TMHP51 Servomechanisms (HT2012)
Lecture 02
Governing Mathematical Model
Stationary Model
Dynamic Model
Magnus Sethson
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1
Servo System Model
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2
Servo System
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3
Stationary
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Mathematical Model (Stationary)
Orifice
r
2
q1 = Cq wxv
(Ps p1 ) (Positive chamber)
⇢
r
2
q2 = Cq wxv
(p2 Pt ) (Negative chamber)
⇢
Continuity
qin
qout = 0 (Non-moving piston)
dV
qout =
(Moving piston)
dt
qin
Force Balance
p 1 A1
p 1 A1
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p 2 A2
p2 A2 = FL (Non-moving piston)
FL = BL ẋp (Moving piston)
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5
Dynamic Model
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6
Dynamic Mathematical Model Principle
Orifice
r
2
q = Cq wxv
(Pa
⇢
Pb )
Continuity
qin
qout
dV
V dp
=
+
dt
e dt
Force Balance
p 1 A1
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P2 A 2
FL
BL ẋp = Mt ẍp
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Simplified Hydraulic Linear Actuator (cont.)
r
2
(Ps
⇢
r
2
q2 = Cq wxv
(p2
⇢
A1 = A2 = Ap
q1 = Cq wxv
q1 = q2 = q
p1 )
Pt )
r
2
q = Cq wxv
(Ps p1 )
⇢
r
2
q = Cq wxv
(p2 Pt )
⇢
Pt ⇡ 0 (Tank pressure) ) p2 = (Ps
pL ⌘ p 1
Orifice
p1 )
p2 (Definition when A1 = A2 )
p2 = p1 Ps + p1 = 2p1 Ps )
Ps + p L
p1 =
2
Ps + p L
Ps p L
p 2 = Ps p 1 = Ps
=
2
2
s
s
2
Ps + p L
(Ps pL )
q = (q1 ) = Cq wxv
(Ps
) = Cq wxv
⇢
2
⇢
s
s
2 Ps p L
(Ps pL )
q = (q2 ) = Cq wxv
(
) = Cq wxv
⇢
2
⇢
p L = p1
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8
Simplified Hydraulic Linear Actuator (cont.)
dV1
V1 dp1
q1 =
+
dt
e dt
dV2
V2 dp2
+
- q2 =
dt
e dt
q1 = q2 (From previous)
Continuity
V1 ⇡ V2 (Simplifying assumption)
V1 + V 2 = V t
dV1
= Ap ẋp
dt
dV2
= Ap ẋp
dt
Ps + p L
p1 =
2
Ps p L
p2 =
2
dp1
1 dpL
)
=
(Ps = const.)
dt
2 dt
dp2
1 dpL
)
=
dt
2 dt
1 Vt dpL
q = (q1 ) = +Ap ẋp +
4 e dt
1 Vt dpL
q = (q2 ) = Ap ẋp
4 e dt
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Simplified Hydraulic Linear Actuator
Force Balance
p 1 A1
p 2 A2
Bp ẋp
FL = Mt ẍp
A1 = A2 = Ap and p1
p L Ap
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Bp ẋp
FL = MT ẍp
p2 = pL )
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Dynamic Mathematical Model
Non-linear model
8
p
A
B
ẋ
F
=
M
ẍ
>
L
p
p
p
L
t
p
>
>
<
q = Ap ẋp + 41 Vet ṗL
>
q
>
>
: q = Cq wxv 1 (Ps pL )
⇢
xv : Input signal
xp : Output signal
FL : External disturbance
xp , ẋp , pL : Internal states
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11
✓
8
pL Ap Bp ẋp FL = Mt ẍp
>
>
>
<
q = Ap ẋp + 41 Vet ṗL
>
q
>
>
: q = Cq wxv 1 (Ps pL )
⇢
Now what?
f
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Two options
t
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Engineering tasks involving the model
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Predict stationary stiffness
Predict system gain
Predict resonance frequencies
Estimate internal damping
Calculate energy consumption
Design a suitable regulator
Identify unstable operating conditions
Regulator stability margins
Step response
Saturation effects
Identify limit cycle conditions
Dynamic response
Bandwidth limits
Rise time
Predict overall system stability
Temperature effects
Wear predictions
Damage spectrum
Viscosity effects
Bulk modulus effects
Obliteration
Cavitation
Noise
Overload conditions
Supply requirements
Modal analysis
Fatigue analysis
Service interval estimation
Condition monitoring
Operational life estimation
Duty cycle
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Frequency and Time domains
f
t
Frequency Time
Classical
Analytical
Algebraic
General behavior
Linearized
Well-defined application
Almost always stable
“Pen & Paper”
Limited complexity
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Modern
Computerized
Numerical
Detailed behavior
Non-linear
Implicit models
Unlimited application
Stability always an issue
“Hacking”
Virtually unlimited complexity
Real-time
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Next Lecture:
Wednesday 7 October, 13:15, P34
Magnus Sethson
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