## Accelerometer Dynamics: Periodic Vibrations

This section concerns the physical properties of acceleration, vibration, and shock measurements in which accelerometers are commonly used. A full understanding of accelerometer dynamics is necessary in relation to characteristics of acceleration, vibration, and shock. The vibrations can be periodic, stationary random, nonstationary random, or transient.

**Periodic Vibrations**

In periodic vibrations, the motion of an object repeats itself in an oscillatory manner. This can be represented by a sinusoidal waveform:

FIGURE 17.2 Logarithmic relationship between acceleration, velocity, and displacement. Velocity at a particular frequency can be obtained by dividing acceleration by a factor proportional to frequency. For displacement, acceleration must be divided by a factor proportional to the square of the frequency. Phase angles need to be determined separately, but they can be neglected in time-averaged measurements.

From the above 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°. The amplitudes of the three quantities are related as a function of frequency, as shown in Figure 17.2.

In nature, vibrations can be periodic but not necessarily sinusoidal. If they are periodic but nonsinusoidal, they can be expressed as a combination of a number of pure sinusoidal curves, described by Fourier analysis as:

The number of terms may be infinite: the higher the number of elements, the better the approximat These elements constitute the frequency spectrum. The vibrations can be represented in time domain frequency domain, both of which are extremely useful in the analysis. As an example, in Figure 17.3, time response of the seismic mass of an accelerometer is given against a rectangular pattern of excitat of the base.

** 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 FIGURE 17.3 Time response of a shock excitation of a single degree-of-freedom system. As the duration of the shock pulse increases, sustained oscillations get shorter in time but larger in amplitude. The maximum system response may be as high as twice the magnitude of the shock pulse.

analysis by taking representative samples. Mathematical tools such as probability distributions, probability densities, frequency spectra, cross- and auto-correlations, Digital Fourier Transforms (DFT), Fast Fourier

Transforms (FFT), auto spectral analysis, RMS values, and digital filter analysis are some of the techniques that can be employed. Interested readers should refer to references for further information.

**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.

**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. A majority of accelerometers described here can be viewed and analyzed as seismic instruments consisting of a mass, a spring, and a damper arrangement, as shown in Figure 17.1. Taking only the mass-spring system, if the system behaves linearly in a time invariant manner, the basic second-order differential equation for the motion of the mass alone under the influence of a force can be written as:

In order to lay a background for further analysis, taking the simple case, the complete solution to Equation 17.5 can be obtained by applying the superposition principle. The superposition principle states that if there are simultaneously superimposed actions on a body, the total effect can be obtained by summing the effects of each individual action.

Equation 17.5 describes essentially a second-order system that can be expressed in Laplace transform as:

As can be seen, in the performance of accelerometers, important parameters are the static sensitivity, the natural frequency, and the damping ratio, which are functions of mass, velocity, and spring constants.

Accelerometers are designed to have different characteristics by suitable selection of these parameters.

Once the response is expressed in the form of Equations 17.7 and 17.8, analysis can be taken further, either in the time domain or in the frequency domain. The time response of a typical second-order system for a unit-step input is given in Figure 17.4. The Bode plot gain phase responses are depicted in Figure 17.5. Detailed discussions about frequency response, damping, damping ratio, and linearity are made in relevant sections, and further information can be obtained in the references.

Systems in which a single structure moves in more than one direction are termed multi-degree-offreedom systems. In this case, the accelerations become functions of dimensions as d2x/dt2, d2y/dt2, and d2z/dt2. Hence, in multichannel vibration tests, multiple transducers must be used to create uniaxial, biaxial, or triaxial sensing points for measurements.

** Mathematically, a linear multidegree-of-freedom**

system can be described by a set of coupled second-order linear differential equations; and when the frequency response is plotted, it normally shows one resonance peak per degree of freedom.

Frequently, acceleration and vibration measurements of thin plates or small masses are required.

Attaching an accelerometer with a comparable mass onto a thin plate or a small test piece can cause “mass loading.” Since acceleration is dependent on the mass, the vibration characteristics of the loaded test piece could be altered, thus yielding wrong measurements. In such cases, a correct interpretation of the results of the measuring instruments must be made.

Some experimental techniques are also available for the correction of the test results in the form of performing repetitive tests conducted by sequentially adding small known masses and by observing the differences.The following sections discuss different types of accelerometers.