Mass and Weight Measurement
Mass and Weight
Mark Fritz
Denver Instrument Company
Emil Hazarian
Denver Instrument Company
Weighing Instruments
Weighing Techniques
Mass and weight are often used interchangeably; however, they are different. Mass is a quantitative
measure of inertia of a body at rest. As a physical quantity, mass is the product of density and volume.
Weight or weight force is the force with which a body is attracted toward the Earth. Weight force is
determined by the product of the mass and the acceleration of gravity.
M =V × D
where M = mass
V = volume
D = density
Note: In most books, the symbol for density is the Greek letter ρ.
W =M ×G
where W = weight
G = gravity
The embodiment of units of mass are called weights; this increases the confusion between mass and
weight. In the International System of Units (SI), the modernized metric measurement system, the unit
for mass is called the kilogram and the unit for force is called the newton. In the United States, the
customary system the unit for mass is called the slug and the unit for force is called the pound. When
using the U.S. customary units of measure, people are using the unit pound to designate the mass of an
object because, in the United States, the pound has been defined in terms of the kilogram since 1893.
20.1 Weighing Instruments
Weighing is one of the oldest known measurements, dating back to before written history. The equal
arm balance was probably the first instrument used for weighing. It is a simple device in which two pans
are suspended from a beam equal distance from a central pivot point. The standard is placed in one pan
and the object to be measured is placed in the second pan; when the beam is level, the unknown is equal
to the standard. This method of measurement is still in wide use throughout the world today. Figure 20.1
shows an Ainsworth equal arm balance.
© 1999 by CRC Press LLC
FIGURE 20.1 Ainsworth FV series equal arm balance. (Courtesy Denver Instrument Company.)
A balance is a measuring instrument used to determine the mass of an object by measuring the force
exerted by the object on its support within the gravitational field of the Earth. One places a standard of
known value on one pan of the balance. One then adds the unknown material to the second pan, until
the gravitational force on the unknown material equals the gravitational force on the standard. This can
be expressed mathematically as:
S × G =U × G
where S = mass of the standard
G = gravity
U = mass of the unknown
Given the short distance between pans, one assumes that the gravitational forces acting on them are
equal. Another assumption is that the two arms of the balance are equal.
Since the gravitational force is equal, it can be removed from the equation and the standard and the
unknown are said to be equal. This leads to one of the characteristics of the equal arm balance as well
as of other weighing devices, the requirement to have a set of standards that allows for every possible
measured value. The balance can only indicate if objects are equal and has a limited capability to determine
how much difference there is between two objects.
© 1999 by CRC Press LLC
Probably, the first attempt to produce direct reading balances was the introduction of the single pan
substitution balance. A substitution balance is, in principle, similar to an equal arm balance. The object
to be measured is placed in the weighing pan; inside the balance are a series of calibrated weights that
can be added to the standard side of the balance, through the use of dials and levers. The standard weights
can be added and subtracted through the use of the balance’s mechanical system, to equal a large variety
of weighing loads. Very small differences between the standard weights and the load are read out on an
optical scale.
The spring scale is probably the least expensive device for making mass measurements. The force of
gravity is once again used as the reference. The scale is placed so that the unknown object is suspended
by the spring and the force of gravity can freely act on the object. The elasticity of the spring is assumed
to be linear and the force required to stretch it is marked on the scale. When the force of gravity and the
elastic force of the spring balance, the force is read from the scale, which has been calibrated in units of
mass. Capacity can be increased by increasing the strength of the spring. However, as the strength of the
spring increases, the resolution of the scale decreases. This decrease in resolution limits spring scales to
relatively small loads of no more than a few kilograms. There are two kinds of springs used: spiral and
cantilevered springs.
The torsion balance is a precise adaptation of the spring concept used to determine the mass indirectly.
The vertical force produced by the load produces a torque on a wire or beam. This torque produces an
angular deflection. As long as the balance is operated in the linear range, the angular deflection is
proportional to the torque. Therefore, the angular deflection is also proportional to the applied load.
When the torsion spring constant is calibrated, the angular deflection can be read as a mass unit. Unlike
the crude spring scales, it is possible to make torsion balances capable of measuring in the microgram
region. The torsion element could be a band, a wire, or a string.
The beam balance is probably the next step in accuracy and cost. The beam balance uses the same
principle of operation as the equal arm balance. However, it normally has only one pan and the beam
is offset. A set of sliding weights are mounted on the beam. As the weights slide out the beam, they gain
a mechanical advantage due to the inequality of the distance from the pivot point of the balance. The
weights move out along the beam until the balance is in equilibrium. Along the beam, there are notched
positions that are marked to correspond to the force applied by the sliding weights. By adding up the
forces indicated by the position of each weight, the mass of the unknown material is determined. Beam
balances and scales are available in a wide range of accuracy’s load capacities. Beam balances are available
to measure in the milligram range and large beam scales are made with capacities to weigh trucks and
trains. Once again, the disadvantage is that as load increases the resolution decreases. Figure 20.2 shows
an example of a two pan beam balance.
The next progression in cost and accuracy is the strain gage load cell. A strain gage is an electrically
resistive wire element that changes resistance when the length of the wire element changes. The gage is
bonded to a steel cylinder that will shorten when compressed or lengthen when stretched. Because the
gage is bonded to the cylinder, the length of the wire will lengthen or contract with the cylinder. The
electrical resistance is proportional to the length of the wire element of the gage. By measuring the
resistance of the strain gage, it is possible to determine the load on the load cell. The electric resistance
is converted into a mass unit readout by the electric circuitry in the readout device.
The force restorative load cell is the heart of an electronic balance, shown in Figure 20.3. The force
restorative load cell uses the principle of the equal arm balance. However, in most cases, the fulcrum is
offset so it is no longer an equal arm balance, as one side is designed to have a mechanical advantage.
This side of the balance is attached to an electric coil. The coil is suspended in a magnetic field. The
other side is still connected to a weighing pan. Attached to the beam is a null indicating device, consisting
of a photodiode and a light-emitting diode (LED) that are used to determine when the balance is in
equilibrium. When a load is placed on the weighing pan, the balance goes out of equilibrium. The LED
photodiode circuit detects that the balance is no longer in equilibrium, and the electric current in the
coil is increased to bring the balance back to equilibrium. The electric current is then measured across
a precision sense resistor and converted into a mass unit reading and displayed on the digital readout.
© 1999 by CRC Press LLC
FIGURE 20.2 Beam balance. (Courtesy Denver Instrument Company.)
FIGURE 20.3 Force restorative load cell. (Courtesy Denver Instrument Company.)
A variation of the latter is the new generation of industrial scales, laboratory balances, and mass comparators. Mass comparators are no longer called balances because they always perform a comparison
between known masses (standards) and unknown masses. These weighing devices are employing the
electromagnetic force compensation principle in conjunction with joint flexures elements replacing the
© 1999 by CRC Press LLC
traditional knife-edge joints. Some of the advantages include a computer interfacing capability and a
maintenance-free feature because there are no moving parts.
Another measuring method used in the weighing technology is the vibrating cord. A wire or cord of
known length, which vibrates transversely, is tensioned by the force F. The vibration frequency changes
in direct proportion to the load F. The piezoelectric effect is also used in weighing technology. Such
weighing devices consist of the presence of an electric voltage at the surface of a crystal when the crystal
is under load. Balances employing the gyroscopic effect are also used. This measuring device uses the
output signal of a gyrodynamic cell similar to the frequency. Balances wherein the weight force of the
load changes the reference distance of the capacitive or inductive converters are also known. As well,
balances using the radioactivity changes of a body as a function of its mass under certain conditions exist.
20.2 Weighing Techniques
When relatively low orders of accuracy are required, reading mass or weight values directly from the
weighing instrument are adequate. Except for the equal arm balance and some torsion balances, most
modern weighing instruments have direct readout capability. For most commercial transactions and for
simple scientific experiments, this direct readout will provide acceptable results.
In the case of equal arm balances, the balance will have a pointer and a scale. When relatively low
accuracy is needed, the pointer and scale are used to indicate when the balance is close to equilibrium.
The same is true when using a torsion balance. However, the equal arm balances of smaller (e.g., 30 g)
or larger (e.g., 900 kg) capacity are also used for high-accuracy applications. Only the new generation of
electronic balances are equal or better in terms of accuracy and benefit from other features.
Weighing is a deceptively simple process. Most people have been making and using weighing measurements for most of their lives. We have all gone to the market and purchased food that is priced by
weight. We have weighed ourselves many times, and most of us have made weight or mass measurements
in school. What could be simpler? One places an object on the weighing pan or platform and reads the
Unfortunately, the weighing process is very susceptible to error. There are errors caused by imperfections in the weighing instrument; errors caused by biases in the standards used; errors caused by the
weighing method; errors caused by the operator; and errors caused by environmental factors. In the case
of the equal arm balance, any difference between the lengths of the arms will result in a bias in the
measurement result. Nearly all weighing devices will have some degree of error caused by small amounts
of nonlinearity in the device. All standards have some amount of bias and uncertainty. Mass is the only
base quantity in the International System of Units (SI) defined in relation with a physical artifact. The
international prototype of the kilogram is kept at Sevres in France, under the custody of the International
Bureau of Weights and Measures. All weighing measurements originate from this international standard.
The international prototype of the kilogram is, by international agreement, exact; however, over the last
century, it has changed in value. What one does not know is the exact magnitude or direction of the
change. Finally, environmental factors such as temperature, barometric pressure, and humidity can affect
the weighing process.
There are many weighing techniques used to reduce the errors in the weighing process. The simplest
technique is substitution weighing. The substitution technique is used to eliminate some of the errors
introduced by the weighing device. The single-substitution technique is one where a known standard
and an unknown object are both weighed on the same device. The weighing device is only used to
determine the difference between the standard and the unknown. First, the standard is weighed and the
weighing device’s indication is noted. (In the case of an equal arm balance, tare weights are added to the
second pan to bring the balance to equilibrium.) The standard is then removed from the weighing device
and the unknown is placed in the same position. Again, the weighing device’s indication is noted. The
first noted indication is subtracted from the second indication. This gives the difference between the
standard and the unknown. The difference is then added to the known value of the standard to calculate
the value of the unknown object. A variation of this technique is to use a small weight of known value
© 1999 by CRC Press LLC
to offset the weighing device by a small amount. The amount of offset is then divided by the known
value of the small weight to calibrate the readout of the weighing device. The weighing results of this
measurement is calculated as follows:
)( ) (O − O )
U = S + O2 − O1 SW
where U
= value of the unknown
= known value of the standard
= small sensitivity weight used to calibrate the scale divisions
= first observation (standard)
= second observation (unknown)
= third observation (unknown + SW)
These techniques remove most of the errors introduced by the weighing device, and are adequate when
results to a few tenths of a gram are considered acceptable.
If results better than a few tenths of a gram are required, environmental factors begin to cause
significant errors in the weighing process. Differences in density between the standard and the unknown
object and air density combine together to cause significant errors in the weighing process.
It is the buoyant force that generates the confusion in weighing. What is called the “true mass” of an
object is the mass determined in vacuum. The terms “true mass” and “mass in vacuum” are referring to
the same notion of inertial mass or mass in the Newtonian sense. In practical life, the measurements are
performed in the surrounding air environment. Therefore, the objects participating in the measurement
process adhere to the Archimedean principle being lifted with a force equal to the weight of the displaced
volume of air. Applying the buoyancy correction to the measurement requires the introduction of the
term “apparent mass.” The “apparent mass” of an object is defined in terms of “normal temperature”
and “normal air density,” conventionally chosen as 20°C and 1.2 mg cm–3, respectively. Because of these
conventional values, the “apparent mass” is also called the “conventional mass.” The reference material
is either brass (8.4 g cm–3) or stainless steel (8.0 g cm–3), for which one obtains an “apparent mass versus
brass” and an “apparent mass versus stainless steel,” respectively. The latter is preferred for reporting the
“apparent mass” of an object.
Calibration reports from the National Institute of Standards and Technology will report mass in three
ways: True Mass, Apparent Mass versus Brass, and Apparent Mass versus Stainless Steel. Conventional
mass is defined as the mass of an object with a density of 8.0 g cm–3, at 20°C, in air with a density of
1.2 mg cm–3. However, most scientific weighings are of materials with densities that are different from
8.0 g cm–3. This results in significant measurement errors.
As an example, use the case of a chemist weighing 1 liter of water. The chemist will first weigh a mass
standard, a 1 kg weight made of stainless steel; then the chemist will weigh the water. The 1 kg mass
standard made of 8.0 g cm–3 stainless steel will have a volume of 125 cm3. The same mass of water will
have a volume approximately equal to 1000 cm3 (Volume = Mass/Density). The mass standard will
displace 125 cm3 of air, which will exert a buoyant force of 150 mg (125 cm3 × 1.2 mg cm–3). However,
the water will displace 1000 cm3 air, which will exert a buoyant force of 1200 mg (1000 cm3 × 1.2 mg cm–3).
Thus, the chemist has introduced a significant error into the measurement by not taking the differing
densities and air buoyancy into consideration.
Using 1.2 mg cm–3 for the density of air is adequate for measurements made close to sea level; it must
be noted that air density decreases with altitude. For example, the air density in Denver, CO, is approximately 0.98 mg cm–3. Therefore, to make accurate mass measurements, one must measure the air density
at the time of the measurement if environmental errors in the measurement are to be reduced.
Air density can be calculated to an acceptable value using the following equations:
ρA ≅ 0.0034848 t + 273.15 P − 0.0037960 × U × es
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where ρA
= air density in mg cm–3
= temperature in °C
= barometric pressure in pascals
= relative humidity in percent
= saturation vapor pressure
es ≅ 1.7526 × 1011 × e
( −5315.56 (273.15 + t ))
where e ≅ 2.7182818
t = temperature in °C
To apply an air buoyancy correction to the single substitution technique, use the following formulae:
) (
M u = ⎛⎝ M s 1 − ρA ρs + O2 − O1 M SW 1 − ρA ρSW
where Mu
) (O − O ))⎞⎠ (1 − ρ
= mass of the unknown (in a vacuum)
= mass of the standard (in a vacuum)
= mass of the sensitivity weight
= air density
= density of the standard
= density of the unknown
= density of the sensitivity weight
= first observation (standard)
= second observation (unknown)
= third observation (unknown + SW)
CM = M u 1 − 0.0012 ρu
) 0.99985
where CM = conventional mass
Mu = mass of the unknown in a vacuum
ρu = density of the unknown
When very precise measurements are needed, the double-substitution technique coupled with an air
buoyancy correction will provide acceptable results for nearly all scientific applications. The doublesubstitution technique is similar to the single-substitution technique using the sensitivity weight. In the
double-substitution technique, the sensitivity weight is weighed with both the mass standard and the
unknown. The main advantage of this technique over single substitution is that any drift in the weighing
device is accounted for in the technique. Because of the precision of this weighing technique, it is only
appropriate to use it on precision balances or mass comparators. As in the case of single substitution,
one places the standard on the balance pan and takes a reading. The standard is then removed and the
unknown object is placed on the balance pan and a second reading is taken. The third step is to add the
small sensitivity weight to the pan with the unknown object and take a third reading. Then remove the
unknown object and return the standard to the pan with the sensitivity weight and take a fourth reading.
The mass is calculated using the following formulae:
Mu =
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(M (1 − ρ ρ ) + (O − O + O − O )) 2 (M ( 1 − ρ
(1 − ρ
) (O − O ))
where Mu
= mass of the unknown (in a vacuum)
= mass of the standard (in a vacuum)
= mass of the sensitivity weight
= air density
= density of the standard
= density of the unknown
= density of the sensitivity weight
= first observation (standard)
= second observation (unknown)
= third observation (unknown + sensitivity weight)
= fourth observation (standard + sensitivity weight)
CM = M u 1 − 0.0012 ρu
) 0.99985
where CM = conventional mass
Mu = mass of the unknown in a vacuum
ρu = density of the unknown
To achieve the highest levels of accuracy, advanced weighing designs have been developed. These
advanced weighing designs incorporate redundant weighing, drift compensation, statistical checks, and
multiple standards. The simplest of these designs is the three-in-one design. It uses two standards to
calibrate one unknown weight. In its simplest form, one would perform three double substitutions. The
first compares the first standard and the unknown weight; the second double substitution compares the
first standard against the second standard, which is called the check standard; and the third and final
comparison compares the second (or check standard) against the unknown weight. These comparisons
would then result in the following:
= reading with standard on the balance
= reading with unknown on the balance
= reading with unknown and sensitivity weight on the balance
= reading with standard and sensitivity weight on the balance
= reading with standard on the balance
= reading with check standard on the balance
= reading with check standard and sensitivity weight on the balance
= reading with standard and sensitivity weight on the balance
= reading with check standard on the balance
= reading with unknown on the balance
= reading with unknown and sensitivity weight on the balance
= reading with check standard and sensitivity weight on the balance
The measured differences are calculated using the following formulae:
) ] [
) 2] × [M (1 − ρ
a = O1 − O2 + O4 − O3 2 × M SW 1 − ρA ρSW
b = O5 − O6 + O8 − O7
) ] [
c = O9 − O10 + O12 − O11 2 × M SW 1 − ρA ρSW
© 1999 by CRC Press LLC
) O −O ]
) O −O ]
− O10
where a
= difference between standard and unknown
= difference between standard and check standard
= difference between check standard and unknown
= mass of sensitivity weight
= air density calculated using Equations 20.5 and 20.6
= density of sensitivity weight
The least-squares measured difference is computed for the unknown from:
du = −2a − b − c 3
Using the least-squares measured difference, the mass of the unknown is computed as:
) ) (1 − ρ
U = S 1 − ρA ρS + du
where U = mass of unknown
S = mass of the standard
du = least-squares measured difference of the unknown
ρA = air density calculated using Equations 20.5 and 20.6
ρS = density of the standard
ρU = density of the unknown
The conventional mass of the unknown is now calculated as:
CU = U 1 − 0.0012 ρU
) 0.99985
where CU = conventional mass
U = mass of unknown
ρU = density of unknown
The least-squares measured difference is now computed for the check standard as:
dCS = −a − 2b − c 3
Using the least-squares measured difference, the mass of the check standard is computed from:
CS = S 1 − ρA ρS + dCS
where CS
) (1 − ρ
= mass of check standard
= mass of the standard
= least-squares measured difference of the check standard
= air density calculated using Equations 20.5 and 20.6
= density of the standard
= density of unknown
The mass of the check standard must lie within the control limits for the check standard. If it is out of
the control limits, the measurement must be repeated.
The short-term standard deviation of the process is now computed:
Short-term standard deviation = 0.577 a − b + c
© 1999 by CRC Press LLC
The short-term standard deviation is divided by the historical pooled short-time standard deviation
to calculate the F-statistic:
F-statistic = short-term standard deviation/pooled short-time standard deviation
The F-statistic must be less than the value obtained from the student t-variant at the 99% confidence
level for the number of degrees of freedom of the historical pooled standard deviation. If this test fails,
the measurement is considered to be out of statistical control and must be repeated.
By measuring a check standard and by computing the short-term standard deviation of the process
and comparing them to historical results, one obtains a high level of confidence in the computed value
of the unknown.
There are many different weighing designs that are valid; the three-in-one (three equal weights) and
four equal weights are the ones that can be easily calculated without the use of a computer. Primary
calibration laboratories — private and government — are using these multiple intercomparisons, stateof-the-art mass calibration methods under the Mass Measurement Assurance Program using the Mass
Code computer program provided by the National Institute of Standards and Technology. A full discussion of these designs can be found in the National Bureau of Standards Technical Note 952.
1. J. K. Taylor and H. V. Oppermann, Handbook for the Quality Assurance of Metrological Measurements, NBS Handbook 145, Washington, D.C.: U.S. Department of Commerce, National Bureau
of Standards, 1986.
2. L. V. Judson, Weights and Measures Standards of the United States, A Brief History, NBS Special
Publication 447, Washington, D.C.: Department of Commerce, National Bureau of Standards, 1976.
3. P. E. Pontius, Mass and Mass Values, NBS Monograph 133, Washington, D.C.: Department of
Commerce, National Bureau of Standards, 1974.
4. K.B. Jaeger and R. S. Davis, A Primer for Mass Metrology, NBS Special Publication 700-1, Washington, D.C.: Department of Commerce, National Bureau of Standards, 1984.
5. J. M. Cameron, M. C. Croarkin, and R. C. Raybold, Designs for the Calibration of Standards of
Mass, NBS Technical Note 952, Washington, D.C.: Department of Commerce, National Bureau of
Standards, 1977.
6. G. L. Harris (Ed.), Selective Publications for the Advanced Mass Measurements Workshop, NISTIR
4941, Washington, D.C.: Department of Commerce, National Institute of Standards and Technology, 1992.
7. Metron Corporation, Physical Measurements, NAVAIR 17-35QAL-2, California: U.S. Navy, 1976.
8. R. S. Cohen, Physical Science, New York: Holt, Rinehart and Winston, 1976.
9. D. B. Prowse, The Calibration of Balances, CSIRO Division of Applied Physics, Australia, 1995.
10. E. Hazarian, Techniques of mass measurement, Southern California Edison Mass Seminar Notebook,
Los Angeles, CA, 1994.
11. E. Hazarian, Analysis of mechanical convertors of electronic balances, Measurement Sci. Conf.,
Anaheim, CA, 1993.
12. B. N. Taylor, Guide for the use of the International System of Units (SI), NIST SP811, 1995.
© 1999 by CRC Press LLC
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Halit Eren. "Density Measurement."
Density Measurement
Halit Eren
Curtin University of Technology
Solid Density
Fluid Density
Pycnometric Densitometers • Buoyancy Type Densitometers •
Hydrometers • Hydrostatic Weighing Densitometers •
Balance-Type Densitometers • Column-Type Densitometers •
Vibrating Element Densitometers • Radioactive
Densitometers • Refractometer and Index of Refraction
Densitometers • Coriolis Densitometers • Absorption-Type
Density is a significant part of measurement and instrumentation. Density measurements are made for
at least two important reasons: (1) for the determination of mass and volume of products, and (2) the
quality of the product. In many industrial applications, density measurement ascertains the value of the
Density is defined as the mass of a given volume of a substance under fixed conditions. However,
ultimate care must be exercised in measurements because density varies with pressure and temperature.
The variation is much greater in gases.
In many modern applications, the densities of products are obtained by sampling techniques. In
measurements, there are two basic concepts: static density measurements and dynamic (on-line) density
measurements. Within each concept, there are many different methods employed. These methods are
based on different physical principles. In many cases, the application itself and the characteristics of the
process determine the best suitable method to be used. Generally, static methods are well developed,
lower in cost, and more accurate. Dynamic samplers are expensive, highly automated, and use microprocessor-based signal processing devices. Nevertheless, nowadays, many static methods are also computerized, offering easy to use, flexible, and self-calibrating features.
There is no single universally applicable density measurement technique. Different methods must be
employed for different products and materials. In many cases, density is normalized under reference
The density of a substance is determined by dividing the density of that substance by the density of a
standard substance obtained under the same conditions. This dimensionless ratio is called the specific
gravity (SG), also termed the relative density. The specific gravities of liquid and gases under reference
conditions are given by:
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Liquid SG = density of liquid density of water
Gas SG = density of gas density of air
Commonly accepted sets of conditions are normal temperature and pressure (NTP) and standard
temperature and pressure (STP). NTP is usually taken as the temperature of 0.00°C and a pressure of
760 mm Hg. The NTP is accepted as 15.00 or 15.56°C and 101.325 kPa.
21.1 Solid Density
If the uniformity is maintained, the determination of density of solids is a simple task. Once the volume
of the solid and its mass are known, the density can be found using the basic ratio: density = mass/volume
(kg m–3).
However, in many applications, solids have different constituents and are made up from different
materials having different ratios. Their volumes can also change often. In these cases, dynamic methods
are employed, such as radioactive absorption types, ultrasonic, and other techniques. Some of these
methods are described below.
