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Unformatted text preview: 18.03 Lecture #17 Oct. 16, 2009: notes The syllabus topic for today continues frequency response . What I want to do is look carefully at the phenomenon of resonance for the springanddashpot model. (The mathematics for studying resonance in an RLC circuit is absolutely identical; this is examined in section 2.7 of the text.) The idea is to understand the response of a system to sinusoidal input. There are two big points to be made. First is to understand what resonance means. Second is to extend our spring anddashpot example to cover electrical circuits. After those two things, I’ll say what I can about the mathematics of resonance. The equation is the one I wrote on Wednesday: a mass m attached to a spring with spring constant k , with a damping c by a dashpot, driven by a purely sinusoidal external force: my ′′ + cy ′ + ky = cos( ωt ) = Re( e iωt ) . (Driven damped spring) In the RLC circuit version, c becomes the resistance, k the inverse of the capacitance, and m the inductance. First, some formal questions. Why not look at a general sinusoidal driving force C cos( ωt − α )? The answer is that nothing mathematically interesting changes. The solution is shifted in time by α/ω , and multiplied by C . (An electrical engineer might object to the claim that there is no interesting difference between an applied voltage of one volt and one of 100 , 000 volts, but I’m not an electrical engineer.) Taking C = 1 and α = 0 simplifies the formulas a bit without obscuring the phenomena I want to look at. Remember that the characteristic polynomial of the differential equation (Driven damped spring) is p ( D ) = mD 2 + cD + k. (Characteristic polynomial) The substitution rule says that p ( D )( e rt ) = p ( r ) e rt , and therefore that p ( D ) parenleftbigg e rt p ( r ) parenrightbigg = e rt ( p ( r ) negationslash = 0) . Taking r = iω and taking real parts, we get p ( D ) parenleftbigg Re parenleftbigg e iωt p ( iω ) parenrightbiggparenrightbigg = cos( ωt ) ( p ( iω ) negationslash = 0) . (Exponential response formula, version 1) Now write the complex number p ( iω...
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 Fall '09
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 RLC, exponential response formula, c2 /2m2, tm ert, c2 /4m2

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