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Overheads for ACS111 Systems Modelling
© University of Sheffield 2009. This work is licensed under a Creative Commons Attribution 2.0 License.
The following resources are taken from the 2009/2010 ‘introduction to modelling’ lecture as forming a part of Systems Modelling for first year engineering undergraduate. For supporting and other documentation for this lecture and others on the course please see http://controleducation.group.shef.ac.uk/OER_index.htm.
The main focus is on electrical and mechanical systems, but there is also some discussion of dc motors, fluids and heat as well as an introduction to time series modelling. The main emphasis is on why modelling is important and how to go about doing this from first principles (e.g. Kirchhoff's laws, Newton's Laws, etc.). Given the focus is on new students arriving at University, there is no attempt to develop models beyond second order.
The resources here include the lecture hand out (pdf) which includes embedded tutorial questions, some powerpoints for structuring lectures , flash animations to step through modelling process for electrical circuits and a large data base of CAA developed on WebCT (here provided in a zip file). The lecture notes also contains a brief overview on usage for lecturing staff.
These were developed at the University of Sheffield and authored by J A Rossiter from the Department of Automatic Control and Systems Engineering.
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Anthony Rossiter
Department of Automatic Control and Systems Engineering
University of Sheffield
www.shef.ac.uk/acse
Week 5
Systems Modelling
1st Order Models
3
• What is a derivative and what does to differentiate mean ? • Example of terminology• Meaning of derivative • Differentiate • Typical derivatives in mechanical and electrical components• 2nd derivatives• Derivatives in electrical components• Derivatives in mechanical components• Derivatives in fluid and heat flow• Properties of components• Rotational systems• 1st order mechanical systems• 1st order rotational systems• 1st order electrical systems• Analogies• Interpreting models• Heat• Fluid flow• Case study• Equivalent electrical circuit• Equivalent circuit• Changing the model• Summary• Parachutist• Elevator or flap on an aircraft• Acceleration of a bike• Challenge activities – year 1• RULES• Credits
Contents
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What is a derivative and what does to differentiate mean ?
We need only:
The notation for derivative
What is a derivative ?
Requires two variables which are related through some function. For instance
x(t) = 4t+3
The full terminology isThe derivative of variable 1 with respect to variable 2
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Example of terminology
Take x = 4t +3, then we could have
• The derivative of x with respect to t (denoted dx/dt)
or
• The derivative of t with respect to x(denoted dt/dx)
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Meaning of derivative
Derivative is a gradient. 1. dx/dt is the gradient of x(t) when t is on the horizontal axis and x on
the vertical axis. 2. dt/dx is the gradient of x(t) [or t(x) = (x-3)/4] when x is on the
horizontal axis and t on the vertical axis COMPLETE BOXES 4A AND 4B
Summary: The derivative of variable 1 w.r.t variable 2 is the gradient of the curve when variable 2 is on the horizontal axis and variable 1 is on the vertical axis.
7
Differentiate
This means, to find the derivative
This module is not concerned with how to differentiate, only with how to interpret the result.
What do dy/dx or dr/dt mean?
8
Typical derivatives in mechanical and electrical components
We usually differentiate w.r.t. time1. Velocity is the derivative of displacement wrt. time
2. Acceleration is the derivative of velocity wrt. time
3. Current is the derivative of charge wrt time
4. Power (W) is the rate of change of energy(E) with time
5. Flow rate (Q) is the rate of change of volume(V) with
time.
dt
dVQ
dt
dEW
dt
dqi
dt
dva
dt
dxv ;;;;
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2nd derivatives
Higher order derivatives have a special notation.
e.g. acceleration is the 2nd derivative of displacement
Do not interpret the superscripts as powers. They are notation which is specific to derivatives.
3
3
2
2
;dt
xd
dt
da
dt
xd
dt
dx
dt
d
dt
dva
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Derivatives in electrical components
1. Lenz’s law: the equation of an inductor (L in henry).
2. Faradays law: the equation of a capacitor (C in farad)
dt
diLv
C
i
dt
dvori
dt
dq
dt
dvCqCv
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Derivatives in mechanical components
1. The equation of a damper
2. The equation of a mass (Newton’s law)
3. The equation of a spring
dt
dxBBvf
2
2
dt
xdM
dt
dvMf
kvdt
dforkxf
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Derivatives in fluid and heat flow
1. The equation of a flow Q through a restriction
2. The equation of heat flow through an object
1. Heat stored in an object
2. Pressure of fluid in a container
Pdt
dVRPQR
2121 ; TTdt
dERTTWR
Wdt
dTCTTCWt );( 21
C
Q
dt
dPPPQCt 21
13
Properties of components
Dissipate heat: Resistor and damper.
Resists velocity
Store energy due to change in state: Spring and capacitor.
Resists displacement
Possess energy if state is moving: inductor and mass.
Resists change in velocity
RiivorBvfvislossheatrate 22:
222:
222 vC
C
qJor
xkJstoredEnergy
22:
22 iLJor
vMJstoredEnergy
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Rotational systems
Analogous to linear mechanical systems
Torsional spring (resilient shaft)
Torsional viscous damping
Rotating Inertia
k
Bdt
dB
dt
dJ
dt
dJ
2
2
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1st order mechanical systems
Consider a mass in parallel with a damper .
Complete BOX 4C
The force is shared between them.
1st order ODE because linear in state and 1st derivative
Bvdt
dvMf
dt
dvMf
Bvffff
M
B
MB
;
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1st order mechanical systems
Consider a spring in parallel with a damper .