21.2 Fluid Density
The measurement of densities of fluids is much more complex than for solids. For fluid densities, many
different techniques are available. This is mainly due to complexities in processes, variations of fluid
densities during the processes, and diverse characteristics of the process and the fluids themselves. Some
of these methods are custom designed and applicable to special cases only. Others are very similar in
principles and technology, and applicable to many different type of fluids. At present, apart from conventional methods, there are many novel and unusual techniques undergoing extensive development and
research. For example, densitometers based on electromagnetic principles [1] can be given as part of an
intelligent instrumentation system.
Fluids can be divided to liquids and gases. Extra care and further considerations are necessary in gas
density measurements. For example, perfect gases contain an equal number of molecules under the same
conditions and volumes. Therefore, molecular weights can be used in density measurements.
Depending on the application, fluid densities can be measured both in static and dynamic forms. In
general, static density measurements of fluids are well developed, precise, and have greater resolution
than most dynamic techniques. Pycnometers and buoyancy are examples of static techniques that can
be adapted to cover small density ranges with a high resolution and precision. Nowadays, many manufacturers offer dynamic instruments previously known to be static. Also, many static density measurement
devices are computerized and come with appropriate hardware and software. In general, static-type
measurements are employed in laboratory conditions, and dynamic methods are employed for real-time
measurements where the properties of a fluid vary from time to time.
In this chapter, the discussion will concentrate on the commonly applied, modern density measuring
devices. These devices include:
Pycnometric densitometers
Buoyancy-type densitometers
Hydrostatic weighing densitometers
Balance-type densitometers
Column-type densitometers
Vibrating element densitometers
Radioactive densitometers
Refractometer and index of reflection densitometers
Coriolis densitometers
Absorption-type densitometers
© 1999 by CRC Press LLC
FIGURE 21.1 A pycnometer. A fixed volume container
is filled with liquid and weighed accurately; capillary is
used to determine the exact volume of the liquid.
Pycnometric Densitometers
Pycnometers are static devices. They are manufactured as fixed volume vessels that can be filled with the
sample liquid. The density of the fluid is measured by weighing the sample. The simplest version consists
of a vessel in the shape of a bottle with a long stopper containing a capillary hole, as shown in Figure 21.1.
The capillary is used to determine the exact volume of the liquid, thus giving high resolution when filling
the pycnometer. The bottle is first weighed empty, and then with distilled-aerated water to determine
the volume of the bottle. The bottle is then filled with the process fluid and weighed again. The density
is determined by dividing the mass by the volume. The specific gravity of the liquid is found by the ratio
of the fluid mass to water mass. When pycnometers are used, for good precision, ultimate care must be
exercised during the measurements; that is, the bottle must be cleaned after each measurement, the
temperature must be kept constant, and precision balances must be used. In some cases, to ensure filling
of the pycnometer, twin capillary tubes are used. The two capillaries, made of glass, are positioned such
that the fluid can be driven into the vessel under vacuum conditions. Accurate filling to graduation marks
on the capillary is then made.
The pycnometers have to be lightweight, strong enough to contain samples, and they need to be
nonmagnetic for accurate weighing to eliminate possible ambient magnetic effects. Very high-resolution
balances must be used to detect small differences in weights of gases and liquids. Although many
pycnometers are made of glass, they are also made of metals to give enough strength for the density
measurements of gases and liquids at extremely high pressures. In many cases, metal pycnometers are
necessary for taking samples from the line of some rugged processes.
Pycnometers have advantages and disadvantages. Advantages are that if used correctly, they are accurate; and they can be used for both density and specific gravity measurements. The disadvantages include:
1. Great care must be exercised for accurate results.
2. The sample must be taken off-line, with consequent time lag in results. This creates problems of
relating samples to the materials that exist in the actual process.
3. High-precision pycnometers are expensive. They require precision weighing scales and controlled
laboratory conditions. Specialized techniques must be employed to take samples in high-pressure
processes and hostile conditions, such as offshore installations.
4. Their good performances might depend on the skill of operator.
Buoyancy-Type Densitometers
The buoyancy method basically uses Archimedes principle. A suspended sinker, with a known mass and
volume attached to a fine wire, is totally immersed in the sample liquid. A precision force balance is used
to measure the force to support the sinker. Once the mass, volume, and supporting weight of the sinker
© 1999 by CRC Press LLC
FIGURE 21.2 Hydrometer. A fixed weight and volume
bulb is placed into the liquid. The bulb sinks in the liquid,
depending on its density. The density is read directly from
the scale. Temperature correction is necessary.
are known, the density of the liquid can be calculated. However, some corrections need to be made for
surface tension on the suspension wire, the cubicle expansion coefficient of the sinker, and the temperature of process. Buoyancy-type densitometers give accurate results and are used for the calibration of
the other liquid density transducers.
One advanced version of the buoyancy technique is the magnetic suspension system. The sinker is
fully enclosed in a pressure vessel, thus eliminating surface tension errors. Their uses can also be extended
to applications such as the specific gravity measurements under low vapor pressures and density measurements of hazardous fluids.
Hydrometers are the most commonly used devices for measurement of the density of liquids. They are
so commonly used that their specifications and procedure of use are described by national and international standards, such as ISO 387. The buoyancy principle is used as the main technique of operation.
The volume of fixed mass is converted to a linear distance by a sealed bulb-shaped glass tube containing
a long stem measurement scale, shown in Figure 21.2. The bulb is ballasted with a lead shot and pitch,
the mass of which depends on the density range of the liquid to be measured. The bulb is simply placed
into the liquid and the density is read from the scale. The scale is graduated in density units such as
kg m–3. However, many alternative scales are offered by manufacturers, such as specific gravity, API gravity,
Brix, Brine, etc. Hydrometers can be calibrated for different ranges for surface tensions and temperatures.
Temperature corrections can be made for set temperature such as 15°C, 20°C, or 25°C. ISO 387 covers
a density range of 600 kg m–3 to 2000 kg m–3. Hydrometers have a number of advantages and disadvantages. The advantages include:
1. Relatively low cost and easy to use
2. Good resolution for small range
3. Traceable to national and international standards
The disadvantages include:
1. They have small span; therefore, a number of meters are required to cover a significant range.
2. They are made from glass and fragile. Metal and plastic versions are not as accurate.
3. The fluid needs to be an off-line sample, not representing the exact conditions of the process.
There are pressure hydrometers for low vapor pressure hydrocarbons, but this adds a need for
accurately determining the pressure too.
4. If good precision is required, they are difficult to use, needing surface tension and temperature
corrections. Further corrections could be required for opaque fluids.
© 1999 by CRC Press LLC
FIGURE 21.3 Hydrostatic weighing. The total weight of a fixed-volume tube containing liquid is determined
accurately. The density is calculated using mass: volume ratio.
Hydrostatic Weighing Densitometers
The most common device using a hydrostatic weighing method consists of a U-tube that is pivoted on
flexible end couplings. A typical example is shown in Figure 21.3. The total weight of the tube changes,
depending on the density of fluid flowing through it. The change in the weight needs to be measured
accurately, and there are a number of methods employed to do this. The most common commercial
meters use a force balance system. The connectors are stainless steel bellows. In some cases, rubber or
PTFE are used, depending on the process fluid characteristics and the accuracy required. There are
temperature and pressure limitations due to bellows, and the structure of the system may lead to a reading
offset. The meter must be securely mounted on a horizontal plane for optimal accuracy.
The advantages of this method include:
1. They give continuous reading and can be calibrated accurately.
2. They are rugged and can be used for two-phase liquids such as slurries, sugar solutions, powders,
The disadvantages of these meters include:
1. They must be installed horizontally on a solid base. These meters are not flexible enough to adapt
to any process; thus, the process must be designed around it.
2. They are bulky and cumbersome to use.
3. They are unsuitable for gas density measurements.
Balance-Type Densitometers
Balance-type densitometers are suitable for liquid and gas density measurements. Manufacturers offer
many different types; four of the most commonly used ones are discussed below.
Balanced-Flow Vessel
A fixed volume vessel as shown in Figure 21.4 is employed for the measurements. While the liquid is
flowing continuously through the vessel, it is weighed automatically by a sensitive scale — a spring balance
system or a pneumatic force balance transmitter. Because the volume and the weight of the liquid are
known, the density or specific gravity can easily be calculated and scaled in respective units. In the design
process, extra care must be exercised for the flexible end connections.
© 1999 by CRC Press LLC
FIGURE 21.4 Balanced flow vessel. An accurate spring balance or force balance system is used to weigh the vessel
as the liquid flows through it.
FIGURE 21.5 Chain balance float. The fixed volume and weight plummet totally suspended in the liquid assumes
equilibrium position, depending on the density. The force exerted by the chains on the plummet is a function of
plummet position; hence, the measured force is proportional to the density of the liquid.
Chain Balanced Float
In this system, a self-centering, fixed-volume, submerged plummet is used for density measurements, as
illustrated in Figure 21.5. The plummet is located entirely under the liquid surface. At balance, the
plummet operates without friction and is not affected by surface contamination. Under steady-state
conditions, the plummet assumes a stable position. The effective weight of the chain on the plummet
varies, depending on the position of the plummet, which in turn is a function of the density of the liquid.
The plummet contains a metallic transformer core that transmits changes in the position to be measured
by a pickup coil. The voltage differential, a function of plummet displacement, is calibrated as a measure
of variations in specific gravity. A resistance thermometer bridge is used for the compensation of temperature effects on density.
Gas Specific Gravity Balance
A tall column of gas is weighed by the floating bottom of the vessel. This weight is translated into the
motion of an indicating pointer, which moves over a scale graduated in units of density or specific gravity.
This method can be employed for any gas density measurement.
© 1999 by CRC Press LLC
FIGURE 21.6 Buoyancy gas balance. The position of the balance beam is adjusted by a set pressure air, air is then
displaced by gas of the same pressure. The difference in the reading of the balance beam gives the SG of the gas. The
pressures are read on the manometer.
FIGURE 21.7 Reference column densitometer. Two identical tubes, having the same distance from the surface, are
placed in water and liquid. Water with known density characteristics is used as the reference. The pressures necessary
to displace the fluids inside the tubes are proportional to the densities of the fluids. The pressure difference at the
differential pressure transmitter is translated into relative densities.
Buoyancy Gas Balance
In this instrument, a displacer is mounted on a balance beam in a vessel, as shown in Figure 21.6. The
displacer is balanced for air, and the manometer reading is noted at the exact balance pressure. The air
is then displaced by gas, and the pressure is adjusted until the same balance is restored. The ratio of the
pressure of air to the pressure of gas is then the density of the gas relative to air. This method is commonly
applied under laboratory conditions and is not suitable for continuous measurements.
Column-Type Densitometers
There are number of different versions of column methods. As a typical example, a reference column
method is illustrated in Figure 21.7. A known head of sample liquid and water from the respective bubbler
pipes are used. A differential pressure measuring device compares the pressure differences, proportional
to relative densities of the liquid and the water. By varying the depth of immersion of the pipes, a wide
© 1999 by CRC Press LLC
FIGURE 21.8 Two-tube column densitometer. The pressure difference at the differential pressure transmitter
depends on the relative positions of the openings of the pipes and the density of liquid. Once the relative positions
are fixed, the pressure difference can be related to the equivalent weight of the liquid column at the openings of the
pipes, hence the density of the liquid.
FIGURE 21.9 Suppression-type, two-tube column densitometer. Operation principle is the same as in Figures 21.7
and 21.8. In this case, the suppression chamber affords greater accuracy in readings.
range of measurements can be obtained. Both columns must be maintained at the same temperature to
avoid the necessity for corrections of temperature effects.
A simpler and more widely used method of density measurement is achieved by the installation of
two bubbler tubes as illustrated in Figure 21.8. The tubes are located in the sample fluid such that the
end of one tube is higher than that of the other. The pressure required to bubble air into the fluid from
both tubes is equal to the pressure of the fluid at the end of the bubbler tubes. The outlet of one tube
higher than the other and the distances of the openings of the tubes are fixed; hence, the difference in
the pressure is the same as the weight of a column of liquid between the ends. Therefore, the differential
pressure measurement is equivalent to the weight of the constant volume of the liquid, and calibrations
can be made that have a direct relationship to the density of the liquid. This method is accurate to within
0.1% to 1% specific gravity. It must be used with liquids that do not crystallize or settle in the measuring
chamber during measurements.
Another version is the range suppression type, which has an additional constant pressure drop chamber
as shown in Figure 21.9. This chamber is in series with the low-pressure side to give advantages in scaling
and accurate readings of densities.
© 1999 by CRC Press LLC
Vibrating Element Densitometers
If a body containing or surrounded by a fluid is set to resonance at its natural frequency, then the
frequency of oscillation of the body will vary as the fluid properties and conditions change. The natural
frequency is directly proportional to the stiffness of the body and inversely proportional to the combined
mass of the body and the fluid. It is also dependent on the shape, size, and elasticity of the material,
induced stress, mass, and mass distribution of the body. Basically, the vibration of the body can be equated
to motion of a mass attached to a mechanical spring. Hence, an expression for the frequency can be
written as:
( (
Resonant frequency = SQRT K M + kρ
where K is the system stiffness, M is the transducer mass, k is the system constant, and ρ is the fluid density.
A factor common to all types of vibrating element densitometers is the problem of setting the element
in vibration and maintaining its natural resonance. There are two drives for the purpose.
Magnetic Drives
Magnetic drives of the vibrating element and the pickup sensors of vibrations are usually achieved using
small coil assemblies. Signals picked up by the sensors are amplified and fed back as a drive to maintain
the disturbing forces on the vibrating body of the meter.
In order to achieve steady drives, the vibrating element sensor can be made from nonmagnetic
materials. In this case, small magnetic armatures are attached.
The main advantage of magnetic drive and pickup systems is they are noncontact methods. They use
conventional copper windings and they are reliable within the temperature range of –200 to +200°C.
Piezoelectric Drives
A wide range of piezoelectric materials are available to meet the requirements of driving vibrating
elements. These materials demonstrate good temperature characteristics as do magnetic drive types. They
also have the advantage of being low in cost. They have high impedance, making the signal conditioning
circuitry relatively easy. They do not require magnetic sensors.
The piezoelectric drives are mechanically fixed on the vibrating element by adhesives. Therefore,
attention must be paid to the careful placement of the mount in order to reduce the strain experienced
by the piezo element due to thermal and pressure stresses while the instrument is in service.
A number of different types of densitometers have been developed that utilize this phenomenon. The
three main commercial types are introduced here.
Vibrating Tube Densitometers
These devices are suitable for highly viscous liquids or slurry applications. The mode of operation of
vibration tube meters is based on the transverse vibration of tubes as shown in Figure 21.10. The tube
and the driving mechanisms are constrained to vibrate on a single plane. As the liquid moves inside the
tube, the density of the entire mass of the liquid is measured. The tube length is approximately 20 times
greater than the tube diameter.
A major design problem with the vibrating tube method is the conflict to limit the vibrating element
to a finite length and accurately fix the nodes. Special attention must be paid to avoid any exchange of
vibrational energy outside the sensory tube. The single tube has the disadvantage of presenting obstruction to the flow, thus experiencing some pressure losses. The twin tube, on the other hand, offers very
small blockage (Figure 21.11) and can easily be inspected and cleaned. Its compact size is another distinct
advantage. In some densitometers, the twin tube is designed to achieve a good dynamic balance, with
the two tubes vibrating in antiphase. Their nodes are fixed at the ends, demonstrating maximum sensitivity to installation defects, clamping, and mass loading.
© 1999 by CRC Press LLC
FIGURE 21.10 Vibrating tube densitometer. Tube containing fluid is vibrated at resonant frequency by electromagnetic vibrators. The resonant frequency, which is a function of the density of the fluid, is measured accurately. The
tube is isolated from the fixtures by carefully designed bellows.
FIGURE 21.11 Two-tube vibrating densitometer. Two tubes are vibrated in antiphase for greater accuracy. Twintube densitometers are compact in size and easy to use.
The main design problems of the vibrating tube sensors are in minimizing the influence of end padding
and overcoming the effects of pressure and temperature. Bellows are used at both ends of the sensor
tubes to isolate the sensors from external vibrations. Bellows also minimize the end loadings due to
differential expansions and installation stresses.
The fluid runs through the tubes; therefore, no pressure balance is required. Nevertheless, in some
applications, the pressure stresses the tubes, resulting in stiffness changes. Some manufacturers modify
the tubes to minimize the pressure effects. In these cases, corrections are necessary only when high
accuracy is mandatory. The changes in the Young’s modulus with temperature can be reduced to nearzero using Ni-span-C materials whenever corrosive properties of fluids permit. Usually, manufacturers
provide pressure and temperature correction coefficients for their products.
It is customary to calibrate each vibration element densitometer against others as a transfer of standards. Often, the buoyancy method is used for calibration purposes. The temperature and pressure
coefficients are normally found by exercising the transducer over a range of temperatures and pressures
on a liquid with well-known properties. Prior to calibration, the vibration tube densitometers are subjected to a programmed burn-in cycle to stabilize them against temperatures and pressures.
Vibrating Cylinder Densitometers
A thin-walled cylinder, with a 3:1 length:diameter ratio, is fixed with stiff ends. The thickness of the
cylinder wall varies from 25 μm to 300 μm, depending on the density range and type of fluid used. The
cylinder can be excited to vibrate in a hoop mode by magnetic drives mounted either in or outside the
For good magnetic properties, the cylinder is made of corrosion-resistant magnetic materials. Steel
such as FV520 is often used for this purpose. Such materials have good corrosion-resistance characteristics;
unfortunately, due to their poor thermoelastic properties, they need extensive temperature corrections.
© 1999 by CRC Press LLC
FIGURE 21.12 Tuning fork densitometer. Twin forks are inserted into the fluid and the natural frequencies are
measured accurately. The natural frequency of the forks is a function of the density of the fluid.
Nickel-iron alloys such as Ni-span-C are often used to avoid temperature effects. Once correctly treated,
the Ni-span-C alloy has near-zero Young’s modulus properties. Because the cylinder is completely
immersed in the fluid, there are no pressure coefficients.
The change in the resonant frequency is determined by the local mass loading of the fluid in contact
with the cylinder. The curve of frequency against density is nonlinear and has a parabolic shape, thus
requiring linearization to obtain practical outputs. The resonant frequency range varies from 2 kHz to
5 kHz, depending on the density range of the instrument. The cylinders need precision manufacturing
and thus are very expensive to construct. Each meter needs to be calibrated individually for different
temperatures and densities to suit specific applications. In the case of gas density applications, gases with
well-known properties (e.g., pure argon or nitrogen) are used for calibrations. In this case, the meters
are subjected to a gas environment with controlled temperature and pressure. The calibration curves are
achieved by repetitions to suit the requirements of individual customers for particular applications. In
the case of liquids, the meters are calibrated with liquids of known density, or they are calibrated against
another standard (e.g., pycnometer or buoyancy type densitometers).
Vibration cylinder-type densitometers have zero pressure coefficients and they are ideal for liquefied
gas products or refined liquids. Due to relatively small clearances between cylinder and housing, they
require regular cleaning. They are not suitable for liquids or slurries with high viscous properties.
Tuning Fork Densitometers
These densitometers make use of the natural frequency of low-mass tuning forks, shown in Figure 21.12.
In some cases, the fluid is taken into a small chamber in which the electromechanically driven forks are
situated. In other cases, the fork is inserted directly into the liquid. Calibration is necessary in each
The advantages of vibrating element meters include:
They are suitable for both liquids and gases with reasonable accuracy.
They can be designed for real-time measurements.
They can easily be interfaced because they operate on frequencies and are inherently digital.
They are relatively robust and easy to install.
Programmable and computerized versions are available. Programmed versions make all the corrections automatically. They provide the output of live density, density at reference conditions,
relative density, specific gravity, concentration, solid contents, etc.
The disadvantages include:
1. They do not relate directly to primary measurements; therefore, they must be calibrated.
2. They all have problems in measuring multiphase liquids.
Radioactive Densitometers
As radioactive isotopes decay, they emit radiation in the form of particles or waves. This physical
phenomenon can be used for the purposes of density measurement. For example, γ rays are passed
© 1999 by CRC Press LLC
through the samples and their rate of arrivals are measured using ion- or scintillation-based detection [2].
Generally, γ-ray mass absorption rate is independent of material composition; hence they can be programmed for a wide range materials. Densitometers based on radiation methods can provide accuracy
up to +0.0001 g mL–1. Many of these devices have self-diagnostic capabilities and are able to compensate
for drift caused by source decay, thus pinpointing any signaling problems.
If γ rays of intensity J0 penetrate a material of a density ρ and thickness d then the intensity of the
radiation after passing through the material can be expressed by:
J = J 0 exp n ρ d
where n is the mass absorption coefficient.
The accuracy of the density measurement depends on the accuracy of the measurement of the intensity
of the radiation and the path length d. A longer path length through the material gives a stronger detection
For accurate operations, there are many arrangements for relative locations of transmitters and detectors, some of which are illustrated in Figures 21.13 and 21.14. Generally, the source is mounted in a lead
container clamped onto the pipe or the container wall. In many applications, the detector is also clamped
onto the wall.
FIGURE 21.13 Fixing radioactive densitometer on an enlarged pipe. The pipe is enlarged to give longer beam length
through the liquid, and hence better attenuation of the radioactive energy.
FIGURE 21.14 Fixing radioactive densitometer on an elongated pipe. Elongated path yields a longer path length
of the radioactive energy through the liquid; hence, a stronger attenuation.
© 1999 by CRC Press LLC
The advantages of using radioactive methods include:
The sensor does not touch the sample; hence, there is no blockage to the path of the liquid.
Multiphase liquids can be measured.
They come in programmable forms and are easy to interface.
They are most suitable in difficult applications, such as mining and heavy process industries.
The disadvantages include:
A radioactive source is needed; hence, there is difficulty in handling.
For reasonable accuracy, a minimum path length is required.
There could be long time constants, making them unsuitable in some applications.
They are suitable only for solid and liquid density measurements.
Refractometer and Index of Refraction Densitometers
Refractometers are essentially optical instruments operating on the principles of refraction of light
traveling in liquid media. Depending on the characteristics of the samples, measurement of refractive
index can be made in a variety of ways (e.g., critical angle, collimation, and displacement techniques).
Usually, an in-line sensing head is employed, whereby a sensing window (commonly known as a prism)
is wetted by the product to be measured. In some versions, the sensing probes must be installed inside
the pipelines or in tanks and vessels. They are most effective in reaction-type process applications where
blending and mixing of liquids take place. For example, refractometers can measure dissolved soluble
solids accurately.
Infrared diodes, lasers, and other lights may be used as sources. However, this measurement technique
is not recommended in applications in processes containing suspended solids, high turbidity, entrained
air, heavy colors, poor transparency and opacity, or extremely high flow rates. The readings are automatically corrected for variations in process temperature. The processing circuitry can include signal
outputs adjustable in both frequency and duration.
Another version of a refractometer is the index of refraction type densitometer. For example, in the
case of position-sensitive detectors, the index of refraction of liquid under test is determined by measuring
the lateral displacement of a laser beam. When the laser beam impinges on the cell at an angle of incidence,
as in Figure 21.15, the axis of the emerging beam is displaced by the cell wall and by the inner liquid.
The lateral displacement can accurately be determined by position-sensitive detectors. For maximum
sensitivity, the devices need to be calibrated with the help of interferometers.
Refractometers are often used for the control of adulteration of liquids of common use (e.g., edible
oils, wines, and gasoline). They also find application in pulp and paper, food and beverage, sugar, dairy,
and other chemical industries.
Coriolis Densitometers
The Coriolis density metering systems are similar to vibrating tube methods, but with slight variations
in the design. They are comprised of a sensor and a signal-processing transmitter. Each sensor consists
of one or two flow tubes enclosed in a sensor housing. They are manufactured in various sizes and shapes
[3]. The sensor tubes are securely anchored at the fluid inlet and outlet points and force is vibrated at
the free end, as shown in Figure 21.16. The sensor operates by applying Newton’s second law of motion
(F = ma).
