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System Modelling: 1st Order Models


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Introduction to physical modelling of low order linear systems. For full slide set and supporting material go to:

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System Modelling: 1st Order Models

  1. 1. 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 . 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 .
  2. 2. Anthony Rossiter Department of Automatic Control and Systems Engineering University of Sheffield Week 5 Systems Modelling 1 st Order Models
  3. 3. <ul><li>What is a derivative and what does to differentiate mean ? </li></ul><ul><li>Example of terminology </li></ul><ul><li>Meaning of derivative </li></ul><ul><li>Differentiate </li></ul><ul><li>Typical derivatives in mechanical and electrical components </li></ul><ul><li>2nd derivatives </li></ul><ul><li>Derivatives in electrical components </li></ul><ul><li>Derivatives in mechanical components </li></ul><ul><li>Derivatives in fluid and heat flow </li></ul><ul><li>Properties of components </li></ul><ul><li>Rotational systems </li></ul><ul><li>1st order mechanical systems </li></ul><ul><li>1st order rotational systems </li></ul><ul><li>1st order electrical systems </li></ul><ul><li>Analogies </li></ul><ul><li>Interpreting models </li></ul><ul><li>Heat </li></ul><ul><li>Fluid flow </li></ul><ul><li>Case study </li></ul><ul><li>Equivalent electrical circuit </li></ul><ul><li>Equivalent circuit </li></ul><ul><li>Changing the model </li></ul><ul><li>Summary </li></ul><ul><li>Parachutist </li></ul><ul><li>Elevator or flap on an aircraft </li></ul><ul><li>Acceleration of a bike </li></ul><ul><li>Challenge activities – year 1 </li></ul><ul><li>RULES </li></ul><ul><li>Credits </li></ul>Contents
  4. 4. What is a derivative and what does to differentiate mean ? <ul><li>We need only: </li></ul><ul><li>The notation for derivative </li></ul><ul><li>What is a derivative ? </li></ul><ul><li>Requires two variables which are related through some function. For instance </li></ul><ul><li>x(t) = 4t+3 </li></ul><ul><li>  The full terminology is </li></ul><ul><li>The derivative of variable 1 with respect to variable 2 </li></ul>
  5. 5. Example of terminology <ul><li>Take x = 4t +3, then we could have </li></ul><ul><li>The derivative of x with respect to t (denoted dx/dt) </li></ul><ul><li>or </li></ul><ul><li>The derivative of t with respect to x (denoted dt/dx) </li></ul>
  6. 6. Meaning of derivative <ul><li>Derivative is a gradient. </li></ul><ul><li>dx/dt is the gradient of x(t) when t is on the horizontal axis and x on the vertical axis. </li></ul><ul><li>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 </li></ul><ul><li>COMPLETE BOXES 4A AND 4B </li></ul><ul><li>Summary: </li></ul><ul><li>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. </li></ul>
  7. 7. Differentiate <ul><li>This means, to find the derivative </li></ul><ul><li>This module is not concerned with how to differentiate, only with how to interpret the result. </li></ul><ul><li>What do dy/dx or dr/dt mean? </li></ul>
  8. 8. Typical derivatives in mechanical and electrical components <ul><li>We usually differentiate w.r.t. time </li></ul><ul><li>Velocity is the derivative of displacement wrt. time </li></ul><ul><li>Acceleration is the derivative of velocity wrt. time </li></ul><ul><li>Current is the derivative of charge wrt time </li></ul><ul><li>Power (W) is the rate of change of energy(E) with time </li></ul><ul><li>Flow rate (Q) is the rate of change of volume(V) with time. </li></ul>
  9. 9. 2 nd derivatives <ul><li>Higher order derivatives have a special notation. </li></ul><ul><li>e.g. acceleration is the 2 nd derivative of displacement </li></ul><ul><li>Do not interpret the superscripts as powers. They are notation which is specific to derivatives. </li></ul>
