Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 5 / e

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- Solution e

Solution

For the critically damped case, causality reduces the convolution to

y (t) = \int_{0}^{\infty} s e^{- κ s} cos [κ (t - s)] d s .

Although the forcing is not integrable on the full real line, this integral converges absolutely because the Green's function is integrable. Thus the calculation is valid directly; the corresponding Fourier transform can be interpreted as a tempered distribution.

Use the complex exponential and

\int_{0}^{\infty} s e^{- a s} d s = a^{- 2}

for

Re a > 0

:

y (t) = Re [e^{iκ t} \int_{0}^{\infty} s e^{- κ (1 + i) s} d s] = Re [\frac{e ^{iκ t}}{κ ^{2} ( 1 + i ) ^{2}}] .

Since

(1 + i)^{2} = 2 i

, the convolution response is

y (t) = \frac{sin ( κ t )}{2 κ ^{2}} .

This is the bounded steady response to the stated forcing acting for all time. Without an initial condition or the convolution selection, the general solution additionally contains

(C_{1} + C_{2} t) e^{- κ t}

. No switch-on at

t = 0

was specified, so zero initial displacement and velocity should not be imposed on this all-time forcing.

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