제어응용연구실 1 modeling of mechanical systems Ⅰ. 제어응용연구실 2 contents ▶...
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제어응용연구실 1
Modeling of Mechanical Modeling of Mechanical systems Ⅰsystems Ⅰ
제어응용연구실 2
CONTENTSCONTENTS
▶ ▶ Equations of Mechanical SystemEquations of Mechanical System
▶ ▶ Modeling of Mechanical System ElementsModeling of Mechanical System Elements
제어응용연구실 3
IntroductionIntroduction
PointPoint
1. 선형 / 비선형 모형과 시변 / 시불변 모형의 형태와 물리적인 의 미를 학습
2. Translational motion, Rotation Motion 의 운동 해석
제어응용연구실 4
Modeling of Modeling of Mechanical System ElementsMechanical System Elements
제어응용연구실 5
Translational MotionTranslational Motion
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
• MassMass
Ma forces
: An element that stores the kinetic energy of translational motion
g
wm ▶ ( g is the gravitational acceleration constant )
2222 in./s386ft/s2.32s/cm981s/m81.9 g
※ ※ Mass Mass is analogous to inductance inductance of electric networks
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Modeling of Mechanical System ElementsModeling of Mechanical System Elements
※ ※ Conversion TableConversion Table
제어응용연구실 7
※ ※ Conversion TableConversion Table
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
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Modeling of Mechanical System ElementsModeling of Mechanical System Elements
Force : 1 N = 0.2248 lb (force) = 3.5969 oz (force)
Mass : 1kg = 1000g = 2.2046 lb (mass) = 35.274 oz (mass) = 0.06852 slug
Distance : 1 m = 3.2808 ft = 39.37 in. 1 in. = 25.4 mm 1 ft = 0.3048 m
dt
tdvM
dt
tydMtMatf
)()()()(
2
2
Figure. Force-mass system
M
y (t)
f (t)
The force equationThe force equation
―
―
―
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Modeling of Mechanical System ElementsModeling of Mechanical System Elements
• Linear SpringLinear Spring
( K : the spring constant, or simply stiffness )( K : the spring constant, or simply stiffness )
Figure. Force-spring system
)()( tKytf The force equationThe force equation
: An element that stores potential energy
※ ※ Linear Spring Linear Spring is analogous to a capacitor capacitor in electric networks
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Modeling of Mechanical System ElementsModeling of Mechanical System Elements
• Friction Friction
― Viscous Friction
Figure. Dashpot for viscous friction
dt
tdyBtf
)()(
The force equationThe force equation
: A retarding force that is a linear relationship between the applied force and velocity.
f
y
BSlope( B : Viscous frictional coefficient )( B : Viscous frictional coefficient )
0
제어응용연구실 11
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
― Static Friction
: A retarding force that tends to prevent motion from beginning.
f
y0
sF
0)()(
ysFtf
― Coulomb Friction
: A retarding force that has a constant amplitude with respect to the change of velocity.
f
y0
cF
dttdy
dttdyFtf c /)(
/)()(
cF
제어응용연구실 12
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
Rotational MotionRotational Motion
J torques
• InertiaInertia
: An element that stores the kinetic energy of rotational motion
Figure. Torque-inertia system
2
2 )()()()(
dt
tdJ
dt
tdJtJtT
221 MrJ
For instance, the inertia of a circular disk or shaft about its geometric axis is given byFor instance, the inertia of a circular disk or shaft about its geometric axis is given by
The torque equationThe torque equation
제어응용연구실 13
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
22
22
222
25-
cm-g 980sec-cm-g 1
in-oz 386 sec-in.-oz 1
ft-lb 32.2sec-in.-oz 192 sec-ft-lb 1
sec-in.-oz 10 1.417 cm-g 1 : Inertia
ft-lb 0.00521 in.-oz 1
in.-oz 192 ft -lb 1
in.-oz 0.0139 cm-g 1 : Torque
deg/sec 6 rpm 1
rad/sec 1047.060
2 rpm 1 : Velocity Angular
deg 3.57180 rad 1 :nt DisplacemeAngular
―
―
―
―
제어응용연구실 14
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
• Torsional SpringTorsional Spring
Figure. Torque-torsional spring system
The torque equationThe torque equation
)()( tKtT
• Friction for Rotational MotionFriction for Rotational Motion
dt
tdBtT
)()( : friction Viscous
0)()( : friction Static
sFtT
dttd
dttdFtT c /)(
/)()( : friction Coulomb
제어응용연구실 15
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
Conversion ( Conversion ( Between Translational and Rotational MotionsBetween Translational and Rotational Motions ))
Figure. Rotary-to-linear motion Control system ( lead screw)
2
2
L
g
WJ
• A load may be controlled to move along a straight line through a rotary motor-and-screw assembly.
제어응용연구실 16
Figure. Rotary-to-linear motion Control system ( rack and pinion)
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
Figure. Rotary-to-linear motion Control system ( belt and pulley)
• A similar situation in which a rack -and-pinion is used as a mechanical linkage.
• The control of a mass through a pulley by a rotary motor, such as the control of a printwheel in an electric typewriter.
