AN ABSTRACT OF THE THESIS OF

HARLAN BRADFORD SMITH for the MASTER OF SCIENCE(Name) (Degree)

in MECHANICAL ENGINEERING presented on(Major)

3uht 741971(Dater)

Title: DESIGN CONSIDERATIONS FOnSTRAIN GAGE TRANSDUCERS

Abstract approved: Redacted for PrivacyIHansp. Dahlke

Design considerations are presented which apply to all types of

electrical resistance strain gage transducers. While the concepts

apply to mass produced transducers, the information should be most

useful for the engineer or technician who designs one-of-a-kind or

"home-made" transducers.

Errors which affect the accuracy of strain gage transducers are

presented together with the usual techniques of eliminating or minimiz-

ing these errors. Procedures are detailed for temperature compen-

sation and for modulus compensation. References are given which

discuss each of the errors or problems in detail.

Nomographs are presented which calculate strain levels,

maximum excitation, and output signal for torsion, axial force, beam,

ring, and diaphragm type transducers. Five transducer examples

illustrate the use of the nomographs.

Design Considerations for StrainGage Transducers

by

Harlan Bradford Smith

A THESIS

submitted to

Oregon State University

in partial fulfillment ofthe requirements for the

degree of

Master of Science

June 1972

APPROVED:

Redacted for Privacy

Assiociatl Professor of Mechanical Engineeringin charge of major

Redacted for Privacy

Head of De rtmjntof Mechanical and Nuclear Engineering

Redacted for Privacy

Dean of Graduate School

Date thesis is presented Juht 1) 1971

Typed by Mary Jo Stratton for Harlan Bradford Smith

TABLE OF CONTENTS

INTRODUCTION

Page

1

MEASURED PARAMETER VS. STRAININ ELASTIC ELEMENT 4

Configuration 4Deviations from Simple Elastic Theory 4Material Selection 4Temperature Problems 5

Machining 5

Supporting Structure 6

Alignment 6

Natural Frequency 6

Trans ient Measurements 7

Modification of Measured Parameter 7

STRAIN IN ELASTIC ELEMENT VS.STRAIN IN GAGES 9

Stiffening Effect 9

Gage Misalignment 9

Gage Thickness 9

Gage Size 10Adhesives 10

STRAIN IN GAGES VS. RESISTANCE CHANGES 11

Gage Factor Tolerance 11

Temperature Changes, Gage Heating 11

Pressure Sensitivity 13Compensation for Temperature Induced

Zero Shift 14Modulus Compensation 16

GAGE RESISTANCE CHANGES VS. SIGNALAT READOUT 20

Transducer Output 20Switches and Slip Rings 20Lead Wire 21

Page

Moisture 22

Noise 22

CALIBRATION: NATURE OF ERRORS 23

TRANSDUCER EXAMPLES 24

Cantilever Bending 24

Axial Force 26

Square Ring 29Diaphragm Pressure Transducer 29

Torsion 31

CONCLUSION 33

BIBLIOGRAPHY 34

APPENDICES 36

LIST OF FIGURES

Figure Page

1 Temperature compensation. 15

2 Modulus compensation. 17

3 Standardization resistors. 19

4 Strain gage anemometer. 25

5 Axial force pressure gage. 27

6 Square ring. 30

7 Diaphragm pressure transducer. 30

8 Torsion transducer. 32

9 Index to nomographs. 44

10 Beam and ring constants. 45

11 Moment of inertia nomograph. 46

12 Beam and ring bending nomograph. 47

13 Natural frequency of transducers. 48

14 Axial force transducers. 49

15 Maximum bridge excitation. 50

16 Output signal vs. strain. 51

17 Natural frequency of diaphragm pressuretransducers with fixed edges. 52

18 Diaphragm pressure transducersensitivity (fixed edges). 53

19 Torsion transducer sensitivity anddeflection. 54

Figure Page

20 Torsional properties of squaresand circles. 55

21 Torsional frequency. 56

Table

1

2

LIST OF TABLES

List of symbols.

pals

37

Properties of some transducermaterials. 43

DESIGN CONSIDERATIONS FOR STRAIN GAGE TRANSDUCERS

INTRODUCTION

Electrical resistance strain gages are in common use today to

determine stresses on structural members. The ease with which

strain gages can be attached to metals has led to their widespread use

in industry. Also, refinement of readout equipment has made oscillo-

scopes and strain indicators readily available.

The need to measure parameters such as acceleration, displace-

ment, force, pressure and torque often leads to the development of

special transducers. The cost of a transducer is usually much less

than its readout equipment. Where strain gage signal conditioning

equipment is already available, the relative low cost and versatility of

strain gage transducers make them attractive.

A bonded strain gage transducer consists basically of an elastic

structure which will strain linearly with the parameter being mea-

sured, and strain gages attached to the structure which convert strain

to electrical signal.

The design of the elastic structure involves deflections, stiff-

ness, forces, and natural frequency of the structure. Nomograph

solutions are presented in the appendix for some common configura-

tions.

Values computed from nomographs have limited accuracy due to

2

the problems of printing, straight edge manipulation and drafting. To

insure dimensional accuracy, these nomographs were drawn by a

computer plotter. The computer program is included in the appendix.

However, extreme accuracy is not needed in most calculations.

Roark (12, p. 58) states:

No calculated value of stress, strength, or deformation canbe regarded as exact. The formulas used are based on certainassumptions as to properties of materials, regularity of form,and boundary conditions that are only approximately true, andthey are derived by mathematical procedures that often involvefurther approximations. In general, therefore, great precisionin numerical work is not justified.

The following presents some of the problems which have been

encountered by strain gage transducer manufacturers. Their common

methods of avoiding or minimizing problems are presented. Refer-

ences are given which treat each of the problems in greater detail.

However, the design of readout equipment is beyond the scope of this

thesis.

3

GENERAL CONCEPTS

A strain gage transducer converts some measured parameter to

a resistance change. A structure is designed to deform elastically

when subjected to some physical quantity. Strain gages mounted on

the elastic element are elongated (or compressed) by strain in the

metal. The gages change resistance with elongation. A voltage change

accompanies the resistance change. The voltage change is measured by

a suitable readout device; an oscilloscope, a strain gage indicator,

a galvanometer, etc. Errors can affect the transmission of informa-

tion at any of these locations in a strain gage transducer system.

A good strain gage transducer must respond only to the measured

parameter and not to spurious inputs such as temperature, humidity,

side load, or vibration. In general, it must have low hysteresis,

good linearity, good repeatability and low zero shift.'

1 See glossary of terms (p. 40) for definition of technical terms.

4

MEASURED PARAMETER VS. STRAININ ELASTIC ELEMENT

Factors affecting the relationship between measured parameter

and strain in an elastic element include:

Configuration

The single, most important relationship between strain and

measured parameter is the configuration of the transducer. The

effects of dimensions, type of transducer, and material properties are

presented in nomographs in the appendix for numerous simple trans-

ducer designs. The nomographs are illustrated by examples later in

the text. For transducers not covered in the appendix, see references

such as Roark's Formulas for Stress and Strain (12).

Deviations from Simple Elastic Theory

Deviations from the simple elastic theory usually cause only

small non-linearity. Shortening of cantilevers as they deflect,

stretching of diaphragms and other non-linearities are limited by

limiting deflection. See nomographs Figures 10 and 18.

Material Selection

Material properties such as Young's modulus, modulus of

rigidity and Poisson's ratio vary with heat treatment, alloying

5

proportions, and cold work to name just a few factors. Stein says that

handbook values will vary 5% (13, p. 447).

Stein discusses the metallurgical problems associated with

transducers. Unless a transducer must have precise zero return or

meet other special requirements, one of the metals in Table 2 will be

adequate (13, p. 410).

Temperature Problems

Two additional problems with most materials are the variation

of the modulus of elasticity with temperature and the expansion of

metals with increasing temperature. These problems will be discussed

under Temperature Changes, Gage Heating.

Machining

Machining tolerances and dimensional uncertainties will affect

sensitivity of a transducer. For instance, machining a 1 /8 inch by

1/8 inch cross section to 0.124 by 0.124 inch will increase strain

2-1/2% when the transducer is used in bending as a load cell.

Residual stresses caused by rolling, drawing, stretching or

very large overloads will cause hysteresis (noncoincidence of loading

and unloading curves). Stein says:

Exercise works out these localized stresses, whereas astress relieving operation allows the stresses to work them-selves into uniformity. A hundred cycles or so are all that

6

need to be carried out, but the loads imposed should be 50%higher than normal (10, p. 417).

Supporting Structure

Imperfections of the supporting structure will, at best, change

only the sensitivity from the calculated value. Pinned bearings,

clamped beams or bolted joints cause mechanical hysteresis or friction.

For precision transducers, Stein recommends welding, brazing, the

use of flexures and "fancy, 3-dimensional machining from a solid

piece. . " (13, p. 419).

Alignment

Additional problems arise in the application of transducers.

