# Chapter8 - CHAPTER 8 SPECTRUM ANALYSIS I NTRODUCTION We have seen that the frequency response function T j of a system characterizes the amplitude

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CHAPTER 8 SPECTRUM ANALYSIS INTRODUCTION We have seen that the frequency response function T(j ) of a system characterizes the amplitude and phase of the output signal relative to that of the input signal for purely harmonic (sine or cosine) inputs. We also know from linear system theory that if the input to the system is a sum of sines and cosines, we can calculate the steady-state response of each sine and cosine separately and sum up the results to give the total response of the system. Hence if the input is: k 10 0 k k k k1 A x(t) B sin t 2 (1) then the steady state output is: k 10 0 k k k k k A y(t) T(j0) B T(j ) sin t T(j ) 2 (2) Note that the constant term, a term of zero frequency, is found from multiplying the constant term in the input by the frequency response function evaluated at ω = 0 rad/s. So having a sum of sines and cosines representation of an input signal, we can easily predict the steady state response of the system to that input. The problem is how to put our signal in that sum of sines and cosines form. For a periodic signal, one that repeats exactly every, say, T seconds, there is a decomposition that we can use, called a Fourier Series decomposition, to put the signal in this form. If the signals are not periodic we can extend the Fourier Series approach and do another type of spectral decomposition of a signal called a Fourier Transform. In this chapter much of the emphasis is on Fourier Series because an understanding of the Fourier Series decomposition of a signal is important if you wish to go on and study other spectral techniques. This Fourier theory is used extensively in industry for the analysis of signals. Spectrum analyzers that automatically calculate many of the functions we discuss here are readily available from hardware and software companies. See for example, the advertisements in the IEEE Signal Processing Magazine. Spectral analysis is popular because examination of the frequency content in a signal is often useful when trying to understand what physical components are contributing to a signal. Physical quantities, such as machine rotation rates, structural resonances and effects of material treatments, often have an easily recognizable effect on the frequency representation of the signal. The blade passage rate of rotors and fans in helicopters and turbomachinery will show up as a series of peaks in the spectrum at multiples of the blade passage frequency. Resonance phenomena, that can be related to natural

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8-2 frequencies of plates, beams and shells or of acoustical spaces in machines, will show up as elevated regions in the spectrum. Damping material in an acoustic space will give rise to a high frequency roll off in the spectrum, and a broadening of resonance phenomena.
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## This note was uploaded on 12/26/2011 for the course ME 365 taught by Professor Merkle during the Fall '07 term at Purdue University-West Lafayette.

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Chapter8 - CHAPTER 8 SPECTRUM ANALYSIS I NTRODUCTION We have seen that the frequency response function T j of a system characterizes the amplitude

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