Guide to Measurement and Instrumentation -- Translational Motion, Vibration, and Shock Measurement (part 2)

Velocity

Translational velocity cannot be measured directly and therefore must be calculated indirectly by other means as set out here.

Differentiation of Displacement Measurements

Differentiation of position measurements obtained from any of the translational displacement transducers described in Section 2 can be used to produce a translational velocity signal.

Unfortunately, the process of differentiation always amplifies noise in a measurement system.

Therefore, if this method has to be used, a low-noise instrument such as a d.c.-excited carbon film potentiometer or laser interferometer should be chosen. In the case of potentiometers, a.c. excitation must be avoided because of the problem that harmonics in the power supply would cause.

Integration of Output of an Accelerometer

Where an accelerometer is already included within a system, integration of its output can be performed to yield a velocity signal. The process of integration attenuates rather than amplifies measurement noise, and this is therefore an acceptable technique in terms of measurement accuracy.

Conversion to Rotational Velocity

Conversion from translational to rotational velocity is the final measurement technique open to the system designer and is the one used most commonly. This conversion enables any of the rotational velocity-measuring instruments described in Section20tobeapplied.

Calibration of Velocity Measurement Systems

Because translational velocity is never measured directly, the calibration procedure used depends on the system used for velocity measurement. If a velocity measurement is being calculated from a displacement or acceleration measurement, the traceability of system calibration requires that the associated displacement or acceleration transducer used is calibrated correctly. The only other measurement technique is conversion of the translational velocity into rotational velocity, in which case the system calibration depends on the calibration of the rotational velocity transducer used.

Figure 14: Accelerometer case Accelerating body Displacement transducer

Acceleration

The only class of device available for measuring acceleration is the accelerometer. These are available in a wide variety of types and ranges designed to meet particular measurement requirements. They have a frequency response between zero and a high value, and have a form of output that can be integrated readily to give displacement and velocity measurements.

The frequency response of accelerometers can be improved by altering the level of damping in the instrument. Such adjustment must be done carefully, however, because frequency response improvements are only achieved at the expense of degrading the measurement sensitivity.

In addition to their use for general-purpose motion measurement, accelerometers are widely used to measure mechanical shocks and vibrations.

Most forms of accelerometer consist of a mass suspended by a spring and damper inside a housing, as shown in Figure 14. The accelerometer is fastened rigidly to the body undergoing acceleration. Any acceleration of the body causes a force, Fa, on the mass, M, given by Fa = M x.

This force is opposed by the restraining effect, Fs, of a spring with spring constant K, and the net result is that mass is displaced by a distance, x, from its starting position such that Fs =Kx.

In steady state, when the mass inside is accelerating at the same rate as the case of the accelerometer, Fa =Fs, and so...

This is the equation of motion of a second-order system, and, in the absence of damping, the output of the accelerometer would consist of non-decaying oscillations. A damper is therefore included within the instrument, which produces a damping force, Fd, proportional to the velocity of the mass, M, given by Fd = B_ x. This modifies the previous equation of motion [Equation (4)] to the following:

[...]

One important characteristic of accelerometers is their sensitivity to accelerations at right angles to the sensing axis (the direction along which the instrument is designed to measure acceleration). This is defined as cross-sensitivity and is specified in terms of the output, expressed as a percentage of full-scale output, when an acceleration of some specified magnitude (e.g., 30g) is applied at 90_ to the sensing axis.

The acceleration reading is obtained from the instrument by measurement of the displacement of the mass within the accelerometer. Many different displacement-measuring techniques are used in the various types of accelerometers available commercially. Different types of accelerometers also vary in terms of the type of spring element and form of damping used.

Resistive potentiometers are one such displacement-measuring instrument used in accelerometers.

These are used mainly for measuring slowly varying accelerations and low-frequency vibrations in the range of 0_50g. The measurement resolution obtainable is about 1 in 400 and typical values of cross-sensitivity are _1%. Inaccuracy is about _1% and life expectancy is quoted at two million reversals. A typical size and weight are 125 cm3 and 500 gram.

Strain gauges and piezoresistive sensors are also used in accelerometers for measuring accelerations up to 200g. These serve as the spring element as well as measuring mass displacement, thus simplifying the instrument's construction. Their typical characteristics are a resolution of 1 in 1000, inaccuracy of _1%, and cross-sensitivity of 2%. They have a major advantage over potentiometer-based accelerometers in terms of their much smaller size and weight (3 cm^3 and 25-gram).

Another displacement transducer found in accelerometers is the LVDT. This device can measure accelerations up to 700g with a typical inaccuracy of _1% of full scale. They are of a similar physical size to potentiometer-based instruments but are lighter in weight (100 gram).