Inside the housing, the tubes are vibrated in their natural frequencies using drive coils and a feedback
circuit. This resonant frequency of the assembly is a function of the geometry of the element, material
of construction, and mass of the tube assembly. The tube mass comprises two parts: the mass of the tube
itself and the mass of the fluid inside the tube. The mass of the tube is fixed for a given sensor. The mass
of fluid in the tube is equal to the fluid density multiplied by volume. Because the tube volume is constant,
the frequency of oscillation can be related directly to the fluid density. Therefore, for a given geometry
© 1999 by CRC Press LLC
FIGURE 21.15 Index of refraction-type densitometer. The angle of refraction of the beam depends on the shape,
size, and thickness of the container, and the density of fluid in the container. Because the container has the fixed
characteristics, the position of the beam can be related to density of the fluid. Accurate measurement of the position
of the beam is necessary.
FIGURE 21.16 Coriolis densitometer. Vibration of the tube is detected and related to the mass and flow rate of the
fluid. Further calibrations and calculations must be made to determine the densities.
of tube and the material of the construction, the density of the fluid can be determined by measuring
the resonant frequency of vibration. Temperature sensors are used for overcoming the effects of changes
in modulus of elasticity of the tube. The fluid density is calculated using a linear relationship between
the density and the tube period and calibration constants.
Special peripherals, based on microprocessors, are offered by various manufacturers for a variety of
measurements. However, all density peripherals employ the natural frequency of the sensor coupled with
the sensor temperature to calculate on-line density of process fluid. Optional communication, interfacing
facilities, and appropriate software are also offered.
Absorption-Type Densitometers
Absorption techniques are also used for density measurements in specific applications. X-rays, visible
light, UV light, and sonic absorptions are typical examples of this method. Essentially, attenuation and
© 1999 by CRC Press LLC
phase shift of a generated beam going through the sample is sensed and related to the density of the
sample. Most absorption-type densitometers are custom designed for applications having particular
characteristics. Two typical examples are: (1) UV absorption or X-ray absorptions are used for determining the local densities of mercury deposits in arc discharge lamps, and (2) ultrasonic density sensors
are used in connection with difficult density measurements (e.g., density measurement of slurries). The
lime slurry, for example, is a very difficult material to handle. It has a strong tendency to settle out and
coat all equipment with which it comes in contact. An ultrasonic density control sensor can fully be
emerged into an agitated slurry, thus avoiding the problems of coating and clogging. Inasmuch as the
attenuation of the ultrasonic beam is proportional to the suspended solids, the resultant electronic signal
is proportional to the specific gravity of the slurry. Such devices can give accuracy up to 0.01%. The
ultrasonic device measures the percentage of the suspended solids in the slurry by providing a close
approximation of the specific gravity.
1. H. Eren, Particle concentration characteristics and density measurements of slurries using electromagnetic flowmeters, IEEE Trans. Instr. Meas., 44, 783-786, 1995.
2. Micro Motion Product Catalogue, Mount Prospect, IL: Fisher-Rosemount, 1995.
3. Kay-Ray, Solution for Process Measurement, Mount Prospect, IL: Fisher-Rosemount, 1995.
List of Manufacturers
ABB K-Flow Inc.
Drawer M Box 849
Millville, NJ 08332
Tel: (800) 825-3569
American Density Materials Inc.
Rd. 2, Box 38E
Belvidere, J 07823
Tel: (908) 475-2373
Anton Paar U.S.A.
13, Maple Leaf Ct.
Ashland, VA 23005
Tel: (800) 221-0174
Arco Instrument Company, Inc.
1745 Production Circle
Riverside, CA 92509
Tel: (909) 788-2823
Fax: (909) 788-2409
Cambridge Applied Systems, Inc.
196 Boston Avenue
Medford, MA 02155
Tel: (617) 393-6500
© 1999 by CRC Press LLC
Automation Products, Inc.
3032 Max Roy Street
Houston, TX 77008
Tel: (800) 231-2062
Fax: (713) 869-7332
Kay-Ray/Sensall, Fisher-Rosemount
1400 Business Center Dr.
Mount Prospect, IL 60056
Tel: (708) 803-5100
Fax: (708) 803-5466
McGee Engineering Co., Inc.
Tujunga Canyon Blvd.
Tujunga, CA 91042
Tel: (800) 353-6675
Porous Materials, Inc.
Cornell Business & Technology Park
Ithaca, NY 14850
Tel: (800) 825-5764
Princo Instruments Inc
1020 Industrial Hwy., Dept L
Southampton, PA 18966-4095
Tel: (800) 496-5343
Quantachrome Corp.
1900-T Corporate Drive
Boynton Beach, FL 33426
Tel: (800) 966-1238
Tricor Systems, Inc.
400-T River Ridge Rd.
Elgin, IL 60123
Tel: (800) 575-0161
X-rite, Inc.
3100-T 44th St. S.W
Grandville, MI 49418
Tel: (800) 545-0694
© 1999 by CRC Press LLC
Copyright 2000 CRC Press LLC. <>.
Christopher S. Lynch. "Strain Measurement."
Strain Measurement
Christopher S. Lynch
The Georgia Institute of Technology
Fundamental Defintions of Strain
Principles of Operation of Strain Sensors
Piezoresistive Foil Gages • Piezoresistive Semiconducting
Gages • Piezoelectric Gages • Fiber Optic Strain Gages •
Birefringent Film Strain Sensing • Moiré Strain Sensing
This chapter begins with a review of the fundamental definitions of strain and ways it can be measured.
This is followed by a review of the many types of strain sensors and their application, and sources for
strain sensors and signal conditioners. Next, a more detailed look is taken at operating principles of
various strain measurement techniques and the associated signal conditioning.
22.1 Fundamental Definitions of Strain
Stress and strain are defined in many elementary textbooks about the mechanics of deformable bodies
[1, 2]. The terms stress and strain are used to describe loads on and deformations of solid materials. The
simplest types of solids to describe are homogeneous and isotropic. Homogeneous means the material
properties are the same at different locations and isotropic means the material properties are independent
of direction in the material. An annealed steel bar is homogeneous and isotropic, whereas a human femur
is not homogeneous because the marrow has very different properties from the bone, and it is not
isotropic because its properties are different along the length and along the cross-section.
The concepts of stress and strain are introduced in the context of a long homogeneous isotropic bar
subjected to a tensile load (Figure 22.1). The stress σ, is the applied force F, divided by the cross-sectional
area A. The resulting strain ε, is the length change ΔL, divided by the initial length L. The bar elongates
in the direction the force is pulling (longitudinal strain εL ) and contracts in the direction perpendicular
to the force (transverse strain εt ).
When the strain is not too large, many solid materials behave like linear springs; that is, the displacement is proportional to the applied force. If the same force is applied to a thicker piece of material, the
spring is stiffer and the displacement is smaller. This leads to a relation between force and displacement
that depends on the dimensions of the material. Material properties, such as the density and specific
heat, must be defined in a manner that is independent of the shape and size of the specimen. Elastic
material properties are defined in terms of stress and strain. In the linear range of material response, the
stress is proportional to the strain (Figure 22.2). The ratio of stress to strain for the bar under tension is
an elastic constant called the Young’s modulus E. The negative ratio of the transverse strain to longitudinal
strain is the Poisson’s ratio v.
Forces can be applied to a material in a manner that will cause distortion rather than elongation
(Figure 22.3). A force applied tangent to a surface divided by the cross-sectional area is described as a
shear stress τ. The distortion can be measured by the angle change produced. This is the shear strain γ
© 1999 by CRC Press LLC
FIGURE 22.1 When a homogeneous isotropic bar is stretched by a uniaxial force, it elongates in the direction of
the force and contracts perpendicular to the force. The relative elongation and contraction are defined as the
longitudinal and transverse strains, respectively.
FIGURE 22.2 The uniaxial force shown in Figure 22.1 produces uniaxial stress in the bar. When the material
response is linear, the slope of the stress vs. strain curve is the Young’s modulus. The negative ratio of the transverse
to longitudinal strain is the Poisson’s ratio.
when the angle change is small. When the relation between shear stress and shear strain is linear, the
ratio of the shear stress to shear strain is the shear modulus G.
Temperature change also induces strain. This is thermal expansion. In most materials, thermal strain
increases with temperature. Over a limited temperature range, the relationship between thermal strain
© 1999 by CRC Press LLC
FIGURE 22.3 When a block of material is subjected to forces parallel to the sides as shown, it distorts. The force
per unit area is the shear stress τ, and the angle change is the shear strain γ.
FIGURE 22.4 Some elements of a bar under uniaxial tension undergo elongation and contraction. These elements
lie in principal directions. Other elements undergo distortion as well.
and temperature is linear. In this case, the strain divided by the temperature change is the thermal
expansion coefficient α. In isotropic materials, thermal expansion only produces elongation strain, no
shear strain.
Principal directions in a material are directions that undergo elongation but no shear. On any particular
surface of a solid, there are always at least two principal directions in which the strain is purely elongation.
This is seen if two squares are drawn on the bar under uniform tension (Figure 22.4). When the bar is
stretched, the square aligned with the load is elongated, whereas the square at 45° is distorted (angles
have changed) and elongated. If the principal directions are known, as with the bar under tension, then
strain gages can be applied in these directions. If the principal directions are not known, such as near a
hole or notch, in an anisotropic specimen, or in a structure with complicated geometry, then additional
strain gages are needed to fully characterize the strain state.
The elastic and thermal properties can be combined to give Hooke’s law, Equations 22.1 to 22.6.
© 1999 by CRC Press LLC
ε xx =
ε yy =
ε xx =
σ xx v
− σ yy + σ zz
σ yy
σ xx + σ zz
σ zz v
− σ xx + σ yy
γ xy =
γ xz =
γ yz =
τ xy
τ xz
τ yz
Several types of sensors are used to measure strain. These include piezoresistive gages (foil or wire
strain gages and semiconductor strain gages), piezoelectric gages (polyvinylidene fluoride (PVDF) film
and quartz), fiber optic gages, birefringent films and materials, and Moiré grids. Each type of sensor
requires its own specialized signal conditioning. Selection of the best strain sensor for a given measurement is based on many factors, including specimen geometry, temperature, strain rate, frequency, magnitude, as well as cost, complexity, accuracy, spatial resolution, time resolution, sensitivity to transverse
strain, sensitivity to temperature, and complexity of signal conditioning. Table 22.1 describes typical
characteristics of several sensors. Table 22.2 lists some manufacturers and the approximate cost of the
sensors and associated signal conditioning electronics.
The data in Table 22.1 are to be taken as illustrative and by no means complete. The sensor description
section describes only the type of sensor, not the many sizes and shapes. The longitudinal strain sensitivity
is given as sensor output per unit longitudinal strain in the sensor direction. If the signal conditioning
is included, the sensitivities can all be given in volts out per unit strain [3, 4], but this is a function of
amplification and the quality of the signal conditioner. The temperature sensitivity is given as output
change due to a temperature change. In many cases, higher strain resolution can be achieved, but resolving
smaller strain is more difficult and may require vibration and thermal isolation. For the Moiré technique,
the strain resolution is a function of the length of the viewing area. This technique can resolve a
displacement of 100 nm (1/4 fringe order). This is divided by the viewing length to obtain the strain
resolution. The spatial resolution corresponds to the gage length for most of the sensor types. The
measurable strain range listed is the upper limit for the various sensors. Accuracy and reliability are
usually reduced when sensors are used at the upper limit of their capability.
Manufacturers of the various sensors provide technical information that includes details of using the
sensors, complete calibration or characterization data, and details of signal conditioning. The extensive
technical notes and technical tips provided by Measurements Group, Inc. address such issues as thermal
effects [5], transverse sensitivity corrections [6], soldering techniques [7], Rosettes [8], and gage
fatigue [9]. Strain gage catalogs include information about gage materials, sizes, and selection. Manufacturers of other sensors provide similar information.
© 1999 by CRC Press LLC
© 1999 by CRC Press LLC
ΔR/R/Δεt = <0.02
ΔR/R/Δεt = <0.02
ΔR/R/Δεt = ???
ΔQ/A/Δεt = 60 nC/m2/με
ΔR/R/ΔεL = 2.1
ΔR/R/ΔεL = 2.1
ΔR/R/ΔεL = 150
ΔQ/A/ΔεL = 120 nC/m2/με
ΔQ/A/ΔεL = 150 nC/m2/με
bonded to steel
2 to 1000 μstrain/V
K e = 0.15–0.002
1 fringe order/417 nm displ.
1 fringe order/417 nm
Near zero
Transverse strain
Longitudinal strain
Not defined
ΔR/R/ΔΤ = 2 × 10–6/°C
ΔR/R/ΔΤ = 2 × 10–6/°C
ΔR/R/ΔΤ = 1.7 × 10–3/°C
ΔQ/A/ΔΤ = –27 μC/m2/°C
ΔQ/A/ΔΤ = 0
Temperature sensitivity
41.7 με over
10 mm
<1 μstraina
<11 μstrain
<0.1 μstrain
1–10 μstrain
<0.01 μstrain
20 mm gage
<1 μstrain
With good signal conditioning.
Equal to grid area.
c Gage response is within 100 ns. Most signal conditioning limits response time to far less than this.
d Annealed foil has a low yield stress and a large strain to failure. It also has hysteresis in the unload and a zero shift under cyclic load.
e This technique measures a difference in principal strains. ε – ε = Nλ/2tK
f Approximately the film thickness.
g The spatial strain resolution depends on the strain level. This is a displacement measurement technique.
Fiber optic Fabry–Perot
Birefringent Film
Piezoresistive constantan foil
Annealed constantan foild
Piezoresistive semiconductor
Piezoelectric PVDF
Piezoelectric quartz
TABLE 22.1 Comparison of Strain Sensors
2–10 mm
0.5 mmf
full fieldg
5–100 mmb
5–100 mm
1–15 mm
Gage size
Gage size
<20 μs
<5 μs
Limited by signal
<1 μsc
<1 μs
<1 μs
<1 μs
<10 μs
Time resolution
strain range
TABLE 22.2 Sources and Prices of Strain Sensors
Omega Engineering
Dynasen, Inc.
Entran Sensors and
Amp Inc.
Kistler Instrument
F&S Inc.
Sensor Types
Sensor Cost
P.O. Box 27777
Raleigh, NC
P.O. Box 2618
College Station, TX
Piezoresistive foil
Birefringent film
From $5.00
From $10.00
Wheatstone bridge
From $500
$5000 to 10,000
Piezoresistive foil
and wire
Load cells
From $5.00
P.O. Box 4047
Stamford, CT
20 Arnold Pl.
Goleta, CA
Piezoresistive foil
From $5.00
Strain meter
Wheatstone bridge
From $550
From $2700
Piezoresistive foil
Specialty gages for
shock wave
Piezoelectric PVDF
From $55.00
2-Channel pulsed
Passive charge
Piezoelectric PVDF
(not calibrated)
From $5.00
Entran Devices, Inc.
10 Washington Ave.
Fairfield, NJ
Piezo Film Sensors
P.O. Box 799
Valley Forge, PA
Amherst, NY
Fiber and Sensor
P.O. Box 11704,
Blacksburg, VA
512 Princeton Dr.
Vestal, NY
From $55.00
From $15.00
Piezoelectric quartz
Fabry–Perot strain
From $75
Charge amplifier
From $3500.00
22.2 Principles of Operation of Strain Sensors
Piezoresistive Foil Gages
Piezoresistive foil and wire gages comprise a thin insulating substrate (usually polyimide film), a foil or
wire grid (usually constantan) bonded to the substrate, lead wires to connect the grid to a resistance
measuring circuit, and often an insulating encapsulation (another sheet of polyimide film) (Figure 22.5).
The grid is laid out in a single direction so that strain will stretch the legs of the grid in the length
direction. The gages are designed so that strain in the width or transverse direction separates the legs of
the grid without straining them. This makes the gage sensitive to strain only along its length. There is
always some sensitivity to transverse strain, and almost no sensitivity to shear strain. In most cases, the
transverse sensitivity can be neglected.
When piezoresistive foil or wire strain gages are bonded to a specimen and the specimen is strained,
the gage strains as well. The resistance change is related to the strain by a gage factor, Equation 22.7.
© 1999 by CRC Press LLC
Gage construction of a foil or wire piezoresistive gage.
= GL ε L
where ΔR/R = Relative resistance change
= Gage factor
= Strain
These gages respond to the average strain over the area covered by the grid [10]. The resistance change
is also sensitive to temperature. If the temperature changes during the measurement period, a correction
must be made to distinguish the strain response from the thermal response. The gage response to
longitudinal strain, transverse strain, and temperature change is given by Equation 22.8.
= GL ε L + Gt ε t + GT ΔT
where GL, Gt , and G T are the longitudinal, transverse, and temperature sensitivity, respectively. Micromeasurements, Inc. uses a different notation. Their gage data is provided as GL = FG, Gt = K t FG , GT = βg .
When a strain gage is bonded to a specimen and the temperature changes, the strain used in Equation 22.8
is the total strain, thermal plus stress induced, as given by Equation 22.7.
The temperature contribution to gage output must be removed if the gages are used in tests where
the temperature changes. A scheme referred to as self-temperature compensation (STC) can be used.
This is accomplished by selecting a piezoresistive material whose thermal output can be canceled by the
strain induced by thermal expansion of the test specimen. Gage manufacturers specify STC numbers that
match the thermal expansion coefficients of common specimen materials.
Strain of piezoresistive materials produces a relative resistance change. The resistance change is the
result of changes in resistivity and dimensional changes. Consider a single leg of the grid of a strain gage
with a rectangular cross-section (Figure 22.6). The resistance is given by Equation 22.9.
where R
= Resistance
= Resistivity
= Length
= Area of the cross-section
© 1999 by CRC Press LLC
FIGURE 22.6 A single leg of a piezoresistive gage is used to explain the source of the relative resistance change that
occurs in response to strain.
A small change in resistance is given by the first-order terms of a Taylor’s series expansion,
Equation 22.10.
ΔR =
Δρ +
ΔL +
Differentiating Equation 22.9 to obtain each term of Equation 22.10 and then dividing by the initial
resistance leads to Equation 22.11.
R0 ρ0 L0 A0
The relative resistance change is due to a change in resistivity, a change in length strain, and a change
in area strain.
The strain gage is a composite material. The metal in a strain gage is like a metal fiber in a polymer
matrix. When the polymer matrix is deformed, the metal is dragged along in the length direction; but
in the width and thickness directions, the strain is not passed to the metal. This results in a stress state
called uniaxial stress. This state was discussed in the examples above. The mathematical details involve
an inclusion problem [11, 12]. Accepting that the stress state is uniaxial, the relationship between the
area change and the length change in Equation 22.11 is found from the Poisson’s ratio. The area strain
is the sum of the width and thickness strain, Equation 22.12.
ΔA Δw Δt
A0 w0 t 0
The definition of the Poisson’s ratio gives Equation 22.13.
= −2v
where Δw/w = width strain and Δt/t = thickness strain. Substitution of Equation 22.13 into Equation 22.11
gives Equation 22.14 for the relative resistance change.
1 + 2v
R0 ρ0
The relative resistivity changes in response to stress. The resistivity is a second-order tensor [13], and
the contribution to the overall resistance change can be found in terms of strain using the elastic
© 1999 by CRC Press LLC
constitutive law [14]. The results lead to an elastic gage factor just over 2 for constantan gages. If the
strain is large, the foil or wire in the gage will experience plastic deformation. When the deformation is
plastic, the resistivity change is negligible and the dimensional change dominates. In this case, Poisson’s
ratio is 0.5 and the gage factor is 2. This effect is utilized in manufacturing gages for measuring strains
larger than 1.5%. In this case, annealed foil is used. The annealed foil undergoes plastic deformation
without failure. These gages are capable of measuring strain in excess of 10%. When metals undergo
plastic deformation, they do not unload to the initial strain. This shows up as hysteresis in the gage
response, that is, on unload, the resistance does not return to its initial value.
Foil and wire strain gages can be obtained in several configurations. They can be constructed with
different backing materials, and left open faced or fully encapsulated. Backing materials include polyimide
and glass fiber-reinforced phenolic resin. Gages can be obtained with solder tabs for attaching lead wires,
or with lead wires attached. They come in many sizes, and in multiple gage configurations called rosettes.
Strain gages are mounted to test specimens with adhesives using a procedure that is suitable for bonding
most types of strain sensors. This is accomplished in a step-by-step procedure [15] that starts with surface
preparation. An overview of the procedure is briefly described. To successfully mount strain gages, the
surface is first degreased. The surface is abraded with a fine emery cloth or 400 grit paper to remove any
loose paint, rust, or deposits. Gage layout lines are drawn (not scribed) on the surface in a cross pattern
with pen or pencil, one line in the grid direction and one in the transverse direction. The surface is then
cleaned with isopropyl alcohol. This can be done with an ultrasonic cleaner or with wipes. If wiped, the
paper or gauze wipe should be folded and a small amount of alcohol applied. The surface should be
wiped with one pass and the wipe discarded. This should be repeated, wiping in the other direction. The
final step is to neutralize the surface, bringing the alkalinity to a pH of 7 to 7.5. A surface neutralizer is
available from most adhesive suppliers. The final step is to apply the gage.
Gage application is accomplished with cellophane tape, quick-set glue, and a catalyst. The gage is
placed on a clean glass or plastic surface with bonding side down, using tweezers. (Never touch the gage.
Oils from skin prevent proper adhesion.) The gage is then taped down with a 100 mm piece of cellophane
tape. The tape is then peeled up with the gage attached. The gage can now be taped onto its desired
location on the test specimen. Once the gage has been properly aligned, the tape is peeled back from one
side, lifting the gage from the surface. The tape should remain adhered to the surface about 1 cm from
the gage. Note that one side of the tape is still attached to the specimen so that the gage can be easily
returned to its desired position. A thin coating of catalyst is applied to the exposed gage surface. A drop
of glue is placed at the joint of the tape and the specimen. Holding the tape at about a 30° angle from
the surface, the tape can be slowly wiped down onto the surface. This moves the glue line forward. After
the glue line has passed the gage, the gage should be pressed in place and held for approximately 1 min.
The tape can now be peeled back to expose the gage, and lead wires can be attached.
The relative resistance change of piezoresistive gages is usually measured using a Wheatstone bridge
[16]. This allows a small change of resistance to be measured relative to an initial zero value, rather than
relative to a large resistance value, with a corresponding increase in sensitivity and resolution. The
Wheatstone bridge is a combination of four resistors and a voltage source (Figure 22.7). One to four of
the resistors in the bridge can be strain gages. The output of the bridge is the difference between the
voltage at points B and D. Paths ABC and ADC are voltage dividers so that VB and VD are given by
Equations 22.15a and b.
© 1999 by CRC Press LLC
VB = Vin
R1 + R2
VD = Vin
R3 + R4
The Wheatstone bridge is used to measure the relative resistance change of piezoresistive strain gages.
The bridge output, Equation 22.16, is zero when the balance condition, Equation 22.17, is met.
V0 = VB − VD
R1R3 = R2 R4
Wheatstone bridge signal conditioners are constructed with a way to “balance” the bridge by adjusting
the ratio of the resistances so that the bridge output is initially zero.
The balance condition is no longer met if the resistance values undergo small changes ΔR1, ΔR2 , ΔR3 ,
ΔR4 . If the values R1 + ΔR1, etc. are substituted into Equation 22.15, the results substituted into
Equation 22.16, condition (22.17) used, and the higher order terms neglected, the result is Equation 22.18
for the bridge output.
Vout = Vin
(R + R )(R + R )
⎛ ΔR1 ΔR2 ΔR3 ΔR4 ⎞
⎜− R + R − R + R ⎟
4 ⎠
The Wheatstone bridge can be used to directly cancel the effect of thermal drift. If R1 is a strain gage
bonded to a specimen and R2 is a strain gage held onto a specimen with heat sink compound (a thermally
conductive grease available at any electronics store), then R1 will respond to strain plus temperature, and
R2 will only respond to temperature. Since the bridge subtracts the output of R1 from that of R2 , the
temperature effect cancels.
The sensitivity of a measuring system is the output per unit change in the quantity to be measured.