Misalignment of transducers with the direction of load usually causes

only small errors since transducer load varies with the cosine of the

misalignment However, large errors can result from cross axis

sensitivity, especially if forces are to be resolved into components

and the smaller of the components is to be measured (as in the case of

wind tunnel balances).

Natural Frequency

Undamped transducers such as diaphragm pressure gages and

accelerometers exhibit inaccuracies of 10% when subjected to sinusoidal

7

excitation of about 0.3 times their natural frequency (16, p. 76). When

excited at 0.1 times their natural frequency, the error is down to 1%.

The useful range can be extended to 0.4 times the natural frequency if

a suitable damping oil or electrical damping is added. However, the

addition of oil can decrease the natural frequency (16, p. 34). Dove

and Adams discuss mounting techniques and their effect on natural

frequencies of accelerometers (6, p. 482). Natural frequencies are

given in the appendix for the more common transducers.

Transient Measurements

Transient motion measurements such as wave propagation and

mechanical shock increase dynamic requirements of transducers. An

undamped transducer overshoots a square wave input by 100%. If the

natural frequency can be made high enough so that five or more cycles

of the transducer correspond to the rise time of the input (all "square"

waves have a finite rise time), the transient motion will be measured

fairly accurately. The addition of damping improves the transient

response of most transducers by removing transducer natural

frequencies from the output (8).

Modification of Measured Parameter

Any measuring system will affect the phenomenon being mea-

sured. Every measuring system transfers energy in order that

8

information can be passed.

The amount of energy drawn from the source system in theprocess of measurement must be small compared to the totalamount of energy available in the source system at the pointof disturbance (13, p. 47).

A pressure transducer requires a volume change; a force trans-

ducer requires displacement; a displacement transducer exerts a force

on the "source system." An accelerometer must have a much smaller

mass than the object to which it is attached; volume change of a

pressure transducer must be small compared to the combustion

chamber of an engine; the force required to deflect a cantilever which

measures displacement of a concrete structure must be small

compared to the forces deflecting the structure.

9

STRAIN IN ELASTIC ELEMENT VS.STRAIN IN GAGES

Elongation in the strain gage is considered representative of the

calculated strain in the metal. Factors affecting this relationship

include:

Stiffening Effect

On thin sections, especially in bending, the gages, their cement

and waterproof covering tend to carry an appreciable portion of the load,

stiffen the section, and give a low reading. For metal sections larger

than 1/8 by 1/8 inch the effect is usually small (6, p. 217-221).

Gage Misalignment

Gage misalignment is a serious problem only in the torsion

transducer where gages are applied at 45 degrees to the axis of a

shaft. A small error in gage orientation will result in rather large

sensitivity to bending or tension. Failure to mount gages on the center-

line of axial force or bending transducers results in cross axis

sensitivity, especially in narrow, deep sections.

Gage Thickness

Gages have a finite thickness and are located above the surface

of the metal. In bending of thin sections, the strain in the gages will

10

therefore be larger than the strain in the metal. This effect tends to

cancel the stiffening effect and is small for metal thickness larger

than 1 /8 inch.

Gage Size

All strain gages have a finite size. As a result, they average

strain over their length and cannot always be placed on the area of

maximum strain. The resulting loss in sensitivity can be as much

as 30%.

Adhesives

Cement creep gives the strain gage an apparent strain. Factors

affecting cement creep include temperature, moisture, curing cycle,

type of cement, and thickness of glue line. Manufacturers' recom-

mendations should be relied upon for limitations of cements, curing

method and technique (9, p. A129-131, A137). The temperature of a

precision transducer should be kept well below the curing temperature

of the adhesive.

11

STRAIN IN GAGES VS. RESISTANCE CHANGES

Gage resistance change versus elongation is the gage factor:

GF L/LAR /R

Strain gage manufacturers vary the cold work, the alloy composition,

etc. to arrive at a gage factor. Gages are mounted on a specimen and

the gage factor is determined experimentally. Some factors affecting

resistance versus elongation are:

Gage Factor Tolerance

Manufacturers depend upon similarity and quality control to

limit gage factor variations. As a result, gage factor tolerance is

usually specified about 1%. Therefore, use identical gages, from the

same package and lot number.

Temperature Changes, Gage Heating

Temperature changes will cause spurious zero shift resistance

changes due to differential expansion between gage and specimen and

due to thermal coefficient of resistivity. "Zero shift" due to tempera-

ture can usually be reduced to less than 1% of full output between 50

and 80°F by the following:

Use strain gages temperature-compensated to match the metal

12

used for the elastic element. For transducers used at temperatures

near room temperature, the normal gage compensation should be

used. It should be emphasized that over a wide temperature range,

any "temperature compensated" strain gage will show temperature

sensitivity. Manufacturers publish families of "apparent strain"

curves--readout versus temperature for zero mechanical strain. For

transducers operating at other temperature ranges (say from -80 to

-150 oF), a compensation should be selected for the elastic element

which gives the lowest apparent strain change over the desired

temperature span. Thus a gage which is "compensated for steel" at

room temperature may be the best gage to use on an aluminum

transducer used near 250°F.

Use a four-arm bridge with all four gages mounted on the

elastic element.

Subject electrically adjacent pairs of gages (i. e., gages C and

T, Figure 1) to similar temperature conditions. Maintain symmetry of

heat transfer; be sure that protective coatings are the same thickness

on all gages; eliminate thermal gradients where possible; shield a

bending transducer from thermal radiation.

Design the transducer with a high output (use high strength

elastic element). While this does nothing to reduce temperature

effects, it will make the temperature sensitivity a smaller percent of

the measured parameter (6, p. 106).

13

Because strain gages are resistors, the transducer must dissi-

pate heat. Maximum voltage which can be applied to a transducer

is limited by the heat dissipation capability to the mounting surface.

Excessive temperature differences between the strain gage grid and

the elastic element cause loss of temperature compensation, hysteresis

and cement creep. Bubbles and voids in the cement cause local "hot

spots" which degrade transducer performance (9, p. TN- 127). Figure

15 shows recommended excitation levels for gages mounted on various

surfaces (9, p. TN- 127). Stein discusses a pulsed excitation method

in which a high voltage bridge supply does not cause high gage heating.

The system also has other advantages (14).

The maximum voltage which can be supplied to a strain gage

transducer can be determined experimentally. At zero load, slowly

increase excitation voltage until a zero shift occurs. The last stable

voltage is the maximum which can be supplied (9, p. TN- 127).

The problems of drift and temperature compensation are greatly

minimized if the transducer can be dynamically tested or used in a

rapid manner.

Pressure Sensitivity

Strain gages subjected to large pressures can exhibit large

errors (the basis of the Bridgman pressure gage (7, p. 166)).

Pressure applied to the grid of a strain gage compresses the gage and

14

increases its resistance. A pressure of 100 pounds per square inch

applied to one gage of a transducer will cause a change of about 10

microstrain (10'5 inch/inch).

Compensation for TemperatureInduced Zero Shift

Wider temperature variations or requirements for better

accuracy require consideration of additional problems.

After following the recommendations under Temperature

Changes, Gage Heating, the transducer will still have some tempera-

ture sensitivity; the bridge balance point will still vary somewhat with

temperature. The following procedures usually involve a large amount

of trial and error work. They should be used only where high accuracy

is needed. They must be done before final waterproofing.

Hook up the transducer to a strain indicator and balance it.

Increase the temperature of the transducer 100 degrees (say from 40

to 140 °F); record the balance point shift. For each microstrain the

balance point increases (120 ohm gages), add 0.05 inch of No. 34

copper wire to a compressive arm of the bridge. (For 350 ohm gages,

add 0.14 inch per microstrain. ) If the balance point decreases, put

the temperature sensitive copper "resistor" in the tension arm (see

Figure 1). Then repeat the temperature test. If the balance shift is

sufficiently minimized, add a constantan resistor (RBAL) in an arm

15

10

5

050

cA.1.3,11te

veaa1.11-%

Change in reading after RGC

100

Temperature (oF)

t150

RBAL RGC 0.60" loop of#34 copper wire

120 ohm gages

Power

Figure 1. Temperature compensation.

15

16

adjacent to the copper wire to bring the bridge back into balance (3).

Modulus Compensation

Gage factor, elastic constants, and dimensions change with

temperature. To compensate for the usual increase in sensitivity with

temperature due to these factors, a resistor is placed in the power

leads to decrease current to the Wheatstone bridge. Average values

of the compensating resistor have been tabulated by strain gage manu-

facturers and range from 2 to 50 ohms.

To determine the value needed for the compensating resistor, it

is first necessary to calibrate the transducer at various temperatures.

The transducer must be calibrated in the same manner as it will be

loaded in use. In calibration for modulus compensation, the spring

constant of the loading structure is just as important as the spring

constant of the transducer. Stein says, "System responses to force

governed and displacement governed loading systems are entirely

different!" (15, p. 465). Opposite extremes in spring constant of the

loading structure are dead weights with almost zero spring constant

and gage blocks with almost infinite spring constant.