22 rg
WMrJ
제어응용연구실 17
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
Gear Trains, Levers, and Timing BeltsGear Trains, Levers, and Timing Belts
Figure. Gear train
2211
2211
1221
.3
.2
.1
TT
rr
NrNr
2
1
1
2
2
1
1
2
2
1 r
r
N
N
T
T
제어응용연구실 18
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
Figure. Gear train with friction and inertia
1. The torque equation for gear 2
2
22
222
22
22
)()()(
cFdt
tdB
dt
tdJtT
2. The torque equation on the side of gear 1
)()()(
)( 11
11
112
12
1 tTFdt
tdB
dt
tdJtT c
• Example of torque equationExample of torque equation
제어응용연구실 19
Modeling of Mechanical System ElementsModeling of Mechanical System Elements
3. Converte
2
22
2
112
2
2
12
12
2
2
2
12
2
11
)()()()(
cFN
N
dt
tdB
N
N
dt
tdJ
N
NtT
N
NtT
2
22
2
12
2
1
22
12
2
2
1
22
12
2
2
1
: orquefriction t Coulomb : Torque
:locity Angular ve :t coefficienfriction Visous
:nt displacemeAngular : Inertia
cFN
NT
N
N
N
NB
N
N
N
NJ
N
N
※ The following quantities are obtained when reflecting from gear 2 to gear 1 :
제어응용연구실 20
Equations of Mechanical SystemEquations of Mechanical System
제어응용연구실 21
Equations of Mechanical SystemEquations of Mechanical System
Equations of Mechanical SystemEquations of Mechanical System
Figure. Mass-spring-friction system Figure. Free-body diagram
)()()(
)(2
2
tKydt
tdyB
dt
tydMtf
)(1
)()()(
2
2
tfM
tyM
K
dt
tdy
M
B
dt
tyd
• Example ofExample of Mechanical system Mechanical system
제어응용연구실 22
)()(
21 txdt
tdx
)(1
)()()(
212 tf
Mtx
M
Btx
M
K
dt
tdx
KBsMssF
sY
2
1
)(
)(
Equations of Mechanical SystemEquations of Mechanical System
제어응용연구실 23
• Example ofExample of Mechanical system Mechanical system
Equations of Mechanical SystemEquations of Mechanical System
제어응용연구실 24
)()()()()(
)( 21121
21
2
11 tytyKdt
tdy
dt
tdyB
dt
tydMgMtf
)()()(
)()()()(
2222
2
222
2
221121
12 tyKdt
tydB
dt
tydMtytyK
dt
tdy
dt
tdyBgM
)()( 11 tytx
dt
tdx
dt
tdytx
)()()( 11
2
)()( 23 tytx
dt
tdx
dt
tdytx
)()()( 32
4
Equations of Mechanical SystemEquations of Mechanical System
― Force equations for the system
― State variables
제어응용연구실 25
)()(
21 txdt
tdx
gtfM
txtxM
Btxtx
M
K
dt
tdx )(
1)]()([)]()([
)(
142
1
131
1
12
)()(
43 txdt
tdx
gtxBBM
txM
KKtx
M
Btx
M
K
dt
tdx
)()(
1)()()(
)(421
23
2
212
2
11
2
14
)()( 11 txty
)()( 32 txty
Equations of Mechanical SystemEquations of Mechanical System
― State equations
― Output equations
제어응용연구실 26
Equations of Mechanical SystemEquations of Mechanical System
― State diagram
제어응용연구실 27
s
gKMKMKMsBMBMBMsMM
sFKKsBBsM
gsY
1221111221112
21
21212
21
)(
)()()(
)(
s
gMMKsMMBsMMsF
KsBsY
)()(
)()( 2112112
21112
2121212
2121211
321211
421
)(])([
])([
KKsKBBKsBBMKKKM
sMBBBMsMM
Equations of Mechanical SystemEquations of Mechanical System
― Applying the gain formula to the state diagram
제어응용연구실 28
• Example ofExample of Mechanical system Mechanical system
uemotor torq)( tTm
tcoefficienfriction ousmotor viscmB
ntdisplaceme laod)( tLshaft theofconstant springK
velocitylaod)( tL
inertiamotor mJ
inertia loadLJ
citymotor velo)( tmntdisplacememotor m
▲
▲
▲
▲
▲
▲
▲
▲
▲
Figure. Motor-load system
Equations of Mechanical SystemEquations of Mechanical System
제어응용연구실 29
)(1
)]()([)()(
2
2
tTJ
ttJ
K
dt
td
J
B
dt
tdm
mLm
m
m
m
mm
2
2 )()]()([
dt
tdJttK L
LLm
)(1
)]()([)()(
2
2
tTJ
ttJ
K
dt
td
J
B
dt
tdm
mLm
m
m
m
mm
)]()([)(
2
2
ttJ
K
dt
tdLm
L
L
Equations of Mechanical SystemEquations of Mechanical System
― Torque equations for the system
제어응용연구실 30
Equations of Mechanical SystemEquations of Mechanical System
)()()(
231 txtxdt
tdx
)()(
12 tx
J
K
dt
tdx
L
)(1
)()()(
313 tT
Jtx
J
Btx
J
K
dt
tdxm
mm
m
m
― State equations
― State diagram
제어응용연구실 31
])([ )(
)(
)(
)(23
23
KBsJJKsJBsJJs
KsJ
ssT
sX
sT
s
mLmLmLm
L
mm
m
])([ )(
)(
)(
)(23
2
KBsJJKsJBsJJs
K
ssT
sX
sT
s
mLmLmLmmm
m
Equations of Mechanical SystemEquations of Mechanical System
― Applying the gain formula to the state diagram