Alciatore D.G., Histand M.B. Introduction to Mechatronics and Measurement Systems

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Figure 9.27 Biaxial stress in a long, thin-walled pressure vessel.

strain

gage

strain

gage

internal pressure

wall thickness t

9.3 Stress and Strain Measurement 401

Solving for the stress components gives

–

--------------

νε

+()=

(9.29)

1 ν

–

--------------

νε

+()=

(9.30)

For a thin-walled pressure vessel (i.e., t / r < 1/10), the stresses are approxi-

mated by

-----

=σ

-----

(9.31)

where p is the internal pressure, t is the wall thickness, and r is the radius of the ves-

sel. The stress ␴

is the transverse or hoop stress, and ␴

is the axial or longitudinal

stress. Either Equation 9.29 or 9.30 can be used to compute the pressure in the vessel

based on the strain gage measurements, yielding

tσ

-------

r(1 ν

)–

---------------------

νε

+()==

(9.32)

2tσ

----------

2tE

r(1 ν

)–

---------------------

νε

+()==

(9.33)

Either expression would yield the correct pressure value for an ideal thin-walled

vessel and error-free measurements. In this example, the strain gages are serving as

a pressure transducer.

For uniaxial and biaxial loading, we already know the directions of principal

stresses in the component; hence, we need only one or two gages, respectively, to

determine the stress magnitudes. However, when the loading is more complex or

when the geometry is more complex, which is often the case in mechanical design,

we have to use three gages mounted in three different directions as illustrated in

Figure 9.28 . This assembly of strain gages is referred to as a strain gage rosette.

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Figure 9.28 General state of planar stress on the surface of a component.

strain gage

rosette

(ε

, ε

)

Figure 9.29 Assortment of different strain gage and rosette

configurations. (Courtesy of Measurements Group Inc., Raleigh, NC)

402 CHAPTER 9 Sensors

There is a wide variety of commercially available rosettes with two or more grid

patterns accurately oriented on a single backing in close proximity. An assortment

of rosettes and single-element gages illustrating the variety of shapes and sizes is

shown in Figure 9.29 .

The most common rosette patterns for measuring a general state of planar stress

are illustrated in Figure 9.30 , where the grids are shown as single lines labeled by let-

ters. Of these, the rectangular strain gage is the most common configuration, where

the strain gages are positioned 45 ⬚ apart (see Figure 9.31 ). Figure 9.32 shows several

commercially available three-gage rosettes.

Using principles of solid mechanics, the magnitude and direction of the princi-

pal stresses can be determined directly from three simultaneous strain measurements

using any of the rosette patterns shown in Figure 9.30 . The magnitude and direction

of the principal stresses for the rectangular rosette are

max, min

---

ε+

1 ν–

---------------

1 ν+

------------

2 ε

ε–

()

2 ε

ε–

()

+±=

(9.34)

max

21 ν+()

--------------------

2 ε

ε–

()

2 ε

ε–

()

(9.35)

2θ

tan

2ε

ε––

–

-----------------------------=

(9.36)

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Figure 9.30 Most common strain gage rosette configurations.

45°

rectangular

equiangular (delta)

T-delta

maximum

principal

stress

direction

Figure 9.31 Rectangular strain gage rosette.

45° 45°

Figure 9.32 Various three-gage commercial rosettes. (Courtesy of

Measurements Group Inc., Raleigh, NC)

Three-Element 60°(Delta) Rosettes

Three-Element 45°(Rectangular) Rosettes

Stacked Type 250WR

Single-Plane Type 250UR

Single-Plane Type 250YAStacked Type 250WY

9.3 Stress and Strain Measurement 403

where ε

, ε

, and ε

are the strains in each of the rosette gages, and ␪

is the angle

from gage “a” to the direction of maximum principal stress. When using Equa-

tion 9.36 to calculate ␪

, you must use a quadrant-sensitive inverse tangent. If the

numerator is positive (ε

> (ε

⫹ ε

)/2), 2 ␪

lies in the first or second quadrant

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404 CHAPTER 9 Sensors

so 0 < ␪

<90

⬚ . Otherwise, 2 ␪

lies in the third or fourth quadrant resulting in

⫺ 90

⬚ < ␪

<0.

The relations for the equiangular (delta) rosette are

(9.37)

max

31 ν+

()

--------------------

2 ε

ε–

()

2 ε

ε–

()

2 ε

ε–

()

++=

(9.38)

2θ

tan

3 ε

–()

2ε

– ε

–

-----------------------------=

(9.39)

If the numerator in Equation 9.39 is positive (ε

> ε

), 2 ␪

lies in the first or second

quadrant so 0 < ␪

< 90

⬚ . Otherwise, 2 ␪

lies in the third or fourth quadrant resulting

in ⫺ 90

⬚ < ␪

< 0.