Load the transducer at various temperatures to obtain the curve

in Figure 2. (If the procedure in Compensation for Temperature

Induced Zero Shift were followed, the readout should not change for

zero load. ) Place a nickel resistor in the power supply diagonal of the

10

102101

10O "0.."4 9kO Cl.)

k0

-I

0 10

0

-1- 2- 3

1000/0 fa" toad

65 Fahrenheit degrees

`No load

s-30 60 90 120Temperature (0F)

RBAL

150

RGC

RMOD/2 RMOD/2

Power

Figure 2. Modulus compensation.

180

17

18

bridge circuit. To find its resistance value, find the sensitivity

change over a 65°F span. Multiply the sensitivity change by four times

the bridge resistance. Thus in Figure 2, the bridge resistance is

350 ohms, add 35 ohms or each RMOD/2 equals 17.5 ohms. After

adding the resistors, repeat the procedure to check remaining

temperature sensitivity. 1 (The sensitivity of this transducer will be

lowered 10% as a result of the resistor. )

The purpose of using two identical resistors (RMOD/2) is to

maintain symmetry and to simplify electrical calibration (see

Calibration, Nature of Errors). Once the RMOD/2 resistors have been

added to a transducer, it is imperative that the proper leads always be

attached to the power supply.

Two additional resistors which are not temperature sensitive are

sometimes put in transducers to standardize the sensitivity and the

input resistance. RSEN in Figure 3 reduces the sensitivity of the

transducer to some standard value. To avoid changing the modulus

temperature compensation, it must be very small compared to the

bridge resistance. RES reduces the input resistance to a standard

value, it must be very large compared to the bridge resistance.

1 This is a simplified procedure which assumes that the bridge resis-tance is large compared to the internal resistance of the power supply,and that the temperature sensitivity of nickel (26% over 65 Fahrenheitdegrees) is large compared to the temperature-sensitivity of thebridge. For a more exact analysis, see (11).

19

Figure 3. Standardization resistors.

These two resistors should only be used to standardize a family of

transducers and are not needed in "one of a kind" transducers (15,

18).

20

GAGE RESISTANCE CHANGES VS.SIGNAL AT READOUT

The signal measured at the readout apparatus can be modified

from the resistance signal at the compensated bridge. Several factors

are:

Transducer Output

The transducer output, expressed in millivolts per volt, is

proportional to strain in the metal, the gage factor, and the number of

effective arms. The nomograph in Figure 16 illustrates this relation-

ship. For an unbalance measuring system, the reading is proportional

to the supply voltage. The millivolt output is the important design

consideration, and applying a high voltage to the transducer is just as

important as the millivolt/volt sensitivity. (For the maximum voltage

which can be supplied, see Figure 15.) For null-balance measuring

systems, the voltage which can be supplied is only of secondary

importance; the millivolt/volt sensitivity is the design criteria (10).

The supply voltage of null balance systems should be checked to

prevent overheating (see Temperature Changes, Gage Heating. )

Switches and Slip Rings

Switches and slip rings in strain gage circuits are sources of

randomly varying resistance. If switches or slip rings must be used,

21

they should be placed in the leads, outside the Wheatstone bridge

circuit. Other techniques are available for further minimizing slip

ring and contact errors (5, p. 689; 6, p. 117).

Lead Wire

Lead wire resistances cause additional problems. Adding

resistance in the form of long wires to the power leads reduces the

sensitivity of a transducer. The problem can be minimized by using

short, low resistance cables, high resistance gages, and by always

using the same length cables. Cables subjected to unusually high

pressure or mechanical strain behave as strain gages themselves and

contribute errors (13, p. 170).

Lead wire resistance usually increases with temperature. While

the use of high resistance gages, short cables, and full bridges

minimize the problem, extreme temperature variations may require

the use of special wire whose resistance changes little with tempera-

ture (such as constantan).

Large capacitance between long leads may cause capacitive

balancing problems in alternating current bridges. Many strain

indicators and oscilloscopes have built-in capacitance balance or

capacitance compensation features (6, p. 150).

22

Moisture

Moisture causes erratic variations in resistances, producing

changes which will be interpreted as strain. Underwater use of trans-

ducers requires special consideration of leakage paths. Strain gage

manufacturers publish recommended coatings for various environ-

ments (9, p. A134).

Permeable cables which allow moisture to penetrate the outer

cover should be avoided. Even though a cable may be specified water-

proof for ordinary use, it may not be suitable for use with strain gage

transducers. Leakage of cables can be checked by measuring open

circuit resistance between leads with a Megger. If the resistance

falls when several loops of cable (not the ends) are immersed in water,

the cable should not be used. Special provision should be made to

waterproof the ends of cables; moisture can travel down the cable and

change resistance values (19).

Noise

Electrostatic and magnetic noise can interfere with strain

signals. Stein discusses experimental noise hunting techniques and

discusses methods of reducing or avoiding them (13, p. 220-h).

23

CALIBRATION; NATURE OF ERRORS

Many of the preceding errors and problems are of a constant

nature. Variations in material properties, location of gages, machin-

ing inaccuracies, rigidity of supports, and gage factor variations will

change the sensitivity of the transducer from that which is calculated

by as much as 50%. Differences of 5 to 15% are common. These

differences will not change the sensitivity with time. Therefore, when

an overall system calibration is performed, relating readout change to

primary parameter, the strain gage transducer can easily be made

accurate within 1% of full scale (2).

"Electrical calibration" of transducers is a method of relating

readout of an electrical device to a calibration resistor. It does nothing

to insure that the measured parameter will actually cause such a

resistance change. It cannot be performed in place of overall system

calibration. This calibration method most often involves producing a

known resistance change by means of parallel resistors, temporarily

placed across one or more gages of a Wheatstone bridge (6, p. 99).

In using electrical calibration with strain gage transducers, there are

numerous problems, especially when the transducer has RMOD, RES

and RSEN (Figure 3) resistors added (1).

24

TRANSDUCER EXAMPLES

Nomographs were constructed with the aid of a computer plotter

for some common types of strain gage transducers. The examples

which follow illustrate their use.

Table 1 (Appendix I) is a list of variables used in the nomo-

graphs. Figure 9 explains which nomographs to use for each type of

transducer. For example, axial-force-transducer strain, and deflec-

tion, Y, are found from nomograph Figure 14. The natural frequency

is found from Figure 13. The maximum bridge excitation and the out-

put signal are found from Figures 15 and 16 respectively. The nomo-

graphs can be used to find any one of the unknowns in the equations

they represent. The diaphragm pressure sensor nomograph, Figure

18, can be used to find STRAIN if the material and dimensions are

known; if the material and STRAIN are known and thickness is given,

radius can be found.

Cantilever Bending

The strain gage anemometer (Figure 4) is a roughened sphere

mounted on a cantilever beam. The cantilever is made of aluminum

and its width is equal to its thickness since both components of wind

direction are to be measured. Determine length, L, deflection, Y,

and natural frequency.

25

From Table 2, use 7075 T-6 aluminum with E = 1.05 x 107 and

design to only 1000 microstrain in the metal. From Figure 10 a

cantilever has a strain sensitivity constant of 0.333 and a deflection

constant of 3.0. Therefore, "KE" is 3.15 x 10 7 psi.

Enter Moment of Inertia nomograph, Figure 11, with B= 0.4 inch

and H = 0.4 inch; determine I = 2.1 x 10-3 inches 4,

Enter Beam and Ring Bending nomograph, Figure 12, with:

KE = 3.15 x 107 psiFORCE = 10 pounds

I = 2.1 x 10-3 inchesSTRAIN = 1000 microstrain

H = 0.4 inchK2 = 0,333

Find Y = 0.22 in; L = 11.2 in.

Wind force-10 to +10

lbs.

0.4"Gages 1 & 2(gages 3 & 4

opposite)

4

Roughened sphere 2.6 lb. mass

L

Figure 4. Strain gage anemometer.

26

Next, determine the natural frequency. The total mass is

determined from Figure 10 as the concentrated mass plus 0.246 times

the mass of the beam. The mass of the beam is 0.24 pounds. The

total mass is therefore 2. 66 pounds mass. (The mass of the beam was

negligible compared to the sphere. ) Enter Natural Frequency of

Transducers, Figure 13, with Force = 10 poundsY = 0.22 inch

Total mass = 2.66 pounds mass

and determine frequency of 13 cycles per second.

Axial Force

An axial force or a P/A transducer is used to measure pressure

fluctuations of 1000 psi. The transducer (Figure 5) is machined from

one piece of Beryllium copper. The force varies from 0 to 1230

pounds.