If the resistance change is from a strain gage, the sensitivity of the Wheatstone bridge system is proportional to the input voltage. Increasing the voltage increases the sensitivity. There is a practical limitation
to increasing the voltage to large values. The power dissipated (heat) in the gage is P = I2R, where I, the
current through the gage, can be found from the input voltage and the bridge resistances. This heat must
go somewhere or the temperature of the gage will continuously rise and the resistance will change due
to heating. If the gage is mounted on a good thermal conductor, more power can be conducted away
than if the gage is mounted on an insulator. The specimen must act as a heat sink.
Heat sinking ability is proportional to the thermal conductivity of the specimen material. A reasonable
temperature gradient to allow the gage to induce in a material is 40°C per meter (about 1°C per 25 mm).
© 1999 by CRC Press LLC
For thick specimens (thickness several times the largest gage dimension), this can be conducted away to
the grips or convected to the surrounding atmosphere. If the four bridge resistances are approximately
equal, the power to the gage in terms of the bridge voltage is given by Equation 22.19.
Pg =
The power per unit grid area, Ag, or power density to the gage can be equated to the thermal
conductivity of the specimen and the allowable temperature gradient in the specimen by Equation 22.20.
= K ∇T
4 RAg
Thermal conductivities of most materials can be found in tables or on the Web. Some typical values
are Al: K = 204 W m–1 °C–1, steel: K = 20 to 70 W m–1 °C–1, glass: K = 0.78 W m–1 °C–1 [17].
The acceptable bridge voltage can be calculated from Equation 22.21.
Vin = K ∇T 4 RAg
A sample calculation shows that for a 0.010 m × 0.010 m 120 Ω grid bonded to a thick piece of
aluminum with a thermal conductivity of 204 W m–1 °C–1 and an acceptable temperature gradient of
40°C per meter, the maximum bridge voltage is 19 V. If thin specimens are used, the allowable temperature
gradient will be smaller. If smaller gages are used for better spatial resolution, the bridge excitation voltage
must be reduced with a corresponding reduction in sensitivity.
A considerably higher bridge voltage can be used if the bridge voltage is pulsed for a short duration.
This dissipates substantially less energy in the gage and thus increases the sensitivity by a factor of 10 to
100. Wheatstone bridge pulse power supplies with variable pulse width from 10 μs and excitation of 350 V
are commercially available [18].
The strain measurement required is often in a complex loading situation where the directions of
principal strain are not known. In this case, three strain gages must be bonded to the test specimen at
three angles. This is called a strain rosette. The angle between the rosette and the principal directions,
as well as the magnitude of the principal strains, can be determined from the output of the rosette gages.
This is most easily accomplished using the construct of a Mohr’s circle (Figure 22.8).
A common rosette is the 0–45–90° pattern. The rosette is bonded to the specimen with relative rotations
of 0°, 45°, and 90°. These will be referred to as the x, x ′, and y directions. The principal directions are
labeled the 1 and 2 directions. The unknown angle between the x direction and the 1 direction is labeled
θ. The Mohr’s circle is drawn with the elongational strain on the horizontal axis and the shear strain on
the vertical axis. The center of the circle is labeled C and the radius R. The principal directions correspond
to zero shear strain. The principal values are given by Equations 22.22 and 22.23.
ε1 = C + R
ε2 = C − R
A rotation through an angle 2θ on the Mohr’s circle corresponds to a rotation of the rosette of θ
relative to the principal directions. The center of the circle is given by Equation 22.24 and the output of
the strain gages is given by Equations 22.25 to 22.27.
© 1999 by CRC Press LLC
FIGURE 22.8 A three-element rosette is used to measure strain when the principal directions are not known. The
Mohr’s circle is used to find the principal directions and the principal strain values.
ε xx + ε yy
ε xx − C = Rcos2θ
ε x′ x′ − C = − R sin 2θ
ε yy − C = − R cos 2θ
Dividing Equation 22.25 by Equation 22.26 leads to θ and then to R, Equations 22.28 and 22.29.
) (
R2 = ε xx − C + ε x′ x′ − C
tan2θ =
C − ε x′ x′
ε xx − C
The principal directions and principal strain values have been found from the output of the three
rosette gages.
Piezoresistive Semiconducting Gages
Piezoresistive semiconductor strain gages, like piezoresistive foil and wire gages, undergo a resistance
change in response to strain, but with nearly an order of magnitude larger gage factor [19]. The coupling
between resistance change and temperature is very large, so these gages have to be temperature compensated. The change of resistance with a small temperature change can be an order of magnitude larger
than that induced by strain. Semiconductor strain gages are typically used to manufacture transducers
such as load cells. They are fragile and require great care in their application.
© 1999 by CRC Press LLC
Typical gage construction of a piezoelectric gage.
Piezoelectric Gages
Piezoelectric strain gages are, effectively, parallel plate capacitors whose dielectric changes polarization
in response to strain [14]. When the polarization changes, a charge proportional to the strain is produced
on the electrodes. PVDF film strain gages are inexpensive, but not very accurate and subject to depoling
by moderate temperature. They make good sensors for dynamic measurements such as frequency and
logarithm decrement, but not for quantitative measurements of strain. When used for quasistatic measurements, the charge tends to drain through the measuring instrument. This causes the signal to decay
with a time constant dependent on the input impedance of the measuring instrument. Quartz gages are
very accurate, but also lose charge through the measuring instrument. Time constants can be relatively
long (seconds to hours) with electrometers or charge amplifiers.
The PVDF gage consists of a thin piezoelectric film with metal electrodes (Figure 22.9). Lead wires
connect the electrodes to a charge measuring circuit. Gages can be obtained with the electrodes encapsulated between insulating layers of polyimide.
The gage output can be described in terms of a net dipole moment per unit volume. If the net dipole
moment is the total charge, Q, on the electrodes multiplied by spacing, d, between the electrodes, then
the polarization is given by Equation 22.30.
From Equation 22.30, it is seen that the polarization P (approximately equal to the electric displacement D)
is the charge per unit electrode area (Figure 22.10).
A Taylor’s series expansion of Equation 22.30 gives Equation 22.31.
ΔP =
ΔV +
∂ Qd
( )
Which, after differentiating Equation 22.30 and substituting becomes Equation 22.31.
ΔP =
© 1999 by CRC Press LLC
FIGURE 22.10 A representative cross-section of a piezoelectric material formed into a parallel plate capacitor. The
piezoelectric material is polarized. This results in charge on the electrodes. When the material is strained, the
polarization changes and charge flows.
For PVDF film, the second term in Equation 22.32 dominates. The output is proportional to the
remanent polarization P0 . The remanent polarization slowly decays with time, has a strong dependence
on temperature, and decays rapidly at temperatures around 50°C. This makes accuracy a problem. If the
sensors are kept at low temperature, accuracy can be maintained within ±3%.
Strain sensors can also be constructed from piezoelectric ceramics like lead zirconate titanate (PZT)
or barium titanate. Ceramics are brittle and can be depoled by strain so should only be used at strains
less than 200 microstrain. PZT loses some of its polarization with time and thus has accuracy problems,
but remains polar to temperatures of 150°C or higher. “Hard” PZT (usually iron doped) is the best
composition for polarization stability and low hysteresis. Quartz has the best accuracy. It is not polar,
but polarization is induced by strain. Quartz has excellent resolution and accuracy over a broad temperature range but is limited to low strain levels. It is also brittle, so is limited to small strain.
Two circuits are commonly used for piezoelectric signal conditioning: the electrometer and the charge
amplifier (Figure 22.11). In the electrometer circuit, the piezoelectric sensor is connected to a capacitor
with a capacitance value C, at least 1000 times that of the sensor Cg . There is always some resistance in
FIGURE 22.11 The electrometer and charge amplifier are the most common circuits used to measure charge from
piezoelectric transducers.
© 1999 by CRC Press LLC
FIGURE 22.12 A schematic of the Fabry–Perot fiber optic strain gage. When the cavity elongates, alternating
constructive and destructive interference occur.
the cable that connects the sensor to the capacitor. The circuit is simply two capacitors in parallel
connected by a resistance. The charge equilibrates with a time constant given by RgCg . This time constant
limits the fastest risetime that can be resolved to about 50 ns, effectively instantaneous for most applications. The charge is measured by measuring the voltage on the capacitor, then using Equation 22.33.
Q = CV
The difficulty is that measuring devices drain the charge, causing a time decay with a time constant
RC. This causes the signal to be lost rapidly if conventional op amps are used. FET input op amps have
a very high input impedance and can extend this time constant to many hours. The charge amplifier is
another circuit used to measure charge. This is usually an FET input op amp with a capacitor feedback.
This does not really amplify charge, but produces a voltage proportional to the input charge. Again, the
time constant can be many hours, allowing use of piezoelectric sensors for near static measurements.
High input impedance electrometer and charge amplifier signal conditioners for near static measurements are commercially available [20] as well as low-cost capacitive terminators for high-frequency (high
kilohertz to megahertz) measurements [18]. An advantage of piezoelectric sensors is that they are active
sensors that do not require any external energy source.
Fiber Optic Strain Gages
Fiber optic strain gages are miniature interferometers [21, 22]. Many commercially available sensors are
based on the Fabry–Perot interferometer. The Fabry–Perot interferometer measures the change in the
size of a very small cavity.
Fabry–Perot strain sensors (Figure 22.12) comprise a laser light source, single-mode optical fibers, a
coupler (the fiber optic equivalent of a beam splitter), a cavity that senses strain, and a photodetector.
Light leaves the laser diode. It passes down the fiber, through the coupler, and to the cavity. The end of
the fiber is the equivalent of a partially silvered mirror. Some of the light is reflected back up the fiber
and some is transmitted. The transmitted light crosses the cavity and then is reflected from the opposite
end back into the fiber where it recombines with the first reflected beam. The two beams have a phase
difference related to twice the cavity length. The recombined beam passes through the coupler to the
photodetector. If the two reflected beams are in phase, there will be constructive interference. If the two
© 1999 by CRC Press LLC
beams are out of phase, there will be destructive interference. The cavity is bonded to a specimen. When
the specimen is strained, the cavity stretches. This results in a phase change of the cavity beam, causing
a cycling between constructive and destructive interference. For a 1.3 μm light source, each peak in output
corresponds to a 650 nm gap displacement. The gap displacement divided by the gap length gives the
strain. The output is continuous between peaks so that a 3 mm gage can resolve 1 μstrain.
Birefringent Film Strain Sensing
Birefringent film strain sensors give a full field measurement of strain. A nice demonstration of this effect
can be achieved with two sheets of inexpensive Polaroid film, a 6 mm thick, 25 mm × 200 mm bar of
Plexiglas (polymethylmethacrylate or PMMA), and an overhead projector. Place the two Polaroid sheets
at 90° to one another so that the light is blocked. Place the PMMA between the Polaroid sheets. Apply
a bending moment to the bar and color fringes will appear. Birefringent materials have a different speed
of light in different directions. This means that if light is polarized in a particular direction and passed
through a birefringent specimen, if the fast direction is aligned with the electric field vector, the light
passes through faster than if the slow direction is aligned with the electric field vector. This effect can be
used to produce optical interference. In some materials, birefringence is induced by strain. The fast and
slow directions correspond to the directions of principal strain, and the amount of birefringence corresponds to the magnitude of the strain. One component of the electric field vector travels through the
specimen faster than the other. They emerge with a phase difference. This changes the relative amplitude
and thus rotates the polarization of the light. If there is no birefringence, no light passes through the
second polarizer. As the birefringence increases with strain, light passes through. As it further increases,
the polarization rotation will be a full 180° and again no light will pass through. This produces a fringe
that corresponds to a constant difference in principal strains. The difference in principal strains is given
by Equation 22.34.
ε 2 − ε1 =
where ε1, ε2
= Principal strains
= Fringe order
= Wavelength
= Specimen thickness
= Strain-optical coefficient of the photoelastic material
A similar technique can be used with a birefringent plastic film with a silvered backing laminated to
the surface of a specimen. Polarized light is passed through the film; it reflects from the backing, passes
back through the film, and through the second polarizer. In this case, because light passes twice through
the film, the equation governing the difference in principal strains is Equation 22.35.
ε 2 − ε1 =
If the polarizers align with principal strain directions, no birefringence is observed. Rotation of both
polarizers allows the principal directions to be found at various locations on the test specimen. If a full
view of the fringes is desired, quarter wave plates are used (Figure 22.13). In this arrangement, light is
passed through the first polarizer, resulting in plane polarization; through the quarter wave plate, resulting
in circular polarization; through the test specimen, resulting in phase changes; through the second quarter
wave plate to return to plane polarization; and then through the final polarizer.
The optical systems for viewing birefringence are commercially available as “Polariscopes” [23]. Optical
components to construct custom systems are available from many optical components suppliers.
© 1999 by CRC Press LLC
FIGURE 22.13 A schematic of the polariscope, a system for measuring birefringence. This technique gives a full
field measure of the difference in principal strains.
Moiré Strain Sensing
Moiré interference is another technique that gives a full field measurement, but it measures displacement
rather than strain. The strain field must be computed from the displacement field. This technique is
based on the interference obtained when two transparent plates are covered with equally spaced stripes.
If the plates are held over one another, they can be aligned so that no light will pass through or so that
all light will pass through. If one of the plates is stretched, the spacing of the lines is wider on the stretched
plate. Now, if one plate is placed over the other, in some regions light will pass through and in some
regions it will not (Figure 22.14). The dark and light bands produced give information about the displacement field.
Moiré is defined as a series of broad dark and light patterns formed by the superposition of two regular
gratings [24]. The dark or light regions are called fringes. Examples of pure extension and pure rotation
are shown. In both cases, some of the light that would emerge from the first grating is obstructed by the
superimposed grating. At the centers of the dark fringes, the bar of one grating covers the space of the
other and no light comes through. The emergent intensity, I, is zero. Proceeding from there toward the
next dark fringe, the amount of obstruction diminishes linearly and the amount of light increases linearly
until the bar of one grating falls above the bar of the other. There, the maximum amount of light passes
through the gratings.
Both geometric interference and optical interference are used. This discussion is restricted to geometric
interference. Geometric moiré takes advantage of the interference of two gratings to determine displacements and rotations in the plane of view. In-plane moiré is typically conducted with two gratings, one
applied to the specimen (specimen grating) and the other put in contact with the specimen grating
(reference grating). When the specimen is strained, interference patterns or fringes occur. N is the moiré
fringe order. Each fringe corresponds to an increase or decrease of specimen displacement by one grating
pitch. The relationship between displacement and fringes is δ = gN, where δ is component of the
displacement perpendicular to the reference grating lines, g is reference grating pitch, and N is the fringe
For convenience, a zero-order fringe is designated assuming the displacement there is zero. With the
reference grating at 0° and 90°, the fringe orders Nx and Ny are obtained. The displacements in x, y
directions are then obtained from Equations 22.36 and 22.37.
( )
( )
( )
( )
ux x , y = gN x x , y
uy x , y = gN y x , y
Differentiation of Equations 22.36 and 22.37 gives the strains, Equations 22.38 through 22.40.
© 1999 by CRC Press LLC
FIGURE 22.14 A demonstration of moiré fringes formed by overlapping gratings. The fringes are the result of
stretching and relative rotation of the gratings. The fringe patterns are used to determine displacement fields.
εx =
∂N x
∂u ⎞ 1 ⎛ ∂N y
∂N x ⎞
1 ⎛ ∂u
ε xy = ⎜ y + x ⎟ = ⎜ g
∂y ⎠ 2 ⎝ ∂x
∂y ⎠
2 ⎝ ∂x
εy =
∂N y
In most cases, the sensitivity of geometric moiré is not adequate for determination of strain distributions. Strain analysis should be conducted with high-sensitivity measurement of displacement using
moiré interferometry [24, 25]. Moiré interferometers are commercially available [26]. Out-of-plane measurement can be conducted with one grating (the reference grating). The reference grating is made to
interfere with either its reflection or its shadow [27, 28].
© 1999 by CRC Press LLC
1. N. E. Dowling, Mechanical Behavior of Materials, Englewood Cliffs, NJ: Prentice-Hall, 1993, 99-108.
2. R. C. Craig, Mechanics of Materials, New York: John Wiley & Sons, 1996.
3. A. Vengsarkar, Fiber optic sensors: a comparative evaluation, The Photonics Design and Applications
Handbook, 1991, 114-116.
4. H. U. Eisenhut, Force measurement on presses with piezoelectric strain transducers and their static
calibration up to 5 MN, New Industrial Applications of the Piezoelectric Measurement Principle, July
1992, 1-16.
5. TN-501-4, Strain Gauge Temperature Effects, Measurements Group, Inc., Raleigh, NC 27611.
6. TN-509, Transverse Sensitivity Errors, Measurements Group, Inc., Raleigh, NC 27611.
7. TT-609, Soldering Techniques, Measurements Group, Inc., Raleigh, NC 27611.
8. TN-515, Strain Gage Rosettes, Measurements Group, Inc., Raleigh, NC 27611.
9. TN-08-1, Fatigue of Strain Gages, Measurements Group, Inc., Raleigh, NC 27611.
10. C. C. Perry and H. R. Lissner, The Strain Gage Primer, New York: McGraw-Hill, 1962.
11. J. B. Aidun and Y. M. Gupta, Analysis of Lugrangian gauge measurements of simple and nonsimple
plain waves, J. Appl. Phys., 69, 6998-7014, 1991.
12. Y. M. Gupta, Stress measurement using piezoresistance gauges: modeling the gauge as an elasticplastic inclusion, J. Appl. Phys., 54, 6256-6266, 1983.
13. D. Y. Chen, Y. M. Gupta, and M. H. Miles, Quasistatic experiments to determine material constants
for the piezoresistance foils used in shock wave experiments, J. Appl. Phys., 55, 3984, 1984.
14. C. S. Lynch, Strain compensated thin film stress gauges for stress wave measurements in the
presence of lateral strain, Rev. Sci. Instrum., 66(11), 1-8, 1995.
15. B-129-7 M-Line Accessories Instruction Bulletin, Measurements Group, Inc., Raleigh, NC 27611.
16. J. W. Dally, W. F. Riley, and K. G. McConnell, Instrumentation for Engineering Measurements,
2nd ed., New York: John Wiley & Sons, 1993.
17. J. P. Holman, Heat Transfer, 7th ed., New York: McGraw Hill, 1990.
18. Dynasen, Inc. 20 Arnold Pl., Goleta, CA 93117.
19. M. Dean (ed.) and R. D. Douglas (assoc. ed.), Semiconductor and Conventional Strain Gages, New
York: Academic Press, 1962.
20. Kistler, Instruments Corp., Amhurst, NY, 14228-2171.
21. J. S. Sirkis, Unified approach to phase strain temperature models for smart structure interferometric
optical fiber sensors. 1. Development, Opt. Eng., 32(4), 752-761, 1993.
22. J. S. Sirkis, Unified approach to phase strain temperature models for smart structure interferometric
optical fiber sensors. 2. Applications, Optical Engineering, 32(4), 762-773, 1993.
23. Photoelastic Division, Measurements Group, Inc., P.O. Box 27777, Raleigh, NC 27611.
24. T. Valis, D. Hogg, and R. M. Measures, Composite material embedded fiber-optic Fabry-Perot
strain rosette, SPIE, 1370, 154-161, 1990.
25. D. Post, B. Han, and P. Lfju, High Sensitivity Moiré, New York: Springer-Verlag, 1994.
26. V. J. Parks, Geometric Moiré, Handbook on Experimental Mechanics, A. S. Kobayashi, Ed., VCH
Publisher, Inc., 1993.
27. Photomechanics, Inc. 512 Princeton Dr. Vestal, NY, 13850-2912.
28. T. Y. Kao and F. P. Chiang, Family of grating techniques of slope and curvature measurements for
static and dynamic flexure of plates, Opt. Eng., 21, 721-742, 1982.
29. D. R. Andrews, Shadow moiré contouring of impact craters, Opt. Eng., 21, 650-654, 1982.
© 1999 by CRC Press LLC
Copyright 2000 CRC Press LLC. <>.
M. A. Elbestawi. "Force Measurement."
Force Measurement
General Considerations
Hooke’s Law
Force Sensors
Basic Methods of Force Measurement
M. A. Elbestawi
McMaster University
Strain Gage Load Cell • Piezoelectric Methods • Capacitive
Force Transducer • Force Sensing Resistors (Conductive
Polymers) • Magnetoresistive Force Sensors • Magnetoelastic
Force Sensors • Torsional Balances
Force, which is a vector quantity, can be defined as an action that will cause an acceleration or a certain
reaction of a body. This chapter will outline the methods that can be employed to determine the
magnitude of these forces.
23.1 General Considerations
The determination or measurement of forces must yield to the following considerations: if the forces acting
on a body do not produce any acceleration, they must form a system of forces in equilibrium. The system
is then considered to be in static equilibrium. The forces experienced by a body can be classified into two
categories: internal, where the individual particles of a body act on each other, and external otherwise. If
a body is supported by other bodies while subject to the action of forces, deformations and/or displacements will be produced at the points of support or contact. The internal forces will be distributed
throughout the body until equilibrium is established, and then the body is said to be in a state of tension,
compression, or shear. In considering a body at a definite section, it is evident that all the internal forces
act in pairs, the two forces being equal and opposite, whereas the external forces act singly.
23.2 Hooke’s Law
The basis for force measurement results from the physical behavior of a body under external forces.
Therefore, it is useful to review briefly the mechanical behavior of materials. When a metal is loaded in
uniaxial tension, uniaxial compression, or simple shear (Figure 23.1), it will behave elastically until a
critical value of normal stress (S) or shear stress (τ) is reached, and then it will deform plastically [1].
In the elastic region, the atoms are temporarily displaced but return to their equilibrium positions when
the load is removed. Stress (S or τ) and strain (e or γ) in the elastic region are defined as indicated in
Figure 23.2.
© 1999 by CRC Press LLC
FIGURE 23.1 When a metal is loaded in uniaxial tension (a) uniaxial compression (b), or simple shear (c), it will
behave elastically until a critical value of normal stress or shear stress is reached.
S = p / A0
e = (l / l0
S = p / A0
e = (l / l0
X= F / A0
K = (x / y
FIGURE 23.2 Elastic stress and strain for: (a) uniaxial tension; (b) uniaxial compression; (c) simple shear [1].
Poisson’s ratio (v) is the ratio of transverse (e2) to direct (e1) strain in tension or compression. In the
elastic region, v is between 1/4 and 1/3 for metals. The relation between stress and strain in the elastic
region is given by Hooke’s law:
S ! E e tension or compression
X ! GK simple shear
where E and G are the Young’s and shear modulus of elasticity, respectively. A small change in specific
volume ((Vol/Vol) can be related to the elastic deformation, which is shown to be as follows for an
isotropic material (same properties in all directions).
! e1 1 2v
The bulk modulus (K = reciprocal of compressibility) is defined as follows:
¨ (Vol ¸
K ! (p ©
ª Vol º
© 1999 by CRC Press LLC
where Δp is the pressure acting at a particular point. For an elastic solid loaded in uniaxial compression
⎛ ΔVol ⎞
K =S ⎜
1 − 2v
⎝ Vol ⎠ e1 1 − 2v
Thus, an elastic solid is compressible as long as v is less than 1/2, which is normally the case for metals.
Hooke’s law (Equation 23.2) for uniaxial tension can be generalized for a three-dimensional elastic
The theory of elasticity is well established and is used as a basis for force measuring techniques. Note
that the measurement of forces in separate engineering applications is very application specific, and care
must be taken in the selection of the measuring techniques outlined below.
Basic Methods of Force Measurement
An unknown force may be measured by the following means:
1. Balancing the unknown force against a standard mass through a system of levers.
2. Measuring the acceleration of a known mass.
3. Equalizing it to a magnetic force generated by the interaction of a current-carrying coil and a
4. Distributing the force on a specific area to generate pressure, and then measuring the pressure.
5. Converting the applied force into the deformation of an elastic element.
The aforementioned methods used for measuring forces yield a variety of designs of measuring
equipment. The challenge involved with the task of measuring force resides primarily in sensor design.