The relations for the T-delta rosette, which has four gages, are

(9.40)

max

21 ν+()

--------------------

ε–

()

-- -

ε–

()

(9.41)

2θ

tan

2 ε

–()

3 ε

–()

----------------------------

(9.42)

If the numerator in Equation 9.42 is positive (ε

> ε

), 2 ␪

lies in the first or second

quadrant so 0 < ␪

< 90

⬚ . Otherwise, 2 ␪

lies in the third or fourth quadrant resulting

in ⫺ 90

⬚ < ␪

< 0.

Lab Exercise 13 explores how to use a strain gage rosette and commercial strain

gage instrumentation to make strain measurements. These measurements are used to

compute corresponding stresses that can be compared to expected theoretical results.

Video Demo 9.10 shows a demonstration of the experiment, Internet Link 9.4 points

to a PDF file containing the underlying analysis with the theoretical results, and

Video Demo 9.11 discusses the results of the analysis.

max, min

---

ε+

1 ν–

--------------------------

1 ν+

------------

2 ε

ε–

()

2 ε

ε–

()

2 ε

ε–

()

++±=

max, min

---

ε+

1 ν–

---------------

1 ν+

------------

ε–

()

-- -

ε–

()

+±=

Lab Exercise

Lab 13Strain

gages

■ CLASS DISCUSSION ITEM 9.11

Strain Gage Bond Effects

Does a strain gage bonded to a component have any influence on the stresses that

are being measured? If so, how? In what cases are these effects significant?

Video Demo

9.10Strain gage

rosette experiment

9.11Strain gage

rosette experiment

analysis

discussion

Internet Lin

9.4 Strain gage

rosette experiment

analysis

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9.3 Stress and Strain Measurement 405

9.3.4 Force Measurement with Load Cells

A load cell is a sensor used to measure force. It contains an internal flexural ele-

ment, usually with several strain gages mounted on its surface. The flexural ele-

ment’s shape is designed so that the strain gage outputs can be easily related to the

applied force. The load cell is usually connected to a bridge circuit to yield a volt-

age proportional to the load. Two commercial load cells, which are used to measure

uniaxial force, are shown in Figure 9.33 . An example of the application of load cells

is in commercially available laboratory materials testing machines for measuring

forces applied to a test specimen. Load cells are also used in weight scales, and they

are sometimes included as integral parts of mechanical structures to monitor forces

in the structures.

Figure 9.33 Typical axial load cells.

(b) Courtesy of Transducer Techniques, Temecula, CA

(a) Courtesy of MTS Systems Corp., Minneapolis, MN

(continued )

Orthopedic biomechanics is a field that deals with the analysis of the loading of the skeletal

system, engineering approaches to understanding the mechanical properties of biological

tissues, and choosing appropriate systems to replace the tissues when they fail. It is a major

growth industry in the health care field and offers many opportunities to engineers interested

A Strain Gage Load Cell for an Exteriorized Skeletal Fixator

DESIGN

EXAMPLE 9.1

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406 CHAPTER 9 Sensors

in problems that have a biological or medical flavor. Many interesting engineering design

problems are associated with the choice of implantable materials, the long-term strength of

materials in the hostile environment of the body, and the specification of engineered materi-

als that must attach securely to biological tissues. Replacement joints are among the most

successful designs to date, and you probably know someone who has had a joint replaced

with a stainless steel or titanium prosthesis.

We have been involved in a number of exciting bioengineering research projects. One

of the most interesting involved the analysis of loading in the limbs in patients who had

suffered severely broken bones. When one suffers a multiple fracture in a leg bone, the

application of a simple plaster or fiberglass cast is not sufficient to allow the bone to heal.

Bone is a living tissue constantly being naturally remodeled or replaced. Interestingly, stress

is important in initiating the healing process and in maintaining healthy bones. As a case in

point, astronauts in a weightless space environment for extended periods exhibit bone mass

loss due to the lack of a gravitational stress.

