30079_29.pdf

CHAPTER 29

MEASUREMENTS

E. L. Hixson

E. A. Ripperger

University of Texas

Austin, Texas

29.3.3 Use of Normal Distribution

to Calculate the Probable

Error in X

29.1 STANDARDS AND

ACCURACY 917

29.1.1 Standards 917

29.1.2 Accuracy and Precision 918

29. 1 .3 Sensitivity or Resolution 9 1 8

29.1.4 Linearity

924

29.3.4 External Estimates

925

919

29.4 APPENDIX 928

29.4.1 Vibration Measurement 928

29.4.2 Acceleration Measurement 928

29.4.3 Shock Measurement

29.2 IMPEDANCE CONCEPTS 919

928

29.4.4 Sound Measurement

928

29.3 ERROR ANALYSIS

923

29.3.1 Introduction

923

29.3.2 Internal Estimates

923

29.1 STANDARDS AND ACCURACY

29.1.1 Standards

Measurement is the process by which a quantitative comparison is made between a standard and a

measurand. The measurand is the particular quantity of interest—the thing that is to be quantified.

The standard of comparison is of the same character as the measurand and, so far as mechanical

engineering is concerned, the standards are defined by law and maintained by the National Institute

of Science and Technology (NIST).* The four independent standards that have been defined are

length, time, mass and temperature.1 All other standards are derived from these four. Before 1960,

the standard for length was the international prototype meter, kept at Sevres, France. In 1960, the

meter was redefined as 1,650,763.73 wavelengths of krypton light. Then, in 1983, the 17th General

Conference on Weights and Measures, adopted and immediately put into effect a new standard:

"meter is the distance traveled in a vacuum by light in 1/299,792,458 seconds."2 However, there is

a copy of the international prototype meter, known as the National Prototype Meter, kept at the

National Institute of Science and Technology. Below that level there are several bars known as

National Reference Standards and below that there are the working standards. Interlaboratory stan-

dards in factories and laboratories are sent to the National Institute of Science and Technology for

comparison with the working standards. These interlaboratory standards are the ones usually available

to engineers.

Standards for the other three basic quantities have also been adopted by the National Institute of

Science and Technology and accurate measuring devices for those quantities should be calibrated

against those standards.

The standard mass is a cylinder of platinum-iridium, the international kilogram, also kept at

Sevres, France. It is the only one of the basic standards that is still established by a prototype. In

the United States, the basic unit of mass is the U.S. basic prototype kilogram No. 20. There are

working copies of this standard that are used to determine the accuracy of interlaboratory standards.

Force is not one of the fundamental quantities, but in the United States the standard unit of force is

^Formerly known as the "National Bureau of Standards."

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

the pound, defined as the gravitational attraction for a certain platinum mass at sea level and 45°

latitude.

Absolute time, or the time when some event occurred in history, is not of much interest to

engineers. They are more likely to need to measure time intervals, that is, the time between two

events. At one time the second, the basic unit for time measurements, was defined as 1/86400 of

the average period of rotation of the earth on its axis, but that is not a practical standard. The period

varies and the earth is slowing down. Consequently, a new standard based on the oscillations asso-

ciated with a certain transition within the cesium atom has been defined and adopted. The second is

now "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between

two hyperfine levels of the fundamental state of cesium 133." 3 Thus, the cesium "clock" is the basic

frequency standard, but tuning forks, crystals, electronic oscillators, and so on may be used as sec-

ondary standards. For the convenience of anyone who requires a time signal of a high order of

accuracy, the National Institute of Science and Technology broadcasts continuously time signals of

different frequencies from stations WWV, WWVB, and WWVL, located in Fort Collins, Colorado,

and WWVH, located in Hawaii. Other nations also broadcast timing signals. For details on the time

signal broadcasts, potential users should consult the National Institute of Science and Technology.4

Temperature is one of four fundamental quantities in the international measuring system. Tem-

perature is fundamentally different in nature from length, time, and mass. It is an intensive quantity,

whereas the others are extensive. Join two bodies that have the same temperature together and you

will have a larger body at that same temperature. Join two bodies that have a certain mass and you

will have one body of twice the mass of the original body. Two bodies are said to be at the same

temperature if they are in thermal equilibrium. The International Practical Temperature Scale (IPTS-

68), adopted in 1968 by the International Committee on Weights and Measurement,5 is the one now

in effect and the one with which engineers are primarily concerned. In this system, the kelvin (K) is

the basic unit of temperature. It is 1/273.16 of the temperature at the triple point of water, which is

the temperature at which the solid, liquid, and vapor phases of water exist in equilibrium. Degrees

celsius (°C) are related to degrees kelvin by the equation

t = T - 273.15

where t = degrees celsius

T = degrees kelvin

Zero celsius is the temperature established between pure ice and air-saturated pure water at normal

atmospheric pressure. The IPTS-68 established six primary fixed reference temperatures and proce-

dures for interpolating between them. These are the temperatures and procedures used for calibrating

precise temperature-measuring devices.

