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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 spring-and-dashpot 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- and-dashpot 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
- RLC, exponential response formula, c2 /2m2, tm ert, c2 /4m2