With a severely broken bone, one that has been fractured into a number of pieces, an

interesting biomechanical invention is often used to help the healing process. It is known as

an exteriorized skeletal fixator. As illustrated in the following figure, it consists of a struc-

tural bar outside the body fastened to stainless steel pins that pierce and hold large bone

fragments in place until the bone heals. The engineered structure therefore carries most of

the load of the body when the animal or person walks.

knee

fractured

femur

strain gage

rosette

stainless

steel pins

stainless

steel bar

It is important to understand how the fixator is loaded while the patient walks. This

information is necessary in order to size the fixator so that it does not deform too much and

prevent bone healing. One subtle feature of the healing process is that a very small amount

of bone stress is necessary for healing. If all the stress is removed from the bone and no

relative motion occurs between the fragments, the bone will not heal. But too much relative

motion would hinder healing.

To measure the magnitude of loading for a verification study, we need to design a load

cell as part of a skeletal fixator to monitor the complex loading profiles that occur during

walking. We are interested in measuring the load profiles in the cylindrical stainless steel bar

that supports the broken leg. Bone loads during walking are fairly complex and consist of

axial stress, bending stress, and torsional stress.

Our objective is to design a load cell that allows us to easily and reliably determine the

axial, bending, and torsional loads while the subject is walking. We can do this by mounting

(continued )

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9.4 Temperature Measurement 407

three rectangular strain gage rosettes on the bar 90 ⬚ apart. The center ( b ) gage of each rosette

(see Figure 9.31 ) will be aligned with the axis of the bar.

If the principal stresses are calculated at each rosette, the axial, bending, and torsion

loads can be determined. The principal stresses in the axial direction can be used to deter-

mine the axial load and bending loads in two directions. The maximum shear stresses at each

rosette can be averaged to determine the torsional load. Thus, the nine strain gages (from three

rosettes) yield simultaneous measurements of axial load, bending load, and torsional load.

To obtain accurate measurements, the nine strain gages are now connected to strain

gage bridge circuits. A bridge console such as the Measurements Group 2100 provides up to

10 bridges for simultaneous measurement of strain. The analog outputs of the bridge console

can be digitized at an appropriate sampling frequency to yield the real-time stress profiles.

■ CLASS DISCUSSION ITEM 9.12

Sampling Rate Fixator Strain Gages

What is an appropriate sampling rate for the fixator strain gages in Design

Example 9.1 assuming a typical human walking gait?

9.4 TEMPERATURE MEASUREMENT

Because temperature is such an important variable in many engineering systems,

an engineer should be familiar with the basic methods of measuring it. Tempera-

ture sensors appear in buildings, chemical process plants, engines, transportation

vehicles, appliances, computers, and many other devices that require monitoring and

control of temperature.

Because many physical phenomena depend on temperature, we can use this

dependence to indirectly measure temperature by measuring quantities such as pres-

sure, volume, electrical resistance, and strain and then convert the values using the

physical relationship between the quantity and temperature.

The temperature scales used to express temperature are

■ Celsius ( ⬚ C): Common SI unit of relative temperature.

■ Kelvin (K): Standard SI unit of absolute thermodynamic temperature. Note the

absence of the degree symbol.

■ Fahrenheit ( ⬚ F): English system unit of relative temperature.

■ Rankine ( ⬚ R): English system unit of absolute thermodynamic temperature.

The relationships between these scales are summarized here:

= T

− 273.1

(9.43)

= (9Ⲑ

+ 32

(9.44)

= T

+ 4

(9.45)

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Figure 9.34 Bimetallic strip.

metal A

metal B

δ = δ(T, A, B)

408 CHAPTER 9 Sensors

where T

is temperature in degrees Celsius, T

is temperature in Kelvin, T

is tem-

perature in degrees Fahrenheit, and T

is temperature in degrees Rankine.

The following subsections introduce common devices used for measuring tem-

perature. They include the liquid-in-gas thermometer, bimetallic strip, resistance

temperature device, thermistor, and thermocouple.

9.4.1 Liquid-in-Glass Thermometer

A simple nonelectrical temperature-measuring device is the liquid-in-glass ther-

mometer. It typically uses alcohol or mercury as the working fluid, which expands

and contracts relative to the glass container. The upper range is usually on the order

of 600

⬚ F.

When making measurements in a liquid, the depth of immersion is impor-

tant, as it can result in different measurements. Because readings are made visually,

and there can be a meniscus at the top of the working fluid, measurements must be

made carefully and consistently

9.4.2 Bimetallic Strip

Another nonelectrical temperature-measuring device used in simple control sys-

tems is the bimetallic strip. As illustrated in Figure 9.34 , it is composed of two

or more metal layers having different coefficients of thermal expansion. The strip

can be straight, as shown in the figure, or coiled for a more compact design (e.g.,

in older thermostats such as that shown in Video Demo 9.4). Because these layers

are permanently bonded together, the structure will deform when the temperature

changes. This is due to the difference in the thermal expansions of the two metal

layers. The deflection ␦ can be related to the temperature of the strip. Bimetal-

lic strips are used in household and industrial thermostats where the mechanical

motion of the strip makes or breaks an electrical contact to turn a heating or cooling

system on or off.

