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Unformatted text preview: Chapter 2 Dynamic Models Figure 2.39
Mechanical systems A A 11. Why do we approxi
a linear model? 12. Give the relationships for
(a) heat ﬂow acros
(b) heat st
13. Name Problems for Section 2.1: Dyn . . . for
differentlal equations '
e ink the system W1 zero initial con 2.1 Write th and (b), state whether youth
at all, givon that there are non« for your answer. No friction orage in a substance. and give the equations for mate a physical model of the pl s a substance, and the three relationships governin ant (which is always nonli g ﬂuid ﬂow. PRﬁngMS Friction (b) armies of Mechanical Systems
. n in Fig. 2.39. For (a) i ' l s stems . I
the mewamca y ecay so that it has no motion show ll eventually d v
ditions for both masses, and give .172 2.2 Write the differential you think the system wil . .
tial conditions f0 motion for the double are non—zero im . . f
Write the equations 0
2.3 displacement an horizontal. T Assume that the
the spring is always
and the springs are equation for th attached thre —pendulurn system all en
les of the pendulums are sm
i6 pendulum rods are taken to be mass e—fourths of the way down. i Fig. 2.40. State whether at it has no motion at all, give
ason for your answer. shown in Fig.
ough to ensure that near) with _ a T838011 n that there
2.41 . less, of length l. Figure 2.4.0 Mechanical system for
Problem 2.2 figure 2.4a
Double pendulum 2.4 2.5 2.6 2.7 2.8 No friction No friction Write the equations of motion of a pendulum consisting of a thin, 4 kg stick of length l
suspended from a pivot. How long should the rod be in order for the period to be exactly
2 sec? (The inertia I of a thin stick about an end point is %m12. Assume that 9 is small
enough that sine % 9.) Why do you think grandfather clocks are typically about 6 ft
high? For the car suspension discussed in Example 2.2, plot the position of the car and the
wheel after the car hits a “unit bump”(i.e., r is a unit step) using MAT LAB. Assume that
m1 = 10 kg, m2 2 350 kg, KW : 500,000 N/m, Ks = 10,000 N/m. Find the value of b
that you would prefer if you were a passenger in the car. Write the equations of motion for a body of mass M suspended from a ﬁxed point by a
spring with a constant k. Carefully deﬁne where the body’s displacement is zero. Automobile manufacturers are contemplating building active suspension systems. The
simplest change is to make shock absorbers with a changeable damping, b(u 1). It is also
possible to make a device to be placed in parallel with the springs that has the ability to
supply an equal force, M27 in opposite directions on the wheel axle and the car body. (a) Modify the equations of motion in Example 2.2 to include such control inputs.
(b) Is the resulting system linear? (c) Is it possible to use the forcer L12 to completely replace the springs and shock absorber?
Is this a good idea? Modify the equation of motion for the cruise control in Example 21, Eq. (2.4), so that it
has a control law; that is, let u = K(vr — v), (2.89) where
v, 2 reference speed, (2.90)
K = constant. (291) This is a “proportional”control law in which the difference between vr and the actual
speed is used as a signal to speed the engine up or slow it down. Revise the equations
of motion with vr as the input and v as the output and find the transfer function. Assume
that m = lOOO kg and b = 50 N~sec/m, and ﬁnd the response for a unit step in vr using Figure 2.44 Chapter 2 Dynamic Models
Circuit for Problem 2.11 that you think would result in a control or, ﬁnd a value of K
ible to the reference speed MATLAB. Using trial and err
system in which the actual speed converges as quickly as poss with no objectionable behavior. 2.9 In many mechanical positioning syste and another. An example is shown in Fig. 2. Fig. 2.42 depicts such a situation, where a etween one part of the system
ﬂexibility of the solar panels.
the mass M and another ften modeled by a spring ms there is ﬂexibility b
7 where there is
force 14 is applied to
n the objects is 0 mass m is connected to it. The coupling betwee
constant k with a damping coefficient 19, although the actual situation is usually much
more complicated than this. Figure 2.45
Circuit for Problem 2.12 f motion governing this system.
tion between the control input u and the output y. (3) Write the equations 0
(b) Find the transfer func “agate 2A2
Schematic of a system
with flexibility
2.13 A common CO .
nnection for a motor . .
haVe the motor c Power amphﬁer 13 Shown in F’  .
ampliﬁer. Assumlelrltrhialtt tfliilzw the input voltage, and the connectlii'nzijiallltel: Idea 18 to
. ense resistor r is a Current
reSistor R, l ‘ . S Very small 0 ~
When Rf ”a:: find the transfer function from Vin to Ia A18: 3122:2631: t1th £11176 feedbaCk
, w  ' rans er function
Problems for Section 2.2: Models ofEiccrric Circuits Figure; 2345
2.10 A ﬁrst step toward a realistic model of an opdamp is given by the following equations and . OP'amp Circuit for is shown in Fig. 2.43: Pmblem 2.13
V — 107 [V V ]
out —— S + 1 + ~ .
