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First Order Systems: Dynamic Systems

First Order Systems: Dynamic Systems. ISAT 300 Spring 1999. Static vs. Dynamic. Static means that the system doesn’t change with time Dynamic means that the system is changing with time. Example Static System: Force Transducer. Example Dynamic System: Cooling of a Cake.

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First Order Systems: Dynamic Systems

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  1. First Order Systems:Dynamic Systems ISAT 300 Spring 1999

  2. Static vs. Dynamic • Static means that the system doesn’t change with time • Dynamic means that the system is changing with time

  3. Example Static System: Force Transducer

  4. Example Dynamic System: Cooling of a Cake

  5. Example Dynamic System: Bacteria Growth

  6. Mathematics of Bacterial Growth The equation is derived from the concept of Conservation of Mass The change in the number of bacteria at any time Is proportional to the number of bacteria present at any time (Don’t confuse with K the sensitivity) Proportionality Constant

  7. Solve the differential equation for bacteria growth

  8. Why are 1st order equations important for instrumentation • Many instruments exhibit a 1st order response • Provides a parameter, called the time constant, for choosing an instrument. This parameter lets you know if the instrument will respond quick enough to capture changes in the system.

  9. Time Constant • The parameter for characterizing the response time of an instrument is the Time Constant, t

  10. Time Constant • The Time Constant is the time it takes a first order system to reach 63.3% (0.633) of its final value in response to a step change in the system 1.0 0.633 Output t Time

  11. System Behavior Output Time Instrument Response

  12. Anatomy of a Differential Equation Homogeneous Diff. Eq. Initial Condition Homogeneous Solution (General Solution)

  13. Anatomy of a Diff. Eq. Cont’d. Non Homogeneous Diff. Eq. Forcing Function Initial Condition Complete Solution General+Particular

  14. Anatomy of a Diff. Eq. Cont’d. Apply Initial Condition to Complete Solution

  15. Types of Forcing Functions (inputs) • Step • Ramp Solution Solution

  16. Types of Forcing Functions (inputs) • Sinusoidal Solution

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