Vibration from a gearbox contains two close harmonics at and

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Electronics, Instrumentation & Communication · 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

Vibration from a gearbox contains two close harmonics at

300.0\,\mathrm{Hz}

and

300.6\,\mathrm{Hz}

, recorded for

T = 1.0\,\mathrm{s}

. In the resulting spectrum the two lines appear as a single merged peak. An engineer proposes to append

7.0\,\mathrm{s}

of zeros to the record before transforming it (a total padded length of

8.0\,\mathrm{s}

), arguing that the eight-fold increase in the number of transform points, and hence in the number of frequency bins, will separate the two lines. ((a)) State the discrete transform's bin spacing before and after the proposed zero-padding. ((b)) Decide whether the padding actually separates the two tones, referring to the shape of

|W(\nu)|

that the true record length produces. ((c)) State the record length actually needed to resolve a

0.6\,\mathrm{Hz}

separation.

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