Complete box 4D
The force is shared between them.
1st order ODE because linear in state and 1st derivative
kxdt
dxBf
kxfdt
dxBBvffff
k
BkB
;
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1st order rotational systems
By analogy one can form models like
Computation of torque – complete box 4E.
kdt
dBB
dt
dJ ;
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1st order electrical systems
Complete boxes 4F-4G, that is find models for
1. Resistor in series with an inductor
2. Resistor in series with a capacitor
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Analogies
What analogies are there between 1st order electrical and mechanical systems?
Resistor+capacitor (in series) Damper+spring (parallel)
Resistor+inductor (in series) Damper +mass (parallel)
Link this to:
Analogies between variables
Kirchhoff or force balance
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Reminders of analogies
Force
Velocity
Displacement
Spring
Mass
Damper
Parallel
Series
Voltage
Current
Charge
Capacitance
Inductance
Resistance
Series
Parallel
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Interpreting models
Same model implies same behaviour!If you understand behaviour of mechanical
systems, you also understand that of electrical systems.
Note:
This is exponential convergence to a fixed point.
ateafxa
ftxaxdt
dxf ))0(()(
22
Heat
A block of metal at T1o C is placed in an environment at T2o C. The rate of heat transfer from the metal to the environment is given by W = k1(T1-T2). The metal has specific heat k2 J/degree. Find an equation for the temperature of the metal.
)( 1211
2 TTkdt
dTk
23
Fluid flow
Complete Box 4H
NOTE: Both fluid and heat flow are also given by 1st order ODE – same behaviour again!
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Case study
Explain why an arrangement of two tanks connected in series with a high/low pressure supply coming into one, is equivalent to a resistor/capacitor circuit.
First do a single tank:
)(
);(.
tantan
tantan
kink
kkin
PPAR
g
A
gFlow
dt
dP
ghPPPFlowR
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Equivalent electrical circuit
)(
);(.
tantan
tantan
kink
kkin
PPAR
g
A
gFlow
dt
dP
ghPPPFlowR
RC
vv
C
i
dt
dv
idtC
vvviR
cinc
ccin
)(
1);(.
11
11
Resistor andcapacitor in series
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Add a second tank
Now flow is leaving tank 1 as well as entering. Tank 2 also has dynamics.
)(
)(
2
2tan1tan
2
2tan
2
2tan1tan1tan1tan
2
2tan1tan1tan
R
PP
A
g
dt
dP
R
PP
R
PP
A
g
A
gFlow
dt
dP
R
PP
R
PPFlow
kkk
kkkink
kkkin
27
Equivalent electrical circuit
)(
)(
2
2tan1tan
2
2tan
2
2tan1tan1tan1tan
2
2tan1tan1tan
R
PP
A
g
dt
dP
R
PP
R
PP
A
g
A
gFlow
dt
dP
R
PP
R
PPFlow
kkk
kkkink
kkkin
22
212
1
11
22
2111
;RC
vv
dt
dv
C
i
dt
dv
iiR
vv
R
vvi
cccc
ccc
This is a resistor/capacitor with an extra parallel loop just around theCapacitor.
28
Equivalent circuit
Can you see extension to a 3rd tank?
R1
R2
vc1
vc2C1
C2
V
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Changing the model
How would the modelling change if the input was a flow rate (say from a tap) rather than a pressure?
Can you simplify these models and hence simulate their behaviour as studied in ACS112?
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Summary
• You should have almost finished self assessment 1.
• You should consider overlaps with ACS112. • We model so as to understand behaviour and
hence do design. Can you choose parameters to get desired behaviour?
• Questions?
ParachutistWhat attributes would a model have?1.Gravitational force.2.Damping from wind resistance?3.Any spring effects?
Elevator or flap on an aircraft
How does wing force depend on angle of flap?How does actuator force apply?What form of actuator, motor, pneumatic, manual, … ?
MAIN WING
Force actuator on wing
WIND FORCE
ACTUATORFLAP
Acceleration of a bike
How does the force on the pedal translate to acceleration?
What is the impact of braking?
What do the gears do?
Bike accelerationIs there enough information here to
model the bike?What about friction – where does this
occur?
Pedal force
Force on road
Front gear diameter
Pedal arm length Rear gear
diameter
Wheel diameter
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Challenge activities – year 1
Covers ACS111, 112, 123, 108
By Anthony Rossiter
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RULES• Answers should be uploaded to the relevant folder on
discussions in the ACS108 MOLE site.• Where figures are required, these should be either:
– in jpg as separate attachments with the solution in text, – or incorporated into a word or powerpoint or pdf document
with the solution.• The judging will be based on a combination of
accuracy and timing. A nearly correct early submission may outscore a perfect late submission.
• Solutions and prizes will be presented in lectures.
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Challenge for weeks 4-6A flow system has a model equivalent to two parallel resistances:
one path has resistance R1=1 and the other path has an element of resistance 1 and two more series elements of resistance: (i) R2 which is the greater of 0 and cos(2-/4) and (ii) R3 which is the greater of 0 and sin(2-/4) respectively.
1. Find all the values of such that the overall resistance is a maximum (i.e. derivative is zero).
2. Plot a MATLAB graph showing how the overall resistance varies with to validate your answer.
1
R1
R2 R3
This resource was created by the University of Sheffield and released as an open educational resource through the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme.
© University of Sheffield 2009
This work is licensed under a Creative Commons Attribution 2.0 License.
Where Matlab® screenshots are included, they appear courtesy of The MathWorks, Inc.
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