The basics of sensor design can be resolved into two problems:
1. Primary geometric, or physical constraints, governed by the application of the force sensor device.
2. The means by which the force can be converted into a workable signal form (such as electronic
signals or graduated displacements).
The remaining sections will discuss the types of devices used for force to signal conversion and finally
illustrate some examples of applications of these devices for measuring forces.
23.3 Force Sensors
Force sensors are required for a basic understanding of the response of a system. For example, cutting
forces generated by a machining process can be monitored to detect a tool failure or to diagnose the
causes of this failure in controlling the process parameters, and in evaluating the quality of the surface
produced. Force sensors are used to monitor impact forces in the automotive industry. Robotic handling
and assembly tasks are controlled by detecting the forces generated at the end effector. Direct measurement of forces is useful in controlling many mechanical systems.
Some types of force sensors are based on measuring a deflection caused by the force. Relatively high
deflections (typically, several micrometers) would be necessary for this technique to be feasible. The
excellent elastic properties of helical springs make it possible to apply them successfully as force sensors
that transform the load to be measured into a deflection. The relation between force and deflection in
the elastic region is demonstrated by Hooke’s law. Force sensors that employ strain gage elements or
piezoelectric (quartz) crystals with built-in microelectronics are common. Both impulsive forces and
slowly varying forces can be monitored using these sensors.
Of the available force measuring techniques, a general subgroup can be defined as that of load cells.
Load cells are comprised generally of a rigid outer structure, some medium that is used for measuring
© 1999 by CRC Press LLC
Thin Diaphragm
Air supply Ps
Hydralic load cell
Pneumatic load cell
FIGURE 23.3 Different types of load cells [2].
the applied force, and the measuring gage. Load cells are used for sensing large, static or slowly varying
forces with little deflection and are a relatively accurate means of sensing forces. Typical accuracies are
of the order of 0.1% of the full-scale readings. Various strategies can be employed for measuring forces
that are strongly dependent on the design of the load cell. For example, Figure 23.3 illustrates different
types of load cells that can be employed in sensing large forces for relatively little cost. The hydraulic
load cell employs a very stiff outer structure with an internal cavity filled with a fluid. Application of a
load increases the oil pressure, which can be read off an accurate gage.
Other sensing techniques can be utilized to monitor forces, such as piezoelectric transducers for quicker
response of varying loads, pneumatic methods, strain gages, etc. The proper sensing technique needs
special consideration based on the conditions required for monitoring.
Strain Gage Load Cell
The strain gage load cell consists of a structure that elastically deforms when subjected to a force and a
strain gage network that produces an electrical signal proportional to this deformation. Examples of this
are beam and ring types of load cells.
Strain Gages
Strain gages use a length of gage wire to produce the desired resistance (which is usually about 120 ;)
in the form of a flat coil. This coil is then cemented (bonded) between two thin insulating sheets of paper
or plastic. Such a gage cannot be used directly to measure deflection. It has to be first fixed properly to
a member to be strained. After bonding the gage to the member, they are baked at about 195rF (90rC)
to remove moisture. Coating the unit with wax or resin will provide some mechanical protection. The
resistance between the member under test and the gage itself must be at least 50 M;. The total area of
all conductors must remain small so that the cement can easily transmit the force necessary to deform
the wire. As the member is stressed, the resulting strain deforms the strain gage and the cross-sectional
area diminishes. This causes an increase in resistivity of the gage that is easily determined. In order to
measure very small strains, it is necessary to measure small changes of the resistance per unit resistance
((R/R). The change in the resistance of a bonded strain gage is usually less than 0.5%. A wide variety of
gage sizes and grid shapes are available, and typical examples are shown in Figure 23.4.
The use of strain gages to measure force requires careful consideration with respect to rigidity and
environment. By virtue of their design, strain gages of shorter length generally possess higher response
frequencies (examples: 660 kHz for a gage of 0.2 mm and 20 kHz for a gage of 60 mm in length). The
environmental considerations focus mainly on the temperature of the gage. It is well known that resistance
is a function of temperature and, thus, strain gages are susceptible to variations in temperature. Thus, if
it is known that the temperature of the gage will vary due to any influence, temperature compensation
is required in order to ensure that the force measurement is accurate.
A Wheatstone bridge (Figure 23.5) is usually used to measure this small order of magnitude. In
Figure 23.5, no current will flow through the galvanometer (G) if the four resistances satisfy a certain
© 1999 by CRC Press LLC
FIGURE 23.4 Configuration of metal-foil resistance strain gages: (a) single element; (b) two element; and (c) three
FIGURE 23.5 The Wheatstone bridge.
condition. In order to demonstrate how a Wheatstone bridge operates [3], a voltage scale has been drawn
at points C and D of Figure 23.5. Assume that R1 is a bonded gage and that initially Equation 23.7 is
satisfied. If R1 is now stretched so that its resistance increases by one unit (+ΔR), the voltage at point D
will be increased from zero to plus one unit of voltage (+ΔV), and there will be a voltage difference of
one unit between C and D that will give rise to a current through C. If R4 is also a bonded gage, and at
the same time that R1 changes by +ΔR, R4 changes by –ΔR, the voltage at D will move to +2ΔV. Also, if
at the same time, R2 changes by –ΔR, and R3 changes by +ΔR, then the voltage of point C will move to
–2ΔV, and the voltage difference between C and D will now be 4ΔV. It is then apparent that although a
single gage can be used, the sensitivity can be increased fourfold if two gages are used in tension while
two others are used in compression.
R1 R2
R4 R3
The grid configuration of the metal-foil resistance strain gages is formed by a photo-etching process.
The shortest gage available is 0.20 mm; the longest is 102 mm. Standard gage resistance are 120 Ω and
350 Ω. A strain gage exhibits a resistance change ΔR/R that is related to the strain in the direction of the
grid lines by the expression in Equation 23.8 (where Sg is the gage factor or calibration constant for the
= Sg ε
© 1999 by CRC Press LLC
Axial gages
1 and 3
Axial gages
2 and 4 on the
bottom sufaces
+ -
FIGURE 23.6 Beam-type load cells: (a) a selection of beam-type load cells (elastic element with strain gages); and
(b) gage positions in the Wheatstone bridge.
Beam-Type Load Cell
Beam-type load cells are commonly employed for measuring low-level loads [3]. A simple cantilever
beam (see Figure 23.6(a)) with four strain gages, two on the top surface and two on the bottom surface
(all oriented along the axis of the beam) is used as the elastic member (sensor) for the load cell. The
gages are wired into a Wheatstone bridge as shown in Figure 23.6(b). The load P produces a moment
M = Px at the gage location (x) that results in the following strains:
6 Px
I1 ! I 2 ! I 3 ! I 4 ! 2 !
where b is the width of the cross-section of the beam and h is the height of the cross-section of the beam.
Thus, the response of the strain gages is obtained from Equation 23.10.
6S Px
! 2 ! 3 ! 4 ! g 2
The output voltage Eo from the Wheatstone bridge, resulting from application of the load P, is obtained
from Equation 23.11. If the four strain gages on the beam are assumed to be identical, then Equation 23.11
Eo !
6Sg PxE1
The range and sensitivity of a beam-type load cell depends on the shape of the cross-section of the beam,
the location of the point of application of the load, and the fatigue strength of the material from which
the beam is fabricated.
Ring-Type Load Cell
Ring-type load cells incorporate a proving ring (see Figure 23.7) as the elastic element. The ring element
can be designed to cover a very wide range of loads by varying the diameter D, the thickness t, or the
depth w of the ring. Either strain gages or a linear variable-differential transformer (LVDT) can be used
as the sensor.
The load P is linearly proportional to the output voltage Eo . The sensitivity of the ring-type load cell
with an LVDT sensor depends on the geometry of the ring (R, t, and w), the material from which the
ring is fabricated (E), and the characteristics of the LVDT (S and Ei ). The range of a ring-type load cell
is controlled by the strength of the material used in fabricating the ring.
© 1999 by CRC Press LLC
Gage 2
+ 4
Gage 3
FIGURE 23.7 Ring-type load cells: (a) elastic element with strain-gage sensors; (b) gage positions in the Wheatstone
bridge; and (c) elastic element with an LVDT sensor.
Piezoelectric Methods
A piezoelectric material exhibits a phenomenon known as the piezoelectric effect. This effect states that
when asymmetrical, elastic crystals are deformed by a force, an electrical potential will be developed
within the distorted crystal lattice. This effect is reversible. That is, if a potential is applied between the
surfaces of the crystal, it will change its physical dimensions [4]. Elements exhibiting piezoelectric qualities
are sometimes known as electrorestrictive elements.
The magnitude and polarity of the induced surface charges are proportional to the magnitude and
direction of the applied force [4]:
Q ! dF
where d is the charge sensitivity (a constant for a given crystal) of the crystal in C/N. The force F causes
a thickness variation (t meters of the crystal:
where a is area of crystal, t is thickness of crystal, and Y is Young’s modulus.
strain a(t
The charge at the electrodes gives rise to a voltage E0 = Q/C, where C is capacitance in farads between
the electrodes and C = Ia/t where I is the absolute permittivity.
© 1999 by CRC Press LLC
FIGURE 23.8 Modes of operation for a simple plate as a piezoelectric device [4].
FIGURE 23.9 Curvature of “twister” and “bender” piezoelectric transducers when voltage applied [4].
Eo =
dF d tF
C ε a
The voltage sensitivity = g = d/ε in volt m/N can be obtained as:
Eo = g F = gtP
The piezoelectric materials used are quartz, tourmaline, Rochelle salt, ammonium dihydrogen phosphate
(ADP), lithium sulfate, barium titanate, and lead zirconate titanate (PZT) [4]. Quartz and other earthly
piezoelectric crystals are naturally polarized. However, synthetic piezoelectric materials, such as barium
titanate ceramic, are made by baking small crystallites under pressure and then placing the resultant
material in a strong dc electric field [4]. After that, the crystal is polarized, along the axis on which the
force will be applied, to exhibit piezoelectric properties. Artificial piezoelectric elements are free from
the limitations imposed by the crystal structure and can be molded into any size and shape. The direction
of polarization is designated during their production process.
The different modes of operation of a piezoelectric device for a simple plate are shown in Figure 23.8
[4]. By adhering two crystals together so that their electrical axes are perpendicular, bending moments
or torque can be applied to the piezoelectric transducer and a voltage output can be produced
(Figure 23.9) [4]. The range of forces that can be measured using piezoelectric transducers are from 1 to
200 kN and at a ratio of 2 × 105.
© 1999 by CRC Press LLC
Piezoelectric crystals can also be used in measuring an instantaneous change in the force (dynamic
forces). A thin plate of quartz can be used as an electronic oscillator. The frequency of these oscillations
will be dominated by the natural frequency of the thin plate. Any distortion in the shape of the plate
caused by an external force, alters the oscillation frequency. Hence, a dynamic force can be measured by
the change in frequency of the oscillator.
Resistive Method
The resistive method employs the fact that when the multiple contact area between semiconducting
particles (usually carbon) and the distance between the particles are changed, the total resistance is altered.
The design of such transducers yields a very small displacement when a force is applied. A transducer
might consist of 2 to 60 thin carbon disks mounted between a fixed and a movable electrode. When a
force is applied to the movable electrode and the carbon disks move together by 5 to 250 μm per interface,
the transfer function of their resistance against the applied force is approximately hyperbolic, that is,
highly nonlinear. The device is also subject to large hysteresis and drift together with a high transverse
In order to reduce hysteresis and drift, rings are used instead of disks. The rings are mounted on an
insulated rigid core and prestressed. This almost completely eliminates any transverse sensitivity error.
The core’s resonant frequency is high and can occur at a frequency as high as 10 kHz. The possible
measuring range of such a transducer is from 0.1 kg to 10 kg. The accuracy and linear sensitivity of this
transducer is very poor.
Inductive Method
The inductive method utilizes the fact that a change in mechanical stress of a ferromagnetic material
causes its permeability to alter. The changes in magnetic flux are converted into induced voltages in the
pickup coils as the movement takes place. This phenomenon is known as the Villari effect or magnetostriction. It is known to be particularly strong in nickel–iron alloys.
Transducers utilizing the Villari effect consist of a coil wound on a core of magnetostrictive material.
The force to be measured is applied on this core, stressing it and causing a change in its permeability
and inductance. This change can be monitored and used for determining the force.
The applicable range for this type of transducer is a function of the cross-sectional area of the core.
The accuracy of the device is determined by a calibration process. This transducer has poor linearity and
is subject to hysteresis. The permeability of a magnetostrictive material increases when it is subjected to
pure torsion, regardless of direction. A flat frequency response is obtained over a wide range from 150 Hz
to 15,000 Hz.
Piezotransistor Method
Devices that utilize anisotropic stress effects are described as piezotransistors. In this effect, if the upper
surface of a p–n diode is subjected to a localized stress, a significant reversible change occurs in the
current across the junction. These transistors are usually silicon nonplanar type, with an emitter base
junction. This junction is mechanically connected to a diaphragm positioned on the upper surface of a
typical TO-type can [4]. When a pressure or a force is applied to the diaphragm, an electronic charge is
produced. It is advisable to use these force-measuring devices at a constant temperature by virtue of the
fact that semiconducting materials also change their electric properties with temperature variations. The
attractive characteristic of piezotransistors is that they can withstand a 500% overload.
Multicomponent Dynamometers Using Quartz Crystals As Sensing Elements
The Piezoelectric Effects in Quartz.
For force measurements, the direct piezoelectric effect is utilized. The direct longitudinal effect measures
compressive force; the direct shear effect measures shear force in one direction. For example, if a disk of
crystalline quartz (SiO2) cut normally to the crystallographic x-axis is loaded by a compression force, it
will yield an electric charge, nominally 2.26 pC/N. If a disk of crystalline quartz is cut normally to the
© 1999 by CRC Press LLC
FIGURE 23.10 Three-component force transducer.
FIGURE 23.11 Force measuring system to determine the tool-related cutting forces in five-axis milling [6].
crystallographic y-axis, it will yield an electric charge (4.52 pC/N) if loaded by a shear force in one specific
direction. Forces applied in the other directions will not generate any output [5].
A charge amplifier is used to convert the charge yielded by a quartz crystal element into a proportional
voltage. The range of a charge amplifier with respect to its conversion factor is determined by a feedback
capacitor. Adjustment to mechanical units is obtained by additional operational amplifiers with variable
The Design of Quartz Multicomponent Dynamometers.
The main element for designing multicomponent dynamometers is the three-component force transducer
(Figure 23.10). It contains a pair of X-cut quartz disks for the normal force component and a pair of
Y-cut quartz disks (shear-sensitive) for each shear force component.
Three-component dynamometers can be used for measuring cutting forces during machining. Four
three-component force transducers sandwiched between a base plate and a top plate are shown in
Figure 23.10. The force transducer is subjected to a preload as shear forces are transmitted by friction.
The four force transducers experience a drastic change in their load, depending on the type and position
of force application. An overhanging introduction of the force develops a tensile force for some transducers, thus reducing the preload. Bending of the dynamometer top plate causes bending and shearing
stresses. The measuring ranges of a dynamometer depend not only on the individual forces, but also on
the individual bending stresses.
Measuring Signals Transmitted by Telemetry.
Figure 23.11 shows the newly designed force measuring system RCD (rotating cutting force dynamometer). A ring-shaped sensor (1) is fitted in a steep angle taper socket (2) and a base ring (3) allowing
sensing of the three force components Fx , Fy and Fz at the cutting edge as well as the moment Mz . The
© 1999 by CRC Press LLC
Photo not available.
FIGURE 23.12 Capacitive force transducer.
physical operating principle of this measuring cell is based on the piezoelectric effect in quartz plates.
The quartz plates incorporated in the sensor are aligned so that the maximum cross-sensitivity between
the force components is 1%. As a result of the rigid design of the sensor, the resonant frequencies of the
force measuring system range from 1200 Hz to 3000 Hz and the measuring ranges cover a maximum of
10 kN [6].
Force-proportional charges produced at the surfaces of the quartz plates are converted into voltages
by four miniature charge amplifiers (7) in hybrid construction. These signals are then filtered by specific
electrical circuitry to prevent aliasing effects, and digitized with 8 bit resolution using a high sampling
rate (pulse-code modulation). The digitized signals are transmitted by a telemetric unit consisting of a
receiver and transmitter module, an antenna at the top of the rotating force measuring system (8), as
well as a fixed antenna (9) on the splash cover of the two-axis milling head (10). The electrical components, charge amplifier, and transmitter module are mounted on the circumference of the force measuring
system [6].
The cutting forces and the moment measured are digitized with the force measuring system described
above. They are modulated on an FM carrier and transmitted by the rotating transmitter to the stationary
receiver. The signals transmitted are fed to an external measured-variable conditioning unit.
Measuring Dynamic Forces.
Any mechanical system can be considered in the first approximation as a weakly damped oscillator
consisting of a spring and a mass. If a mechanical system has more than one resonant frequency, the
lowest one must be taken into consideration. As long as the test frequency remains below 10% of the
resonant frequency of the reference transducer (used for calibration), the difference between the dynamic
sensitivity obtained from static calibration will be less than 1%. The above considerations assume a
sinusoidal force signal. The static calibration of a reference transducer is also valid for dynamic calibration
purposes if the test frequency is much lower (at least 10 times lower) than the resonant frequency of the
Capacitive Force Transducer
A transducer that uses capacitance variation can be used to measure force. The force is directed onto a
membrane whose elastic deflection is detected by a capacitance variation. A highly sensitive force transducer can be constructed because the capacitive transducer senses very small deflections accurately. An
electronic circuit converts the capacitance variations into dc-voltage variations [7].
The capacitance sensor illustrated in Figure 23.12 consists of two metal plates separated by an air gap.
The capacitance C between terminals is given by the expression:
C ! IoI r
© 1999 by CRC Press LLC
FIGURE 23.13 Diagram of a typical force sensing resistor (FSR).
FIGURE 23.14 Resistance as a function of force for a typical force sensing resistor.
where C
= Capacitance in farads (F)
= Dielectric constant of free space
= Relative dielectric constant of the insulator
= Overlapping area for the two plates
= Thickness of the gap between the two plates
The sensitivity of capacitance-type sensors is inherently low. Theoretically, decreasing the gap h should
increase the sensitivity; however, there are practical electrical and mechanical conditions that preclude
high sensitivities. One of the main advantages of the capacitive transducer is that moving of one of its
plate relative to the other requires an extremely small force to be applied. A second advantage is stability
and the sensitivity of the sensor is not influenced by pressure or temperature of the environment.
Force Sensing Resistors (Conductive Polymers)
Force sensing resistors (FSRs) utilize the fact that certain polymer thick-film devices exhibit decreasing
resistance with the increase of an applied force. A force sensing resistor is made up of two parts. The first
is a resistive material applied to a film. The second is a set of digitating contacts applied to another film.
Figure 23.13 shows this configuration. The resistive material completes the electrical circuit between the
two sets of conductors on the other film. When a force is applied to this sensor, a better connection is
made between the contacts; hence, the conductivity is increased. Over a wide range of forces, it turns
out that the conductivity is approximately a linear function of force. Figure 23.14 shows the resistance
of the sensor as a function of force. It is important to note that there are three possible regions for the
sensor to operate. The first abrupt transition occurs somewhere in the vicinity of 10 g of force. In this
© 1999 by CRC Press LLC
region, the resistance changes very rapidly. This behavior is useful when one is designing switches using
force sensing resistors.
FSRs should not be used for accurate measurements of force because sensor parts may exhibit 15%
to 25% variation in resistance between each other. However, FSRs exhibit little hysteresis and are considered far less costly than other sensing devices. Compared to piezofilm, the FSR is far less sensitive to
vibration and heat.
Magnetoresistive Force Sensors
The principle of magnetoresistive force sensors is based on the fact that metals, when cooled to low
temperatures, show a change of resistivity when subjected to an applied magnetic field. Bismuth, in
particular, is quite sensitive in this respect. In practice, these devices are severely limited because of their
high sensitivity to ambient temperature changes.
Magnetoelastic Force Sensors
Magnetoelastic transducer devices operate based on the Joule effect; that is, a ferromagnetic material is
dimensionally altered when subjected to a magnetic field. The principle of operation is as follows: Initially,
a current pulse is applied to the conductor within the waveguide. This sets up a magnetic field circumference-wise around the waveguide over its entire length. There is another magnetic field generated by
the permanent magnet that exists only where the magnet is located. This field has a longitudinal component. These two fields join vectorally to form a helical field near the magnet which, in turn, causes
the waveguide to experience a minute torsional strain or twist only at the location of the magnet. This
twist effect is known as the Wiedemann effect [8].
Magnetoelastic force transducers have a high frequency response (on the order of 20 kHz). Some of
the materials that exhibit magnetoelastic include Monel metal, Permalloy, Cekas, Alfer, and a number of
nickel–iron alloys. Disadvantages of these transducers include: (1) the fact that excessive stress and aging
may cause permanent changes, (2) zero drift and sensitivity changes due to temperature sensitivity, and
(3) hysteresis errors.
Torsional Balances
Balancing devices that utilize the deflection of a spring may also be used to determine forces. Torsional
balances are equal arm scale force measuring devices. They are comprised of horizontal steel bands instead
of pivots and bearings. The principle of operation is based on force application on one of the arms that
will deflect the torsional spring (within its design limits) in proportion to the applied force. This type
of instrument is susceptible to hysteresis and temperature errors and therefore is not used for precise
Tactile Sensors
Tactile sensors are usually interpreted as a touch sensing technique. Tactile sensors cannot be considered
as simple touch sensors, where very few discrete force measurements are made. In tactile sensing, a force
“distribution” is measured using a closely spaced array of force sensors.
Tactile sensing is important in both grasping and object identification operations. Grasping an object
must be done in a stable manner so that the object is not allowed to slip or damaged. Object identification
includes recognizing the shape, location, and orientation of a product, as well as identifying surface
properties and defects. Ideally, these tasks would require two types of sensing [9]:
1. Continuous sensing of contact forces
2. Sensing of the surface deformation profile
These two types of data are generally related through stress–strain relations of the tactile sensor. As a
result, almost continuous variable sensing of tactile forces (the sensing of the tactile deflection profile)
is achieved.
© 1999 by CRC Press LLC
FIGURE 23.15 Tactile array sensor.
Tactile Sensor Requirements.
Significant advances in tactile sensing are taking place in the robotics area. Applications include automated
inspection of surface profiles, material handling or parts transfer, parts assembly, and parts identification
and gaging in manufacturing applications and fine-manipulation tasks. Some of these applications may
need only simple touch (force–torque) sensing if the parts being grasped are properly oriented and if
adequate information about the process is already available.
Naturally, the main design objective for tactile sensing devices has been to mimic the capabilities of
human fingers [9]. Typical specifications for an industrial tactile sensor include:
Spatial resolution of about 2 mm
Force resolution (sensitivity) of about 2 g
Maximum touch force of about 1 kg
Low response time of 5 ms
Low hysteresis
Durability under extremely difficult working conditions
Insensitivity to change in environmental conditions (temperature, dust, humidity, vibration, etc.)
Ability to monitor slip
Tactile Array Sensor.
Tactile array sensors (Figure 23.15) consist of a regular pattern of sensing elements to measure the
distribution of pressure across the finger tip of a Robot. The 8 × 8 array of elements at 2 mm spacing in
each direction, provides 64 force sensitive elements. Table 23.1 outlines some of the characteristics of
early tactile array sensors. The sensor is composed of two crossed layers of copper strips separated by
strips of thin silicone rubber. The sensor forms a thin, compliant layer that can be easily attached to a
variety of finger-tip shapes and sizes. The entire array is sampled by computer.
