Spring. Ky. F = )s(Ky. )s(F = Friction. Viscous friction. Static friction. Coulumb
friction. Reference: Kuo, B.C. et al. (2004). Automatic control systems, Chapter 4.
Modelling of electromechanical systems
Modelling for control • Modelling principles
Structure of discussion:
• Modelling of electromechanical systems
• • • • • •
• Modelling of other physical systems • State-space modelling • Web-based learning: Transfer function modelling, state space modelling, PID control • Statistical modelling
Mechanical translational elements Mechanical rotational elements Sensors and encoders Electromechanical system models Questions and Answers Further Reading
1
1. Mechanical translational elements Spring Mass System
2
Mechanical elements Reference: Kuo, B.C. et al. (2004). Automatic control systems, Chapter 4.
Mass
F=m
•• d2y = my 2 dt
F=B
Dashpot
F = Ky
Spring
F(s) = ms 2 y(s)
• dy = B y F(s) = Bsy(s) dt
F(s) = Ky(s)
Friction
3
Reference: Murray, R.M. (2003). Principles of Feedback and Control, Caltech.
Viscous friction
Static friction
4
Coulumb friction
Examples
5
6
7
8
9
2. Mechanical rotational elements
10
Conversion between translational and rotational motion
Reference: Kuo, B.C. et al. (2004). Automatic control systems, Chapter 4.
Inertia
•• d 2θ T = J 2 = Jθ dt
Torsional spring
T = Kθ
Lead screw
Rack and pinion
T (s) = Js θ(s) 2
T (s) = Kθ(s)
Belt and pulley The three types of friction for translational motion carry over to rotational • motion e.g. for viscous friction dθ T = B = B θ T (s) = Bsθ(s) dt
Gear Train
r1 N 2 = r2 N 1
θ1r1 = θ 2 r2
T1r1 = T2 r2
Thus
T1 θ 2 N1 ω2 r1 = = = = T2 θ1 N 2 ω1 r2
11
12
Examples
Gear non-linearity - backlash
13
14
15
16
17
18
19
20
3. Sensors and encoders
Potentiometers
Reference: Kuo, B.C. et al. (2004). Automatic control systems, Chapter 4.
(a) Potentiometer used as a position indicator. (b) Two potentiometers used to sense the positions of two shafts. Linear motion potentiometer
Electrical circuit representation
21
22
Potentiometers
Potentiometers
(a) A dc-motor, position-control system with potentiometers as error sensors. Typical waveforms of signals in the control system of part (a).
(a) An ac-control system with potentiometers as error detectors. Typical waveforms of signals in the control system of part (a).
23
24
Tachometer
Incremental encoder
Velocity controlled system with tachometer feedback
Typical rotary incremental encoder
Position controlled system with tachometer feedback
25
4. Electromechanical system models
(a) Typical rectangular output waveform of a single-channel encoder device (bidirectional). Typical dual-channel encoder signals in quadrature (bidirectional).
26
Model of DC motor
Reference: Dorf, R.C. and Bishop, R.H. (2005). Modern control systems, Chapter 2.
(a) DC motor wiring diagram
(b) DC motor sketch
27
28
Model of DC motor
Armature controlled DC motor
29
5. Questions and Answers
30
(a)
31
32
(b)
Question
33
34
35
36
Answer (a)
(b)
37
Question
38
Answer
39
40
Question
41
Answer
42
Answer
43
44
6. Further Reading Many control textbooks deal with this topic in detail. Four examples are: • Dorf, R.C. and Bishop, R.H. (2005). Modern Control Systems, Chapter 2. • Kuo, B.C. and Golnaraghi, F. (2003). Automatic Control Systems, Chapter 4. • Messner, W.C. and Tilbury, D. (1999). Control tutorials for MATLAB and Simulink: a Web-based approach, Prentice-Hall. • Franklin, G.F., Powell, J.D. and Emami-Naeini, A. (2006). Feedback control of dynamic systems (5th Edition), Pearson Education Inc., Chapter 2. Trade magazines (e.g. Control Engineering) sometimes have webaccessible tutorial articles on aspects of modelling. One example of these articles is : 1.VanDoren, V. (2001). “Mathematical models aid process control”, Control Engineering, http://www.controleng.com/article/CA84928.html
More advanced application: Astrom, K.J., Klein, R.E. and Lennartsson, A. (2005). “Bicycle dynamics and control”, IEEE Control Systems Magazine, August, 45 pp. 26-47.
