Page 395 - The Mechatronics Handbook

From the preceding equations it can be seen that the basic form and the period of vibration remains
the same in acceleration, velocity, and displacement. But velocity leads displacement by a phase angle of
90∞ and acceleration leads velocity by another 90∞.
In nature, vibrations can be periodic, but not necessarily sinusoidal. If they are periodic but nonsinu-
soidal, they can be expressed as a combination of a number of pure sinusoidal curves, determined by
Fourier analysis as

..
xt() = X 0 + X 1 sin ( ω 1 t + Φ 1 ) + X 2 sin ( ω 2 t + Φ 2 ) ++ X n sin ( ω n t + Φ n ) (19.11)
where ω 1 , ω 2 ,…, ω n are the frequencies (rad/s), X 0 , X 1 ,…, X n are the maximum amplitudes of respective
frequencies, and φ 1 , φ 2 ,…, φ n are the phase angles.
The number of terms may be infinite, and the higher the number of elements, the better the approx-
imation. These elements constitute the frequency spectrum. The vibrations can be represented in the
time domain or frequency domain, both of which are extremely useful in analysis.
Stationary Random Vibrations
Random vibrations are often met in nature, where they constitute irregular cycles of motion that never
repeat themselves exactly. Theoretically, an infinitely long time record is necessary to obtain a complete
description of these vibrations. However, statistical methods and probability theory can be used for the
analysis by taking representative samples. Mathematical tools such as probability distributions, probability
densities, frequency spectra, cross-correlations, auto-correlations, digital Fourier transforms (DFTs), fast
Fourier transforms (FFTs), auto-spectral-analysis, root mean squared (rms) values, and digital filter analysis
are some of the techniques that can be employed.
Nonstationary Random Vibrations
In this case, the statistical properties of vibrations vary in time. Methods such as time averaging and
other statistical techniques can be employed.
Transients and Shocks
Often, short duration and sudden occurrence vibrations need to be measured. Shock and transient
vibrations may be described in terms of force, acceleration, velocity, or displacement. As in the case of
random transients and shocks, statistical methods and Fourier transforms are used in the analysis.

Typical Error Sources and Error Modeling

Acceleration measurement errors occur due to four primary reasons: sensors, acquisition electronics,
signal processing, and application specific errors. In the direct acceleration measurements, the main error
sources are the sensors and data acquisition electronics. These errors will be discussed in the biasing
section and in some cases, sensor and acquisition electronic errors may be as high as 5%. Apart from
these errors, sampling and A/D converters introduce the usual errors, which are inherent in them and
exist in all computerized data acquisition systems. However, the errors may be minimized by the careful
selection of multiplexers, sample-and-hold circuits, and A/D converters.
When direct measurements are made, ultimate care must be exercised for the selection of the correct
accelerometer to meet the requirements of a particular application. In order to reduce the errors, once
the characteristics of the motion are studied, the following particulars of the accelerometers need to be
considered: the transient response or cross-axis sensitivity; frequency range; sensitivity, mass and dynamic
range; cross-axis response; and environmental conditions, such as temperature, cable noise, stability of
bias, scale factor, and misalignment, etc.
Sensitivity of Accelerometers

During measurements, the transverse motions of the system affect most accelerometers and the sensitivity
to these motions plays a major role in the accuracy of the measurement. The transverse, also known as
cross-axis, sensitivity of an accelerometer is its response to acceleration in a plane perpendicular to the

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