A typical tactile sensor array can consist of several sensing elements. Each element or taxel
(Figure 23.16) is used to sense the forces present. Since tactile sensors are implemented in applications
where sensitivity providing semblance to human touch is desired, an elastomer is utilized to mimic the
human skin. The elastomer is generally a conductive material whose electrical conductivity changes locally
when pressure is applied. The sensor itself consists of three layers: a protective covering, a sheet of
conductive elastomer, and a printed circuit board. The printed circuit board consists of two rows of two
“bullseyes,” each with conductive inner and outer rings that compromise the taxels of the sensor. The
outer rings are connected together and to a column-select transistor. The inner rings are connected to
diodes (D) in Figure 23.16. Once the column in the array is selected, the current flows through the diodes,
through the elastomer, and thence through a transistor to ground. As such, it is generally not possible
to excite just one taxel because the pressure applied causes a local deformation in neighboring taxels.
This situation is called crosstalk and is eliminated by the diodes [10].
Tactile array sensor signals are used to provide information about the contact kinematics. Several
feature parameters, such as contact location, object shape, and the pressure distribution, can be obtained.
© 1999 by CRC Press LLC
FIGURE 23.16 Typical taxel sensor array.
FIGURE 23.17 General arrangement of an intelligent sensor array system [9].
TABLE 23.1 Summary of Some of the Characteristics
of Early Tactile Arrays Sensors
Size of array
Device parameter
Cell spacing (mm)
Zero-pressure capacitance (fF)
Rupture force (N)
Max. linear capacitance (fF)
Max. output voltage (V)
Max. resolution (bit)
Readout (access) time (μs)
(4 × 4)
(8 × 8)
(16 × 16)
©IEEE 1985.
The general layout of a sensor array system can be seen in Figure 23.17. An example of this is a contact
and force sensing finger. This tactile finger has four contact sensors made of piezoelectric polymer strips
on the surface of the fingertip that provide dynamic contact information. A strain gage force sensor
provides static grasp force information.
© 1999 by CRC Press LLC
1. M. C. Shaw, Metal Cutting Principles, Oxford: Oxford Science Publications: Clarendon Press, 1989.
2. E. O. Doebelin, Measurement Systems, Application and Design, 4th ed., New York: McGraw-Hill,
3. J. W. Dally, W. F. Riley, and K. G. McConnel, Instrumentation for Engineering Measurements, New
York: John Wiley & Sons, 1984.
4. P. H. Mansfield, Electrical Transducers for Industrial Measurement, London: The Butterworth Group,
5. K. H. Martini, Multicomponent dynamometers using quartz crystals as sensing elements, ISA
Trans., 22(1), 1983.
6. G. Spur, S. J. Al-Badrawy, and J. Stirnimann, Measuring the Cutting Force in Five-Axis Milling,
Translated paper “Zerpankraftmessung bei der funfachsigen Frasbearbeitung”, Zeitschrift fur
wirtschaftliche Fertigung und Automatisierung 9/93 Carl Hanser, Munchen, Kistler Piezo-Instrumentation, 20.162e 9.94.
7. C. L. Nachtigal, Instrumentation and Control, Fundamentals and Applications, Wiley Series in
Mechanical Engineering Practice, New York: Wiley Interscience, John Wiley & Sons, 1990.
8. C. W. DeSilva, Control Sensors and Actuators, Englewood Cliffs, NJ: Prentice-Hall, 1989.
9. J. W. Gardner, Microsensors Principles and Applications, New York: John Wiley & Sons, 1995.
10. W. Stadler, Analytical Robotics and Mechatronics, New York: McGraw-Hill, 1995.
Further Information
C. P. Wright, Applied Measurement Engineering, How to Design Effective Mechanical Measurement Systems,
Englewood Cliffs, NJ: Prentice-Hall, 1995.
E. E. Herceg, Handbook of Measurement and Control, Pennsauken, NJ: Schavitz Engineering, 1972.
D. M. Considine, Encyclopedia of Instrumentation and Control, New York: McGraw-Hill, 1971.
H. N. Norton, Sensor and Analyzer Handbook, Englewood Cliffs, NJ: Prentice Hall, 1982.
S. M. Sze, Semiconductor Sensors, New York: John Wiley & Sons, 1994.
B. Lindberg and B. Lindstrom, Measurements of the segmentation frequency in the chip formation
process, Ann. CIRP, 32(1), 1983.
J. Tlusty and G. C Andrews, A critical review of sensors for unmanned machining, Ann. CIRP, 32(2), 1983.
© 1999 by CRC Press LLC
Copyright 2000 CRC Press LLC. <>.
Ivan J. Garshelis. "Torque and Power Measurement."
Torque and
Power Measurement
Fundamental Concepts
Angular Displacement, Velocity, and Acceleration • Force,
Torque, and Equilibrium • Stress, Rigidity, and Strain • Work,
Energy, and Power
Arrangements of Apparatus for Torque and
Power Measurement
Torque Transducer Technologies
Surface Strain • Twist Angle • Stress
Torque Transducer Construction, Operation, and
Mechanical Considerations • Electrical Considerations • Costs
and Options
Ivan J. Garshelis
Magnova, Inc.
Apparatus for Power Measurement
Absorption Dynamometers • Driving and Universal
Dynamometers • Measurement Accuracy • Costs
Torque, speed, and power are the defining mechanical variables associated with the functional performance of rotating machinery. The ability to accurately measure these quantities is essential for determining a machine’s efficiency and for establishing operating regimes that are both safe and conducive
to long and reliable services. On-line measurements of these quantities enable real-time control, help to
ensure consistency in product quality, and can provide early indications of impending problems. Torque
and power measurements are used in testing advanced designs of new machines and in the development
of new machine components. Torque measurements also provide a well-established basis for controlling
and verifying the tightness of many types of threaded fasteners. This chapter describes the basic concepts
as well as the various methods and apparati in current use for the measurement of torque and power;
the measurement of speed, or more precisely, angular velocity, is discussed elsewhere in this handbook [1].
24.1 Fundamental Concepts
Angular Displacement, Velocity, and Acceleration
The concept of rotational motion is readily formalized: all points within a rotating rigid body move in
parallel or coincident planes while remaining at fixed distances from a line called the axis. In a perfectly
rigid body, all points also remain at fixed distances from each other. Rotation is perceived as a change
in the angular position of a reference point on the body, i.e., as its angular displacement, Δθ, over some
time interval, Δt. The motion of that point, and therefore of the whole body, is characterized by its
© 1999 by CRC Press LLC
FIGURE 24.1 (a) The off-axis force F at P produces a torque T = (F cos β)l tending to rotate the body in the CW
direction. (b) Transmitting torque T over length L twists the shaft through angle φ.
clockwise (CW) or counterclockwise (CCW) direction and by its angular velocity, ω = Δθ/Δt. If during
a time interval Δt, the velocity changes by Δω, the body is undergoing an angular acceleration, α = Δω/Δt.
With angles measured in radians, and time in seconds, units of ω become radians per second (rad s–1)
and of α, radians per second per second (rad s–2). Angular velocity is often referred to as rotational speed
and measured in numbers of complete revolutions per minute (rpm) or per second (rps).
Force, Torque, and Equilibrium
Rotational motion, as with motion in general, is controlled by forces in accordance with Newton’s laws.
Because a force directly affects only that component of motion in its line of action, forces or components
of forces acting in any plane that includes the axis produce no tendency for rotation about that axis.
Rotation can be initiated, altered in velocity, or terminated only by a tangential force Ft acting at a finite
radial distance l from the axis. The effectiveness of such forces increases with both Ft and l; hence, their
product, called a moment, is the activating quantity for rotational motion. A moment about the rotational
axis constitutes a torque. Figure 24.1(a) shows a force F acting at an angle β to the tangent at a point P,
distant l (the moment arm) from the axis. The torque T is found from the tangential component of F as:
T = Ft l = F cos β l
The combined effect, known as the resultant, of any number of torques acting at different locations along
a body is found from their algebraic sum, wherein torques tending to cause rotation in CW and CCW
directions are assigned opposite signs. Forces, hence torques, arise from physical contact with other solid
bodies, motional interaction with fluids, or via gravitational (including inertial), electric, or magnetic
force fields. The source of each such torque is subjected to an equal, but oppositely directed, reaction
torque. With force measured in newtons and distance in meters, Equation 24.1 shows the unit of torque
to be a Newton meter (N·m).
A nonzero resultant torque will cause the body to undergo a proportional angular acceleration, found,
by application of Newton’s second law, from:
Tr = Iα
where I, having units of kilogram meter2 (kg m2), is the moment of inertia of the body around the axis
(i.e., its polar moment of inertia). Equation 24.2 is applicable to any body regardless of its state of motion.
© 1999 by CRC Press LLC
When α = 0, Equation 24.2 shows that Tr is also zero; the body is said to be in equilibrium. For a body
to be in equilibrium, there must be either more than one applied torque, or none at all.
Stress, Rigidity, and Strain
Any portion of a rigid body in equilibrium is also in equilibrium; hence, as a condition for equilibrium
of the portion, any torques applied thereto from external sources must be balanced by equal and directionally opposite internal torques from adjoining portions of the body. Internal torques are transmitted
between adjoining portions by the collective action of stresses over their common cross-sections. In a
solid body having a round cross-section (e.g., a typical shaft), the shear stress τ varies linearly from zero
at the axis to a maximum value at the surface. The shear stress, τm, at the surface of a shaft of diameter,
d, transmitting a torque, T, is found from:
τm =
πd 3
Real materials are not perfectly rigid but have instead a modulus of rigidity, G, which expresses the
finite ratio between τ and shear strain, γ. The maximum strain in a solid round shaft therefore also exists
at its surface and can be found from:
γm =
τm 16T
G πd 3G
Figure 24.1(b) shows the manifestation of shear strain as an angular displacement between axially
separated cross-sections. Over the length L, the solid round shaft shown will be twisted by the torque
through an angle φ found from:
πd 4G
Work, Energy, and Power
If during the time of application of a torque, T, the body rotates through some angle θ, mechanical work:
W = Tθ
is performed. If the torque acts in the same CW or CCW sense as the displacement, the work is said to
be done on the body, or else it is done by the body. Work done on the body causes it to accelerate, thereby
appearing as an increase in kinetic energy (KE = Iω2/2). Work done by the body causes deceleration with
a corresponding decrease in kinetic energy. If the body is not accelerating, any work done on it at one
location must be done by it at another location. Work and energy are each measured in units called a
joule (J). Equation 24.6 shows that 1 J is equivalent to 1 N·m rad, which, since a radian is a dimensionless
ratio, ≡ 1 N·m. To avoid confusion with torque, it is preferable to quantify mechanical work in units of
m·N, or better yet, in J.
The rate at which work is performed is termed power, P. If a torque T acts over a small interval of time
Δt, during which there is an angular displacement Δθ, work equal to TΔθ is performed at the rate TΔθ/Δt.
Replacing Δθ/Δt by ω, power is found simply as:
P = Tω
© 1999 by CRC Press LLC
FIGURE 24.2 Schematic arrangement of devices used for the measurement of torque and power.
The unit of power follows from its definition and is given the special name watt (W). 1 W = 1 J s–1 =
1 m·N s–1. Historically, power has also been measured in horsepower (Hp), where 1 Hp = 746 W. Rotating
bodies effectively transmit power between locations where torques from external sources are applied.
24.2 Arrangements of Apparatus for Torque
and Power Measurement
Equations 24.1 through 24.7 express the physical bases for torque and power measurement. Figure 24.2
illustrates a generalized measurement arrangement. The actual apparatus used is selected to fulfill the
specific measurement purposes. In general, a driving torque originating within a device at one location
(B in Figure 24.2), is resisted by an opposing torque developed by a different device at another location
(F). The driving torque (from, e.g., an electric motor, a gasoline engine, a steam turbine, muscular effort,
etc.) is coupled through connecting members C, transmitting region D, and additional couplings E, to
the driven device (an electric generator, a pump, a machine tool, mated threaded fasteners, etc.) within
which the resisting torque is met at F. The torque at B or F is the quantity to be measured. These torques
may be indirectly determined from a correlated physical quantity, e.g., an electrical current or fluid
pressure associated with the operation of the driving or driven device, or more directly by measuring
either the reaction torque at A or G, or the transmitted torque through D. It follows from the cause-andeffect relationship between torque and rotational motion that most interest in transmitted torque will
involve rotating bodies.
To the extent that the frames of the driving and driven devices and their mountings to the “Earth” are
perfectly rigid, the reaction at A will at every instant equal the torque at B, as will the reaction at G equal
the torque at F. Under equilibrium conditions, these equalities are independent of the compliance of any
member. Also under equilibrium conditions, and except for usually minor parasitic torques (due, e.g.,
to bearing friction and air drag over rapidly moving surfaces), the driving torque at B will equal the
resisting torque at F.
Reaction torque at A or G is often determined, using Equation 24.1, from measurements of the forces
acting at known distances fixed by the apparatus. Transmitted torque is determined from measurements,
on a suitable member within region D, of τm, γm, or φ and applying Equations 24.3, 24.4, or 24.5 (or
analogous expressions for members having other than solid round cross-sections [2]). Calibration, the
measurement of the stress, strain, or twist angle resulting from the application of a known torque, makes
it unnecessary to know any details about the member within D. When α ≠ 0, and is measurable, T may
also be determined from Equation 24.2. Requiring only noninvasive, observational measurements, this
method is especially useful for determining transitory torques; for example those associated with firing
events in multicylinder internal combustion engines [3].
Equations 24.6 and 24.7 are applicable only during rotation because, in the absence of motion, no work
is done and power transfer is zero. Equation 24.6 can be used to determine average torque from calorimetric
© 1999 by CRC Press LLC
measurements of the heat generated (equal to the mechanical work W) during a totalized number of
revolutions (≡ θ/2π). Equation 24.7 is routinely applied in power measurement, wherein T is determined
by methods based on Equations 24.1, 24.3, 24.4, or 24.5, and ω is measured by any suitable means [4].
F, T, and φ are sometimes measured by simple mechanical methods. For example, a “torque wrench”
is often used for the controlled tightening of threaded fasteners. In these devices, torque is indicated by
the position of a needle moving over a calibrated scale in response to the elastic deflection of a spring
member, in the simplest case, the bending of the wrench handle [5]. More generally, instruments,
variously called sensors or transducers, are used to convert the desired (torque or speed related) quantity
into a linearly proportional electrical signal. (Force sensors are also known as load cells.) The determination of P most usually requires multiplication of the two signals from separate sensors of T and ω. A
transducer, wherein the amplitude of a single signal proportional to the power being transmitted along
a shaft, has also been described [6].
24.3 Torque Transducer Technologies
Various physical interactions serve to convert F, τ, γ, or φ into proportional electrical signals. Each requires
that some axial portion of the shaft be dedicated to the torque sensing function. Figure 24.3 shows typical
features of sensing regions for four sensing technologies in present use.
Surface Strain
Figure 24.3(a) illustrates a sensing region configured to convert surface strain (γm) into an electric signal
proportional to the transmitted torque. Surface strain became the key basis for measuring both force
and torque following the invention of bonded wire strain gages by E. E. Simmons, Jr. and Arthur C. Ruge
in 1938 [7]. A modern strain gage consists simply of an elongated electrical conductor, generally formed
in a serpentine pattern in a very thin foil or film, bonded to a thin insulating carrier. The carrier is
attached, usually with an adhesive, to the surface of the load carrying member. Strain is sensed as a
change in gage resistance. These changes are generally too small to be accurately measured directly and
so it is common to employ two to four gages arranged in a Wheatstone bridge circuit. Independence
from axial and bending loads as well as from temperature variations are obtained by using a four-gage
bridge comprised of two diametrically opposite pairs of matched strain gages, each aligned along a
principal strain direction. In round shafts (and other shapes used to transmit torque), tensile and
compressive principal strains occur at 45° angles to the axis. Limiting strains, as determined from
Equation 24.4 (with τm equal to the shear proportional limit of the shaft material), rarely exceed a few
parts in 103. Typical practice is to increase the compliance of the sensing region (e.g., by reducing its
diameter or with hollow or specially shaped sections) in order to attain the limiting strain at the highest
value of the torque to be measured. This maximizes the measurement sensitivity.
Twist Angle
If the shaft is slender enough (e.g., L > 5 d) φ, at limiting values of τm for typical shaft materials, can
exceed 1°, enough to be resolved with sufficient accuracy for practical torque measurements (φ at τm can
be found by manipulating Equations 24.3, 24.4, and 24.5). Figure 24.3(b) shows a common arrangement
wherein torque is determined from the difference in tooth-space phasing between two identical “toothed”
wheels attached at opposite ends of a compliant “torsion bar.” The phase displacement of the periodic
electrical signals from the two “pickups” is proportional to the peripheral displacement of salient features
on the two wheels, and hence to the twist angle of the torsion bar and thus to the torque. These features
are chosen to be sensible by any of a variety of noncontacting magnetic, optical, or capacitive techniques.
With more elaborate pickups, the relative angular position of the two wheels appears as the amplitude
of a single electrical signal, thus providing for the measurement of torque even on a stationary shaft (e.g.,
[13–15]). In still other constructions, a shaft-mounted variable displacement transformer or a related
type of electric device is used to provide speed independent output signals proportional to φ.
© 1999 by CRC Press LLC
FIGURE 24.3 Four techniques in present use for measuring transmitted torque. (a) Torsional strain in the shaft
alters the electrical resistance for four strain gages (two not seen) connected in a Wheatstone bridge circuit. In the
embodiment shown, electrical connections are made to the bridge through slip rings and brushes. (b) Twist of the
torsion section causes angular displacement of the surface features on the toothed wheels. This creates a phase
difference in the signals from the two pickups. (c) The permeabilities of the two grooved regions of the shaft change
oppositely with torsional stress. This is sensed as a difference in the output voltages of the two sense windings.
(d) Torsional stress causes the initially circumferential magnetizations in the ring (solid arrows) to tilt (dashed arrows).
These helical magnetizations cause magnetic poles to appear at the domain wall and ring ends. The resulting magnetic
field is sensed by the field sensor.
In addition to elastic strain, the stresses by which torque is transmitted are manifested by changes in the
magnetic properties of ferromagnetic shaft materials. This “magnetoelastic interaction” [8] provides an
inherently noncontacting basis for measuring torque. Two types of magnetoelastic (sometimes called
magnetostrictive) torque transducers are in present use: Type 1 derive output signals from torque-induced
variations in magnetic circuit permeances; Type 2 create a magnetic field in response to torque. Type 1
transducers typically employ “branch,” “cross,” or “solenoidal” constructions [9]. In branch and cross
designs, torque is detected as an imbalance in the permeabilities along orthogonal 45° helical paths (the
principal stress directions) on the shaft surface or on the surface of an ad hoc material attached to the
shaft. In solenoidal constructions torque is detected by differences in the axial permeabilities of two
adjacent surface regions, preendowed with symmetrical magnetic “easy” axes (typically along the 45°
principal stress directions). While branch and cross type sensors are readily miniaturized [10], local
variations in magnetic properties of typical shaft surfaces limit their accuracy. Solenoidal designs, illustrated in Figure 24.3(c), avoid this pitfall by effectively averaging these variations. Type 2 transducers are
generally constructed with a ring of magnetoelastically active material rigidly attached to the shaft. The
ring is magnetized during manufacture of the transducer, usually with each axial half polarized in an
© 1999 by CRC Press LLC
FIGURE 24.4 Modular torque transducer showing generic features and alternative arrangements for free floating
or rigid mounting. Bearings* are used only on rotational models. Shaft extensions have keyways or other features to
facilitate torque coupling.
opposite circumferential direction as indicated by the solid arrows in Figure 24.3(d) [11]. When torque
is applied, the magnetizations tilt into helical directions (dashed arrows), causing magnetic poles to
develop at the central domain wall and (of opposite polarity) at the ring end faces. Torque is determined
from the output signal of one or more magnetic field sensors (e.g., Hall effect, magnetoresistive, or flux
gate devices) mounted so as to sense the intensity and polarity of the magnetic field that arises in the
space near the ring.
24.4 Torque Transducer Construction, Operation,
and Application
Although a torque sensing region can be created directly on a desired shaft, it is more usual to install a
preassembled modular torque transducer into the driveline. Transducers of this type are available with
capacities from 0.001 N·m to 200,000 N·m. Operating principle descriptions and detailed installation
and operating instructions can be found in the catalogs and literature of the various manufactures
[12–20]. Tradenames often identify specific type of transducers; for example, Torquemeters [13] refers to
a family of noncontact strain gage models; Torkducer® [18] identifies a line of Type 1 magnetoelastic
transducers; Torqstar™ [12] identifies a line of Type 2 magnetoelastic transducers; Torquetronic [16] is a
class of transducers using wrap-around twist angle sensors; and TorXimitor™ [20] identifies optoelectronic based, noncontact, strain gage transducers. Many of these devices show generic similarities transcending their specific sensing technology as well as their range. Figure 24.4 illustrates many of these
common features.
Mechanical Considerations
Maximum operating speeds vary widely; upper limits depend on the size, operating principle, type of
bearings, lubrication, and dynamic balance of the rotating assembly. Ball bearings, lubricated by grease,
oil, or oil mist, are typical. Parasitic torques associated with bearing lubricants and seals limit the accuracy
of low-end torque measurements. (Minute capacity units have no bearings [15]). Forced lubrication can
© 1999 by CRC Press LLC
allow operation up to 80,000 rpm [16]. High-speed operation requires careful consideration of the effects
of centrifugal stresses on the sensed quantity as well as of critical (vibration inducing) speed ranges.
Torsional oscillations associated with resonances of the shaft elasticity (characterized by its spring constant) with the rotational inertia of coupled masses can corrupt the measurement, damage the transducer
by dynamic excursions above its rated overload torque, and even be physically dangerous.
Housings either float on the shaft bearings or are rigidly mounted. Free floating housings are restrained
from rotating by such “soft” means as a cable, spring, or compliant bracket, or by an eccentric external
feature simply resting against a fixed surface. In free floating installations, the axes of the driving and
driven shafts must be carefully aligned. Torsionally rigid “flexible” couplings at each shaft end are used
to accommodate small angular and/or radial misalignments. Alternatively, the use of dual flexible couplings at one end will allow direct coupling of the other end. Rigidly mounted housings are equipped
with mounting feet or lugs similar to those found on the frame of electric motors. Free-floating models
are sometimes rigidly mounted using adapter plates fastened to the housing. Rigid mountings are
preferred when it is difficult or impractical to align the driving and driven shafts, as for example when
driving or driven machines are changed often. Rigidly mounted housings require the use of dual flexible
couplings at both shaft ends.
Modular transducers designed for zero or limited rotation applications have no need for bearings. To
ensure that all of the torque applied at the ends is sensed, it is important in such “reaction”-type torque
transducers to limit attachment of the housing to the shaft to only one side of the sensing region. Whether
rotating or stationary, the external shaft ends generally include such torque coupling details as flats,
keyways, splines, tapers, flanges, male/female squares drives, etc.
Electrical Considerations
By their very nature, transducers require some electrical input power or excitation. The “raw” output
signal of the actual sensing device also generally requires “conditioning” into a level and format appropriate for display on a digital or analog meter or to meet the input requirements of data acquisition
equipment. Excitation and signal conditioning are supplied by electronic circuits designed to match the
characteristics of the specific sensing technology. For example, strain gage bridges are typically powered
with 10 V to 20 V (dc or ac) and have outputs in the range of 1.5 mV to 3.0 mV per volt of excitation
at the rated load. Raising these millivolt signals to more usable levels requires amplifiers having gains of
100 or more. With ac excitation, oscillators, demodulators (or rectifiers) are also needed. Circuit elements
of these types are normal when inductive elements are used either as a necessary part of the sensor or
simply to implement noncontact constructions.
Strain gages, differential transformers, and related sensing technologies require that electrical components be mounted on the torqued member. Bringing electrical power to and output signals from these
components on rotating shafts require special methods. The most direct and common approach is to
use conductive means wherein brushes (typically of silver graphite) bear against (silver) slip rings. Useful
life is extended by providing means to lift the brushes off the rotating rings when measurements are not
being made. Several “noncontacting” methods are also used. For example, power can be supplied via
inductive coupling between stationary and rotating transformer windings [12–15], by the illumination
of shaft mounted photovoltaic cells [20], or even by batteries strapped to the shaft [21] (limited by
centrifugal force to relatively low speeds). Output signals are coupled off the shaft through rotary
transformers, by frequency-modulated (infrared) LEDs [19, 20], or by radio-frequency (FM) telemetry
[21]. Where shaft rotation is limited to no more than a few full rotations, as in steering gear, valve
actuators or oscillating mechanisms, hard wiring both power and signal circuits is often suitable. Flexible
cabling minimizes incidental torques and makes for a long and reliable service life. All such wiring
considerations are avoided when noncontact technologies or constructions are used.