  10. 10. Derivatives in electrical components <ul><li>Lenz’s law: the equation of an inductor (L in henry). </li></ul><ul><li>Faradays law: the equation of a capacitor (C in farad) </li></ul>
  11. 11. Derivatives in mechanical components <ul><li>The equation of a damper </li></ul><ul><li>The equation of a mass (Newton’s law) </li></ul><ul><li>The equation of a spring </li></ul>
  12. 12. Derivatives in fluid and heat flow <ul><li>The equation of a flow Q through a restriction </li></ul><ul><li>The equation of heat flow through an object </li></ul><ul><li>Heat stored in an object </li></ul><ul><li>Pressure of fluid in a container </li></ul>
  13. 13. Properties of components <ul><li>Dissipate heat: Resistor and damper. </li></ul><ul><li>Resists velocity </li></ul><ul><li>Store energy due to change in state: Spring and capacitor. </li></ul><ul><li>Resists displacement </li></ul><ul><li>Possess energy if state is moving: inductor and mass. </li></ul><ul><li>Resists change in velocity </li></ul>
  14. 14. Rotational systems <ul><li>Analogous to linear mechanical systems </li></ul><ul><li>Torsional spring (resilient shaft) </li></ul><ul><li>Torsional viscous damping </li></ul><ul><li>Rotating Inertia </li></ul>
  15. 15. 1 st order mechanical systems <ul><li>Consider a mass in parallel with a damper . </li></ul><ul><li>Complete BOX 4C </li></ul><ul><li>The force is shared between them. </li></ul><ul><li>1 st order ODE because linear in state and 1 st derivative </li></ul>
  16. 16. 1 st order mechanical systems <ul><li>Consider a spring in parallel with a damper . </li></ul><ul><li>Complete box 4D </li></ul><ul><li>The force is shared between them. </li></ul><ul><li>1 st order ODE because linear in state and 1 st derivative </li></ul>
  17. 17. 1 st order rotational systems <ul><li>By analogy one can form models like </li></ul><ul><li>Computation of torque – complete box 4E. </li></ul>
  18. 18. 1 st order electrical systems <ul><li>Complete boxes 4F-4G, that is find models for </li></ul><ul><li>Resistor in series with an inductor </li></ul><ul><li>Resistor in series with a capacitor </li></ul>
  19. 19. Analogies <ul><li>What analogies are there between 1 st order electrical and mechanical systems? </li></ul><ul><li>Resistor+capacitor (in series) Damper+spring (parallel) </li></ul><ul><li>Resistor+inductor (in series) Damper +mass (parallel) </li></ul><ul><li>Link this to: </li></ul><ul><li>Analogies between variables </li></ul><ul><li>Kirchhoff or force balance </li></ul>
  20. 20. Reminders of analogies <ul><li>Force </li></ul><ul><li>Velocity </li></ul><ul><li>Displacement </li></ul><ul><li>Spring </li></ul><ul><li>Mass </li></ul><ul><li>Damper </li></ul><ul><li>Parallel </li></ul><ul><li>Series </li></ul><ul><li>Voltage </li></ul><ul><li>Current </li></ul><ul><li>Charge </li></ul><ul><li>Capacitance </li></ul><ul><li>Inductance </li></ul><ul><li>Resistance </li></ul><ul><li>Series </li></ul><ul><li>Parallel </li></ul>
  21. 21. Interpreting models <ul><li>Same model implies same behaviour! </li></ul><ul><li>If you understand behaviour of mechanical systems, you also understand that of electrical systems. </li></ul><ul><li>Note: </li></ul><ul><li>This is exponential convergence to a fixed point. </li></ul>
  22. 22. Heat <ul><li>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. </li></ul>
  23. 23. Fluid flow <ul><li>Complete Box 4H </li></ul><ul><li>NOTE: Both fluid and heat flow are also given by 1 st order ODE – same behaviour again! </li></ul>
  24. 24. Case study <ul><li>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. </li></ul><ul><li>First do a single tank: </li></ul>
  25. 25. Equivalent electrical circuit Resistor and capacitor in series
  26. 26. Add a second tank <ul><li>Now flow is leaving tank 1 as well as entering. Tank 2 also has dynamics. </li></ul>
  27. 27. Equivalent electrical circuit This is a resistor/capacitor with an extra parallel loop just around the Capacitor.
