} Vts}. for each of the networks Figure Pu Show" In Figure P14
1 H 1 o 1 F
H I i
I Q 19
r m e _ . +
'
(a) m (aural 18. Find the transfer functions, C(s) = vom mm, for each of the networks
O—h— .
Figure stgl'ﬂ" shown in Figure P25 Solve the problem using mesh analysis. 19. Repeat Problem 18 using nodal equations 20. 3. Write, but do not solve, the mesh and nodal equations for the netWOrk of
Figure P16. Problems 1 13 Figure P16
Iymballc "an. I). Use MATLAB, the Symbolic Math Toolboxl and the equations found In {at to
solve for the transfer function, Gist = Vols) Vis}. Use both the mesh and nodal
equahons and show that either set yields the same transfer tunctlon
21. Find the transfer function, 0(5) = Vote} Vim, for each of the operational
ampliﬁer circuits shown in Figure P17.
Figure P2.7ir 100 kﬂ lUOkQ I}!!! —’\/\/\/—i on
IUO kﬂ l ,uF (b) 1 14 Chapter 2 Modeling In the Frequency Domain 22. Find the transfer function Gm) = V45) 14(3), for each of the operational
ampliﬁer circuits shown in Figure P2.8 Figure P23 [00 kg 1 p F 100 kg (5) 23. Find the transfer function, 0(3) = X1 (s‘); Ffs}, for the translational mechani
cal system shown in Figure P29 Figure P23 Problems 115 24. Find the transfer function, Gm = X20”) 'Ffs). for the translational mechani
cal network shown in Figure P2 [0. ﬁgure P210 Fnctlonless m 25. Find the transfer function, 6(5) = X20.) Fm. for the translational mechani—
"""" cal system shown in Figure P1] 1. (Hint: Place a zero mass at 1:20).) Flgure P2.11 ., ‘20]
m — +HE
2 me 5 N—sfrn 2 N—sfm 26. For the system of Figure P112 ﬁnd the transfer function. Gfsl = X; {5] Fiat).
Flgure P112 I  27. Find the transfer function, 0(3) = Xﬁs) F (s), for the translational mechani cal system shown in Figure P113. Figure P213 1 N—sfm Fricllonless 28. Find the transfer funetion, 33(3) 3H3), for each of the systems shown in
Figure P214. 116 Chapter 2 Modeling In the Frequency Domain Figure P114 Figure P215 M, =4kg 4 N—sfm 3 Nim FI’iL‘llDﬂlBEb Frictionlcss
/ 29. Write, but do not solve, the equations of motion for the translational mechan
ical system shown in Figure P2.15 Problems 11?!' 31]. For each of the rotational mechanical systems shown in Figure P116. write,
but do not solve, the equations of motion. 3“) Tm 5'2") Figure P116 2 N ms.l' rad 8 Nmfrad
(a)
no DI D2
0 W O NH O
Kl K2 l I I I
K1
(5} . tuml 31. For the rotational mechanical system shown in Figure P117. ﬁnd the tl'ﬂDSfBF
5"'"" function 6(5) = (92(5) HS} Figure F'EJZ'r
l Nrn—s had
I Nms Irad
32. For the rotational mechanical system with gears shown in Figure P2 18, ﬁnd
the transfer function, 6(5) = 03(5),: NS). The gears have inertia and bearing
friction as shown.
Figure P118 TH] E E  NI
JlDl
N2  ' N3
12. D2. Jr3 D: 93“; 118 Chapter 2 Modeling in the Frequency Domain 33. For the rotational syswm shown In Figure P2.19, ﬁnd the transfer function. 6(5) = 92(3) TL?) Figure PZJQ Tm N] = 4
3
D] = I N—msfrad
N2: 12 “93‘” N3=4
a
D; = 2 N—m—sirad N4: D3 = 32 N—m—sfrad 34. Find the transfer function, 6(5) = 62(5) “'Tts). for the rotational mechanical
system shown in Figure P220. 1000 N ms.’rad N3 = 25 ’ zoo legml
Figure P220 250 N mf rad Eililii ' ' . [alllnl 35. Find the transfer function, 015} = 94(5)! HS}, fur the rotational system
5""I" shown in Figure P221. Tm I910] 94!? r] 25 Nmsfrad 61(1) 6’3th N; = 100 W N3 = 20
I N—mi'rad 36. For the rotational system shown in Figure P222. ﬁnd the transfer function.
(3(3) = 6:15) This). Figure P121 Figure P222 Figure P2 23 Figure P224 Problems 119I l Nm—slrad l Nmi'rad no i 9H“
— 10 N4: 10 3
I 0.04 N—msfrad 37. For the rotational system shown in Figure P223, write the equations of
motion from which the transfer function, 6(5) = 61(53) T{s), can be found. Tm Sltn " N
W 0' 1
Jrl 38. Given the rorational system shown in Figure P224, ﬁnd the transfer
function, 6(5) = 95(5) 61(3). 3M} 12’ D h D Tm eﬁm  i _
N4 1
0
14.9 “H ‘ D
K: 39. In the system shown in Figure P225, the Inertia, J', of radius, r, is
constrained to move only about the stationary axis A. A viscous damping
force of translational value 1;. exists between the bodies I and M. If an
external force, ﬁr), is applied to the mass, ﬁnd the transfer function.
