system modelling: 1st order models

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?

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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

12

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

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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

;

17

1st order rotational systems

By analogy one can form models like

Computation of torque – complete box 4E.

kdt

dBB

dt

dJ ;

18

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(()(

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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.

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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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