29.1.2 Accuracy and Precision

In measurement practice, four terms are frequently used to describe an instrument: accuracy, preci-

sion, sensitivity, and linearity. Accuracy, as applied to an instrument, is the closeness with which a

reading approaches the true value. Since there is some error in every reading, the "true value" is

never known. In the discussion of error analysis that follows later, methods of estimating the "close-

ness" with which the determination of a measured value approaches the true value will be presented.

Precision is the degree to which readings agree among themselves. If the same value is measured

many times and all the measurements agree very closely, the instrument is said to have a high degree

of precision. It may not, however, be a very accurate instrument. Accurate calibration is necessary

for accurate measurement. Measuring instruments must, for accuracy, be compared to a standard from

time to time. These will usually be laboratory or company standards, which are in turn compared

from time to time with a working standard at the National Institute of Science and Technology. This

chain can be thought of as the pedigree of the instrument, and the calibration of the instrument is

said to be traceable to NIST.

29.1.3 Sensitivity or Resolution

These two terms, as applied to a measuring instrument, refer to the smallest change in the measured

quantity to which the instrument responds. Obviously, the accuracy of an instrument will depend to

some extent on the sensitivity. If, for example, the sensitivity of a pressure transducer is one kilo-

pascal, any particular reading of the transducer has a potential error of at least one kilopascal. If the

readings expected are in the range of 100 kilopascals and a possible error of 1% is acceptable, then

the transducer with a sensitivity of one kilopascal may be acceptable, depending upon what other

sources of error may be present in the measurement. A highly sensitive instrument is difficult to use.

Therefore, an instrument with a sensitivity significantly greater than that necessary to obtain the

desired accuracy is no more desirable than one with insufficient sensitivity.

Many instruments in use today have digital readouts. For such instruments the concepts of sen-

sitivity and resolution are defined somewhat differently than they are for analog-type instruments.

For example, the resolution of a digital voltmeter depends on the "bit" specification and the voltage

range. The relationship between the two is expressed by the equation6

e - V/2n

where V = voltage range

n = number of bits

Thus, an 8-bit instrument on a one-volt scale would have a resolution of 1/256, or 0.004 volts. On

a ten-volt scale that would increase to 0.04 volts. As in analog instruments, the higher the resolution,

the more difficult it is to use the instrument, so if the choice is available, one should take the

instrument which just gives the desired resolution and no more.

29.1.4 Linearity

The calibration curve for an instrument does not have to be a straight line. However, conversion from

a scale reading to the corresponding measured value is most convenient if it can be done by multi-

plying by a constant rather than by referring to a nonlinear calibration curve, or by computing from

an equation. Consequently, instrument manufacturers generally try to produce instruments with a

linear readout, and the degree to which an instrument approaches this ideal is indicated by its "li-

nearity." Several definitions of "linearity" are used in instrument-specification practice.7 So-called

"independent linearity" is probably the most commonly used in specifications. For this definition,

the data for the instrument readout versus the input are plotted and then a "best straight line" fit is

made using the method of least squares. Linearity is then a measure of the maximum deviation of

any of the calibration points from this straight line. This deviation can be expressed as a percentage

of the actual reading or a percentage of the full scale reading. The latter is probably the most

commonly used, but it may make an instrument appear to be much more linear than it actually is.

A better specification is a combination of the two. Thus, linearity = ±A% of reading or ±B% of

full scale, whichever is greater.

Sometimes the term independent linearity is used to describe linearity limits based on actual

readings. Since both are given in terms of a fixed percentage, an instrument with A% proportional

linearity is much more accurate at low reading values than an instrument with A% independent

linearity.

It should be noted that although specifications may refer to an instrument as having A% linearity,

what is really meant is A% nonlinearity. If the linearity is specified as independent linearity, the user

of the instrument should try to minimize the error in readings by selecting a scale, if that option is

available, such that the actual reading is close to full scale. Never take a reading near the low end

of a scale if it can possibly be avoided.

For instruments that use digital processing, linearity is still an issue since the analog to digital

converter used can be nonlinear. Thus linearity specifications are still essential.