(3(5) = 9159):? (s). 120 Chapter 2 Modeling In the Frequency Domam Figure P225
40. For the combined translational and rotatlonal system shown In Figure P226.
ﬁnd the transfer function, 6(3) = X(s) H5}.
Figure F226 1
J' = l kg—rn
Tit] l Nrnsa'rad ’ l lagm2
N4 = 60
N2 = N3 = 02— l N—rnslrad l 41 Given the combined translational and rotational system shown in Figure
P227, ﬁnd the transfer function, G(.\) = X0.) Tfs} Figure P227 Tm Figure P223 9 (it) Figure P229 Problems 121 42. For the motor, load, and torquespeed curve shown in Figure P228, ﬁnd the
transfer function, 6(5) = 6,15), 150(5). NI =50 D] = 2 Nrnsfrad 61: r N2 = [50 "
a :2 = 13 kgmz
D; = 36 N—msftad T(Nm) 100
50 V a) (radial
150 43. The motor whose torquespeed characteristics are shown [11 Figure P229
drives the load shown in the diagram. Some of the gears have inertia. Find
the transfer function. Gts} = 62(5). E45]. £11”
_ J'l: I kg_m2
N2 = 20  ‘NJ = 10
_ _ 2 = — 2'
:1 i 2 kg m 13 2 k3 '“ 53") D— 32 h—m—sfrad N4=20 If”!
3 J4: 113—1112 T(N—m} 5V RPM
600 If 44. A dc motor develops 50 Nm of torque at a speed of 500 rad s when 10 volts
are applied. It stalls out at this voltage with IOU N—rn of torque. If the inertia
and damping of the armature are 5 kg—n'i2 and l Nm—s; rad, respectively, 122 Chapter 2 Modeling in theI Frequency Domarn ﬁnd the transfer function, 6(3) = ﬂashEGG}, of this motor if it drives an
inertia load of 100 kgm2 through a gear train, as shown in Figure P230. Figure 92.30 9...") + "1
'5111'”! Nl= N4=SU ’ Load w 45. In this chapter We derived the transfer function of a dc motor relating the
mm" angular diaplacement output to the armature voltage input. Often we want
to control the output torque rather than the displacement. Derive the transfer
function of the motor that relates output torque to input armature voltage. 46. Find the transfer function. 6(5) = X(s),"Ea(s), for the system shown in
Figure P231. Figure P231 1
fall} Motor NI = ID
' .r: 1 kgmz o=1 N—tn—slrad
N2 = 20 l—  Radius = 2 In For the motor: 1,, = l ltgm2 Da = l N—m—slrad Ra=ln ﬁ=lNsl'm
Kb = I V—sl'rad K, = l Nmm 47. Find the series and parallel analogs for the translational mechanical system
shown in Figure 220 in the text. 48. Find the series and parallel analogs for the rotational mechanical systems
shown in Figure P2.ll5(b) in the problems. 49. 50. . :IIIIIIII 51_
o "in Figure P232 52. 53. Problems 123 A system’s output, c, is related to the system's input, r, by the straightline
relationship. c = 51' + '7. Is the system linear? Consider the differential equation
(I 2.x d): where f (x) is the input and is a function of the output, I. If f (x) = sin I,
linearize the differential equation for small excursions near
a. x = 0
b. x = 17
Consider the differential equation
:13): dzx dx where f {x} is the input and is a function of the output. )5. lfﬂx} — e“,
linearize the differential equation for 1: near 0. Many systems are piecewise linear. That is. over a large range of variable
values, the system can be described linearly. A system with ampliﬁer satura
tion is one such example Given the differential equation dzx dx
E + + 50X assume that f (x) is as shown In Figure P232. Write the differential equation
for each of the following ranges of x: a. —00<x<'.—2
h. —2<x<2
c. 2<x<m For the translational mechanical system with a nonlinear spring shown
in Figure P233, ﬁnd the transfer function, 6(5) = X(s) 'F(s), for small 124 Chapter 2 Modehng tn the Frequency Domaln excursrons around f0!) = l. The spring is deﬁned by x3“) = l — 3—15").
where x,(t) is the spring displacement and 1;“) is the spring force. Fiﬂure P233 Nonlinear —I f0] 54. Consider the restaurant plate dispenser shown in Figure P234, which con
sists of a vertical stack of dishes supported by a compressed spring. As each
plate is removed, the reduced weight on the dispenser causes the remaining
plates to rise. Assume that the mass of the system minus the top plate is M,
the viscous friction between the piston and the sides of the cylinder is j; , the
spring constant is K, and the weight of a single plate is Wp. Find the transfer
function. 113) Fts). where F(s) is the step reduction in force felt when the
top plate is removed, and H5) is the vertical displacement of the dispenser in
an upward direction. Figure P2 34
Plate dispenser ' Piston Progressive Analysis and Design Problem 55. Highspeed rail pantograph. Problem I? in Chapter I discusses active con
trol of a pantograph mechanism for highspeed rail systems. The diagram for
the pantograph and catenaty coupling is shown in Figure P2.35(a}. Assume the ...
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