Determine:

1. Strain in the metal

2. Deflection of the transducer

3. Natural frequency

4. Maximum voltage which can be applied to the bridge

5. Output signal when this voltage is applied to the bridge

Enter Axial Force nomograph, Figure 14, with:

A = 0.12 square inchesFORCE = 1230 pounds

E = 18.5 x 106 psi (from Table 2)L = 2.0 inches

1230 lb

00.--4

00N

Axial gages

Output signal

Poisson's gages

1.25 inch diameter

"0" ring slots

Poisson's gages

Axial gages

0. 3"

0.4"

Lead wires

....-----

27

Material: Berylliumcopper

Gages: 350 OHM2. 10 gage

factorO. 187 by O. 187

grid

Figure 5. Axial force pressure gage.

28

Determine strain of 550 microstrain (note that this is smaller than the

allowable 2150 microstrain from Table 2 since all dimensions and

loads were fixed). Also, find deflection of 0.0011 inch.

The "head" or the upper portion of the transducer has a mass

of 0.37 pounds. The necked portion or the elastic element has a mass

of 0.074 pounds. From Figure 10, the total mass is W + 0.333 M or

0.395 pounds.

Enter Natural Frequency of Transducers (Figure 13) and find a

natural frequency of 6000 cycles per second (the example is not

shown on the nomograph to avoid confusion. Also, the "PIVOT" must

be extended).

The maximum voltage which can be applied to the bridge is

determined from Figure 15. Use a value of 10 watts per square inch

and 350 ohm gages and determine bridge excitation voltage of 21 volts.

The Poisson's ratio of Beryllium-Copper is 0.24 (Table 2).

Therefore, the number of active arms is 2.48. 1 Enter Output Signal

nomograph, Figure 16, with 2.48 active arms, 550 microstrain, and

gage factor of 2.10 and find 0.73 millivolts per volt. Thus the output

signal is 0.73 millivolts /volt x 21 volts or 15.3 millivolts.

1 A Wheatstone bridge with four gages arranged to measure both tensionand compression as in the bending transducers will have four activearms. In this example, each Poisson's gage lengthens 0.24 times thecalculated strain, each axial gage shortens according to the calcu-lated strain. Thus the number of active arms is 2 + 2 (Poisson'sRatio).

29

Square Ring

The section of square tubing is used (Figure 6) to weigh loads

lifted by a shop crane. Determine the maximum load which can be

lifted and the relative deflection of the two shackles. Assume that the

square tube is low strength steel, not one of the steels in Table 2,

with a maximum allowable strain of only 600 microstrain.

From Figure 10, K = 24 and K2 = 0.236. The Elastic modulus of

steel is approximately 30 x 106 psi, so KE is 7. 2 x 108 psi. From the

Moment of Inertia nomograph, Figure 11, the moment of inertia is

9 x 10-3. Enter Beam and Ring Bending nomograph, Figure 12, with

KE = 7.2 x 108 psiL = 5.6 inches (approximately)

K2 = O. 2361 9 x 1 0 -3 inches 4

H = 0.375 inchSTRAIN = 600 microstrain

And find FORCE = 900 poundsY = O. 025 inch

Diaphragm. Pressure Transducer

A diaphragm type pressure gage is made of phosphor-bronze

(Figure 7) (6, p. 388). Determine the strain at 10 psi and the natural

frequency. Check the deflection to insure linearity.

Enter Pressure Transducer nomographs (Figures 17 and 18)

with

End shackles

B = 2.00 inches

Compressive gages (C)

Tensile gages (T)

Figure 6. Square ring.

E = 16 x 106

Density = 0.32 lb/cu. in.

P= 0 to 10 psi

L 1.85" ..1

30

= 0.045"

Figure 7. Diaphragm pressure transducer.

31

E= 16 x 106 psiP = 10.0 poundsR = 1.85 inchesZ = 0.045 inch

DENSITY = 0.32 pounds/cu. in.

Find frequency of 930 cps, strain of 400 microstrain and deflection

of 0.014 inches. (The deflection of 0.3 times the thickness causes a

non-linearity of about 0.7 percent. )

Tors ion

A square steel shaft is used to measure torque between a motor

armature and a large flywheel (Figure 8). The motor develops 500

inch pounds upon starting. Determine shaft size if strain in the steel

is to be limited to 1000 microstrain. Also determine the torsional

deflection and the natural frequency.

In Figure 19, connect G of 1.06 x 10 7 with TORQUE of 500 to

determine a point on pivot. Connect STRAIN and pivot to determine

Q of 0.026 inches cubed.

To find the size of the shaft, in Figure 20, connect "square"

and Q to find D of 0.5 inches. J is then 0.01 inches 4.

Then re-entering Figure 19 with J = 0.01 inches4 and L = 6

inches, determine THETA = 1.7 degrees or 0.030 radians.

To find the natural frequency of the armature-shaft, enter

Torsional Frequency nomograph, Figure 21, with

32

Motor armatureWeight: 15 lbs.Radius of gyration: 2.5 inches

Figure 8. Torsion transducer.

THETA = 0.030 radianTORQUE = 500 inch pounds

RADIUS OF GYRATION = 2.5 inchesMASS AT END OF SHAFT = 15 pound mass

Then determine FREQ = 40 cycles per second. (The polar moment of

the armature is assumed to be much lower than that of the flywheel. )

33

CONCLUSION

Numerous authors have presented both experimental and

theoretical work related to the measurement of strain on structures.

Others have discussed one particular aspect of strain gage transducer

design. Some (Stein) have presented extensive theoretical work on

strain gage transducers.

The experimental work of strain measurements on structures has

been modified and adapted to apply to strain gage transducers. The

theoretical work pertaining to strain gage transducers has been pre-

sented in a simplified, easily understood manner. Nomographic

solutions for strain, frequency, signal, and input voltage are rapid

and simple.

It is hoped that this information will be useful to the technician

who is familiar with gage application techniques but not with stress

analysis formulae and their manipulation. It is also hoped that the

engineer who designs "hurry up" transducers will benefit. For the

transducer designer, the nomographs should help pick preliminary

configurations and dimensions.

34

BIBLIOGRAPHY

1. Allegany Instrument Company, Cumberland, Maryland.Electrical system calibration.

2. . Load cell calibration.

3. Bean strain gage seminar, session 4, transducer design. In:W. T. Bean Catalog, Detroit, Michigan, W. T. Bean, Inc.1969.

4. Calibration of strain gage transducers with modulus compensatingresistors. Strain Gage Readings Vol. 1, No. 2:7-12. June-July, 1958.

5. Doebelin, Ernest 0. Measurement systems: application anddesign. New York, McGraw-Hill, 1966. 743 p.

6. Dove, Richard C. and Paul H. Adams. Experimental stressanalysis and motion measurement. Columbus, Ohio, CharlesE. Merrill Books, 1964. 515 p.

7. Holman, J. P. Experimental methods for engineers. NewYork, McGraw-Hill, 1966. 412 p.

8. Kearns, R. W. Velocities and displacements associated withtransient response of accelerometers. Paper No. 534.Society of Experimental Stress Analysis. May, 1961.

9. Micro-Measurements Catalog and Technical Data Binder.Romulus, Michigan, Micro-Measurements, 1970.

10. Millivolt or millivolt /volt - -the race for output or should it besensitivity? Strain Gage Readings Vol. 1, No. 4:43-45. Oct-Nov, 1958.

11. Modulus temperature compensation. Strain Gage Readings Vol.1, No. 4:3-6. Oct-Nov, 1958.

12. Roark, Raymond J. Formulas for stress and strain. 4th ed.New York, McGraw-Hill, 1965. 432 p.

13. Stein, Peter K. Measurement engineering. Vol. 1. 4th ed.Phoenix, Arizona, Stein Engineering Services, 1964. 745 p.

35

14. Stein, Peter K. Pulsing strain gage circuits. Instruments andControl Systems, Vol. 38, No. 2:128-134. Feb, 1965.

15. Strain gage bridge systems, null balance, unbalance, referencebridges. Strain Gage Readings Vol. 1, No. 6:15-22. Feb-Mar,1959.

16. Thomson, William T. Vibration theory and applications. 3ded. Englewood Cliffs, New Jersey, Prentice-Hall, 1965. 384 p.

17. Transducers, Inc., Santa Fe Springs, California. Load cells.

18. Varying the sensitivity of strain gages. Strain Gage ReadingsVol. 1, No. 5:23-25. Dec. 1958-Jan. 1959.

19. Whitehead, Bob. Strain gage workshop. Thunderbird MotorInn, Portland, Oregon. Feb. 17, 1971.

APPENDICES

36

APPENDIX I

Table of Symbols

37

Table 1. List of symbols.

