chapter 3 mathematical models of systems · 2020. 10. 8. · 1) introduction a mathematical model...
TRANSCRIPT
CHAPTER 3
Mathematical Models of Systems
Contents
1) Introduction
2) Differential Equations of Physical Systems
3) Transfer Function of Linear Systems
1) Introduction
A mathematical model is a set of equations (usually differential
equations) that represents the dynamics of systems.
In practice, the complexity of the system requires some assumptions
in the determination model.
The equations of the mathematical model may be solved using
mathematical tools such as the Laplace Transform.
Before solving the equations, we usually need to linearize them.
2) Differential Equations
Physical law of the process Differential Equation
Electrical system (Kirchhoff’s laws)
Mechanical system (Newton’s laws)
How do we obtain the equations?
Examples:
i.
ii.
2) Differential Equations cont..
Example: Springer-mass-damper system
Assumption: Wall friction is a viscous force.
The time function of
r(t) sometimes
called forcing
function
Linearly proportional
to the velocity)()( tbvtf
2) Differential Equations cont..
Example: Springer-mass-damper system
Newton’s 2nd Law:
)()()()( tMatrtkytbv
)()()()(
2
2
trtkydt
tdyb
dt
tydM
2) Differential Equations cont..
Example: RLC Circuit
t
tvdiC
tRidt
tdiL
0
)()(1
)()(
0)( cLR VVVtv
3) The Transfer Function
The transfer function of a linear system is the ratio of the Laplace
Transform of the output to the Laplace Transform of the input variable.
The transfer function is given by the following.
)(
)()(
sInput
sOutputsG
Y(s)R(s)kbsMs 2
1
kbsMssR
sYsG
2
1
)(
)()(
3) The Transfer Function cont..
(A) Electrical Network Transfer Function
Component V-I I-V V-Q Impedance Admittance
3) The Transfer Function cont..
(i) One Loop Electrical Network.
Problem: Obtain the transfer function for the following RC network.
3) The Transfer Function cont..
Problem: Obtain the transfer function for the following RLC network.
Answer:
3) The Transfer Function cont..
1
11
2
22
1
11
2
22
1
1
2
2
1
2
2
22
1
1
1
1
2
11
1
1
1
1
1
1
)(
)(
)(
)(
)(
)(
)1(
1)(
1
1)(
)1()(
)(
)(
)(
R
sCR
sC
sRC
R
sCR
sC
sRC
RsC
sCR
sV
sV
sZ
sZ
sV
sV
from
sCRsZ
RsC
sZ
sZ
sZ
sV
sV
i
o
i
o
i
o
Solution:
(B) Mechanical System Relationship
3) The Transfer Function cont..
3) The Transfer Function cont..
Free Body Diagram Equivalent.
Equation of motion.
(1) positive direction of motion to the right.
(2) use Newton’s Law, summing all the force & setting the sum equal to zero.
constant spring
scousitydamping/vi
k
fv
Figure : (a) Free-body diagram, (b) Laplace Transformed
3) The Transfer Function cont..
Derive the transfer function for the mechanical displacement.
(a) Free body diagram equivalent.
Where,
Fa is the applied force (N)y, mass displacement (m)f, viscous friction (N.m.rad-1.s-1)k, spring constant (N.m-1)
(b) The differential equation. Newton’s law; F = ma
where m is a mass (kg), a acceleration (m.s-2) and F force (N).
From Lenz’s law,
(c) Laplace Transform- Transfer function.
Taking Laplace transform and assume zero initial condition
Daya, F
m
Anjakan, y
Daspot
a
f
aFkydt
dyf
dt
ydm
2
2
)()()()(2 sFskYssYfsYms a
3) The Transfer Function cont..
Rearrange the previous equation,
ksfmssF
sY
sFksfmssY
sFskYssYfsYms
a
a
a
2
2
2
1
)(
)(
)()(
)()()()(
Figure : (a) Mass, spring, and damper system; (b) Block Diagram