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Understanding Second Order Systems: Natural Response and Damping Analysis

This lecture reviews the natural response of second order systems, focusing on mathematical solutions and qualitative interpretations. It covers various types of damping (overdamped, critically damped, and underdamped) and their respective characteristics. The governing equation for an unforced second order system is introduced, along with the impacts of the damping ratio (ζ) and natural frequency (ωn). The lecture also examines a practical example involving an electrical circuit to illustrate how to determine system behavior and damping conditions.

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Understanding Second Order Systems: Natural Response and Damping Analysis

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  1. Lecture 22 Second order system natural response Review Mathematical form of solutions Qualitative interpretation Second order system step response Related educational modules: Section 2.5.4, 2.5.5

  2. Second order input-output equations • Governing equation for a second order unforced system: • Where •  is the damping ratio (  0) • n is the natural frequency (n  0)

  3. Homogeneous solution – continued • Solution is of the form: • With two initial conditions: ,

  4. Damping ratio and natural frequency • System is often classified by its damping ratio, : •  > 1  System is overdamped (the response has two time constants, may decay slowly if  is large) •  = 1  System is critically damped (the response has a single time constant; decays “faster” than any overdamped response) •  < 1  System is underdamped (the response oscillates) • Underdamped system responses oscillate

  5. Overdamped system natural response • >1: • We are more interested in qualitative behavior than mathematical expression

  6. Overdamped system – qualitative response • The response contains two decaying exponentials with different time constants • For high , the response decays very slowly • As  increases, the response dies out more rapidly

  7. Critically damped system natural response • =1: • System has only a single time constant • Response dies out more rapidly than any over-damped system

  8. Underdamped system natural response • <1: • Note: solution contains sinusoids with frequency d

  9. Underdamped system – qualitative response • The response contains exponentially decaying sinusoids • Decreasing  increases the amount of overshoot in the solution

  10. Example • For the circuit shown, find: • The equation governing vc(t) • n, d, and  if L=1H, R=200, and C=1F • Whether the system is under, over, or critically damped • R to make  = 1 • Initial conditions if vc(0-)=1V and iL(0-)=0.01A

  11. Part 1: find the equation governing vc(t)

  12. Part 2: find n, d, and  if L=1H, R=200 and C=1F

  13. Part 3: Is the system under-, over-, or critically damped? • In part 2, we found that  = 0.2

  14. Part 4: Find R to make the system critically damped

  15. Part 5: Initial conditions if vc(0-)=1V and iL(0-)=0.01A

  16. Simulated Response

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