Green's Function from Damped Harmonic Oscillator's Response to Pulse-Force, Exams of Classical Mechanics

Download Green's Function from Damped Harmonic Oscillator's Response to Pulse-Force and more Exams Classical Mechanics in PDF only on Docsity! 3. (40 pts) Now consider a pulse-force applied at t0 with width τ . a) Write down the response of the oscillator after the pulse in terms of the pulse start (t0) and the pulse width τ . This is similar to problem 2 except there are no numbers involved. You get x(t) = a ω0 ( 1 − e−β(t−t0) { cos[ω1(t − t0)] + β ω1 sin[ω1(t − t0)] }) − a ω0 ( 1 − e−β(t−t1) { cos[ω1(t − t1)] + β ω1 sin[ω1(t − t1)] }) Then, make the substitution that τ = t1 − t0 to obtain x(t) = a ω0 ( 1 − e−β(t−t0) { cos[ω1(t − t0)] + β ω1 sin[ω1(t − t0)] }) − a ω0 ( 1 − e−β(t−t0−τ) { cos[ω1(t − t0 − τ)] + β ω1 sin[ω1(t − t0 − τ)] }) Separating out the τ parts of the arguments, we obtain at last x(t) = a ω0 ( 1 − e−β(t−t0) { cos[ω1(t − t0)] + β ω1 sin[ω1(t − t0)] }) − a ω0 ( 1 − e−β(t−t0)eβτ) { cos[ω1(t − t0) − ω1τ)] + β ω1 sin[ω1(t − t0) − ω1τ)] }) You could go even further, but that’s probably good enough for now. b) Explain the mathematical process by which Green’s function is obtained from your result in (a). You would see what happens in the limit τ → 0, but keeping the impulse of the force constant, or the product of aτ = b where b is constant. First, rewrite the sin and cos terms with τ in the argument as products of sin and cos to separate the τ completely. Then there will be terms with cos(ω1τ), sin(ω1τ), and eβτ . Since τ → 0, we can expand these three in Taylor series and keep only the terms up to order τ1. Finally, simplify everything to obtain the response to a constant impulse infinitely skinny spike-force. c) Derive the Taylor series expansion of eβτ around τ = 0. f(x) = f(a) + x f ′(x)|a + x2 f ′′(x)|a 2! + x3 f ′′′(x)|a 3! + · · · So, for eβτ we have eβτ = e(β)(0) + βτe(β)(0) + β2τ2e(β)(0) 2 + β2τ3e(β)(0) 6 + · · · eβτ = 1 + βτ + β2τ2 2 + β3τ3 6 + · · · d) Explain why, in keeping with your response to (b), only the first two terms of this expansion are used in the subsequent derivation. We were looking at what happens with τ → 0. When a very small number is raised to a large power, the result is even smaller. So if τ is close to zero, τ2 is even closer to zero. Each increase in the power reduces the size of the subsequent terms. Thus we neglect terms of the order τ2 and higher.