MIT18_03S10_c14 - 18.03 Class 14, March 5, 2010 Complex...

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18.03 Class 14 , March 5, 2010 Complex gain 1. Recap 2. Phase lag 3. Driving via the dashpot 4. Complex gain [1] The story so far: We're studying solutions of linear constant coefficient equations a_n x^(n) + . .. + a_1 x + a_0 = q(t) (*) A key is the characteristic polynomial p(s) = a_n s^n + . .. + a_1 s + a_0 For the homogeneous case, a_n x^(n) + . .. + a_1 x + a_0 = 0 (*)_h we found that the roots of p(s) give the exponents in exponential solutions, and that the general solution is a linear combination of these or (these times a power of t in case there are repeated roots). Euler's formula shows that |e^{(a+bi)t}| = e^{at} so: [Slide] Transience Theorem: All homogeneous solutions of (*)_h decay to zero provided that all the roots of p(s) have negative real parts. In this case the solutions to (*)_h are called "transients," By superposition, all solutions to (*) converge together as t gets large, and we say that the equation is "stable." If we have a system modeled by a stable equation, and we are only interested in what it looks like after the transients have died down, we can eliminate the initial condition: ____________ input | | steady state ---------------->| System |-------------------------> signal |____________| output signal x_p So we look for a particular solution x_p . Sinusoidal input signals are of particular importance. Experiments indicate that sinusoidal in gives sinusoidal out. We decide to set our clock so that the input signal is input = A cos(omega t)
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Experiments indicate that the steady state output signal is again sinusoidal, of the same circular frequency: output = x = B cos (omega t - phi)
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This note was uploaded on 01/18/2012 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c14 - 18.03 Class 14, March 5, 2010 Complex...

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