figure 2.1 (a) torsional spring-mass system. (b) spring...

Ta

m

Ta

(a) (b)

va

Ts

vs

FIGURE 2.1

(a) Torsional spring-mass system. (b) Spring element.

Dorf/BishopModern Control Systems 9/E

© 2000 by Prentice Hall, Upper Saddle River, NJ.

y

by ky

kWallfriction, b

MassM

(a) (b)

r (t)Force

r (t)

yM

FIGURE 2.2

(a) Spring-mass-damper system. (b) Free-body diagram.



Voltagev (t)

0Time

e�a2t

2(p /b2)

FIGURE 2.4

Typical voltage response for underdamped RLC circuit.



(a) (b)

p

2

�p

p

T

up

2�

Length L

Mass M

u

FIGURE 2.6

Pendulum oscillator.



jv

0

s1

s2

u � cos�1z

�2zvn �zvn

jvn�1 � z2

�jvn�1 � z2

vn

s

FIGURE 2.9

An s-plane plot of the poles and zeros of Y(s).



jvn

z increasing

z � 1z � 1

z � 1

z � 1

z � 0 jv

0s

vn

FIGURE 2.10

The locus of roots as z varies with vn constant.



�

Armature

Rf

Ra

Vf

La

Lf

iaia

if (t)Field

v, uInertia � JFriction � b

Load

Statorwinding

Rotor windings

Brush

Bearings

Shaft

BrushCommutator

Inertiaload

Angleu

(a) (b)

�

N

S

FIGURE 2.17

A dc motor (a) wiring diagram and (b) sketch.



FIGURE 2.18

A pancake dc motor with a flat-wound armature and a permanentmagnet rotor. These motors are capable of providing high torquewith a low rotor inertia. A typical mechanical time constant is in

the range of 15 ms. (Courtesy of Mavilor Motors.)



�

�

Tm(s)If (s)Vf (s) s

11

R f � L f s

DisturbanceTd (s)

TL(s)Speedv (s) Position

u (s)Output

Field

1

Js � b

Load

Km

FIGURE 2.19

Block diagram model of field-controlled dc motor.



�

�

Tm(s)Va(s)

Js � b1

s1Km

Ra � Las

Kb

�

�

DisturbanceTd (s)

TL(s)

Speedv (s) Position

u (s)

Back emf

Armature

FIGURE 2.20

Armature-controlled dc motor.



Va(s)

Ra

La

Ia

If

u, v J, bVb

�

�

�

�

Table 2.5

Transfer Functions of Dynamic Elements and Networks6. dc motor, armature-controlled, rotational actuator



V2(s)

Shaft

u (s), v (s)�

�



Table 2.5

Transfer Functions of Dynamic Elements and Networks13. Tachometer, velocity sensor

Vf (s)Output

u (s)Km

s(Js� b)(Lf s� Rf)G(s) �

FIGURE 2.22

Block diagram of dc motor.



Y1(s)

Y2(s)

R1(s)

R2(s)System

Inputs Outputs

FIGURE 2.23

General block representation of two-input, two-output system.



Y1(s)R1(s) G11(s)�

�

G 12(s)

G21 (s)

Y2(s)R2(s) G22(s)�

�

FIGURE 2.24

Block diagram of interconnected system.



Ta

m

Ta

(a) (b)

va

Ts

vs

FIGURE 2.1

(a) Torsional spring-mass system. (b) Spring element.



y

by ky

kWallfriction, b

MassM

(a) (b)

r (t)Force

r (t)

yM

FIGURE 2.2

(a) Spring-mass-damper system. (b) Free-body diagram.



Voltagev (t)

0Time

e�a2t

2(p /b2)

FIGURE 2.4

Typical voltage response for underdamped RLC circuit.



MassM

Nonlinearspring

f � y2

Springforce

Equilibrium(operating point)

f0

f

dfdy y � y0

�

(a) (b)

y0

y

y

FIGURE 2.5

(a) A mass sitting on a nonlinear spring. (b) The spring force versus y.



(a) (b)

p

2

�p

p

T

up

2�

Length L

Mass M

u

FIGURE 2.6

Pendulum oscillator.



�3 �2 �1 0s

� pole� zero

jv

FIGURE 2.7

An s-plane pole and zero plot.



jv

0

s1

s2

u � cos�1z

�2zvn �zvn

jvn�1 � z2

�jvn�1 � z2

vn

s

FIGURE 2.9

An s-plane plot of the poles and zeros of Y(s).



jvn

z increasing

z � 1z � 1

z � 1

z � 1

z � 0 jv

0s

vn

FIGURE 2.10

The locus of roots as z varies with vn constant.



y(t)

e�zvnt envelope

y0Overdamped case

Underdamped case

Time0

FIGURE 2.12

Response of the spring-mass-damper system.



(a) (b)

Currentr(t)

v1(t) v2(t)R1

C1 R2 C2 L

Friction b2

Friction b1

k

Velocityv2(t)

Velocityv1(t)

Force r(t)

M2

M1

FIGURE 2.16

(a) Two-mass mechanical system. (b) Two-node electric circuitanalog C1 = M1, C2 = M2, L = 1/k, R1 = 1/b1, R2 = 1/b2.



�

Armature

Rf

Ra

Vf

La

Lf

iaia

if (t)Field

v, uInertia � JFriction � b

Load

Statorwinding

Rotor windings

Brush

Bearings

Shaft

BrushCommutator

Inertiaload

Angleu

(a) (b)

�

N

S

FIGURE 2.17

A dc motor (a) wiring diagram and (b) sketch.



FIGURE 2.18

A pancake dc motor with a flat-wound armature and a permanentmagnet rotor. These motors are capable of providing high torquewith a low rotor inertia. A typical mechanical time constant is in

the range of 15 ms. (Courtesy of Mavilor Motors.)



�

�

Tm(s)If (s)Vf (s) s

11

R f � L f s

DisturbanceTd (s)

TL(s)Speedv (s) Position

u (s)Output

Field

1

Js � b

Load

Km

FIGURE 2.19

Block diagram model of field-controlled dc motor.



�

�

Tm(s)Va(s)

Js � b1

s1Km

Ra � Las

Kb

�

�

DisturbanceTd (s)

TL(s)

Speedv (s) Position

u (s)

Back emf

Armature

FIGURE 2.20

Armature-controlled dc motor.



Vf (s)

Rf

Lf

Ia

If u, v

J, b

�

�

Table 2.5

Transfer Functions of Dynamic Elements and Networks5. dc motor, field-controlled, rotational actuator



Va(s)

Ra

La

Ia

If

u, v J, bVb

�

�

�

�

Table 2.5

Transfer Functions of Dynamic Elements and Networks6. dc motor, armature-controlled, rotational actuator



V2(s)

Shaft

u (s), v (s)�

�



Table 2.5

Transfer Functions of Dynamic Elements and Networks13. Tachometer, velocity sensor

Vf (s)Output

u (s)Km

s(Js� b)(Lf s� Rf)G(s) �

FIGURE 2.22

Block diagram of dc motor.



figure 2.1 (a) torsional spring-mass system. (b) spring...

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