A Cross sectional area, square inches

B Width of beam, inches

C Compressive strain gage

D Diameter or width of torsional cross section, inches

E Modulus of elasticity, psi

FORCE Load on transducer, pounds

FREQ Natural frequency of transducer and load, cycles persecond or cycles per minute

G Modulus of rigidity, psi

H Height or thickness of beam, inches

I Moment of inertia, inches to the fourth

J Polar moment of inertia, inches to the fourth

K Deflection constant in the equation:

Y K(E)(I)(FORCE) L 3

K2 Constant in equation:

2.0 (K2) L2 (Strain) = Y H

L Length or characteristic dimension of a transducer,inches

M Mass of elastic element, pound mass

P Pressure difference across a diaphragm, psi

Q Constant relating TORQUE to shear stress:

shear stress =

(Continued on next page)

TORQUEQ

38

Table 1. (Continued)

R

RBAL

RES

RGC

RMOD

RSEN

S

STRAIN

Radius of pressure transducer, inches

Constantan resistor to bring Wheatstone bridge intobalance

Parallel resistor which reduces input resistance toa standard

Compensating resistor for temperature induced zeroshift

Compensating resistor which lowers sensitivity withincreasing temperature

Series resistor which reduces sensitivity of a trans-ducer to a standard value

Section modulus, inches cubed

Strain in metal, microinches per inchALL

T Strain gage in tension

THETA Angle of twist of torsional transducer, degrees orradius

TORQUE Twisting moment on torsional transducer, inch pounds

TOTAL MASS Total mass used to find natural frequency. The sumof W and a portion of M (see Figure 10)

W Concentrated mass on a transducer, pound mass

Y Deflection of transducer, inches

Z Thickness of pressure diaphragm, thickness of thinwalled tube, inches

39

APPENDIX II

Glossary of Terms

40

APPENDIX II

Glossary of Terms

Accuracy - Ratio of error to full scale output (usually expressed inpercent)

Ambient Conditions - Conditions of pressure, temperature, humidityof the medium surrounding a transducer

Calibration - A test procedure in which known values of measuredparameters are applied to a transducer and correspond-ing output readings are recorded

Compensation - Provision of a supplementary device or specialmaterial to counteract known sources of error

Drift - Inability of a transducer to hold a constant output over someinterval of time

Elastic Element - The portion of a transducer which strains uni-formly as the measured parameter is applied

Error - Difference between the indicated value and the true value of themeasured parameter

Gage Factor - The ratio of the relative change in resistance to therelative change in length of a strain gage:

A R /RA L /L

Hysteresis - The maximum difference in output at any value ofmeasured parameter when approached first from increas-ing then from decreasing measured parameter

Linearity - The closeness of a calibration curve to a specified straightline

Measured Parameter - A physical quantity, property or conditionwhich is measured

Modulus of Elasticity - Ratio of elastic stress to axial strain in atensile test (Young's Modulus)

41

Modulus of Rigidity - Ratio of elastic shear stress to strain angle inshear (Shear Modulus)

Moment of Inertia - The second moment of area. The integral ofY2 dA where Y is the distance from the neutral axis. Fora rectangle,

BH312

Natural Frequency - Frequency of free vibration of a system.Frequency of measured parameter at which output becomesmuch larger than measured parameter.

Output - The electrical quantity produced by a transducer which is afunction of measured parameter

Poisson's Ratio - Ratio of lateral strain to axial strain in a tensiletest

Repeatability - Ability to reproduce output values when the samemeasured parameter is applied from the same direction

Resolution - Smallest change in measured parameter which producesa detectable change in output

Self Heating - Internal heating of a transducer due to electricalheating by strain gages

Sensitivity - Ratio of change in output to change in measuredparameter

Strain - Ratio of increment in gage length to the gage length:LL

Zero Return - Difference in output at zero load, before and afterapplication of 100 percent of measured parameter

Zero Shift - An error characterized by a parallel displacement of theentire calibration curve.

Reference: (17)

42

APPENDIX III

Transducer Design

TABLE II

PROPERTIES OF SOME TRANSDUCER METALS

MATERIAL

ELASTICMODULUS

E

PSI

MODULUSOF RIGIDITY

G

PSI

POISSON'SRATIO

ELASTICLIMIT,

KSI

STATICSTRAINDESIGNLIMIT

MICROSTRAIN

FATIGUESTRAINDESIGN

LIMITMICROSTRAIN

COEFFOF

THERMALEXPANSION

PER °FSTEELS (BHN 400+)

SAE 4340 6x10-6410 SS 2.9x107 1.06x107 0.30 Above 90 3,000 1,500 TORDS TOOL STEEL

9x106ARMCO 17-4 PHSS

ALUMINUM ALLOYS(BHN 130+)2024 T-81 62014 T-6 1.05x107 4x10 0.33 Above 40 3,800 1,900 13x10-6

7075 T-6X-2020

BERYLLIUM-COPPERBERYLCO 25 HT 1.85x107 7.5x106 0.24 130 7,000 2,150 9x10-6

BEAM & RING BENDING AXIAL FORCE TORSION PRESSURE

MATERIAL PROPERTIES: SEE TABLE II, AND HANDBOOK VALUESLENGTH, DIAM., WEIGHT, GEOMETRY: FROM SPACE CONSIDERATIONS

MOMENTOF INERTIA

FIGURE 11

BEAM & RINGCONSTANTS

FIGURE 10

3:4

45

TORSIONALPROPERTIES

FIGURE 20

0-n

4O

0mzcn

BEAM & RINGBENDING

FIGURE 12

AXIAL FORCETRANSDUCER

FIGURE 14

[TORSIONSENSITIVITY

FIGURE 19

DIAPHRAGMPRESSURE

SENSITIVITY

FIGURE 18

z

4

NATURAL FREQUENCY OFTRANSDUCERS

FIGURE 13

TORS ONALFREQUENCY

FIGURE 21

NATURALFREQUENCYDIAGHRAGM

PRESSUREFIGURE 17

OUTPUT SIGNAL VS. STRAIN FIGURE 16

MAXIMUM BRIDGE EXCITATION FIGURE 15

FIGURE 9 INDEX TO NOMOGRAPHS

ITEMTOTALMASS

DEFLCON'ST., K

STRAIN SENSCONST., K2

L FORCE

W+0.236M 3.0 0.333

T

aOFICE

W

In'

1%;

W+0.264M 12.0 0.167111".--t.--L

T2

FORCE

yy,

W+0.264M 192.0

.

0.0417

T1 T1

L

--

FORCE

WM

C W+0.486M 48.0 0.0833

T TL

FORCE

W

W+0.30M 53.7

0.117 FORTi

0.205 FORC & T2

T I

C

T1C I I T2 L

W+0.333M SEE NOMOGRAPH FIGURE 14

L

FORCE

W

W+0.27M

.

38.4

0.139 FORT1

0.417 FORC & T2

T1

C C

T1I T2 L

FORCE

W

M

W+0.33M 24 0.236

APPROX.

FIGURE 10

BEAM AND RING CONSTANTS

45

gim 1 limit, it furl I Illt I [If 111111 T If / flit 1111f1111j 1111111 III10-1 2 3 4 5 6 7 8 9 100 2 3 4 5

WIDTH sF BEAM. B. INCHESSTRAIN

SECTION MODULUS, S. IN HES CUBED0-6 3 45 7 10-8 345 7 10-4 348 7 10-3 3 45 7 0-2 3 48 7 10-1 3 45 7 10° 3'mut i_l1111_ I 1 1111111 I tau 11 mil 11 'lid

B

/ GAGES

I I1119 I I t 11119 I I 111119 I I II ',ft] r ItII111 1 1111111 ssui irivi r 1

10-834 6 10-734 6 10-834 6 10-834 6 10-434 6 10 4 6 10-234 6 10-134 6 103 3MOMENT OF INERTIA. I. INCHES TO THE FOURTH

f f i r i l I t ' , r r r I T I r g t i l t u f 11 11 1'111 1 f t r 11I T11 I elle mil I I ril !gil 111111,1ml10-2 2 3 4 5 7 10-1 2 3 4 5 7 10° 2

THICKNESS OF BEAM. H. INCHES

FIGURE 11

MOMENT OF INERTIA NOMOGRAPH

PRODUCT OF ELASTIC RJD DEFLECTION CONSTANT. E. PSI107 3 4 5 7 10° 3 4 5 7 109

I mt. t 111111 I °1°t 11

LOAD 0 -NSDUCER. FORCE. POUNDS10-234610- 34510' 6101 346102 34610 346 104 346

. .." I 1 1 11 1 1Intant 1 1 11uu1 t tact.' 1 1 111

OT

1 . 1 1 1 1 1 1 1 1 1 1 I I I I 1 1 1 1 1 1 9 1 1 11n11 I I I I...,7 10.'6 345 710-9 345 710-4 345 710-3 5 710-2 345 7 10-' 345 7 111"MOMENT OF INERTIA. I. INCHES THE FOURTH

PIVOT

QIMENSION L. INCHES2 3 4 5 7 10 2 3 itl 7 111 iI olitttltt 111.1.1 1 otlitl t ttltat l 1 t/t1.1 t a tlatt t t t .11.tiI I 1 1 It'll , w 1 . o,ei 1 I 1 vitt] 1 1 1 I 1 r u I , I I 1