© 1999 by CRC Press LLC
Costs and Options
Prices of torque transducers reflect the wide range of available capacities, performance ratings, types,
styles, optional features, and accessories. In general, prices of any one type increase with increasing
capacity. Reaction types cost about half of similarly rated rotating units. A typical foot-mounted, 565 N·m
capacity, strain gage transducer with either slip rings or rotary transformers and integral speed sensor,
specified nonlinearity and hysteresis each within ±0.1%, costs about $4000 (1997). Compatible instrumentation providing transducer excitation, conditioning, and analog output with digital display of torque
and speed costs about $2000. A comparable magnetoelastic transducer with ±0.5% accuracy costs about
$1300. High-capacity transducers for extreme speed service with appropriate lubrication options can
cost more than $50,000. Type 2 magnetoelastic transducers, mass produced for automotive power steering
applications, cost approximately $10.
24.5 Apparatus for Power Measurement
Rotating machinery exists in specific types without limit and can operate at power levels from fractions
of a watt to some tens of megawatts, a range spanning more than 108. Apparatus for power measurement
exists in a similarly wide range of types and sizes. Mechanical power flows from a driver to a load. This
power can be determined directly by application of Equation 24.7, simply by measuring, in addition to
ω, the output torque of the driver or the input torque to the load, whichever is the device under test
(DUT). When the DUT is a driver, measurements are usually required over its full service range of speed
and torque. The test apparatus therefore must act as a controllable load and be able to absorb the delivered
power. Similarly, when the DUT is a pump or fan or other type of load, or one whose function is simply
to alter speed and torque (e.g., a gear box), the test apparatus must include a driver capable of supplying
power over the DUT’s full rated range of torque and speed. Mechanical power can also be determined
indirectly by conversion into (or from) another form of energy (e.g., heat or electricity) and measuring
the relevant calorimetric or electrical quantities. In view of the wide range of readily available methods
and apparatus for accurately measuring both torque and speed, indirect methods need only be considered
when special circumstances make direct methods difficult.
Dynamometer is the special name given to the power-measuring apparatus that includes absorbing or/and
driving means and wherein torque is determined by the reaction forces on a stationary part (the stator).
An effective dynamometer is conveniently assembled by mounting the DUT in such a manner as to allow
measurement of the reaction torque on its frame. Figure 24.5 shows a device designed to facilitate such
measurements. Commercial models (Torque Table® [12]) rated to support DUTs weighing 222 N to 4900 N
are available with torque capacities from 1.3 N·m 226 to N·m. “Torque tubes” [4] or other DUT mounting
arrangements are also used. Other than for possible rotational/elastic resonances, these systems have no
speed limitations. More generally, and especially for large machinery, dynamometers include a specialized
driving or absorbing machine. Such dynamometers are classified according to their function as absorbing
or driving (sometimes motoring). A universal dynamometer can function as either a driver or an absorber.
Absorption Dynamometers
Absorption dynamometers, often called brakes because their operation depends on the creation of a
controllable drag torque, convert mechanical work into heat. A drag torque, as distinguished from an
active torque, can act only to restrain and not to initiate rotational motion. Temperature rise within a
dynamometer is controlled by carrying away the heat energy, usually by transfer to a moving fluid,
typically air or water. Drag torque is created by inherently dissipative processes such as: friction between
rubbing surfaces, shear or turbulence of viscous liquids, the flow of electric current, or magnetic
hysteresis. Gaspard Riche de Prony (1755–1839), in 1821 [22], invented a highly useful form of a friction
brake to meet the needs for testing the steam engines that were then becoming prevalent. Brakes of this
© 1999 by CRC Press LLC
FIGURE 24.5 Support system for measuring the reaction torque of a rotating machine. The axis of the machine
must be accurately set on the “center of rotation.” The holes and keyway in the table facilitate machine mounting
and alignment. Holes in the front upright provide for attaching a lever arm from which calibrating weights may be
hung [4, 11].
FIGURE 24.6 A classical prony brake. This brake embodies the defining features of all absorbing dynamometers:
conversion of mechanical work into heat and determination of power from measured values of reaction torque and
rotational velocity.
type are often used for instructional purposes, for they embody the general principles and major operating
considerations for all types of absorption dynamometers. Figure 24.6 shows the basic form and constructional features of a prony brake. The power that would normally be delivered by the shaft of the driving
engine to the driven load is (for measurement purposes) converted instead into heat via the work done
by the frictional forces between the friction blocks and the flywheel rim. Adjusting the tightness of the
© 1999 by CRC Press LLC
clamping bolts varies the frictional drag torque as required. Heat is removed from the inside surface of
the rim by arrangements (not shown) utilizing either a continuous flow or evaporation of water. There
is no need to know the magnitude of the frictional forces nor even the radius of the flywheel (facts
recognized by Prony), because, while the drag torque tends to rotate the clamped-on apparatus, it is held
stationary by the equal but opposite reaction torque Fr. F at the end of the torque arm of radius r (a fixed
dimension of the apparatus) is monitored by a scale or load cell. The power is found from Equations 24.1
and 24.7 as P = Frω = Fr2πN/60 where N is in rpm.
Uneven retarding forces associated with fluctuating coefficients of friction generally make rubbing
friction a poor way to generate drag torque. Nevertheless, because they can be easily constructed, ad hoc
variations of prony brakes, often using only bare ropes or wooden cleats connected by ropes or straps,
find use in the laboratory or wherever undemanding or infrequent power measurements are to be made.
More sophisticated prony brake constructions are used in standalone dynamometers with self-contained
cooling water tanks in sizes up to 746 kW (1000 Hp) for operation up to 3600 rpm with torques to
5400 N·m [23]. Available in stationary and mobile models, they find use in testing large electric motors
as well as engines and transmissions on agricultural vehicles. Prony brakes allow full drag torque to be
imposed down to zero speed.
William Froude (1810–1879) [24] invented a water brake (1877) that does not depend on rubbing
friction. Drag torque within a Froude brake is developed between the rotor and the stator by the momentum imparted by the rotor to water contained within the brake casing. Rotor rotation forces the water
to circulate between cup-like pockets cast into facing surfaces of both rotor and stator. The rotor is
supported in the stator by bearings that also fix its axial position. Labyrinth-type seals prevent water
leakage while minimizing frictional drag and wear. The stator casing is supported in the dynamometer
frame in cradle fashion by trunnion bearings. The torque that prevents rotation of the stator is measured
by reaction forces in much the same manner as with the prony brake. Drag torque is adjusted by a valve,
controlling either the back pressure in the water outlet piping [25] or the inlet flow rate [26] or sometimes
(to allow very rapid torque changes) with two valves controlling both [27]. In any case, the absorbed
energy is carried away by the continuous water flow. Other types of cradle-mounted water brakes, while
externally similar, have substantially different internal constructions and depend on other principles for
developing the drag torque (e.g., smooth rotors develop viscous drag by shearing and turbulence).
Nevertheless, all hydraulic dynamometers purposefully function as inefficient centrifugal pumps. Regardless of internal design and valve settings, maximum drag torque is low at low speeds (zero at standstill)
but can rise rapidly, typically varying with the square of rotational speed. The irreducible presence of
some water, as well as windage, places a speed-dependent lower limit on the controllable drag torque. In
any one design, wear and vibration caused by cavitation place upper limits on the speed and power level.
Hydraulic dynamometers are available in a wide range of capacities between 300 kW and 25,000 kW,
with some portable units having capacities as low as 75 kW [26]. The largest ever built [27], absorbing
up to about 75,000 kW (100,000 Hp), has been used to test propulsion systems for nuclear submarines.
Maximum speeds match the operating speeds of the prime movers that they are built to test and therefore
generally decrease with increasing capacity. High-speed gas turbine and aerospace engine test equipment
can operate as high as 30,000 rpm [25].
In 1855, Jean B. L. Foucault (1819–1868) [22] demonstrated the conversion of mechanical work into
heat by rotating a copper disk between the poles of an electromagnet. This simple means of developing
drag torque, based on eddy currents, has, since circa 1935, been widely exploited in dynamometers.
Figure 24.7 shows the essential features of this type of brake. Rotation of a toothed or spoked steel rotor
through a spatially uniform magnetic field, created by direct current through coils in the stator, induces
locally circulating (eddy) currents in electrically conductive (copper) portions of the stator. Electromagnetic forces between the rotor, which is magnetized by the uniform field, and the field arising from the
eddy currents, create the drag torque. This torque, and hence the mechanical input power, are controlled
by adjusting the excitation current in the stator coils. Electric input power is less than 1% of the rated
capacity. The dynamometer is effectively an internally short-circuited generator because the power
associated with the resistive losses from the generated eddy currents is dissipated within the machine.
© 1999 by CRC Press LLC
FIGURE 24.7 Cross-section (left) and front view (right) of an eddy current dynamometer. G is a gear wheel and
S is a speed sensor. Hoses carrying cooling water and cable carrying electric power to the stator are not shown.
Being heated by the flow of these currents, the stator must be cooled, sometimes (in smaller capacity
machines) by air supplied by blowers [23], but more often by the continuous flow of water [25, 27, 28].
In dry gap eddy current brakes (the type shown in Figure 24.7), water flow is limited to passages within
the stator. Larger machines are often of the water in gap type, wherein water also circulates around the
rotor [28]. Water in contact with the moving rotor effectively acts as in a water brake, adding a nonelectromagnetic component to the total drag torque, thereby placing a lower limit to the controllable torque.
Windage limits the minimum value of controllable torque in dry gap types. Since drag torque is developed
by the motion of the rotor, it is zero at standstill for any value of excitation current. Initially rising rapidly,
approximately linearly, with speed, torque eventually approaches a current limited saturation value. As
in other cradled machines, the torque required to prevent rotation of the stator is measured by the
reaction force acting at a fixed known distance from the rotation axis. Standard model eddy current
brakes have capacities from less than 1 kW [23, 27] to more than 2000 kW [27, 28], with maximum
speeds from 12,000 rpm in the smaller capacity units to 3600 rpm in the largest units. Special units with
capacities of 3000 Hp (2238 kW) at speeds to 25,000 rpm have been built [28].
Hysteresis brakes [29] develop drag torque via magnetic attractive/repulsive forces between the magnetic poles established in a reticulated stator structure by a current through the field coil, and those
created in a “drag cup” rotor by the stator field gradients. Rotation of the special steel rotor, through the
spatial field pattern established by the stator, results in a cyclical reversal of the polarity of its local
magnetizations. The energy associated with these reversals (proportional to the area of the hysteresis
loop of the rotor material) is converted into heat within the drag cup. Temperature rise is controlled by
forced air cooling from a blower or compressed air source. As with eddy current brakes, the drag torque
of these devices is controlled by the excitation current. In contrast with eddy current brakes, rated drag
torque is available down to zero speed. (Eddy current effects typically add only 1% to the drag torque
for each 1000 rpm). As a result of their smooth surfaced rotating parts, hysteresis brakes exhibit low
parasitic torques and hence cover a dynamic range as high as 200 to 1. Standard models are available
having continuous power capacities up to 6 kW (12 kW with two brakes in tandem cooled by two
blowers). Intermittent capacities per unit (for 5 min or less) are 7 kW. Some low-capacity units are
convection cooled; the smallest has a continuous rating of just 7 W (35 W for 5 min). Maximum speeds
range from 30,000 rpm for the smallest to 10,000 rpm for the largest units. Torque is measured by a
strain gage bridge on a moment arm supporting the machine stator.
© 1999 by CRC Press LLC
Driving and Universal Dynamometers
Electric generators, both ac and dc, offer another means for developing a controllable drag torque and
they are readily adapted for dynamometer service by cradle mounting their stator structures. Moreover,
electric machines of these types can also operate in a motoring mode wherein they can deliver controllable
active torque. When configured to operate selectively in either driving or absorbing modes, the machine
serves as a universal dynamometer. With dc machines in the absorbing mode, the generated power is
typically dissipated in a convection-cooled resistor bank. Air cooling the machine with blowers is usually
adequate, since most of the mechanical power input is dissipated externally. Nevertheless, all of the
mechanical input power is accounted for by the product of the reaction torque and the rotational speed.
In the motoring mode, torque and speed are controlled by adjustment of both field and armature currents.
Modern ac machines utilize regenerative input power converters to allow braking power to be returned
to the utility power line. In the motoring mode, speed is controlled by high-power, solid-state, adjustable
frequency inverters. Internal construction is that of a simple three-phase induction motor, having neither
brushes, slip rings, nor commutators. The absence of rotor windings allows for higher speed operation
than dc machines. Universal dynamometers are “four-quadrant” machines, a term denoting their ability
to produce torque in the same or opposite direction as their rotational velocity. This unique ability allows
the effective drag torque to be reduced to zero at any speed. Universal dynamometers [25, 28] are available
in a relatively limited range of capacities (56 to 450 kW), with commensurate torque (110 to 1900 N·m)
and speed (4500 to 13,500 rpm) ranges, reflecting their principal application in automotive engine
development. Special dynamometers for testing transmissions and other vehicular drive train components
insert the DUT between a diesel engine or electric motor prime mover and a hydraulic or eddy current
brake [30].
Measurement Accuracy
Accuracy of power measurement (see discussion in [4]) is generally limited by the torque measurement
(±0.25% to ±1%) since rotational speed can be measured with almost any desired accuracy. Torque errors
can arise from the application of extraneous (i.e., not indicated) torques from hose and cable connections,
from windage of external parts, and from miscalibration of the load cell. Undetected friction in the
trunnion bearings of cradled dynamometers can compromise the torque measurement accuracy. Ideally,
well-lubricated antifriction bearings make no significant contribution to the restraining torque. In practice, however, the unchanging contact region of the balls or other rolling elements on the bearing races
makes them prone to brinelling (a form of denting) from forces arising from vibration, unsupported
weight of attached devices, or even inadvertently during the alignment of connected machinery. The
problem can be alleviated by periodic rotation of the (primarily outer) bearing races. In some bearingin-bearing constructions, the central races are continuously rotated at low speeds by an electric motor
while still others avoid the problem by supporting the stator on hydrostatic oil lift bearings [28].
The wide range of torque, speed, and power levels, together with the variation in sophistication of
associated instrumentation, is reflected in the very wide range of dynamometer prices. Suspension systems
of the type illustrated in Figure 24.5 (for which the user must supply the rotating machine) cost $4000 to
$6000, increasing with capacity [12]. A 100 Hp (74.6 kW) portable water brake equipped with a strain
gage load cell and a digital readout instrument for torque, speed, and power costs $4500, or $8950 with
more sophisticated data acquisition equipment [26]. Stationary (and some transportable [23]) hydraulic
dynamometers cost from $113/kW in the smaller sizes [25], down to $35/kW for the very largest [27].
Transportation, installation, and instrumentation can add significantly to these costs. Eddy current
dynamometers cost from as little as $57/kW to nearly $700/kW, depending on the rated capacity, type
of control system, and instrumentation [24, 25, 28]. Hysteresis brakes with integral speed sensors cost
© 1999 by CRC Press LLC
from $3300 to $14,000 according to capacity [29]. Compatible controllers, from manual to fully programmable for PC test control and data acquisition via an IEEE-488 interface, vary in price from $500 to
$4200. The flexibility and high performance of ac universal dynamometers is reflected in their comparatively high prices of $670 to $2200/kW [25, 28].
1. Pinney, C. P. and Baker, W. E., Velocity Measurement, The Measurement, Instrumentation and
Sensors Handbook, Webster, J. G., ed., Boca Raton, FL: CRC Press, 1999.
2. S. Timoshenko, Strength of Materials, 3rd ed., New York: Robert E. Kreiger, Part I, 281–290; Part II,
235–250, 1956.
3. S. J. Citron, On-line engine torque and torque fluctuation measurement for engine control utilizing
crankshaft speed fluctuations, U. S. Patent No. 4,697,561, 1987.
4. Supplement to ASME Performance Test Codes, Measurement of Shaft Power, ANSI/ASME PTC
19.7-1980 (Reaffirmed 1988).
5. See, for example, the catalog of torque wrench products of Consolidated Devices, Inc., 19220 San
Jose Ave., City of Industry, CA 91748.
6. I. J. Garshelis, C. R. Conto, and W. S. Fiegel, A single transducer for non-contact measurement of
the power, torque and speed of a rotating shaft, SAE Paper No. 950536, 1995.
7. C. C. Perry and H. R. Lissner, The Strain Gage Primer, 2nd ed., New York: McGraw-Hill, 1962, 9.
(This book covers all phases of strain gage technology.)
8. B. D. Cullity, Introduction to Magnetic Materials, Reading, MA: Addison-Wesley, 1972, Section 8.5,
9. W. J. Fleming, Magnetostrictive torque sensors—comparison of branch, cross and solenoidal
designs, SAE Paper No. 900264, 1990.
10. Y. Nonomura, J. Sugiyama, K. Tsukada, M. Takeuchi, K. Itoh, and T. Konomi, Measurements of
engine torque with the intra-bearing torque sensor, SAE Paper No. 870472, 1987.
11. I. J. Garshelis, Circularly magnetized non-contact torque sensor and method for measuring torque
using same, U.S. Patent 5,351,555, 1994 and 5,520,059, 1996.
12. Lebow® Products, Siebe, plc., 1728 Maplelawn Road, Troy, MI 48099, Transducer Design Fundamentals/Product Listings, Load Cell and Torque Sensor Handbook No. 710, 1997, also: Torqstar™
and Torque Table®.
13. S. Himmelstein & Co., 2490 Pembroke, Hoffman Estates, IL 60195, MCRT® Non-Contact Strain
Gage Torquemeters and Choosing the Right Torque Sensor.
14. Teledyne Brown Engineering, 513 Mill Street, Marion, MA 02738-0288.
15. Staiger, Mohilo & Co. GmbH, Baumwasenstrasse 5, D-7060 Schorndorf, Germany (In the U.S.:
Schlenker Enterprises Ltd., 5143 Electric Ave., Hillside, IL 60162), Torque Measurement.
16. Torquemeters Ltd., Ravensthorpe, Northampton, NN6 8EH, England (In the U.S.: Torquetronics
Inc., P.O. Box 100, Allegheny, NY 14707), Power Measurement.
17. Vibrac Corporation, 16 Columbia Drive, Amherst, NH 03031, Torque Measuring Transducer.
18. GSE, Inc., 23640 Research Drive, Farmington Hills, MI 48335-2621, Torkducer®.
19. Sensor Developments Inc., P.O. Box 290, Lake Orion, MI 48361-0290, 1996 Catalog.
20. Bently Nevada Corporation, P.O. Box 157, Minden, NV 89423, TorXimitor™.
21. Binsfield Engineering Inc., 4571 W. MacFarlane, Maple City, MI 49664.
22. C. C. Gillispie (ed.), Dictionary of Scientific Biography, Vol. XI, New York: Charles Scribner’s Sons,
23. AW Dynamometer, Inc., P.O. Box 428, Colfax, IL 61728, Traction dynamometers: Portable and
stationary dynamometers for motors, engines, vehicle power take-offs.
24. Roy Porter (ed.), The Biographical Dictionary of Scientists, 2nd ed., New York: Oxford University
Press, 1994.
© 1999 by CRC Press LLC
25. Froude-Consine Inc., 39201 Schoolcraft Rd., Livonia, MI 48150, F Range Hydraulic Dynamometers, AG Range Eddy Current Dynamometers, AC Range Dynamometers.
26. Go-Power Systems, 1419 Upfield Drive, Carrollton, TX 75006, Portable Dynamometer System, GoPower Portable Dynamometers.
27. Zöllner GmbH, Postfach 6540, D-2300 Kiel 14, Germany (In the U.S. and Canada: Roland Marine
Inc., 90 Broad St., New York, NY 10004), Hydraulic Dynamometers Type P, High Dynamic Hydraulic Dynamometers.
28. Dynamatic Corporation, 3122 14th Ave., Kenosha, WI 53141-1412, Eddy Current Dynamometer—Torque Measuring Equipment, Adjustable Frequency Dynamometer.
29. Magtrol, Inc., 70 Gardenville Parkway, Buffalo, NY 14224-1322, Hysteresis Absorption Dynamometers.
30. Hicklin Engineering, 3001 NW 104th St., Des Moines, IA 50322, Transdyne™ (transmission test
systems, brake and towed chassis dynamometers).
© 1999 by CRC Press LLC
Copyright 2000 CRC Press LLC. <>.
R. E. Saad, et. al.. "Tactile Sensing."
Tactile Sensing
R. E. Saad
University of Toronto
A. Bonen
University of Toronto
K. C. Smith
Simplified Theory for Tactile Sensing • Requirements for
Tactile Sensors
University of Toronto
B. Benhabib
Sensing Classification
Mechanical Effects of Contact
University of Toronto
Technologies for Tactile Sensing
Resistive • Capacitive • Piezoelectric • Optical • Photoelastic
Robots in industrial settings perform repetitive tasks, such as machine loading, parts assembly, painting,
and welding. Only in rare instances can these autonomous manipulators modify their actions based on
sensory information. Although, thus far, a vast majority of research work in the area of robot sensing
has concentrated on computer vision, contact sensing is an equally important feature for robots and has
received some attention as well. Without tactile-perception capability, a robot cannot be expected to
effectively grasp objects. In this context, robotic tactile sensing is the focus of this chapter.
25.1 Sensing Classification
Robotic sensing can be classified as either of the noncontact or contact type [1]. Noncontact sensing
involves interaction between the robot and its environment by some physical phenomenon, such as
acoustic or electromagnetic waves, that interact without contact. The most important types of robotic
sensors of the noncontact type are vision and proximity sensors. Contact sensing, on the other hand,
implies measurement of the general interaction that takes place when the robot’s end effector is brought
into contact with an object. Contact sensing is further classified into force and tactile sensing.
Force sensing is defined as the measurement of the global mechanical effects of contact, while tactile
sensing implies the detection of a wide range of local parameters affected by contact. Most significant
among those contact-based effects are contact stresses, slippage, heat transfer, and hardness.
The properties of a grasped object that can be derived from tactile sensing can be classified into
geometric and dynamometric types [2]. Among the geometric properties are presence, location in relation
to the end-effector, shape and dimensions, and surface conditions [3–7]. Among the dynamometric
parameters associated with grasping are: force distribution, slippage, elasticity and hardness, and friction
Tactile sensing requires sophisticated transducers; yet the availability of these transducers alone is not
a sufficient condition for successful tactile sensing. It is also necessary to accurately control the modalities
through which the tactile sensor interacts with the explored objects (including contact forces, as well as
end-effector position and orientation) [13–15]. This leads to active tactile sensing, which requires a high
degree of complexity in the acquisition and processing of the tactile data [16].
© 1999 by CRC Press LLC
FIGURE 25.1 An object indenting a compliant layer, where an array of force-sensing elements is placed at a distance
d from the surface.
25.2 Mechanical Effects of Contact
Tactile sensing normally involves a rigid object indenting the compliant cover layer of a tactile sensor
array [17], Figure 25.1. The indentation of a compliant layer due to contact can be analyzed from two
conceptually different points of view [1]. The first one is the measurement of the actual contact stresses
(force distribution) in the layer, which is usually relevant to controlling manipulation tasks. The second
one is the deflection profile of the layer, which is usually important for recognizing geometrical object
features. Depending on the approach adopted, different processing and control algorithms must be
There exists a definite relationship between the local shape of a contacting body and a set of subsurface
strains (or displacements); however, this relationship is quite complex. Thus, it requires the use of the
Theory of Elasticity and Contact Mechanics to model sensor–object interaction [18], and the use of Finite
Element Analysis (FEA) as a practical tool for obtaining a more representative model of the sensor [19].
