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

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First Order Systems: Dynamic Systems ISAT 300 Spring 1999

Static vs. Dynamic Static means that the system doesnt change with time Dynamic means that the system is changing with time

Example Static System: Force Transducer

Example Dynamic System: Cooling of a Cake

Example Dynamic System: Bacteria Growth

Mathematics of Bacterial Growth The change in the number of bacteria at any time Is proportional to the number of bacteria present at any time Proportionality Constant The equation is derived from the concept of Conservation of Mass (Dont confuse with K the sensitivity)

Solve the differential equation for bacteria growth

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.

Time Constant The parameter for characterizing the response time of an instrument is the Time Constant,

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 Time Output 1.0 0.633

Time Output System Behavior Instrument Response

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

Anatomy of a Diff. Eq. Contd. Non Homogeneous Diff. Eq. Forcing Function Initial Condition Complete Solution General+Particular

Anatomy of a Diff. Eq. Contd. Apply Initial Condition to Complete Solution

Types of Forcing Functions (inputs) Step Ramp Solution

Types of Forcing Functions (inputs) Sinusoidal Solution

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