Lt. 7: i“ r: 0.
Find the transfer function of the simple ampliﬁcation circuit shown using this model.
figure 251.43
2.14 An Op_am CO .
_ 1’1 ‘
P nection With feedback to both the negative and the positi t
l Ve erminals is Circuit for Problem 2.10
shown in Fi . 2.47
g i If the op—amp has the nonideal transfer function given in Pr b1 2
o em .10, g h a p 7 — r R,
1V6 l 6 III Xllilum llaer p()SSIble {01 the Oﬁltllle lﬁedback 1:3th 1 Ill tel lllS ()l the negative feedback r3110, N ~— n , f0r the (JICUIt i0 16 all! Stable
Rm+Rf 2.15 Write the d ~
' ynamic e uat‘ ,
ﬁg 248. q ions and ﬁnd the transfer functions for the circuits show '
x n in (a) passive lead circuit 4.: ,1 Vin if the Opaml)
(b) active lead circuit hown in Fig. 2.44 results in Vow
transfer function of op—amp connection s
the transfer function if the opamp has the nonideal 2.11 Show that the
(c) active lag circuit is ideal. Give
Problem 2.10. 2.12 Show that, with the non
shown in Fig. 2.45 is unstable. ((1) passive notch circuit ideal transfer function of Problem 2.10, the op~amp connection Chapter 2 Dynamic Models
sure 2.47 C R
amp circuit for R
)blem 2.14
R1 R R
v,” ’\/\/\z a w» a 0
V1 V2 V3
Ra R
’\/\/\/
Rb
C RC V01”
ﬁgure 2.48 A?“ .
a) Passive lead; (b) + «Ad/V
‘   c active a I
activelead,( ) _ + Ri R2 y Figure $49
Lag; and (d) passwe u e _
 '  O Opamp biquad
notch cwcmts o _
(a)
R _
f (a) Show that if Ra = R, and R}, 2 RC 2 Rd 2 00, the transfer function from Vin to
V0“; can be written as the low—pass ﬁlter
R  V A
R2 1 van _ 0” = _, (2.92)
Vin S2
“‘2‘ + 2; m + l
a)” can
C where
R
A = —,
Rl'
_ l
(Uri — RC,
_ R
g ‘" 2R2’ (b) Using the MATLAB command step, compute and plot on the same graph the step
responses for the biquad of Fig. 2.49 forA = 1, can 2 l, and g“ = 0.1, 0.5, and 1.0. 2.17 Find the equations and transfer function for the biquad Circuit of Fig. 2.49 if Ra = R,
Rd 2 R1, ande = RC = 00. Problems for Section 2.3: Models of Electromechanical Systems 2.18 The torque constant of a motor is the ratio of torque to current and is often given in ounce
inches per ampere. (Ounce—inches have dimension force >< distance, Where an ounce is
1/16 of a pound.) The electric constant of a motor is the ratio of back emf to speed and is
often given in volts per 1000 rpm. In consistent units, the two constants are the same for
a given motor (d) ause its transfer function lynomials‘ By selecting
ow—pass, band—pass, 249 is called a biquad bee
«order or quadratic .po
he circuit can realize a l ' ' ‘ ~ ‘ Fig.
ﬁexxble Circuit shown in
2.16 The very he ratio of two second can be made to be t
different values for Ra. Rb. RC, and Rd. high—pass, or band—reject (a) Show that the units ounceinches per ampere are proportional to volts per 1000 rpm
by reducing both to MKS (SI) units. Chapter 2 Dynamic Models Figure 2.50 Simpliﬁed modelfor
capacitor microphone 2.19 The electro rpm. What is its torque constant in (b) A certain motor has a back emf of 25 V at 1000 ounce—inches per ampere?