In general, the study of tactile sensors comprises two steps: (1) the forward analysis, related to the
acquisition of data from the sensor (changes on the stress or strains, induced by the indentation of an
object on the compliant surface of the transducer); and, (2) the inverse problem, normally related to the
recovery of force distribution or, in some cases, the recovery of the indentor’s shape.
Simplified Theory for Tactile Sensing
For simplicity, the general two-dimensional tactile problem is reduced herein to a one-dimensional one.
Figure 25.2 shows a one-dimensional transducer that consists of a compliant, homogeneous, isotropic,
and linear layer subjected to a normal stress qv(x) created by the indentation of an object.
For modeling purposes, it is assumed that the compliant layer is an elastic half-space. This simplification yields closed-form equations for the analysis and avoids the formation of a more complex problem,
in which the effect of the boundary conditions at xmin and xmax must be taken into account. It has been
proven that the modeling of the sensor by an elastic half-space represents a reasonable approximation
to the real case [18]. Under these conditions, it can be shown that the normal strain, at a depth y = d,
due to the normal stress qv(y) is given by [20]:
© 1999 by CRC Press LLC
Ideal one-dimensional transducer subjected to a normal stress.
( ) ∫ q (x − x )h (x ,d) dx
εz x =
where εz is the strain at x and z = d due to the normal stress on the surface, and
hz x = −
( )[ ( )
2d 1 + v d 2 1 − v − vx 2
πrE x + d
E and v are, respectively, the modulus of elasticity and the Poisson’s coefficient of the compliant layer.
In obtaining Equation 25.2, it is assumed that the analysis is performed under planar strain conditions.
It should be noted that a similar analysis can be performed for tangential contact stresses or strains.
The normal displacement at the surface, w, is given by:
( ) ∫ q (x − x )k(x ) dx
w x =
k x =
−2 1 − v 2
) log x
The singularity at x = 0 is expected due to the singularity of stress at that point. Note that, k(x) is the
deformation of the surface when a singular load of 1 N is applied at x = 0. The constant xa should be
chosen such that at x = xa, the deformation is zero. In this case, zero deformation should occur at x → ∞
(note that it has been assumed that the sensor is modeled by an elastic half space), namely xa → ∞. This
problem is associated with the two-dimensional deformation of an elastic half-space. To eliminate this
difficulty, the boundary conditions of the transducer must be taken into account (i.e., a finite transducer
must be analyzed), which requires, in general, the use of FEA.
© 1999 by CRC Press LLC
One-dimensional transducer with discrete sensing elements located at z = d.
Since measurements of strain (or stress) are usually done by a discrete number of sensing elements,
Equation 25.2 must be discretized (Figure 25.3). Correspondingly, the force distribution must be reconstructed at discrete positions as shown in Figure 25.3. Let Δxq be the distance between points, where the
force distribution must be reconstructed from strain (or stress) measurements carried out by strain (or
stress) sensing elements uniformly distributed at intervals Δxp, at z = d. Also assume, even though it is
not necessary, that Δxq = Δxp = Δx and that the forces are applied at positions immediately above the
sensor elements. One can now define the strain (stress)-sample vector, ζ, whose components are given
by ζi = εx(xi), i = 1, 2, …, n, and the force distribution vector, F, whose components are given by fi =
qv(xj), j = 1, 2, …, n. Then, the discrete form of Equation 25.1 is given by:
ζ = TF
where the elements of the matrix T are given by Tij = kv(xi – xj), i = 1, 2, …, n and j = 1, 2, …, n [23]. A
similar relation to Equation 25.5 can be obtained discretizing Equation 25.3. In the general case, where
Δxq ≠ Δxp, T is not square. Furthermore, in the general case, the vector F comprises both vertical and
tangential components.
Equations 25.1 and 25.3 represent the regular forward problem, while Equation 25.5 represents the
discretized version of the forward problem. The inverse problem, in most cases, consists of recovering the
applied force profile from the measurements of strain, stress, or deflection. (Note that the surface
displacement can also be used to recover the indentor’s shape.)
In [20], it was shown that the inverse problem is ill-posed because the operators h and k, of
Equations 25.1 and 25.3, respectively, are ill-conditioned. Consequently, the inverse problem is susceptible
to noise. To solve this problem, regularization techniques must be utilized [20].
It has been proven that, in order to avoid aliasing in determining the continuous strain (stress) at a
depth d using a discretized transducer, the elements have to be separated by one tenth of the compliant
layer’s thickness. However, good results were obtained, without much aliasing, by separating the sensing
elements by a distance equal to the sensor’s depth [18].
Requirements for Tactile Sensors
In 1980, Harmon conducted a survey to determine general specifications for tactile sensors [21]. Those
specifications have been used subsequently as guidelines by many tactile sensor designers:
© 1999 by CRC Press LLC
General configuration of a resistive transducer.
Spatial resolution of 1 to 2 mm
Array sizes of 5 × 10 to 10 × 20 points
Sensitivity of 0.5 × 10–2 to 1 × 10–2 N for each force-sensing element (tactel)
Dynamic range of 1000:1
Stable behavior and with no hysteresis
Sampling rate of 100 Hz to 1 kHz
Monotonic response, though not necessarily linear
Compliant interface, rugged and inexpensive
While properties (5), (7), and (8) above should apply to any practical sensor, the others are merely
suggestions, particularly with respect to the number of array elements and spatial resolution.
Developments on tactile sensing following [21] have identified additional desirable qualities; namely,
reliability, modularity, speed, and the availability of multisensor support [16].
25.3 Technologies for Tactile Sensing
The technologies associated with tactile sensing are quite diverse: extensive surveys of the state-of-theart of robotic-tactile-transduction technologies have been presented in [2, 3, 16, 17]. Some of these
technologies will be briefly discussed.
The transduction method that has received the most attention in tactile sensor design is concerned with
the change in resistance of a conductive material under applied pressure. A basic configuration of a
resistive transducer is shown in Figure 25.4. Each resistor, whose value changes with the magnitude of
the force, represents a resistive cell of the transducer. Different materials have been utilized to manufacture
the basic cell.
Conductive elastomers were among the first resistive materials used for the development of tactile
sensors. They are insulating, natural or silicone-based rubbers made conductive by adding particles of
conductive or semiconductive materials (e.g., silver or carbon). The changes in resistivity of the elastomers
© 1999 by CRC Press LLC
under pressure are produced basically by two different physical mechanisms. In the first approach, the
change in resistivity of the elastomer under pressure is associated with deformation that alters the particle
density within it. Two typical designs of this kind are given in [22, 23]. In the second approach, while
the bulk resistance of the elastomer changes slightly when it is compressed, the design allows the increase
of the area of contact between the elastomer and an electrode, and correspondingly a change in the
contact resistance. A typical design of this kind is given in [24]. In [25], a newer tactile sensor is reported
with both three-axis force sensing and slippage sensing functions. In the former case, the pressure sensing
function is achieved utilizing arrays of pressure transducers that measure a change in contact resistance
between a specially treated polyimide film and a resistive substrate.
Piezoresistive elements have also been used in several tactile sensors. This technology is specifically
attractive at present because, with micromachining, the piezoresistive elements can be integrated together
with the signal-processing circuits in a single chip [26]. A 32 × 32-element silicon pressure sensor array
incorporating CMOS processing circuits for the detection of a high-resolution pressure distribution was
reported in [8]. The sensor array consists of an x–y-matrix-organized array of pressure cells with a cell
spacing of 250 μm. CMOS processing circuits are formed around the array on the same chip. Fabrication
of the sensor array was carried out using a 3 mm CMOS process combined with silicon micromachining
techniques. The associated diaphragm size is 50 μm × 50 μm. The overall sensor-array chip size is 10 mm ×
10 mm.
In Figure 25.4, a circuit topology, to scan a 3 × 3 array of piezoresistive elements, is shown. The basic
idea was originally proposed in [24] and adapted on several occasions by different researchers. Using this
method, the changes in resistance are converted into voltages at the output. With the connections as
shown in Figure 25.4, the resistance R21 can be determined from:
V0 =
where Vo is the output voltage, Vcc is the bias voltage, and Rf is the feedback resistance of the output
amplifier stage.
One problem with the configuration shown in Figure 25.4 is the difficulty in detecting small changes
in resistance due to the internal resistance of the multiplexer as well changes in the voltage of power
source, which have a great influence at the output. Other methods utilized to scan resistive transducer
arrays are summarized in [3].
When piezoresistors and circuits are fabricated on the same silicon substrate, the sensor array can be
equipped with a complex switching circuit, next to the sensing elements, that allows a better resolution
in the measurements [9].
Tactile sensors within this category are concerned with measuring capacitance, which varies under applied
load. The capacitance of a parallel-plate capacitor depends on the separation of the plates and their areas.
A sensor using an elastomeric separator between the plates provides compliance such that the capacitance
will vary according to the applied normal load, Figure 25.5(a).
Figure 25.5(b) shows the basic configuration of a capacitive tactile sensor. The intersections of rows
and columns of conductor strips form capacitors. Each individual capacitance can be determined by
measuring the corresponding output voltage at the selected row and column. To reduce cross-talk and
electromagnetic interference, the rows and columns that are not connected are grounded. Figure 25.5(c)
shows an equivalent circuit when the sensor is configured to measure the capacitance formed at the
intersection of row i and row j, Cij . Rd is the input resistance of the detector and Cd represents the effects
of the stray capacitances, including the detector-amplifier input capacitance, the stray capacitance due
© 1999 by CRC Press LLC
FIGURE 25.5 (a) Basic cell of a capacitor tactile sensor. (b) Typical configuration of a capacitive tactile sensor.
(c) Equivalent circuit for the measurement of the capacitance Cij .
to the unselected rows and columns, and the capacitance contributed by the cable that connects the
transducer to the detector. Since the stray capacitance due to the unselected rows and columns changes
with the applied forces, the stray capacitance due to the cable is designed to be predominant [18].
The magnitude of voltage at the input of the detector, 冨Vd 冨 is given by:
Vd =
Cij Rd ω
[ (
1 + ωRd Cij + Cd
Assuming that Cd » Cij and ω is sufficiently large,
Vd ≅
When a load is applied to the transducer, the capacitor is deformed as shown in Figure 25.5(a). For
modeling purposes, it is assumed that the plate capacitor is only under compression. When no load is
applied, the capacitance due to the element in the ith row and the jth column, C 0ij , is given by:
Cij0 = ε
where ε is the permittivity of the dielectric, w and l are the width and the length of the plate capacitor,
respectively, and h0 is the distance between plates when no load is applied. The voltage at the input of
the detector for this particular case is indicated by Vd0; then from Equation 25.8, one obtains:
Vd0 ≅
© 1999 by CRC Press LLC
When a load is applied, the capacitor is under compression and the capacitance is given by:
Cij = ε
h0 − Δh
The strain in this case is given by:
ζz ≅
where Δh is the displacement of the top metal plate and Δh « h0 . The strain can be measured by:
Vd − Vd0
= 1−
= 1−
h0 − Δh Δh Δh
≅ ζz
Consequently, the strain at each tactel can be determined by measuring the magnitudes of Vd and Vd0
for each element.
Note that the presence of a tangential force would offset the plates tangentially and change the effective
area of the capacitor plates. An ideal capacitive pressure sensor can quantify basic aspects of touch by
sensing normal forces, and can detect slippage by measuring tangential forces. However, distinguishing
between the two forces at the output of a single sensing element is a difficult task and requires a more
complex transducer than the one presented in Figure 25.5(a) [27].
Micromachined, silicon-based capacitive devices are especially attractive due to their potential for high
accuracy and low drift. A sensor with 1024 elements and a spatial resolution of 0.5 mm was reported in
[28]. Several possible structures for implementing capacitive high-density tactile transducers in silicon
have been reported in [29]. A cylindrical finger-shaped transducer was reported in [18].
The advantages of capacitive transducers include: wide dynamic range, linear response, and robustness.
Their major disadvantages are susceptibility to noise, sensitivity to temperature, and the fact that capacitance decreases with physical size, ultimately limiting the spatial resolution. Research is progressing
toward the development of electronic processing circuits for the measurement of small capacitances using
charge amplifiers [30], and the development of new capacitive structures [29].
A material is called piezoelectric, if, when subjected to a stress or deformation, it produces electricity.
Longitudinal piezoelectric effect occurs when the electricity is produced in the same direction of the
stress, Figure 25.6. In Figure 25.6(a), a normal stress σ (= F/A) is applied along the Direction 3 and the
charges are generated on the surfaces perpendicular to Direction 3. A transversal piezoelectric effect
occurs when the electricity is produced in the direction perpendicular to the stress.
The voltage V generated across the electrodes by the stress σ is given by:
V = d33 σ
where d33 = Piezoelectric constant associated with the longitudinal piezoelectric effect
ε = Permittivity
h = Thickness of the piezoelectric material
© 1999 by CRC Press LLC
FIGURE 25.6 (a) Basic cell of a pizoelectric transducer. (b) Charge amplifier utilized for the measurement of the
applied force.
Since piezoelectric materials are insulators, the transducer shown in Figure 25.6(a), can be considered
as a capacitor, from an electrical point of view. Consequently,
C εA
where Q = Charge induced by the stress σ
C = Capacitance of the parallel capacitor
A = Area of each electrode
A comparison of Equations 25.14 and 25.15 leads to:
Q = d33 A σ
It is concluded that the force applied to the photoelastic material can be determined by finding the
charge Q. Charge amplifiers are usually utilized for determining Q. The basic configuration of a charge
amplifier is shown in Figure 25.6(b). The charge generated in the transducer is transferred to the capacitor
Cf and the output voltage, Vo is given by:
Vo = −
The circuit must periodically discharge the feedback capacitor Cf to avoid saturation of the amplifier
by stray charges generated by the offset voltages and currents of the operational amplifier. This is achieved
by a switch as shown in Figure 25.6(b) or by a resistor parallel to Cf .
The piezoelectric material most widely used in the implementation of tactile transducers is PVF2. It
shows the largest piezoelectric effect of any known material. Its flexibility, small size, sensitivity, and large
electrical output offer many advantages for sensor applications in general, and tactile sensors in particular.
Examples of tactile sensors implemented with this technology can be found in [1, 31].
The major advantages of the piezoelectric technology are its wide dynamic range and durability.
Unfortunately, the response of available materials does not extend down to dc and therefore steady loads
cannot be measured directly. Also, the PVF2 material produces a charge output that is prone to electrical
interference and is temperature dependent.
© 1999 by CRC Press LLC
Current-to-voltage converter.
The possibility of measuring transient phenomenon using piezoelectric material has recently encouraged some researchers to use the piezoelectric effect for detecting vibrations that indicate incipient slip,
occurrence of contact, local change in skin curvature, and estimating friction and hardness of the object
[7, 10, 11]. If the piezoelectric transducer shown in Figure 25.6(a) is connected to an FET-input operational amplifier configured as a current-to-voltage converter as shown in Figure 25.7, the output voltage
is given by:
Vo =
Rf = ARf d33
where Rf is the feedback resistor. Correspondingly, the circuit configuration provides the mean to measure
of changes in the contact stress. A detailed explanation of the behavior of this sensor can be found in [7].
Recent developments in fiber optic technology and solid-state cameras have led to numerous novel tactile
sensor designs [32, 33]. Some of these designs employ flexible membranes incorporating a reflecting
surface, Figure 25.8. Light is introduced into the sensor via a fiber optic cable. A wide cone of light
propagates out of the fiber, reflects back from the membrane, and is collected by a second fiber. When
an external force is applied onto the elastomer, it shortens the distance between the reflective side of the
© 1999 by CRC Press LLC
(a) Reflective transducer. (b) Light-intensity as a function of the distance h.
Tactile transducer based on the principle of internal reflection.
membrane and the fibers, h. Consequently, the light gathered by the receiving fiber changes as a function
of h, Figure 25.8(b). To recover univocally the distance from the light intensity, a monotonic function is
needed. This can be achieved by designing the transducer such it operates for h > hmin, where hmin is
indicated in Figure 25.8(b). (The region h > hmin is preferred to the h < hmin for dynamic range reasons.)
Another optical effect that can be used is that of frustrated total internal reflection [5, 34]. With this
technique, an elastic rubber membrane covers, without touching, a glass plate (waveguide); light entering
the side edge of the glass is totally reflected by the top and bottom surfaces and propagates along it,
Figure 25.9.
The condition for total internal reflection occurs when:
n2 sin α ≤ n1
where n1 = Index of refraction of the medium surrounding the waveguide (in this case air, n1 ≅ 1)
n2 = Index of refraction of the waveguide
α = Angle of incidence at the interface glass-air
Objects in contact with the elastic membrane deform it and induce contact between the bottom part
of the membrane and the top surface of the waveguide, disrupting the total internal reflection. Consequently, the light in the waveguide is scattered at the contact location. Light that escapes through the
bottom surface of the waveguide can be detected by an array of photodiodes, a solid-state sensor, or,
alternatively, transported away from the transducer by fibers [3]. The detected imaged is stored in a
computer for further analysis. A rubber membrane with a flat surface yields a high-resolution binary
(contact or noncontact) image [5]. If the rubber sheet is molded with a textured surface (Figure 25.9),
then an output proportional to the area of contact is obtained and, consequently, the applied forces can
be detected [3]. Shear forces can also be detected using special designs [35]. Sensors based on frustrated
internal reflection can be molded into a finger shape [5] and are capable of forming very high-resolution
tactile images. Such sensors are commercially available. An improved miniaturized version of a similar
sensor was proposed in [34].
Other types of optical transducers use “occluder” devices. One of the few commercially available tactile
sensors uses this kind of transducer [36]. In one of the two available designs, the transducer’s surface is
made of a compliant material, which has on its underside a grid of elongated pins. When force is applied
to the compliant surface, the pins on the underside undergo a mechanical motion normal to the surface,
© 1999 by CRC Press LLC
FIGURE 25.10
Principle of operation of an occluder transducer.
FIGURE 25.11
A four-layer tactile transducer.
blocking the light path of a photoemitter–detector pair. The amount of movement determines the amount
of light reaching the photoreceiver. Correspondingly, the more force applied, the less amount of light is
collected by the photoreceiver, Figure 25.10. The major problems with this specific device are associated
with creep, hysteresis, and temperature variation. This scheme also requires individual calibration of each
photoemitter–photodetector pair.
Fibers have also been used directly as transducers in the design of tactile sensors. Their use is based
on two properties of fiber optic cables: (1) if a fiber is subjected to a significant amount of bending, then
the angle of incidence at the fiber wall can be reduced sufficiently for light to leave the core [37]; and
(2) if two fibers pass close to one another and both have roughened surfaces, then light can pass between
the fibers. Light coupling between adjacent fibers is a function of their separation [3].
An example of an optical fiber tactile sensor, whose sensing mechanism is based on the increase of
light attenuation due to the microbend in the optical fibers, is shown in Figure 25.11 [37]. The transducer
consists of a four-layer, two-dimensional fiber optic array constructed by using two layers of optical fibers
as a corrugation structure, through which microbends are induced in two orthogonal layers of active
fibers. Each active fiber uses an LED as the emitter and a PIN photodiode as a detector. When an object
is forced into contact with the transducer, a light distribution is detected at each detector. This light
distribution is related to the applied force and the shape of the object. Using complex algorithms and
active sensing (moving the object in relation to the transducer), the object position, orientation, size,
and contour information can be retrieved [37]. However, the recovery of the applied force profiles was
not reported in [37].
© 1999 by CRC Press LLC
An emerging technology in optical tactile sensing is the development of photoelastic transducers. When
a light ray propagates into an optically anisotropic medium, it splits into two rays that are linearly
polarized at right angles to each other and propagate at different velocities. This splitting of a ray into
two rays that have mutually perpendicular polarizations results from a physical property of crystalline
material that is called optical birefringence or simply birefringence. The direction in which light propagates
with the higher velocity is called the fast axis; and the one in which it propagates more slowly is called
the slow axis. Some optically isotropic materials — such as glass, celluloid, bakelite, and transparent
plastics in general — become birefringent when they are subjected to a stress field. The birefringent effect
lasts only during the application of loads. Thus, this phenomenon is called temporary or artificial
birefringence or, more commonly, the photoelastic phenomenon.
Figure 25.12(a) shows a photoelastic transducer proposed in [38]. It consists of a fully supported twolayer beam with a mirrored surface sandwiched in between. Normal line forces are applied to the top
surface of the beam at discrete tactels, separated by equal distances, s, along the beam. The upper
compliant layer is for the protection of the mirror, while the lower one is the photoelastic layer.
Circularly polarized monochromatic light, incident along the z-axis, illuminates the bottom surface
of the transducer. The light propagates parallel to the z-axis, passes through the photoelastic layer, and
then reflects back from the mirror. If no force is applied to the transducer, the returning light is circularly
polarized because unstressed photoelastic material is isotropic. If force is applied, stresses are induced in
the photoelastic layer, making the material birefringent. This introduces a certain phase difference
between the components of the electric field associated with the light-wave propagation. The two directions of polarization are in the plane perpendicular to the direction of propagation (in this case, the x–y
plane). As a consequence of this effect, the output light is elliptically polarized, creating a phase difference
distribution, p, between the input light ant the output light at each point in the x–y plane. The phase
difference distribution carries the information of the force distribution applied to the transducer.
A polariscope is a practical method to observe the spatial variation on light intensity (fringes) due to
the effect of induced phase difference distribution. Polariscopes can be either linear or circular, depending
on the required polarization of the light. They can also be characterized as a reflective or a transparent
type, depending on whether the photoelastic transducer reflect or transmits the light.
A circular, reflective polariscope, shown in Figure 25.12(b), is utilized to illuminate the transducer
shown in Figure 25.12(a). The input light is linearly polarized and is directed toward the photoelastic
transducer by a beam splitter. Before reaching the transducer, the light is circularly polarized by a quarterwave plate. The output light is elliptically polarized when a force is applied. This light is directed toward
a detector passing through the quarter-wave plate, the beam splitter, and an analyzer. Finally, it is detected
by a camera linked to a frame grabber connected to a PC, for further data processing. The light that
illuminates the camera consists of a set of fringes from where the force distribution applied to the
transducer must be recovered. A technique for the recovery of the forces from the fringes is described in
[38]. A model of the transducer using FEA is reported in [39].
One of the earlier applications of photoelasticity to tactile sensing dates back to the development phase
of the Utah/MIT dexterous hand [40]. The researchers proposed the use of the photoelastic phenomenon
as a transduction method for the recovery of the force profile applied to the fingers of the hand. They
limited their application to the development of a single-touch transducer, although they claimed that an
array of such devices could be implemented. However, the construction of a large array of their devices
would be difficult. To overcome this difficulty, another research group proposed a different transducer
[41]. Although an analytical model was developed for the sensor, a systematic method for recovering the
two-dimensional force profile from the light intensity distribution was not reported. Thus, the sensor
was used mainly for the study of the forward analysis, namely, observing the light intensity distribution
for different touching objects brought into contact with the sensor. This sensor could eventually be used
for determining some simple geometric properties of a touching object.
© 1999 by CRC Press LLC
FIGURE 25.12
© 1999 by CRC Press LLC
(a) Photoelastic transducer. (b) Circular reflective polariscope.
A tactile sensor reported in [42] is capable of detecting slippage. The output light intensity (the fringe
pattern) is captured by a camera interfaced to a PC. When an object moves across the surface of the
transducer, the light intensity distribution changes. A direct analysis of the fringes is used to detect
movement of the grasped object; a special technique was reported to optimize the comparison process
for detecting differences between two fringe patterns occurring due to the slippage of the object in contact
with the sensor [42]. It is important to note that such an analysis of the fringes does not require the
recovery of the applied force profile.
Photoelasticity offers several attractive properties for the development of tactile sensors: good linearity,
compatibility with vision-base sensing technologies, and high spatial resolution associated with the latter,
that could lead to the development of high-resolution tactile imagers needed for object recognition and
fine manipulation. Also, photoelastic sensors are compatible with fiber optic technology that allows
remote location of electronic processing devices and avoidance of interference problems.
Other technologies for tactile sensing include acoustic, magnetic, and microcavity vacuum sensors
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