1. Dr. Tamer Samy Gaafar Lec. 3 Mathematical Modeling of Dynamic System

2.  www.tsgaafar.faculty.zu.edu.eg www.tsgaafar.faculty.zu.edu.eg  Email: tsgaafar@yahoo.com

3. Quiz (10 mins) Find the transfer function of the following block diagrams 3

4. Solution: 1. Moving pickoff point A behind block 4

5. 2. Eliminate loop I and Simplify Not feedbackfeedback 5

6. 3. Eliminate loop II & IIII 6

7. Static System: If a system does not change with time, it is called a static system. The o/p depends on the i/p at the present time only Dynamic System: If a system changes with time, it is called a dynamic system. The o/p depends on the i/p at the present time and the past time. (integrator, differentiator). 7

8. A system is said to be dynamic if its current output may depend on the past history as well as the present values of the input variables. Mathematically, Example: A moving mass M y u Model: Force=Mass x Acceleration

9. Ways to Study a System 9 System Experiment with a model of the System Experiment with actual System Physical Model Mathematical Model Analytical Solution Simulation Frequency Domain Time DomainHybrid Domain

10. Model A model is a simplified representation or abstraction of reality. Reality is generally too complex to copy exactly. Much of the complexity is actually irrelevant in problem solving. 10

11. A set of mathematical equations (e.g., differential equs.) that describes the input-output behavior of a system. What is a model used for? Simulation Prediction/Forecasting Prognostics/Diagnostics Design/Performance Evaluation Control System Design

12. 1. Electrical Systems

13. Basic Elements of Electrical Systems The time domain expression relating voltage and current for the resistor is given by Ohm’s law i.e. The Laplace transform of the above equation is

14. Basic Elements of Electrical Systems The time domain expression relating voltage and current for the Capacitor is given as: The Laplace transform of the above equation (assuming there is no charge stored in the capacitor) is

15. Basic Elements of Electrical Systems The time domain expression relating voltage and current for the inductor is given as: The Laplace transform of the above equation (assuming there is no energy stored in inductor) is

16. 16 Componen t SymbolV-I RelationI-V Relation Resistor Capacitor Inductor

17.  The two-port network shown in the following figure has v i (t) as the input voltage and v o (t) as the output voltage. Find the transfer function V o (s)/V i (s) of the network. 17 C i (t) v i ( t ) v o (t)

18.  Taking Laplace transform of both equations, considering initial conditions to zero.  Re-arrange both equations as: 18

19.  Substitute I(s) in equation on left 19

20.  The system has one pole at 20

21.  Design an Electrical system that would place a pole at -3 if added to another system.  System has one pole at  Therefore, 21 C i (t) v i ( t ) v 2 (t)

22. 22 i R (t) v R (t) + - I R (S) V R (S) + - Z R = R Transformation

23. 23 i L (t) v L (t) + - I L (S) V L (S) + - Li L (0) Z L =LS

24. 24 i c (t) v c (t) + - I c (S) V c (S) + - Z C (S)=1/CS

25.  Consider following arrangement, find out equivalent transform impedance. 25 L C R

26. 26 C L R

27. 2. Mechanical Systems

28. Part-I: Translational Mechanical System Part-II: Rotational Mechanical System Part-III: Mechanical Linkages 28 Mechanical Systems

29.  Translational ◦ Linear Motion  Rotational ◦ Rotational Motion 29

31. Basic Elements of Translational Mechanical Systems Translational Spring i) Translational Mass ii) Translational Damper iii) 31

32. Translational Spring i) Circuit Symbols  A translational spring is a mechanical element that can be deformed by an external force such that the deformation is directly proportional to the force applied to it. Translational Spring 32

33.  If F is the applied force  Then is the deformation if  Or is the deformation.  The equation of motion is given as  Where is stiffness of spring expressed in N/m 33

34. Translational Mass ii) Translational Mass is an inertia element. A mechanical system without mass does not exist. If a force F is applied to a mass and it is displaced to x meters then the relation b/w force and displacements is given by Newton’s law. M 34

35. Translational Damper iii) When the viscosity or drag is not negligible in a system, we often model them with the damping force. All the materials exhibit the property of damping to some extent. If damping in the system is not enough then extra elements (e.g. Dashpot) are added to increase damping. 35

36. Common Uses of Dashpots Door Stoppers Vehicle Suspension Bridge Suspension Flyover Suspension 36

37. Translational Damper Where b is damping coefficient (N/ms -1 ). 37 b b

38.  Example-1: Consider a simple horizontal spring-mass system on a frictionless surface, as shown in figure below. or 38

39.  Consider the following system (friction is negligible) 39 Free Body Diagram M Where and are force applied by the spring and inertial force respectively.

40. 40 Then the differential equation of the system is: Taking the Laplace Transform of both sides and ignoring initial conditions we get M

41. 41 The transfer function of the system is if Example-2

42. 42 The pole-zero map of the system is Example-2

43.  Consider the following system 43 Mechanical Network ↑ M

44. 44 Mechanical Network ↑ M At node

45. 45

46. 46

47. 47 Taking Laplace Transform of the equation (2)

48. 48 Car Body Bogie-2 Bogie Frame Bogie-1 Wheelsets Primary Suspension Secondary Suspension

49. 49