  28. 28. Equivalent circuit <ul><li>Can you see extension to a 3 rd tank? </li></ul>R1 R2 vc1 vc2 C1 C2 V
  29. 29. Changing the model <ul><li>How would the modelling change if the input was a flow rate (say from a tap) rather than a pressure? </li></ul><ul><li>Can you simplify these models and hence simulate their behaviour as studied in ACS112? </li></ul>
  30. 30. Summary <ul><li>You should have almost finished self assessment 1. </li></ul><ul><li>You should consider overlaps with ACS112. </li></ul><ul><li>We model so as to understand behaviour and hence do design. Can you choose parameters to get desired behaviour? </li></ul><ul><li>Questions? </li></ul>
  31. 31. Parachutist <ul><li>What attributes would a model have? </li></ul><ul><li>Gravitational force. </li></ul><ul><li>Damping from wind resistance? </li></ul><ul><li>Any spring effects? </li></ul>
  32. 32. Elevator or flap on an aircraft <ul><li>How does wing force depend on angle of flap? </li></ul><ul><li>How does actuator force apply? </li></ul><ul><li>What form of actuator, motor, pneumatic, manual, … ? </li></ul>MAIN WING Force actuator on wing WIND FORCE ACTUATOR FLAP
  33. 33. Acceleration of a bike <ul><li>How does the force on the pedal translate to acceleration? </li></ul><ul><li>What is the impact of braking? </li></ul><ul><li>What do the gears do? </li></ul>
  34. 34. Bike acceleration <ul><li>Is there enough information here to model the bike? </li></ul><ul><li>What about friction – where does this occur? </li></ul>Pedal force Force on road Front gear diameter Pedal arm length Rear gear diameter Wheel diameter
  35. 35. Challenge activities – year 1 Covers ACS111, 112, 123, 108 By Anthony Rossiter
  36. 36. RULES <ul><li>Answers should be uploaded to the relevant folder on discussions in the ACS108 MOLE site. </li></ul><ul><li>Where figures are required, these should be either: </li></ul><ul><ul><li>in jpg as separate attachments with the solution in text, </li></ul></ul><ul><ul><li>or incorporated into a word or powerpoint or pdf document with the solution. </li></ul></ul><ul><li>The judging will be based on a combination of accuracy and timing. A nearly correct early submission may outscore a perfect late submission. </li></ul><ul><li>Solutions and prizes will be presented in lectures. </li></ul>
  37. 37. Challenge for weeks 4-6 <ul><li>A 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. </li></ul><ul><li>Find all the values of  such that the overall resistance is a maximum (i.e. derivative is zero). </li></ul><ul><li>Plot a MATLAB graph showing how the overall resistance varies with  to validate your answer. </li></ul>1 R1 R2 R3
  38. 38. <ul><li>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. </li></ul><ul><li>© University of Sheffield 2009 </li></ul><ul><li>                 This work is licensed under a Creative Commons Attribution 2.0 License . </li></ul><ul><li>Where Matlab® screenshots are included, they appear courtesy of The MathWorks, Inc. </li></ul><ul><li>The name of the University of Sheffield and the logo are the name and registered marks of the University of Sheffield. To the fullest extent permitted by law the University of Sheffield reserves all its rights in its name and marks which may not be used except with its written permission. </li></ul><ul><li>The JISC logo is licensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK: England & Wales Licence.  All reproductions must comply with the terms of that licence. </li></ul><ul><li>The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original publisher. </li></ul>