(a) A vibration sensor on a machine housing records a ringing mode for . The plotted…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Modes, Packets and Media, Ch The continuum limit, Fourier series and the spectrum of a measured signal

Problem

(a) A vibration sensor on a machine housing records a ringing mode for

T=2.0\,\mathrm{s}

. The plotted power spectrum shows a single peak centred at

\nu_0=12.50\,\mathrm{Hz}

with a full width at half maximum, read directly off the graph, of

\Delta\nu_{\rm meas}=0.50\,\mathrm{Hz}

. State whether this width can be attributed to the mode's own damping or is instead set by the finite record, and give the tightest lower bound this measurement places on the mode's quality factor

Q

. (b) A bat call is analysed as a sequence of overlapping segments, each of length

T_{\rm s}=10\,\mathrm{ms}

; the spectral peak found in the segment starting at

t=0

10

20

30

and

40\,\mathrm{ms}

lies at

80.0

76.0

72.0

68.0

and

64.0\,\mathrm{kHz}

respectively. Estimate the sweep rate

\dot\nu

of the call from these values. (c) Using the time-frequency product bound for a linear chirp, decide whether

T_{\rm s}=10\,\mathrm{ms}

was close to the best choice of segment length; if not, give the segment length that would have been better and the resulting total frequency spread at that choice. Take the total frequency spread of a segment of length

T_{\rm s}

to be the sum of its resolution width

1/T_{\rm s}

and the sweep

|\dot\nu|T_{\rm s}

across it.

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