29.2 IMPEDANCE CONCEPTS7

A basic question that must be considered when any measurement is made is how the measured

quantity has been affected by the instrument used to measure it. Is the quantity the same as it would

have been had the instrument not been there? If the answer to the question is no, the effect of the

instrument is called "loading." To characterize the loading, the concepts of "stiffness" and "input

impedance" are used. At the input of each component in a measuring system there exists a variable

qtl, which is the one we are primarily concerned with in the transmission of information. At the

same point, however, there is associated with qtl another variable qi2 such that the product qtl qi2 has

the dimensions of power and represents the rate at which energy is being withdrawn from the system.

When these two quantities are identified, the generalized input impedance Zgi can be defined by

Zsi = qnlqa

(29.1)

if qn is an "effort variable." The effort variable is also sometimes called the "across variable." The

quantity qa is called the "flow variable" or "through variable."

The application of these concepts is illustrated by the example in Fig. 29.1. The output of the

linear network in blackbox (a) is the open circuit voltage EQ until the load ZL is attached across the

terminals A-B. If Thevenin's theorem is applied after the load ZL is attached, the system in Fig. 29.16

is obtained. For that system the current is given by

im = E0/[ZAB + ZJ

(29.2)

and the voltage EL across ZL is

Fig. 29.1 Application of Thevenin's theorem.

EL = imZL = E«ZL/[ZAB + ZJ

EL = E0/[l + ZAB/ZL]

(29.3)

In a measurement situation, EL would be voltage indicated by the voltmeter, ZL would be the

input impedance of the voltmeter, and ZAB would be the output impedance of the linear network. The

true output voltage, E0, has been reduced by the voltmeter, but it can be computed from the voltmeter

reading if ZAB and ZL are known. From Eq. (29.3) it is seen that the effect of the voltmeter on the

reading is minimized by making ZL as large as possible.

If the generalized input and output impedances Zgi and Zgo are defined for nonelectrical systems

as well as electrical systems, Eq. (29.3) can be generalized to

qim = qj\\ + Zgo/Zgl]

(29 A)

where qim is the measured value of the effort variable and qiu is the undisturbed value of the effort

variable. The output impedance Zgo is not always defined or easy to determine; consequently, Zgi

should be large. If it is large enough, knowing Zgo is unimportant. However, Zgo and Zgi can be

measured8 and Eq. 29.4 can be applied.

If qn is a flow variable rather than an effort variable (current is a flow variable, voltage an effort

variable), it is better to define an input admittance

ygi = 4«'4a

(29.5)

rather than the generalized input impedance

Zgi = effort variable/flow variable

The power drain of the instrument is

P = «nfe = A'Ysi

(29.6)

Hence, to minimize power drain, Ygi must be large. For an electrical circuit

/„ = 1J[\ + Y0/Yi

(29.7)

where Im = measured current

4 = actual current

Y0 = output admittance of the circuit

Yf = input admittance of the meter

When the power drain is zero, as in structures in equilibrium—as, for example, when deflection

is to be measured—the concepts of impedance and admittance are replaced with the concepts of

"static stiffness" and "static compliance." Consider the idealized structure in Fig. 29.2.

To measure the force in member K2, an elastic link with a spring constant Km is inserted in series

with K2. This link would undergo a deformation proportional to the force in K2. If the link is very

soft in comparison with Kl, no force can be transmitted to K2. On the other hand, if the link is very

stiff, it does not affect the force in K2 but it will not provide a very good measure of the force. The

measured variable is an effort variable and in general when it is measured, it is altered somewhat. To

apply the impedance concept a flow variable whose product with the effort variable gives power is

selected. Thus,

flow variable = power/effort variable

Mechanical impedance is then defined as force divided by velocity, or

Z = force/velocity

This is the equivalent of electrical impedance. However, if the static mechanical impedance is cal-

culated for the application of a constant force, the impossible result

Z = force/0 = o°

is obtained.

This difficulty is overcome if energy rather than power is used in defining the variable associated

with the measured variable. In that case, the static mechanical impedance becomes the "stiffness"

and

stiffness = Sg = effort/J flow dt

In structures,

Sg = effort variable/displacement

When these changes are made the same formulas used for calculating the error caused by the loading

of an instrument in terms of impedances can be used for structures by inserting S for Z. Thus

qim = qJQ + W

<29-8)

where qim — measured value of the effort variable

qiu = undisturbed value of the effort variable

Sgo = static output stiffness of the measured system

Sgi = static stiffness of the measuring system

For an elastic-force-measuring device such as a load cell, Sgi is the spring constant Km. As an

example, consider the problem of measuring the reactive force at the end of a propped cantilever

beam, as in Fig. 29.3.

According to Eq. 29.8, the force indicated by the load cell will be

Fm = FJ(\ + Sgo/Sgl)

Sgi = Km and Sgo = 3£//L3

The latter is obtained by noting that the deflection at the tip of a tip-loaded cantilever is given by

Fig. 29.2 Idealized elastic structure.

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