9.4.3 Electrical Resistance Thermometer

A resistance temperature device (RTD) is constructed of metallic wire wound

around a ceramic or glass core and hermetically sealed. The resistance of the metal-

lic wire increases with temperature. The resistance-temperature relationship is usu-

ally approximated by the following linear expression:

1 α TT

–()+[]=

(9.46)

where T

is a reference temperature, R

is the resistance at the reference tempera-

ture, and ␣ is a calibration constant. The sensitivity (d R /d T ) is R

␣ . The reference

Video Demo

9.4Thermostat

with bimetallic

strip and mercury

switch

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Figure 9.35 Thermoelectric junction.

metal A metal B

junction

Figure 9.36 Thermocouple circuit.

+−

9.4 Temperature Measurement 409

temperature is usually the ice point of water (0 ⬚ C). The most common metal used in

RTDs is platinum because of its high melting point, resistance to oxidation, predict-

able temperature characteristics, and stable calibration values. The operating range

for a typical platinum RTD is ⫺ 220 ⬚ C to 750

⬚

C. Lower cost nickel and copper types

are also available, but they have narrower operating ranges

A thermistor is a semiconductor device, available in probes of different shapes

and sizes, whose resistance changes exponentially with temperature. Its resistance-

temperature relationship is usually expressed in the form

---

-----–

⎝⎠

⎛⎞

(9.47)

where T

is a reference temperature, R

is the resistance at the reference tempera-

ture, and ␤ is a calibration constant called the characteristic temperature of the

material. A well-calibrated thermistor can be accurate to within 0.01

⬚

C or bet-

ter, which is better than typical RTD accuracies. However, thermistors have much

narrower operating ranges than RTDs. Note in Equation 9.47 that a thermistor’s

resistance actually decreases with increasing temperature. This is very differ-

ent from metal conductors that experience increasing resistance with increasing

temperature (see Section 2.2.1).

9.4.4 Thermocouple

Two dissimilar metals in contact (see Figure 9.35 ) form a thermoelectric junction

that produces a voltage proportional to the temperature of the junction. This is

known as the Seebeck effect.

Because an electrical circuit must form a closed loop, thermoelectric junctions

occur in pairs, resulting in what is called a thermocouple. We can represent a ther-

moelectric circuit containing two junctions as illustrated in Figure 9.36 . Here we

have wires of metals A and B forming junctions at different temperatures T

and

, resulting in a potential V that can be measured. The thermocouple voltage V

depends on the metal properties of A and B and the difference between the junction

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Figure 9.37 Law of leadwire temperatures.

+−

Figure 9.38 Law of intermediate leadwire metals.

+–

410 CHAPTER 9 Sensors

temperatures T

and T

. The thermocouple voltage is directly proportional to the

junction temperature difference:

V α T

–()=

(9.48)

where ␣ is called the Seebeck coefficient. As we will see later in this section, the

relationship between voltage and temperature difference is not exactly linear. How-

ever, over a small temperature range, ␣ is nearly constant.

Secondary thermoelectric effects, known as the Peltier and Thompson effects, are

associated with current flow in the thermocouple circuit, but these are usually negli-

gible in measurement systems when compared with the Seebeck effect. However, when

the current is large in a thermocouple circuit, these other effects become significant.

The Peltier effect relates the current flow to heat flow into one junction and out of the

other. This effect forms the basis of a thermoelectric cooler (TEC) or refrigerator.

To properly design systems using thermocouples for temperature measurement,

it is necessary to understand the basic laws that govern their application. The five

basic laws of thermocouple behavior follow.

1 . Law of leadwire temperatures. The thermoelectric voltage due to two junc-

tions in a circuit consisting of two different conducting metals depends only

on the junction temperatures T

and T

. As illustrated in Figure 9.37 , the tem-

perature environment of the leads away from the junctions ( T

, T

) does not

influence the measured voltage. Therefore, we need not be concerned about

shielding the leadwires from environmental conditions.

2 . Law of intermediate leadwire metals. As illustrated in Figure 9.38 , a

third metal C introduced in the circuit constituting the thermocouple has no

influence on the resulting voltage as long as the temperatures of the two new

junctions (A-C and C-A) are the same ( T

⫽ T

). As a consequence of this law,

a voltage measurement device that creates two new junctions can be inserted

into the thermocouple circuit without altering the resulting voltage.

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