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Unformatted text preview: Solving differential equations with Fourier transforms • Consider a damped simple harmonic oscillator with damping γ and natural frequency ω and driving force f ( t ) d 2 y dt 2 + 2 b dy dt + ω 2 y = f ( t ) • At t = 0 the system is at equilibrium y = 0 and at rest so dy dt = 0 • We subject the system to an force acting at t = t , f ( t ) = δ ( t t ), with t > • We take y ( t ) = R ∞∞ g ( ω ) e i ω t d ω and f ( t ) = R ∞∞ f ( ω ) e i ω t d ω Example continued • Substitute into the differential equation and we find ω 2 ω 2 + 2 ib ω g ( ω ) = f ( ω ) • We find also f ( ω ) = 1 2 π R ∞∞ δ ( t t ) e i ω t dt = 1 2 π e i ω t • We find a relationship between the g ( ω ) and f ( ω ), and then we can write for the response g ( ω ) g ( ω ) = 1 2 π e i ω t ω 2 ω 2 + 2 ib ω • Then with y ( t ) = 0 for t < t , we get y ( t ) for t > t y ( t ) = 1 2 π Z ∞∞ e i ω ( t t ) ω 2 ω 2 + 2 ib ω d ω Example continued • The integral is hard to do (we might get to later), but the point is we have reduced the problem to doing an integral • Assume b < ω , then we find for y ( t ) with t > t , y ( t ) = e...
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 Spring '03
 Staff
 Simple Harmonic Motion, dt, Dirac delta function

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