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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, Ill 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( t ) = Re( e it ) . (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( t )? The answer is that nothing mathematically interesting changes. The solution is shifted in time by / , 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 Im not an electrical engineer.) Taking C = 1 and = 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 = i and taking real parts, we get p ( D ) parenleftbigg Re parenleftbigg e it p ( i ) parenrightbiggparenrightbigg = cos( t ) ( p ( i ) negationslash = 0) . (Exponential response formula, version 1) Now write the complex number p ( i...
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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.
- Fall '09