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EE/ME/AE324: Dynamical Systems

EE/ME/AE324: Dynamical Systems. Chapter 9: Developing Linear Models. From there to here. F rom here to there. N onlinearities are everywhere. Linearization of Nonlinear Elements. Linearization of Nonlinear Elements. Linearization of Nonlinear Elements. Ex.9.1: Nonlinear Spring.

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EE/ME/AE324: Dynamical Systems

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  1. EE/ME/AE324:Dynamical Systems Chapter 9: Developing Linear Models From there to here. From here to there. Nonlinearities are everywhere.

  2. Linearization of Nonlinear Elements

  3. Linearization of Nonlinear Elements

  4. Linearization of Nonlinear Elements

  5. Ex.9.1: Nonlinear Spring

  6. Ex.9.1: Nonlinear Spring

  7. Ex.9.2: Nonlinear Torque

  8. Ex.9.2: Nonlinear Torque

  9. Linearization of Nonlinear Systems • The procedure for linearizing a nonlinear system is similar to that used for a single element: • Determine desired system OPs by solving the nonlinear system eqns. at its EQ pts., e.g., State Eqn. = 0 • Rewrite all nonlinear terms in the model as a sum of their nominal and incremental values, noting derivatives of constants, e.g., nominal values, equal zero • Replace all nonlinear terms by the first two terms in their TSE, e.g., constant plus linear terms • Cancel constant terms in the resulting diff. eqns. leaving only linear terms involving the incremental variables • Determine the ICs of all incremental variables in terms of the ICs associated with variables in the nonlinear system

  10. Ex.9.8: Nonlinear Resistor

  11. Ex.9.8: Nonlinear Resistor

  12. Ex.9.8: Nonlinear Resistor

  13. Ex.9.8: Nonlinear Resistor

  14. Ex.9.9: Nonlinear Resistor

  15. Ex.9.8: Nonlinear Resistor

  16. Ex.9.8: Nonlinear Resistor

  17. Ex.9.9: Nonlinear Resistor

  18. Ex.9.9: Nonlinear Resistor

  19. Ex.9.9: Nonlinear Resistor

  20. 0.2

  21. The Wide Variety of Nonlinear Effects

  22. Mechanical Friction Has Many Models

  23. Mechanical Friction

  24. Mechanical Friction Static (stiction) friction

  25. Mechanical Friction

  26. Mechanical Friction Stick-Slip

  27. Mechanical Friction

  28. Mechanical Friction

  29. Backlash, e.g., Gears

  30. Elastic Hysteresis, e.g., Rubber Band

  31. Magnetic Hysteresis, e.g., Transformer The relationship between magnetic field strength (H) and magnetic flux density (B) is nonlinear

  32. Dead Zone, e.g., Hydraulic Valve In hydraulic valves, a dead zone nonlinearity results if the land width is greater than the port width when the spool is at null position (Merrit, 1967)

  33. Hysteresis + Dead Zone, e.g., Relay Circuits A relay is a voltage-controlled switch with a coil that creates a magnetic field that causes the switch contacts to close when the voltage is greater than a turn-on threshold The contacts remain closed until the voltage diminishes to a turn-off value, at which point the switch contacts open

  34. Saturation, e.g., Op-Amp Circuits

  35. Asymmetric Nonlinearity, e.g., Diode Current Voltage

  36. Questions?

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