21.1 Electronic Considerations

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21.1.1 Sensor Linearity

The importance of sensor linearity is often overlooked (see Chapter 15). It is naively assumed that one

can simply employ a lookup table to provide calibration corrections. This assumption can result in

serious misinterpretations of spectral data, especially in a multimode system. A classic example of

artifacts (nonreal signals) that result from a nonlinear sensor is to be found in the ear. The phenomenon,

known as aural harmonics, is well known to musicians and figures in the use of “fortissimo” and

“pianissimo” in orchestral music. In this chapter we describe how the artifacts mentioned in Chapter 20

are generated. Figure 21.1 illustrates differences according to the nature of the nonlinearity.

The only “real” signals in Figure 21.1 are at frequencies f1 and f2: The number and type of other

“unreal” (artifact) signals depends on the type of nonlinearity. The sensor response for the left graph

(quadratic) is of the form V ¼ ax þ bx2; whereas for the right graph V ¼ ax þ bx2 þ cx3: The influence

of terms other than V ¼ ax (ideal, linear output voltage) was generated by (i) simulating the pair of

harmonic signals, (ii) inputting these signals to each simulated sensor, respectively, and (iii) performing a

Fast Fourier Transform (FFT) on the output.

Although it is possible to understand mathematically the various artifacts using trigonometric

identities, the phenomenon is much easier to demonstrate with a computer. For Figure 21.1, all

2f2

2f2

3f2

3f1

2f1

2f1

f1 − f2 f1 − f2

2f1 − f2

2f2 − f1

f1 + f2 f1 + f2

2f1 + f2

f1 + 2f2

f2

f1

f2

f1

Quadratic only

20 dB

Quadratic plus cubic

Simulated harmonic distortion from Sensor Nonlinearity

FIGURE 21.1 Spectral illustration of sum and difference artifact frequencies according to nonlinear sensor type.

21-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

numerical operations were performed with code written by the author using QuickBasic. It was used to

(i) simulate the harmonic signal that was written to a data file, after which it was (ii) read by the FFT

algorithm based on the details supplied in Numerical Recipes (Press et al., 1986).

21.1.2 Frequency Issues

The choice of a sensor depends largely on the frequencies to be measured. For higher frequencies an

excellent instrument for data collection is a digital (storage) oscilloscope, where a microphone can often

be directly connected to the instrument. At lower frequencies, a serial-port analog-to-digital converter

(ADC) is generally adequate and user-friendly. Examples of each will be provided. The majority of

examples considered in this chapter involve low frequencies, where the eigenmode is typically described

not in terms of frequency but rather the period (reciprocal of frequency).

21.1.3 Data Acquisition

In the absence of sophisticated data collection and analysis tools, the true character of damping is not

readily discovered. Proper characterization is important, since a crude estimate of the damping, based on

a single parameter (such as the viscous linear model), may be inappropriate if the oscillator is driven at

places (either frequency or amplitude) other than where the parameter was measured. Some of the

examples from the experiment that follows were selected to demonstrate the importance of nonlinearity.

The probability that an oscillator, selected at random, might have a Q that varies in time is proving to be

more significant than anticipated. Were it not for dramatic improvements in numerical-type technology,

this improved understanding of damping would not have been possible.

As with computer technology in general over the last decade, ADCs have become much more

powerful. The Dataq model 700, for example, is superior (at lower frequencies) to many of the “plug-in”

boards of the previous generation that were several times more expensive. The Dataq ADC operates

through the USB port (Windows 98 and later), has 16-bit resolution and the software support is

excellent. Especially useful for the present purposes are its ability to (i) easily perform data compression

with which to view long records, (ii) quickly compute an FFT according to different, useful options, and

(iii) easily output files to a spreadsheet.

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