Accelerometers based on variable-inductance displacement-measuring devices have extremely good characteristics and are suitable for measuring accelerations up to 40g. Typical specifications of such instruments are inaccuracy _0.25% of full scale, resolution 1 in 10,000, and cross sensitivity of 0.5%. Their physical size and weight are similar to potentiometer-based devices.

Instruments with an output in the form of a varying capacitance also have similar characteristics.

The other common displacement transducer used in accelerometers is the piezoelectric type.

The major advantage of using piezoelectric crystals is that they also act as the spring and damper within the instrument. In consequence, the device is quite small (15 cm3 ) and low mass (50 gram), but because of the nature of a piezoelectric crystal operation, such instruments are not suitable for measuring constant or slowly time-varying accelerations. As the electrical impedance of a piezoelectric crystal is itself high, the output voltage must be measured with a very high-impedance instrument to avoid loading effects. Many recent piezoelectric crystal-based accelerometers incorporate a high impedance charge amplifier within the body of the instrument. This simplifies the signal conditioning requirements external to the accelerometer but can lead to problems in certain operational environments because these internal electronics are exposed to the same environmental hazards as the rest of the accelerometer. Typical measurement resolution of this class of accelerometer is 0.1% of full scale with an inaccuracy of _1%. Individual instruments are available to cover a wide range of measurements from 0.03g full scale up to 1000g full scale. Intelligent accelerometers are also now available that give even better performance through inclusion of processing power to compensate for environmentally induced errors.

Recently, very small microsensors have become available for measuring acceleration.

These consist of a small mass subject to acceleration mounted on a thin silicon membrane.

Displacements are measured either by piezoresistors deposited on the membrane or by etching a variable capacitor plate into the membrane.

Two forms of fiber-optic-based accelerometers also exist. One form measures the effect on light transmission intensity caused by a mass subject to acceleration resting on a multimode fiber. The other form measures the change in phase of light transmitted through a mono mode fiber that has a mass subject to acceleration resting on it.

Selection of Accelerometers

In choosing between the different types of accelerometers for a particular application, the mass of the instrument is particularly important. This should be very much less than that of the body whose motion is being measured in order to avoid loading effects that affect the accuracy of the readings obtained. In this respect, instruments based on strain gauges are best.

Calibration of Accelerometers

The primary method of calibrating accelerometers is to mount them on a table rotating about a vertical axis such that the sensing axis of the accelerometer is pointing toward the axis of rotation of the table. Acceleration, a, is then given by

[...]

where r is the radius of rotation measured from the center of the rotating table to the center of the accelerometer mass and v is the velocity of rotation of the table (in revolutions per second).

This obviously requires that the rotational speed of the table is measured accurately by a calibrated sensor. Provided that this condition is met, various reference acceleration values can be generated by changing the rotational speed of the table.

Vibration

Nature of Vibration

Vibrations are encountered very commonly in the operation of machinery and industrial plants, and therefore measurement of the accelerations associated with such vibrations is extremely important in industrial environments. The peak accelerations involved in such vibrations can be 100g or greater in magnitude, while both the frequency of oscillation and the magnitude of displacements from the equilibrium position in vibrations have a tendency to vary randomly. Vibrations normally consist of linear harmonic motion that can be expressed mathematically as...

...where X is the displacement from the equilibrium position at any general point in time, Xo is the peak displacement from the equilibrium position, and o is the angular frequency of the oscillations. By differentiating Equation (6) with respect to time, an expression for the velocity v of the vibrating body at any general point in time is obtained as....

Differentiating Equation (7) again with respect to time, we obtain an expression for the acceleration, a, of the body at any general point in time as ...

Inspection of Equation (8) shows that peak acceleration is given by...

This square law relationship between peak acceleration and oscillation frequency is the reason why high values of acceleration occur during relatively low-frequency oscillations. For example, an oscillation at 10 Hz produces peak accelerations of 2g.

Example 2: A pipe carrying a fluid vibrates at a frequency of 50 Hz with displacements of 8 mm from the equilibrium position. Calculate the peak acceleration.

Solution [...]

Using the fact that standard acceleration due to gravity, g, is 9.81m/s2 , this answer can be expressed alternatively as a peak = 789.6/9.81 = 80.5g.

Vibration Measurement

It’s apparent that the intensity of vibration can be measured in terms of displacement, velocity, or acceleration. Acceleration is clearly the best parameter to measure at high frequencies. However, because displacements are large at low frequencies according to Equation (9), it would seem that measuring either displacement or velocity would be best at low frequencies. The amplitude of vibrations can be measured by various forms of displacement transducers. Fiber-optic-based devices are particularly attractive and can give measurement resolution as high as 1 mm. Unfortunately, there are considerable practical difficulties in mounting and calibrating displacement and velocity transducers and therefore they are rarely used. Because of this, vibration is usually measured by accelerometers at all frequencies. The most common type of transducer used is the piezo-accelerometer, which has typical inaccuracy levels of _2%.

