Post on 23-Jan-2021
5/4/2016
1
Chapter three
Laith Batarseh
Mathematical modeling
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Mathematical modeling
Example [1]: mass-spring-damper system
Find the transfer function for the mechanical system shown in the fig. assumezero initial conditions and the desired output is v1
Solution:To find the mathematical model, youcan apply Newton’s 2nd law to eachmass. This procedure produce two DE
0.
)(
0
21212
2
211211
1
t
dttvkvvbdt
tdvM
trvbtvbbdt
tdvM
Solved examples
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Mathematical modeling
Now , use Laplace transform to transform these equations into the s-domain.Remember, we assumed zero initial conditions (i.e. v1(0) = v2(0) = 0.
0 )( 2121222112111
s
sVksVsVbssVMsRsVbsVbbssVM
Rearrange these equations
0- )( 21211211211
sV
s
kbsMsVbsRsVbsVbbsM
These are two equations with tow unknowns: V1(s) and V2(s). To solve theseequations we can use Cramer’s rule. First represent this system of equations bymatrix notation
02
1
121
1211 sR
V
V
s
kbsMb
bbbsM
Example [1]: mass-spring-damper system
Mathematical modeling
The solution of V1(s) is calculated as:
To find the transfer function (G(s)): G(s) = V1(s) / R(s)
2
112211
121
bskbsMbbsM
sRskbsMsV
2
112211
12
bskbsMbbsM
skbsMsG
We can represent the transfer function in terms of the displacement x(t) byrepresenting the velocity (v) in terms of (x) : v(t)= dx(t)/dt. Now, transfer it to s-domain: V1(s) =sX1(s). Remember, we assume zero initial conditions. Apply this toG(s) expression:
s
sG
ssR
sV
sR
sX 11
Example [1]: mass-spring-damper system
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Mathematical modeling
Example [2]: DC motor
Find the transfer function for the mechanical system shown in the fig. assumezero initial conditions and the desired output is v1
Mathematical modeling
Example [2]: DC motor
Solution:
The DC motor converts direct current (DC) electrical energy into rotationalmechanical energy. The transfer function of the DC motor will be developed for alinear approximation to an actual motor, and second-order effects, such as hysteresisand the voltage drop across the brushes, will be neglected. The air-gap flux ф of themotor is proportional to the field current, provided the field is unsaturated, so that:
ф = Kfif ---- (1)
The motor torque (Tm) is related to air-gap flux (ф) by:
Tm = K1 ф ia(t) = K1Kfif(t)ia(t) ---- (2)
as you can see, there are two controlling variables (if and ia). So, we can chose one of them as a controlling input.
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Mathematical modeling
Example [2]: DC motor
Solution:
Assume the controlling current is the filed current (if) and so the armature current(ia) is constant. Now, Eq(2) become:
Tm = K1 ф ia(t) = [K1Kfia] if(t) = Kmif(t)---- (3)
Transfer Eq(3) using Laplace transformation to the s-domain
Tm(s) = KmIf(s)---- (4)Where: Km is the motor constant.The motor torque (Tm) is the summation of torque delivered to the load (TL) and thedisturbance torque (Td), mathematically:
Tm(s) = TL(s) + Td(s) ---- (5)
Where the load torque (TL) can be calculated as: . Note that the output in our case is Ө or ω. Assume zero initial conditions and transform it to s-domain:
TL(s) = Js2Ө(s)+bsӨ(s)---- (6)
... bJTL
Mathematical modeling
Example [2]: DC motor
Solution:
We can relate the field current to the field voltage as: . Again
assume zero initial conditions and transform this relation to s-domain:
dt
diLiRv
f
ffff
7
ff
f
fffffsLR
sVsIsIsLRsV
Assume that the disturbance torque is neglected (i.e. Td = 0). Rearrange equations 1-7 to find relation between the input (If(s)) and the output (Ө(s)). This relation is the required transfer function (G(s)):
8//
/
ff
fm
ff
m
LRsJbss
JLK
RsLbJss
K
sV
ssG
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Mathematical modeling
Example [2]: DC motor
Solution:
Now we can represent this mathematical process by the following block diagram
ff sLR
1Km bJs
1
s
1Vf(s) If(s) Tm(s)
_
+
Td(s)
TL(s)
speed
ω(s) Ө(s)
Field Load
This step can be done if take thespeed (ω) as the desired output. Themain difference in calculations willbe using instead of bJTL
.
