Operating Systems Chapter 2 Homework The Resistances Resistance Heat Through

2036 CHAPTER 2. DYNAMIC MODELS

(a) Write differential equations that describe the operation of this sys-

tem. (It is acceptable to leave in nonlinear form.)

(b) Can one get a linear model?

(c) What is the output of the system?

Figure 2.53: Simplified model for capacitor microphone

Solution:

(a) The free body diagram of the capacitor plate b

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20. A very typical problem of electromechanical position control is an electric

motor driving a load that has one dominant vibration mode. The problem

arises in computer-disk-head control, reel-to-reel tape drives, and many

other applications. A schematic diagram is sketched in Fig. 2.54. The

motor has an electrical constant Ke, a torque constant Kt, an armature

inductance La, and a resistance Ra. The rotor has an inertia J1and

a viscous friction B. The load has an inertia J2. The two inertias are

connected by a shaft with a spring constant kand an equivalent viscous

damping b. Write the equations of motion.

Figure 2.54: Motor with a ‡exible load

(a)

Solution:

(a) Rotor:

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Load:

21. For the robot in Fig. 2.45, assume you have command of the torque on a

servo motor that is connected to the drive wheels with gears that have a

2:1 ratio so that the torque on the wheels is increased by a factor of 2 over

that delivered by the servo. Determine the dynamic equations relating

the speed of the robot with respect to the torque command of the servo.

Your equations will require certain quantities, e.g., mass of vehicle, inertia

and radisus of the wheels, etc. Assume you have access to whatever you

need. .

Fig. 2.45 Hospital robot

(a) Solution: First, let’s consider the problem for the case along the

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When you apply a torque to a drive wheel, that torque partly provides

an angular accelation of the wheel and the remainder is tranferred to

the contact point as a friction force that accelerates the mass of the

22. Using Fig. 2.35, derive the transfer function between the applied torque,

Tm, and the output, 2;for the case when there is a spring attached to

the output load. That is, there is a torque applied to the output load,

Ts;where Ts=Ks2

Fig. 2.35 (a) geometry definitions and forces on teeth (b) definitions for the

dynamic analysis.

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Solution: Equation (2.78), repeated, is

Problems and Solutions for Section 2.4

23. A precision-table leveling scheme shown in Fig. 2.57 relies on thermal

expansion of actuators under two corners to level the table by raising or

lowering their respective corners. The parameters are:

Tact = actuator temperature;

Tamb = ambient air temperature;

Rf= heat ow coefficient 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 electric resistance, (2) the

heat ‡ow into the actuator is proportional to the electric power input,

and (3) the motion dis proportional to the difference between Tact and

Tamb due to thermal expansion. Find the differential equations relating

the height of the actuator dversus the applied voltage vi.

Fig. 2.57 (a) Precision table kept level by actuators; (b) side view of one

actuator

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Solution:

24. An air conditioner supplies cold air at the same temperature to each room

on the fourth ‡oor of the high-rise building shown in Fig. 2.58(a). The ‡oor

plan is shown in Fig. 2.58(b). The cold air ‡ow produces an equal amount

of heat ‡ow qout of each room. Write a set of differential equations

governing the temperature in each room, where

To= temperature outside the building;

Ro= resistance to heat ow through the outer walls;

Ri= resistance to heat ow through the inner walls:

Assume that (1) all rooms are perfect squares, (2) there is no heat ‡ow

through the ‡oors or ceilings, and (3) the temperature in each room is

uniform throughout the room. Take advantage of symmetry to reduce the

number of differential equations to three.

Fig. 2.58 Building air conditioning: (a) high-rise building, (b) ‡oor plan

of the fourth ‡oor

Solution:

We can classify 9 rooms to 3 types by the number of outer walls they have.

Let’s redefince the resistances

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25. For the two-tank ‡uid-‡ow system shown in Fig. 2.59, find the differential

equations relating the ‡ow into the first tank to the ‡ow out of the second

tank.

Fig. 2.59 Two-tank ‡uid-‡ow system for Problem 25

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Solution:

This is a variation on the problem solved in Example 2.19 and the defini-

tions of terms is taken from that. From the relation between the height

of the water and mass ‡ow rate, the continuity equations are

26. A laboratory experiment in the ‡ow of water through two tanks is sketched

in Fig. 2.60. Assume that Eq. (2.93) describes ‡ow through the equal-sized

holes at points A, B, or C.

(a) With holes at B and C but none at A, write the equations of motion

for this system in terms of h1and h2. Assume that when h2= 10 cm,

the out‡ow is 200 g/min.

(b) At h1= 30 cm and h2= 10 cm, compute a linearized model and the

transfer function from pump ‡ow (in cubic centimeters per minute)

to h2.

(c) Repeat parts (a) and (b) assuming hole B is closed and hole A is

open. Assume that h3= 20 cm, h1>20 cm, and h2<20:cm.

Fig. 2.60 Two-tank ‡uid-‡ow system for Problem 26

Solution:

(a) Following the solution of Example 2.19, and assuming the area of

both tanks is A; the values given for the heights ensure that the

water will ‡ow according to

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not result.

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(c) With hole B closed and hole A open, the relevant relations are

27. The equations for heating a house are given by Eqs. (2.81) and (2.82)

and, in a particular case can be written with time in hours as

CdTh

dt =Ku ThTo

R

where

(a) Cis the Thermal capacity of the house, BT U=oF

(b) This the temperature in the house, oF

(c) Tois the temperature outside the house, oF

(d) Kis the heat rating of the furnace, = 90;000 BT U=hour

(e) Ris the thermal resistance, oFper BT U=hour

(f) uis the furnace switch, =1 if the furnace is on and =0 if the furnace

is off.

It is measured that, with the outside temperature at 32 oFand the house

at 60 oF, the furnace raises the temperature 2 oFin 6 minutes (0.1

hour). With the furnace off, the house temperature falls 2 oFin 40

minutes. What are the values of Cand Rfor the house?

Solution:

For the first case, the furnace is on which means u= 1.

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