10-9 3 as 7 10-4 3 45 7 10-3 3 45 7 10-2 3 45 7 10-' 3 45 7 10',,

3DEFLECTION OF LOAD. Y. INCHES

PIVOT

STRAIN SENSITIVITY CONSTANT.7 10-1 7 109 2

1 1 1 1 7 11 1 11 11 1 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 .t,11,1,t

THICKNESS OF7 10-2, ? 7

11115 7 1°9

1111111111 1 11111111111111111111111111111 ItIt1111111.11)Iti

PIVOTH. INCHES

1111 1 111111111111111111111111111111111111111111111 II 111111111111111112 3 4 5 7 102 2 3 4 5 7 103 2

STRAIN IN METAL. MICROSTRAIN

FIGURE 12

BEAM AND RING BENDING NOMOGRAPH

TOTAL MASS. GRAMS45 7 101 3 45 7 102 3 45 7 10" 3 45 7 104 3 45 7 105 3 4ge LI,%,$e ,we . , Ail,10-2 3 45 7 10-1 3 45 7 10° 45 7 101 345 7 102 3 45 7 103

TOTAL MASS, FOUNDS MASS

FORE IE(COIRESPONDING TO FLECTION I, PCUNDS10--34610-34610-1346100 346 01 346102 346103413104

a at and at awl a a at Natal a a I waist 1 I II taad i i iIuu1

NATURAL ,101 3 4 5 7 104 3 4 5 7 1

I II a I .% 1111a aI a I aS 1.1

PIVOT

0. RPM OR CYCLES PER MINM1,4

3 45 7 10° 3 4 5 7 1 3 4 5 7 102 3 4 5 7 103 3 4 5 7 104NATURAL EQUENCY, FREQ. CYCLES PER SEC

106'4'44 3 101-1; 44 3 101-27 3 101417 44 3 101-47 44 3 101-5;----DEFLECTION. Y. INCHES

FIGURE 13

NATURAL FREQUENCY OF TRANSDUCERS

I 1 11111111 1111 VITEFTTA WIT! I-I I p nil 1 1 1 1 1 1,11 1 11/111 1 1 1 11 1 1 11111 I1 1 1141111 1

10-'1 2 3 4 5 7 10-4 2 3 4 5 7 10 1 2 5 7 10° 2CROSS SECTIONAL AREA. A. S i ARE IN ES

LO9D ON TUNSOUCER. FORCEA POU A

10-2 343710-i 345710u 3457101 346710' 345/7 3 345714 3III tad I ill old s *II ant ilia aid kill

3

1 r 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I J 1 1 1 1 1 1 1 1 11

1 I 1 1 11 11 1 I r 1 I 1 1 1 1 1 I

106 2 3 4 5 6 7 8 9 107ELASTIC MODULUS. E. PSI

I I I Jinni ipi411111 vi 1111111mq Ispiviri I tri rg I vform' upp Iium! I I Imo thq I it'101 2 3 4 5 7 104 2 3 4 5 -7-103 2 3 4 5 7 04 2 3 4 5 7

STRESS IN METAL. PSI1 1 I 1 I 1 1111 I 1 1 /11willoirt !gyp! 1I111111rI r 11r1rrr/11r1 r.t

101 2 3 4 5 7 102 2 3 5 7 1-111' 2STRAIN IN METAL. MICRO 'AIN

I till 1 I 1 1 1 1 111 I s r 1 1 1 1 1 I 1 1 1 I 1 1 1111 1 I 11115 7 10-5 3 45 7 10-4 3 4 10-3 3 7 10-2 3 45 7 10-1 3 45 7 10o

DEFLECTIO ANGE IN NGTH3. Y. INCHES

1 I 1 1 111111 1 rllnn I 1 1 111/1111111 I 1 1 11111 1. I 11 11111 / I 1 111111111111 I I I 111111 1 AI 11111 1 1 IIII/11115 7 105 2 3 4 5 7 101 2 3 4 5 7 102 2 3 4

LENGTH OF TRANSOUCER. L. INCHES

FIGURE 14

AXIAL FORCE TRANSDUCERS

TYPICAL POWER-DENSITY LEVELS IN WATTS/SW. INCH

HEAT -SINK CONDITIONS

ACCURACY

REQUIREMENTSEXCELLENT

HEAVY ALUMINUMOR COPPERSPECIMENS

GOODTHICK STEEL

SPECIMENS

FAIRTHIN STAINLESS-

STEEL/ORTITANIUM

STATIC

DYNAMIC

HIGH 2.-5. 1.-2. .5-1.MODERATE 5.-10. 2.-5. 1.-2.

LOW 10.-20. 5.-10. 2.-5.

HIGH 5.-20. 5.-10. 2.-10.MODERATE 10.-20. 10.-20. 5.-10.

LOW 20. -50. 20.-50. 10.-20.

REFERENCE: (9, P.TN-1271

I I

10' 9 8 7 6 5 4 2 10k 9 6 4 6614 3'

GAGE ISTRNCE. OHMS

POWER DENSITY. WATTS PER 6.RE INCH10-2 3

110 4, 1I003 1 5 101

13 4 5 7 102

I

PIVOT

BRIDGE EXCITATION. VOLTS10-1 2 3 4 5 7 100 7 3 4 5 7 101 2 4 5 7 102

GRID AREA OF ONE GAGE. SQUAR NCHES3 4 5 7 10-3 2 3 4 5 7 10-2 2 3 ,, 7 10-1 2 3 4 5 7 100

ImiTHyr,wommwi.vpmkTivvmoyv TlwdoyhityN"ITYMYITI2 3 4 5 6 7 8 910-1 2 3 4 5 6 7 8 9100SQUARE GRID GAGE LENGTH OR GRID WIDTH. INCHES

FIGURE 15

MAXIMUM BRIDGE EXCITATION

1 1 1 1 1 1 I I 1 I4 5 6 7 8 9 10° 2

NUMBER OF ACTIVE ARMS

1 1 1 1 11111 I I Itl3 4

OUTPUT SIGNAL/ INPUT VOLTAGE, MI 'VOLTS PER VOLT3 5 10-1 10° 101i t_t_ti tittltitti t I ell ItItl I t sal titt3 t 1 o titittitttE I al11 t I t th tutt3 t 1 1 tLiatatiti t I s'It I a Iti

IVOT

11111111-19M1111111111/111111111111111111/11M9 1111111111111111111 1111111111/1111 1111,19,12 3 4 5 7 102 2 3 4 5 7 103 2 3 4 5 7 104

STRAIN IN METAL, M ROSTRAIN

GAGE FACTOR OF STRAIN GALES. GF°1(111 7 I 7 , 9 . i,...f 3

, . , I 11_1111 , , , i .9

1 t t ( t I L 1

10

FIGURE 16

OUTPUT SIGNAL VS. STRAIN

DENSITY OF

I

DIAPHRAGM MATERIAL. GRAMS PER C. C.101

, ril 'e ' I,' 4 1,7 . ,5 6 7 8 9 10-1 2 3 4DENSITY OF DIAPHRAGM MATERIAL. POUND R CUBIC INCH

I I r r l i r r r l 1 r 1 If T I T if III r I 1-111191I19 I I r I if fiti106 2 3 4 5 7 107 2 3 4 6 7 108

MODULUS OF E CITY. E. PSI

PIVOT

NATURAL FREQUENCY. RPM OR C ALES PER MJN1O 104 100

4 5 7 102 2 3 4 5 7q

2 3 4 5NATURAL FREQUENCY. FREQ. CYCLE FER SEC

PIVOT

I 0 '"'I I10 -2 4 5 7 10-' 3 4 5

THICKNES iF DIAPHRAGM. Z. INCHES

9 100 2 3 4 5RADIUS OF DIAPHRAGM. R. INCHES

FIGURE 17

NATURAL FREQUENCY OFDIAPHRAGM PRESSURE TRANSDUCERS

WITH FIXED EDGES

6 7 9 101

MODULUS OF ELASTIICITY. E. PSI106 7 108

a nittaac 4144.144Ni 141414111Q .41 43 .144tataalL It .414141

PRESSURE [JIFF R110-2, ?1/.7.,P-1, 1,,7Pu tq.