... bJTL
Mathematical modeling
Example [2]: DC motor
Solution:
If the armature current used to control the system:
Tm = K1 ф ia(t) = [K1Kfif] ia(t) = Kmia(t)---- (9)
Transfer Eq(9) using Laplace transformation to the s-domain
Tm(s) = KmIa(s)---- (10)The motor torque (Tm) is the summation of torque delivered to the load (TL) and thedisturbance torque (Td), mathematically:
Tm(s) = TL(s) + Td(s) ---- (10)
Where the load torque (TL) can be calculated as: . Note that the output in our case is Ө or ω. Assume zero initial conditions and transform it to s-domain:
TL(s) = Js2Ө(s)+bsӨ(s)---- (11)
... bJTL
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Mathematical modeling
Example [2]: DC motor
Solution:
We can relate the armature current to the armature voltage as:
Again assume zero initial conditions and transform this relation to s-domain:
Vb(s) is the back electromotive force and is given as :Vb(s) = Kbω(s). Rearrange to find
G(S)
12
b
aaaaa v
dt
diLiRv
13
aa
baabaaaa
sLR
sKsVsIsVsIsLRsV
14
mbaa
m
KKbJssLRs
K
sV
ssG
Mathematical modeling
Example [2]: DC motor
Solution:
14
mbaa
m
KKbJssLRs
K
sV
ssG
fa
m
sLR
K
Kb
bJs
1
s
1Va(s)
Back electromotive force
speed
ω(s) Ө(s)
Armature
Tm(s)
_
+
Td(s)
+
-
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Mathematical modeling
Example [3]: Liquid level system
The governing equations can be derived as:
R
hqo
dtqqCdh oi
iRqhdt
dhRC
Laplace
)()()( sRQsHssHRC
1)(
)(
RCs
R
sQ
sH
i
1
1
)(
)(
RCssQ
sQ
i
o
Where:C: tank capacitance h: head R: pipe resistance q : flow rate
= change in liquid stored/change in head=cross-sectional area of the tank
Mathematical modeling
Exercise
Governing equations
1
211
R
hhq
11
1 qqdt
dhC
2
2
2 qR
h
212
2 qqdt
dhC
Find :1. Q1(s)/Q(s)2. Q2(s)/Q(s)3. Q2(s)/Q1(s)
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Mathematical modeling
Exercise
1
211
R
hhq
11
1 qqdt
dhC
2
2
2 qR
h
212
2 qqdt
dhC
Mathematical modeling
Exercise
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Mathematical modeling
Exercise
Mathematical modeling
Example [4]: thermal system
A simple thermal system is shown in the figure below
For conduction or convection heat transfer
convectionfor
conductionfor
HAK
x
kAK
Where:q: heat flowK: coefficient ∆Ө = temperature difference
Thermal Resistance and Thermal Capacitance.
KR
1
kcal/sec rate, flowheat in change
C,difference turein tempera change o
The thermal resistance for conduction or convection heat transfer is given by
The thermal capacitance C is defined by
mcC C, turein tempera change
kcal stored,heat in changeo
specific heat of substance, kcal/kg °C
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Mathematical modeling
Example [4]: thermal system
Consider the system shown in Figure. If
kcal/sec rate,input heat state-steadyH
Ckcal/ e,capacitanc thermalC
sec/kcal C ,resistance thermalR
C kcal/kg liquid, ofheat specificc
kg in tank, liquid of massM
kg/sec rate, flow liquid state-steadyG
C liquid, outflowing of re temperatustate-steady
C liquid, inflowing of re temperatustate-steady
o
i
Assume that the temperature of the inflowing liquid is kept constant and that the
heat input rate to the system (heat supplied by the heater) is suddenly changed
from to where hi represents a small change in the heat input rate. The
heat outflow rate will then change gradually from to .The temperature of
the out-flowing liquid will also be changed from to .
H ihH H ohH
o o
Mathematical modeling
Example [4]: thermal system
For the pre-described case, ho, C, and R are obtained, respectively, as:
GchR
McC
Gch
o
o
1
The heat-balance equation for this system is
ioioi Rhdt
dRChh
dt
dCdthhCd
Apply Laplace: 1
RCs
R
sH
ssRHssRCs
i
i