(c) What is the torque constant of the motor of part (b) in newto
mechanical system shown in Fig. 250 represents a simpliﬁed model of a
hone. The system consists in part of a parallel plate capacitor connected
cuit. Capacitor plate a is rigidly fastened to the microphone frame.
d exert a force fs(t) on plate b, which has through the mouthpiece an
cted to the frame by a set of springs and dampers. The capacitance distance x between the r , n—meters per ampere? capacitor microp
into an electric Cir
Sound waves pass
mass M and is conne
C is a function of the C05) = 77 where
a :: dielectric constant of the material between the plates, A 2 surface area of the plates. The charge q and the voltage 6 across the plates are related by q = C ()06
The electric ﬁeld in turn produces the following force fe on the movable plate that opposes
its motion:
2
fe =3 L '
28A ribe the operation of this system. (It is acceptable (a) Write differential equations that desc
to leave in nonlinear form.)
model? f the system? (b) Can one get a linear
(c) What is the output 0 MRI" .XM, control is an electric motor driving electromechanical position
ises in computerdisk—head 2.20 A very typical problem of
a load that has one dominant vibration mode. The problem ar
' , and many other applications. A schematic diagram is control, reel~to sketched in Fig. 2.5L ical constant Kg, a torque constant Kt, an armature inductance La, has an inertia J1 and a viscous
The two inertias are connected by a shaft with a ction B. The load has an inertia 12.
an equivalent viscous damping 17. Write the equations of motion. The motor has an electr
and a resistance Ra. The rotor
fri
spring constant k and fignre 2.51 Motor with a flexible
load Figure 2&2 (a) Precision table kept
level by actuators; (b) side view of one
actuator Problems for Section 2.4: Heat and Fluidwflow Model r 2.21 A precisi
on tab — ‘ .
actuators underliizvelmg scheme shown in Fig. 2.52 relies on thermal '
corners Th comers to level the table by raisin or 1 ~ éXpansmn 0f
. 6 parameters are as follows: g 0W€rmg then respective Tact = actuator temperature
Tamb = ambient air temperature
R = h  '
f eat ﬂow coefﬁment between the actuator and the air
C = thermal capacity of the actuator, R = resistance of the heater. Assume that (1)
the actuator acts
 . as a pure electr' ‘ (
actuator 1s r0 0 . 1C TCSIStdnCB, (2 the h  '
to the diffefenge réleipvageto ;he electric power input, and (3) the m)otion é: thg
, n act and T due t h ‘ P0 10nal e uations ‘ . amb 0 t ermal ex ans  . . q relatlng the height of the actuator d versus the apIpliedoilolfmd the dlfferenual age Vi. QC! (a)
(b) 2.22 All 611) CO d [[0 6 Sup 6 C d p
Ii 1 II T p11 5 01 d1}: at the 53.“ 6 tell! Cldtulel Cat (Km! () ll le 11 Ill
0 h
ﬂOOI Oil the lllgoll‘llseolnllldll g Show] HI] 1g. [he ﬂoor plan IS ShOWn 11]} 1g. b .
t m )
1116 Cold d1] W p1 duCeS all equal cl Quill ()l heat ﬂow (2 Out () each “)0 . W 116 a bel
of dlffel elltlal equdtlons gOVCInng the telllpelatme 111 each IOOID, Whele To 2 temperature outside the building
R = .
0 resrstance to heat ﬂow through the outer walls RV = ‘ '
, res1stance to heat ﬂow through the inner walls Chapter 3 Dynamic Response ire 3.58 ty feedback system
Problem 3.25 Figure 3.59 Unity feedback system
for Problem 3.28 Figure 3.60
Desired closed—loop
pole locations for
Problem 3.28 Compensator Plant
J" K g g 100
mm T 5+4; s+25 Problems for Section 3 3.26 Suppose you desire the
Draw the region in the s .4: Time— peak time of a . . ,
speciﬁcation tp < tp. 3.27 A certain servomech
and no ﬁnite zeros. overshoot (M p), and settling time (a) Sketch the region in the s anism s ‘
The time—domain O Yts) plane that corresponds to v ystem has dy nami speci Domain Specification ' ‘ to b
iven second—order system
g alues of the poles that meet the es dominated b _
ﬁcations on the rise time (ts) are given by tr < 0.6 sec, Mp g 17%, ts g 9.2 sec. will meet all three speciﬁcations. (b) Indicate on your smallest rise—time and als
sign a unity feedback ‘ .
