16 frequency response

16 frequency response - 18.03 Lecture #16 Oct. 14, 2009:...

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18.03 Lecture #16 Oct. 14, 2009: notes The syllabus topic for today is frequency response . 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. What’s resonance? The short answer is that many physical systems respond more strongly when they are driven at something like their natural frequencies. A simple physical example is a child’s swing. Left to itself, the swing will oscillate with a certain period. If it’s pushed periodically, it will begin to oscillate at the frequency with which it’s being pushed. If that frequency matches the swing’s natural frequency, then the amplitude of the oscillation will grow and grow. Here’s a mathematical version of this example. Start with the second order equation y ′′ + y = 0 . (Harmonic oscillator) This is the spring-and-dashpot system with mass m = 1, spring constant k = 1, and no damping. This equation appears all over the place in physics, because of examples like the spring and like the pendulum: it describes lots of physics exactly, and it’s a good approximation to lots more. You know that the general solution to the equation is y h ( t ) = A cos( t ) + B sin( t ) = C cos( t - φ ) . The subscript h stands for “homogeneous.” This means that the system oscillates with period 2 π in a purely sinusoidal fashion. If we now drive the spring with a varying external force
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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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16 frequency response - 18.03 Lecture #16 Oct. 14, 2009:...

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