SS DIAPHRAGM. P. PSI101 3 7 102 39 7 103

TOOT

PIVOT

3 4q.7,,P4

APPROX STRAIN IN Mp . MICROURRIN7 101

, .7. , a 4 .7 P.aDEFLECTION OF CENTER. Y. IN HES*

6 10-7346 10-D348 10-5346 10-4346 10-3346234610-1346Intl a III tnel tinl 44441 44441 4.441 a a I

PIVOT

PIVOT

1 1 4 1 1 1 1 1 1 1 1 1 4 1 1 1 114 Itleirmi 1111911114 5 7 10 2 3 4 5 7 101RADIUS 0 IIRPHRRGM. R. INCHES

I r i s I I 4 4 4 4 1 4 1 . 4 1 4 4 4 4 1 1 1 1 1 1 1 1 I t I 1 1 14 liter lio orritioviv41114 44111444f10-2 2 3 4 5 6 7 8 9 10-1 2 3 4 5THICKNESS OF DIAPHPHPAGM . E. INCHES

FOR LINEARITY, DEFLECTION Y SHOULD BELIMITED TO ABOUT 0.3 TIMES THICKNESS Z

FIGURE 18

DIAPHRAGM PRESSURE TRANSDUCER SENSITIVITY(FIXED EDGES)

I I I i I 1 i r r T 1 Fir ri i i 1 1 i i 1 1 t 1 i 1 1 i 1 t 1 1 v 1 1 t i 1 I I 1 1 i 1 r I i 1 r i 1 i 1 v 1 1 1 1 i r , it2 3 4 6 7 106 2 3 4 5 7 X07 2

MODULUS OF RIGIDITY, G. POUNDS PER SQ. IN.Q , I NCHES CUBED

3 45 7 10-4 3 45 7 10-3 3 45 7 10-2 3 45 7 0-1 3 45I 1 I I a l t t i I i I il 1 1 1 1 11111 I 1 1 11111 1 1 1 I\

1 1111111 r 1 I 1 foul III tint r I 1 I I1nl 1 1 1 1 mil 1 1 1 111111 1 1 1 11111 / I 1111111 1 1

345710-3345710 2345710 13457106 3457101 3457102 3 57103 3457104 3TORQUE, INCH POUNDS

PIVOTU. INCHES TO THE F

10 634 6 10 634 6 10 434 6 10 334 6 10 2mil I I 1 111111 I I 1 1 11111134 6 100 34 6 101 34 6 102 3

1 111111 1 1 1 111111 1 1 1 111111 1 1 1 111111 I

STRAIN IN METAL AT 45 DEGREES. MI CR OS TRA I N4 5 6 7 8 9 102 4 5 6 7 8 903 2

I I 1111(11 I t Is I 11 I iItIt I I I limitPIVOT

LENG TRANSDUCER. L. INCHES45 7 101 3 45 7 10210-1 3 45 7II% lit( 3

ANGLE OF TWIST , T 4. I N RADIANS2 3 4 5 '7 10-3 2 3 4 5 7 10 3 4 5 7 10-1 2

1 rWitImiiiiiNIOMisPhIlwlimilWillreolviqlMIA101miliby , ieoltilsy i,s1,\IVIw lived10!= 2 3 4 5 7 10-1 2 3 4 5 7 100 2 3 4 5 7 101

ANGLE OF TWIST, THETA. IN DEGREES

FIGURE 19

TORSION TRANSDUCER SENSITIVITYAND DEFLECTION

SQUARE TUBE, Z/D RATIO

0.02 0.061 I I

0.1t 1 1 1

0.02 0.04 0.08 0.1 0.2 Iv

ROUND TUBE, Z/D RATIO CIRCLE

SOLIDSQUARE

J. INCITES TO10-634 6 10-534 6 1,0-434 6 1,0-34 6

1 1 11111111 1 11111111 1 11111111 I I 1111

FOURTH34 6 10-134 6 10° 34 6 101 34 6 102 3

I I I 111111 I I I Iliad I I 1 111111 I I 1111/11 IIII III I1111111/1111111111111/1111 1 1 11111111 1 1111 1 1 1 111111111111 11 rilivill

10-1 2 3 4 5 7 6 910° 2 3 4 5INCHESDIMENSI

1,1U-5 3

111 11 1

45 710-41 1 111111 I

345 710-511

34511111 I

710-2

H D

111111 1 I 11111111 1 1111111 1

3 45 710-1 345 710° 3 45 7101 340. INCHES CUBED

ROUND TUBE Z/D RATIO0.02 0.04 0.1

SOLIDCIRCLE

0.2

0.02 0.06 0.1

SQUARE TUBE, Z/D RATIO

FIGURE 20

TORSIONAL PROPERTIES OF SQUARES AND CIRCLES

SQUARE

I 1

21 111111111 If 1 1(111111 1 1 rrInnl 1 111j1111111111 11111111rIT1 1

3 4 5 7 10-3 2 3 4 5 7 10-2111111 r 1 11-11)11111 1 I 111j1j11 1 11 1111111

2 3 4 5 7 10-1 2THETA. TOTAL TWIST OF SHAFT. RADI NS

TORQUE (AT THETA) INCH POUNDS34 6 10-3346 10-2346 10-1346 106 34 6 101 346 102 3 6 103 346 104 3'In1 I 1 1 111111 1 1 1 111111 I 1 1111111 1 1 1 111111 1 1 1 111111 11111 1 1 1 111111 1 1

PIVOT

I 1 I

3 4 5 7 100 3 4 5 7 10NATURAL FREQ

i 1 r 1 r 1 r1 r "

3 4 5 7 10' 3 4 5 7 10'Y, FREQ. CYCLES PER SEC

3

PIVOT

US OF GYRATION OF MASS, NcHES3 101 7 3 10u 7 5 4 3 10-' 7 5 4 3

I I I ' 1 1 1 1 1 1 1 1 A 1 1 1 i 11 1 1 1 1 1 1

MASS PT END OF S" T. POUND MASS10-2 3 45 7 10-1 3 4E 7 10° 3 45 01 3 45 7 102 3 45 103I 1 1 al 11111 1 I I 111111 1 1 111111 1 I I I 1 iail 1 I I I it'll

FIGURE 21

TORSIONAL FREQUENCY

57

APPENDIX IV

Computer Program

58

// JOB 0101 01FF 1

*EOUAT(CARDZ,CRDZO)// FOR*IOCS(1403 PRINTER,CARD)*ONE WORD INTEGERS*_ISTALLC - -- X IS X COORDINATE OF STARTING POINT OF NOMOGRAPH

IS Y COORDINATE OF STARTING POINT OF NOMOGRAPHC--- AXLN LENGTH OF AXIS BETWEEN FIRS AND LASTC--- CYLN LENGTH OF ONE CYCLE (THE SUBRO. LOGAX NEEDS ONE OR THE OTHER)C--- FIRS NUMERICAL VALUE OF FIRST NUMBER ON SCALE.C--- LAST NUMERICAL VALUE OF LAST NUMBER ON SCALE.C--- NCR NEG FOR TITLE BELOW AXIS. MAGNITUDE GREATER THAN 998 ENDSC--- NOMOGRAPH. VALUE GREATER THAN 9998 ENDS RUN.

DIMENSION NLMN(40),PIP(3),ICARD(80)DATA IP /'PI','VD','T '/CALL PLOT (20,-8.9-3)CALL PLOT (0.02.5.-3)

5 NCR=+1READ (2,10) NCR.X.Y.AXLN,CYLN,FIRS,VLAST

10 FORMAT (110,4F8.3.F15.0,F23.0)IF (FIRS) 110.110.20

20 CALL PREADREAD (2.100) ICARD

100 FORMAT (80A1)N=LCHNE(ICARD,1,80,16448)N=N/2+1READ (2,101) (NLMN(I),I=1,N)

101 FORMAT (40A2)NCH=3000+2*NINTER=ISIGN(NCH.NCR)CALL LOGAX (X.Y*NLMN.INTERoAXLN.O.O.FIRS.VLAST,CYLN)GO TO 200

110 CALL PLOT (X.Y.3)XX=X+AXLNCALL PLOT (XX.Y.2)XX=X+AXLN*0.5-0.4YY=Y-0.07+SIGN(0.11.FLOAT(NCR))CALL SYMB (XX,YY00.14.1P.0.0,3006)

200 CONTINUEIF (IABS(NCR)-998) 5.210.210

210 IF (CYLN) 215.215.217215 XXX=X+AXLN+9.0

GO TO 219217 XXX=X+9.0+CYLN*0.43429*ABS(ALOG(VLAST/FIRS))219 CALL PLOT (XXX.0.0.-3)

IF (IABS(NCR) 9998) 5,220.220220 CALL EXIT

END/1 DUP*DELETE NOMOG 01FF*STORECI WS UA NOMOG 01FF

// JOB 0101 01FF B(7)=7.// FOR B(8)=8.*ONE WORD INTEGERS 8(9)=9.*LISTALL B(10)=10.