The frequency response of accelerometers is particularly important in vibration measurement in view of the inherently high-frequency characteristics of the measurement situation. The bandwidth of both potentiometer-based accelerometers and accelerometers using variable inductance-type displacement transducers only goes up to 25 Hz. Accelerometers that include either the LVDT or strain gauges can measure frequencies up to 150 Hz, and the latest instruments using piezoresistive strain gauges have bandwidths up to 2 kHz. Finally, inclusion of piezoelectric crystal displacement transducers yields an instrument with a bandwidth that can be as high as 7 kHz.

When measuring vibration, consideration must be given to the fact that attaching an accelerometer to the vibrating body will significantly affect the vibration characteristics if the body has a small mass. The effect of such "loading" of the measured system can be quantified by the following equation:

[...]

where a1 is the acceleration of the body with accelerometer attached, ab, is the acceleration of the body without the accelerometer, ma is the mass of the accelerometer, and mb is the mass of the body. Such considerations emphasize the advantage of piezoaccelerometers for measuring vibration, as these have a lower mass than other forms of accelerometers and so contribute least to this system-loading effect.

As well as an accelerometer, a vibration measurement system requires other elements to translate the accelerometer output into a recorded signal, as shown in Figure 15.The three other necessary elements are a signal conditioning element, a signal analyzer, and a signal recorder. The signal-conditioning element amplifies the relatively weak output signal from the accelerometer and also transforms the high output impedance of the accelerometer to a lower impedance value. The signal analyzer then converts the signal into the form required for output. The output parameter may be displacement, velocity, or acceleration, and this may be expressed as peak value, r.m.s. value, or average absolute value. The final element of the measurement system is the signal recorder.

All elements of the measurement system, especially the signal recorder, must be chosen very carefully to avoid distortion of the vibration waveform. The bandwidth should be such that it’s at least a factor of 10 better than the bandwidth of the vibration frequency components at both ends. Thus its lowest frequency limit should be less than or equal to 0.1 times the fundamental frequency of vibration and its upper frequency limit should be greater than or equal to 10 times the highest significant vibration frequency component.

If the frequency of vibration has to be known, the stroboscope is a suitable instrument to measure this. If the stroboscope is made to direct light pulses at the body at the same frequency as the vibration, the body will apparently stop vibrating.

Figure 15: Accelerometer Signal analyzer Signal-conditioning element; Signal recorder

Calibration of Vibration Sensors

Calibration of the accelerometer used within a vibration measurement system is normally carried out by mounting the accelerometer in a back-to-back configuration with a reference calibrated accelerometer on an electromechanically excited vibrating table.

Shock

Shock describes a type of motion where a moving body is brought suddenly to rest, often because of a collision. This is very common in industrial situations, and usually involves a body being dropped and hitting the floor. Shocks characteristically involve large-magnitude decelerations (e.g., 500g) that last for a very short time (e.g., 5ms). An instrument having a very high frequency response is required for shock measurement, and, for this reason, piezoelectric crystal-based accelerometers are commonly used. Again, other elements for analyzing and recording the signal are required as shown in Figure 16 and described in the last section.

A storage oscilloscope is a suitable instrument for recording the output signal, as this allows time duration as well as acceleration levels in the shock to be measured. Alternatively, if a permanent record is required, the screen of a standard oscilloscope can be photographed.

Figure 16: Suspended hammer Freely suspended anvil Optical sensors; Accelerometer

Example 3:

A body is dropped from a height of 10 m and suffers a shock when it hits the ground. If the duration of the shock is 5 ms, calculate the magnitude of the shock in terms of g.

Solution:

The equation of motion for a body falling under gravity gives the following expression for terminal velocity, v:

[...]

where x is the height through which the body falls. Having calculated v, the average deceleration during the collision can be calculated as a =v/t, where t is the time duration of the shock. Substituting the appropriate numerical values into these expressions:

Calibration of Shock Sensors

Calibration of the accelerometer used within a shock sensor is carried out frequently using a pneumatic shock exciter. This device consists of a piston within a circular tube. High-pressure air is applied to one face of the piston, but it does not move initially because it’s held at the end of the tube by a mechanical latching mechanism. When the latch is released, the piston accelerates at a high rate until it’s brought to rest by a padded anvil at the other end of the tube.

The accelerometer being calibrated and a calibrated reference accelerometer are both mounted on the anvil. By varying the characteristics of the padding, the deceleration level and hence magnitude of the shock produced on the anvil can be varied.

