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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 Ihap’ter 3 Dynamic Response 23.65 diagram of 1 lot re 3.66
ato double (1)) Derive an expression for the step response overshoot, Mp, of this system. (c) For a given value of overshoot, Mp, how do we solve for g and w"? 3.40 The block diagram of an autopilot designed to maintain the pitch attitude 9 of an aircraft
is shown in Fig. 3.65. The transfer function relating the elevator angle 8e and the pitch attitude 9 is
6(s) we + be + 2)
_ G(S) -— .
56(5) (52 + 5.; + 40) (32 + 0.03s + 0.06) where 6 is the pitch attitude in degrees and Se is the elevator angle in degrees. The
autopilot controller uses the pitch attitude error e to adjust the elevator according to the transfer function
~ _ K(s + 3) E(s) T s + 10 '
Using MATLAB, ﬁnd a value of K that will provide an overshoot of less than 10% and a rise time faster than 0.5 see for a unit—step change in 9r. After examining the step response of the system for various values of K . comment on the difﬁculty associated with making rise time and overshoot measurements for complicated systems. Problemsfor Section 3.6: Stability 3.41 A measure of the degree of instability in an unstable aircr
time it takes for the amplitude of the time response to double (see Fig. 3.66). given some aft response is the amount of nonzero initial condition.
(a) For a ﬁrst~order system, show that the time to double is
1'2 = “‘7
where p is the pole location in the RHP.
(b) For a second—order system (with two complex poles in the REF). show that Amplitude Figure 3667 Magnetic levitati on
system figure 3&3 Control system for
Problem 3.47 Problems 167 3.42 Suppose that unity feedback is to be applied around the listed open~loop systems. Use
Routh s stability criterion to determine whether the resulting closed—loop systems will
be stable. _ WM
(3) KG“) “ 5(53—v252+3s+4) (b) Koo) : 522%?) 4‘3 22 x 3.43 Use Routh’s stability criterion to determine how many roots with positive real parts the
following equations have: (a) s4 -- 8S3 + 32s2 + 80s + 100 = 0.
(b) 55 —- 10s4 + 3053 + 80s2 + 344s + 480 = 0.
(c) s4-—2s3 +7s2 ~2s+8 =0.
((1) s3 -- 52 + 205 + 78 = 0.
(e) s4 + 6s2 + 25 = 0.
3.44 Find the range of K for Which all the roots of the following polynomial are in the LHP: s5 + 55* +10s3 +10s2 + 55+K = 0. Use MATLAB to verify your answer by plotting the roots of the polynomial in the
s—plane for various values of K. 3.45 The transfer function of a typical tape-drive system is given by K (s + 4)
s[(s + 0.5)(s + i)(s2 + 0.45 + 4)}’
where time is measured in milliseconds. Using Routh’s stability criterion, determine the range of K for which this system is stable when the characteristic equation is
1 + G(s) = 0. 3.46 Consider the closed~loop magnetic levitation system shown in Fig. 3.67. Determine the
conditions on the system parameters (a, K, 2, p, K0) to guarantee closed-loop system 0(3) .. stability.
RC 2 ﬁ—w» (Y Z) L; » K0 oy
' K(S+p) (52—412) 3.47 Consider the system shown in Fig. 368. (a) Compute the closed-loop characteristic equation. (b) For what values of (T,A) is the system stable? Hint: An approximate answer may
be found using e‘TSEI—Ts ...
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