SUBROUTINE LOGAx(xPAGE.YPAGE.I8CD.NCHAR.AxISL.THETA.FIRST.vLAST. N8=10+CYCLN) TL(1)=1.1

C--- xPAGE.YPAGE-COORDINATES OF STARTING POINT. INCHES TL(2)=I.2C--- IBCD -AXIS TITLE TL(3)=1.3C--- NCHAR -NO. OF CHAR IN TITLE + TIC ON C.C.. - TIC ON C.. TL(4)=1.4C--- AXISL -AXIS LENGTH IN INCHES. TL(5)=1.6C--- CYCLN -LENGTH OF ONE CYCLE IN INCHES. SPECIFY EITHER AXISL OR TL(6)=I.7C - -- -CYCLN TL(71=1.8C--- THETA -ANGLE OF AXIS FROM x-DIRECTION--DEGREES COUNTER C. TL(8)=1.9C--- FIRST.VLAST -FIRST AND LAST VALUES ON LOG AXIS TL(9)=2.2C--- NOTE THAT IF VLAST IS LESS THAN FIRST. THE SCALE IS DRAWN IN THEC--- OPPOSITE DIRECTION. TL(10)=2.4

TL(111=2.6

DIMENSION IBCD(2). NO(11). B(20). TL(50)TL(121=2.8TL(131=3.2THEN=0.0 TL(14)=3.4

CYLEN=CYCLN TL)151=3.6AXLEN=AXISL TL(161=3.8IF (VLAST-FIRST) 1.2.2 TL(17)=4.2

I THEN=180.0 TL(181=4.4FONEY=FIRST TL(191=4.6FIRST=VLAST TL(201=4.8VLAST=FONEY TL(21)=5.5

2 ANGLE=THETA+THEN TL(221=6.53 IF (CYLEN) 6.4.6 TL(231=7.54 CyLEN= AXLEN/(0.4342944*ALOG(VLAST/FIRST)) TL(24)=8.56 AxLEN= CYLENw(0.4342944.ALOG(VLAST/FIRST)) TL(25)=9.5

BRANCH TO VARIOUS DATA BASED ON CYLENNL=25GO TO 75

IF( CYLEN-4.0) 45.45.70 70 NO(I)=245 IF( CYLEN-2.01 50.50.65 NO(2)=350 IF( CYLEN-1.0) 55.55.60 NO(3)=455 NO(I)=3 NO(4)=5

NO(21=4 NO(5)=6NO(3)=6 NO(6)=7NO(4)=10 NO(71=8N=4 NO(8)=9B(I)=10. NO(9)=10N8=1 N=9TL(1)=2. 8(1)=1.5TL(2)=3. 8(2)=2.TL(3)=4. B(3)=2.5TL(4)=5. 8(4)=3.TL(51=6. 8(5)=4.TL(6)=7. B(6)=5.TL(7)=8. 8(7)=6.TL(8)=9. B(8)=7.NL=8 B(9)=8.GO TO 75 8(10)=9.,

60 NO(1)=3 B(111=10.NO(2)=4 N8=11NO(3)=5 TL(1)=1.1NO(4)=7 TL(2)=1.2NO (5)=10 TL(3)=1.3N=5 TL(4)=1.48(1)=5. TL(5)=1.58(2)=10. TL(6)=1.6NB=2 TL(7)=1.7TL(1)=2. TL(B)=1.8TL(2)=3. TL(9)=I.9TL(3)=4. TL(101=2.1TL(4)=6. TL(111=2.2TL(5)=7. TL(12)=2.3TL(6)=8. TL(131=2.4TL(7)=9. TL(14) =2.6NL=7 TL(15)=2.7GO TO 75 TL(16)=2.8

65 NO(1)=2 TL(171=2.9NO(2)=3 TL(18)=3.2NO(3)=4 TL(191=3.4NO(4)=5 TL(201=3.6NO(5)=7 TL(211=3.8NO(6)=10 TL(221=4.2N=6 TL(23)=4.48(1)=1.5 TL(24)=4.6B(2)=2. TL(251=4.8B(3)=3. TL(26)=5.58(4)=4. TL(27)=6.50(5)=5. TL(281=7.5B(6)=6. TL(29)=8.5

60

TL(30)=9.5NL=30

75 CONTINUEC---

3031

FIND

DO 30 1=1.40IF (IFIX(1000.*( FIRST*10.**(1-20)-1.))) 30,30,31CONTINUEFIRS=FIRST*10.**(I-20)NFST=21I

FIRS AND VLAS

IF (IFIX(1000.*( FIRSFLOAT(N0(1))))) 32,33,3332 NFST=20I33 CONTINUE

DO 35 J=1,40IF (IFIX(1000.*( VLAST*10.**(J-20)-1.))) 35.35.36

35 CONTINUE36 VLAS=VLAST*10.**(J-20)

FIND NOFT + F AND NOLT + VDO 80 K=1,NNNN=NK+1IF (IFIX(1000. *( FLOAT(NO(NNN))FIRS)))78,78,80

78 NOFT=NO(NNN)F=CYLEN*0.4343 *ALOG(FIRS/FLOAT(NOFT))GOTO 82

80 CONTINUENOFT=NO(N)F=CYLEN*0.4343 *ALOG(FIRS)

82 CONTINUEDO 90 K=1.NIF (IFIX(I000. *( FLOAT(NO(K))VLAS))) 90,92,92

90 CONTINUE92 NOLT=NO(K)

V=CYLEN*0.4343 *ALOG(FLOAT(NOLT)/VLAS)K=NCHAR/1ABS(NCHAR)H= FLOAT(K)*(1.THEN/90.0)NCHAR=IABS(NCHAR)STH=ANGLE*0.0174533CTH=COS(STH)STH=SIN(STH)XFT=XPAGEF*CTHYFT=YPAGEF*STHXLT=XPAGE+(AXLEN+V)*CTHYLT=YPAGE+(AXLEN+V)*STH

C - --C---

WRITE OUT NUMBERS FOR BIG TICS

KV=196 DO 100 ITIS=1.N

IF (IF1X(1000.*( 10.**(KV-1)*FLOAT(NO(ITIS))FLOAT(NOFT))))+100.98.98

98 R=CYLEN*0.4342944*ALOG(FLOAT(NO(ITIS))/FLOAT(NOFT))+CYLEN*(KV-1.0)FPN=FLOAT(NO(ITIS))IF(IFIX(1000.*(RVFAXLEN))) 99.99,120

99 HT=FLOAT(IFIX(FLOAT(NO(ITIS))/9.9))*0.035+0.105HD =-0.04*(1.THEN/90.0)+(1.THEN/30.)*(-0.05*FLOAT(IFIX(FLOAT(

+NO(ITIS))/9.9)))VH=-0.07*(1.THEN/90.0)+0.19*HXXX=HD*CTHVH*STH+R*CTH+XFTYYY=VH*CTH+HD*STH+R*STH+YFTCALL NUMB (XXX.YYY,HT,FPN,ANGLE+THEN,-1)

100 CONTINUEHD=(1.THEN/180.)*(0.20)VH=0.19*HXX=HD*CTHVH*STH+R*CTH+XFTYY=VH*CTH+HD*STH+R*STH+YFTFPN=FLOAT(NFST+KV-1)CALL NUMB(XX ,YY 90.105.FPN,ANGLE+THEN,-1)KV=KV+1GO TO 96

120 CONTINUEC--- PLOT THE BIG TIC MARKS

CALL PLOT (XLT,YLT.+3)VK=1.0

140 DO 150 ITE=1.NBITLL=NB+1ITE1F(IFIX(1000.*(10.**(1.VK)*B(ITLL )FLOAT(NOLT)))) 145.145,150

145 R=F+V+AXLEN+CYLEN*0.4342944*ALOG(B(ITLL)/FLOAT(NOLT))CYLEN*(VK-1.+0)XX=XFT+R*CTHYY=YFT+R*STHIF(IFIX(1000.*R)) 160,146,146

146 CALL PLOT( XX,YYs+2)

61

CALL PLOT( XX-0.1*H*STH,YY+0.1*H*CTH,+2)CALL PLOT( XX,YY,+2)

150 CONTINUEVK=VK+1.0GO TO 140

160 CONTINUE

C - --LABEL THE AXIS

R=AXLEN*0.5-0.070*(1.-THFN/90.)*(FLOAT(NCHAR)-1000.*(IFIX(FLOAT(N+CHAR)/1000.)))VH=-0.060+0.380*H+0.15*THEN/180.0ANGLE=ANGLE+THENXX=XPAGE+R*CTH-VH*STHYY=YPAGE+R*STH+VH*CTHCALL SYMB(XX,YY, 0.14.1BCD,ANGLE,NCHAR)

C--- FINALLY, PLOT LITTLE TIC MARKSCALL PLOT (XLT,YLT.+3)VK=1.0IF(NL-1) 190.190,170

170 DO 180 ITE=1,NLITLL=NL+1-ITE1F(IFIX(1000.*(10.**(1.0-VK)*TL(ITLL)-FLOAT(NOLT)))) 175,175,180

175 R=F+V+AXLEN+CYLEN*0.4342944*ALOG(TL(ITLL)/FLOAT(NOLT))-CYLEN*(VK-+1.0)XX=XFT+R*CTHYY=YFT+R*STHIF(IFIX(1000.*R)) 190,176,176

176 CALL PLOT (XX,YY,+2)CALL PLOT (XX-0.060*H*STH.YY+0.060*H*CTH,+2)CALL PLOT (XX,YY,+2)

180 CONTINUEVK=VK+1.0GO TO 170

190 CONTINUERETURNEND

// DUP*DELETE LOGAX 01FF*STORE WS UA LOGAX 01FF