07 system response to exponential, sinusoidal input

07 system response to exponential, sinusoidal input - 18.03...

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18.03 Lecture #7 Sept. 23, 2009: notes This lecture is based very closely on sections IR1–3 and IR5. I therefore won’t write so much here, just recalling a few high points. The equation under consideration is dy dt + ky = q ( t ) . (Decay) Here k is a constant; in order for the word “Decay” to be appropriate, k should be positive. This is a linear ODE. The goal is to understand some possible physical meanings of the equation and the terms in it, and to see how the solutions correspond to these physical meanings. Here are two ways that (Decay) can arise. First, suppose we have a (constantly mixed) tank containing (at time t ) y ( t ) units of solute. Assume that solution ±ows out of the tank at a constant rate, in such a way that the tank would be emptied in time 1 /k ; but that pure water ±ows in at the same rate. Under these assumptions, y obeys the di²erential equation (Decay) with q ( t ) = 0. The solution y ( t ) = Ae kt says that the amount of solute present decays exponentially toward zero. Now assume that we add substance to the tank at a variable rate q ( t ). Then y satis³es (Decay). This picture is typical: k has to do with some underlying physical system, and q is something being done to the system more or less by hand. That’s why q is called the input ; in this ³rst model, it just corresponds to a chemical being poured into a tank. A second model has y ( t ) equal to the temperature of something (perhaps hot) sitting in a large cool environment (perhaps a bath). If the environment is at a constant temperature of zero, then Newton’s law of cooling says that y obeys (Decay) with q ( t ) = 0. Again the solution y ( t ) = Ae kt says that the temperature of the hot object decays exponentially toward the temperature 0 of the
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Fall '09 term at MIT.

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07 system response to exponential, sinusoidal input - 18.03...

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