11 learn in Chapter 4, the configurati
troller.) You are to design the contr
wn in Fig. 3.60. .
haded regions in Fig. 3.59? (A Simple 3.28 Suppose you are to de
Fig. 3.59. (As you wr ro ortional—integral con .
p p in the shaded regions she to” and g correspond to the s poles lie with (a) What values of
estimate from t (in) Let Ka z: a.
system he w1 6(1) sketch the speci he ﬁgure is sufﬁci 0 meet the set —plane where the pole ﬁe locations (de .
tling time speciﬁcation exa order plant depicted in
on shown is referred to as a
oller so that the closed—loop em.) 2. Find values for K an thin the shaded regions. controller for a ﬁrst— d K] so that the poles of e less than t}. y a pair of complex poles (t r), percent s could be placed so that the system noted by x) that will have the
ctly. the closedloop .MW (c) Prove that no matter what the values of Ka and a are, the controller provides enough
ﬂexibility to place the poles anywhere in the complex (lefthalf) plane. 3.29 The open—loop transfer function of a unity feedback system is G(S) :: 3(3 + ' The desired system response to a step input is speciﬁed as peak time tp = 1 sec and
overshoot MP 2 5%. (3) Determine whether both speciﬁcations can be met simultaneously by selecting the
right value of K. (b) Sketch the associated region in the splane where both speciﬁcations are met, and
indicate what root locations are possible for some likely values of K. (c) Relax the specifications in part (a) by the same factor and pick a suitable value for
K, and use MATLAB to verify that the new speciﬁcations are satisﬁed. 3.30 The equations of motion for the DC motor shown in Fig. 2.32 were given in Eqs. (2.52—
2.53) as " KK . K
Jmem + (19+ ’ ﬁe... = _tva. Assume that m = 0.01 kng, b = 0.001 Nmsec,
Ke = 0.02 Vsec,
Kr = 0.02 N.m/A’
Ra = 10 S). (a) Find the transfer function between the applied voltage va and the motor speed 9m. (b) What is the steadystate speed of the motor after a voltage Va 2: 10 V has been
applied? (c) Find the transfer function between the applied voltage Va and the shaft angle 9m. (d) Suppose feedback is added to the system in part (c) so that it becomes a position
servo device such that the applied voltage is given by Va : K(9r “' am), where K is the feedback gain. Find the transfer function between 9, and 9,”. (e) What is the maximum value of K that can be used if an overshoot Mp < 20% is
desired? (D What values of K will provide a rise time of less than 4 sec? (Ignore the Mp
constraint.) (g) Use MATLAB to plot the step response of the position servo system for values of
the gain K = 0.5, l. and 2. Find the overshoot and rise time for each of the three step
responses by examining your plots. Are the plots consistent with your calculations
in parts (e) and (f)? 3.31 You wish to control the elevation of the satelliteetracking antenna shown in Figs. 3.61
and 3.62. The antenna and drive parts have a moment of inertia J and a damping B; (c) What is the maximum value of K that can be used if you wish to have an overshoot 1 Chapter 3 Dynamic Response
Mp < 10%? .. 1 . . . .
we: 3 6 _ (d) What values of K W111 provrde a rise time of less than 80 sec? (Ignore the Mp
:elhte—trackmg constraint.)
tenna A (e) Use MATLAB to plot the step response of the antenna system for K = 200, 400,
Irce: Courtesy Space 1000, and 2000. Find the overshoot and rise time of the four step responses by
tems/Loml examining your plots. Do the plots conﬁrm your calculations in parts (c) and (d)? 3.32 Show that the secondorder system
_ y + 2am + wiy = 0, MO) 2 ya. ya» = 0,
has the response
e—ot
y(t) : yo— sin(wdt + cos‘1 g).
t/ 1 — {2
Prove that, for the underdamped case (g < l), the response oscillations decay at a
predictable rate (see Fig. 3.63) called the logarithmic decrement
2 5 lny—0 lne0rd 0rd 7“: M [1  t2
‘ Ayi Ayz
Ft ureBeéﬁ =ln———— ; n——,
g . YI yi
Schematic of antenna
for Problem 331 where
271 27r
17d : — = is the damped natural period of vibration. The damping coefﬁcient in terms of the
logarithmic decrement is then ut mostly from the Figure 3.63 ‘ Deﬁnition of
logarithmic decrement bearing and aerodynamic friction, b
f motion are tent from back emf of the DC drive motor. The equations 0 these arise to some ex 10‘ +30 2 TC, ’ at
where TC is the torque from the drive motor. Assume th J = 600.000 kgm2 B = 20,000 Nmsec. and the antenna angle 6.
rence command 9r NY d the transfer function between the applied torque Tc a) Fin ‘ f
Eb) Suppose the applied torque is computed so that 9 tracks a re e
according to the feedback law
Tc : Kwr " 9)» e e . W d .
h 1‘ K is h edba k g 1D In 11 t dnbfer functlon bet Ben r (in
W i e e (2 a F (l t e 1" W 6 , 0 ...
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 Spring '07
 Balakrishnan

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