Summary

This section has been concerned with the measurement of translational (in a straight line) motion. This can take the form of displacement, velocity (rate of change of displacement), or acceleration (rate of change of velocity. We have looked at sensors for measuring each of these and, in the case of acceleration, we have also studied vibration and shock measurement, as both of these involve acceleration measurement.

Our study of displacement sensors started with the resistive potentiometer, where we learned that potentiometers come in three different forms: wire wound, carbon film, and plastic film.

We then moved on to look at the linear variable differential transformer, variable capacitance, and variable inductance sensors. We noted that strain gauges were used to measure very small displacements (typically up to 50 mm in magnitude). We also noted that force-measuring piezoelectric sensors could also be regarded as displacement sensors, as their mode of operation is to generate an e.m.f. that is proportional to the distance by which it’s compressed by the applied force. We also discussed the nozzle flapper, which measures displacements by converting them into a pressure change. We then moved on to summarize some other techniques for measuring small- and medium-sized displacements, including translating linear motion into rotational motion, integrating the output from velocity and acceleration sensors, and using specialist devices such as the linear inductosyn, laser interferometer, fotonic sensor, and noncontacting optical sensor. Moving the discussion on to the measurement of relatively large displacements, we noted that this could be achieved by several devices commonly called range sensors. We also included some mention of proximity sensors, as these belong properly within the classification of displacement sensors, although they are a special case in that their binary form of output merely indicates whether the sensor is, or is not, within some threshold distance of a boundary.

Finally, before leaving the subject of displacement measurement, we looked at the techniques used to calibrate them.

Our discussion of translational velocity measurement introduced us to the fact that this cannot be measured directly. We then went on to look at the only three ways to measure it, these being differentiation of position measurements, integration of the output of an accelerometer, and conversion from translational to rotational velocity. Finally, we considered how measurements obtained via each of the techniques could be calibrated.

In the case of acceleration measurement, we observed that this could only be measured by some form of accelerometer. We noted that attributes such as frequency response and cross sensitivity were important as well as measurement accuracy in accelerometers. We discovered that almost all accelerometers work on the principle that a mass inside them displaces when subject to acceleration. Accelerometers differ mainly in the technique used to measure this mass displacement, and we looked in turn at devices that use the resistive potentiometer, strain gauge, piezoresistive sensor, LVDT, variable inductance sensor, and variable capacitance sensor, respectively. We then looked at the one exception to the rule that accelerometers contain a moving mass. This is the piezoelectric accelerometer. Finally, we looked at the primary method of calibrating accelerometers using a rotating table.

We then concluded the section by looking at vibration and shock measurement. Both of these involve accelerations, and therefore both need an accelerometer to quantify their magnitude.

Starting with vibration, we noted that this was a common phenomenon, especially in industrial situations. We learned that vibration consists of a linear harmonic in which the peak acceleration can exceed 100g and where the oscillation frequency and peak amplitude can vary randomly.

We noted that the amplitude of vibration could be calculated from a measurement of the peak acceleration, and we went on to look at the suitability of various forms of accelerometers for such measurement.

Finally, we considered shock measurement. This revealed that very large magnitude decelerations are involved in the phenomenon of shock, which typically occurs when a falling body hits the floor or a collision occurs between two solid objects. A high-frequency response is particularly important in shock measurement, and the most suitable device to measure this is a piezoelectric crystal-based accelerometer.

QUIZ and PROBLEMS:

1. Discuss the mode of operation and characteristics of a linear motion potentiometer.

2. What is an LVDT? How does it work? 3. Explain how the following two instruments work and discuss their main operating characteristics and uses: (a) variable capacitance transducer and (b) variable inductance transducer.

4. Sketch a linear inductosyn. How does it work? What are its main characteristics? 5. What is a laser interferometer and what are its principal characteristics? Explain how it works with the aid of a sketch.

6. What are range sensors? Describe two main types of range sensors.

7. Discuss the main types of proximity sensors available, mentioning particularly their suitability for operation in harsh environments.

8. What are the main considerations in choosing a translational motion transducer for a particular application? Give examples of some types of translational motion transducers and the applications that they are suitable for.

9. Discuss the usual calibration procedures for translational motion transducers.

10. What are the main ways of measuring translational velocities? How are such measurements calibrated? 11. What are the principles of operation of a linear motion accelerometer? What features would you expect to see in a high-quality accelerometer?

12. What types of displacement sensors are used within accelerometers? What are the relative merits of these alternative displacement sensors?

13. Write down a mathematical equation that describes the phenomenon of vibration.

Explain briefly the three main ways of measuring vibration.

14. When an accelerometer is attached to a vibrating body, it has a loading effect that alters the characteristics of the vibration. Write down a mathematical equation that describes this loading effect. How can this loading effect be minimized?

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