Chapter 2 - Measurement Errors

Basic Dimensional Metrology

Chapter 2. Measurement Errors 2.1 Introduction The main objective of the quality control department in a manufacturing company is to inspect the manufactured parts and ensure that they are made within the tolerance limits specified in the drawing or a blueprint. Close tolerance will enable the parts to be assembled without difficulty to build the final product. Precision measurement plays an important role in the process of design and manufacture of the product by ensuring that all the parts are manufactured within the design specifications and tolerances required. The accuracy of measurement needed for a part depends mainly on the function of the part. The accuracy can be affected by errors present in the measuring instrument as well as in the measurement process. The sources of all these errors must be understood and steps must be taken to minimize or eliminate the errors so that high accuracy measurement can be achieved. 2.2 Errors in measurement The main objective of carrying out measurement is to enable someone to make a decision about a product, for example, whether to accept the product, reject the product completely or send it back for rework. This type of measurement service will not be complete if the measurement is not carried out to the desired level of accuracy. However, no measurement can be said to be exact due to the limited accuracy of all measuring instruments. Therefore, in any measurement we not only have to state the dimension measured but also mention the accuracy with which the measurement has been carried out. For instance, if a shaft of nominal diameter 30 mm is measured using a micrometer of accuracy ±0.01 mm and the micrometer shows a reading of 30.2 mm, then the measurement should be expressed as 30.2 ± 0.01 mm. The errors in measurement should be minimized as far as possible and the magnitude of error still present in the measurement should be stated, e.g. Measured diameter = 30.2 ± 0.01 mm Error in measurement = − 0.02 mm Actual diameter = (30.2 + 0.02) ± 0.01 mm = 30.4 ± 0.01 mm Errors in measurement can generally be divided into two categories: (a) Controllable errors and (c) random errors. Controllable errors are errors that can be eliminated by careful measurement. Such errors can also be removed by correcting the measurement so that the effect of the error can be cancelled. There are several types of controllable errors, such as catastrophic errors, alignment errors, errors due to ambient condition and errors due to elastic deformation. These errors can be reduced by taking the necessary precautions during the measurement. Random errors are errors that are contained within the measurement process itself and are difficult to be removed. Examples of random errors are scale errors, reading errors, random fluctuations in readings due to condition of part surface etc. Because random errors cannot be eliminated, statistical methods are necessary to analyze these errors and obtain meaningful results. 2.3. Types of controllable errors 2.3.1 Catastrophic errors Catastrophic errors are large magnitude errors that result from errors in reading an instrument. For instance, a reading displayed as 5.78 mm may be misread as 5.28 mm. Such errors could also result from careless mistakes in calculations or simply by pressing the wrong button on a calculator. 8

Mani Maran Ratnam (2009)


Catastrophic error can be easily detected when the measurement is repeated and new readings are compared with the other readings. For instance, assume that a measurement process results in the following dimensions on a component: 20.2 mm, 20.1 mm, 19.9 mm, 20.1 mm, 23.0 mm and 20.0 mm. In this set of measurement it is obvious that the reading having the value 23.0 mm is significantly different from the others and this could be due to errors made by the instrument reader. This reading can, therefore, be removed from the set of measurements before further analysis is carried out, such as determining the average dimension. 2.3.2 Alignment error Alignment error occurs because the instrument is wrongly aligned relative to the workpiece or part being measured. An example of a situation that causes alignment error in the use of a dial indicator is shown in Figure 2.1. The dimension of interest is D, but due to misalignment the dial indicator gives a reading L which is larger than D. The error e in the measurement due to the misalignment is, therefore, Error , e = L − D = L − L cos θ = L (1 − cos θ )

(2.1)

For example, if the angle due to misalignment θ = 10° and the dial indicator shows a reading on 4.52 mm, then the error e = 4.52(1 – cos10°) = 0.069 mm. Since this type of error is a function of the cosine of the angle of misalignment, hence it is also known as cosine error. θ Dial indicator

a

c

L θ D b a

c

D b

Figure 2.1. Misalignment error.

2.3.3 Calibration error The actual dimensions of standards used for calibration, such as block gages, angle blocks, length bars etc., are usually slightly different from the nominal dimensions. For instance, a 30 mm nominal size block gage may have an actual dimension of 30.015 mm. If the deviation of the actual dimension from the nominal dimension is known, corrections can be made to the measurement carried out using the standard or an instrument calibrated using the standard. These deviations can be obtained from calibrations certificates that are supplied with the standards.

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2.3.4 Error due to ambient condition Changes in the ambient conditions during measurement from those of international standard conditions, i.e. temperature of 20°C, barometric pressure of 760 mmHg and humidity of 10 mmHg, can cause errors in measurement. Among these factors, temperature has the most serious effect on the measurement taken. Thus, changes in temperature from the international standard temperature must be taken into account in most precision measurements. The change in the dimension measured can be caused by temperature changes in both the part being measured and the instrument used, and is given by

δL = L (α 1δt 1 − α 2δt 2 )

(2.2)

where, L is the dimension measured, t1 is the temperature of part being measured, t2 is the temperature of measuring instrument, α1 is the thermal expansion coefficient of part material, α2 is the thermal expansion coefficient of instrument material, δt1 = t1 − 20°C δt2 = t2 − 20°C

Contraction (µm)

From eq.(2.2) we can see that in order to eliminate error δL in measurement both the part and instrument temperatures must be 20°C. In this case, δt1 = δt2 = 0 and, therefore, δL = 0. This requires the part and the instrument to be kept in a controlled laboratory maintained at 20°C. However, if the temperature is not 20°C but is at some arbitrary but equal value, e.g. t1=t2=t, then δL = 0 if α1 = α2. This implies that both the part and instrument must be made from the same material since they have the same expansion coefficient. In practice, the part and the instrument are usually made from different materials and their temperatures are difficult to measure. Neither is it possible to maintain their temperatures at exactly 20°C. Thus, the most effective way of overcoming error due to the temperature changes is to leave both the part and instrument to achieve stable ambient temperature. Then, the effect of the deviation of the actual temperature from 20°C is taken into account in the evaluating the error. As an example the effect of temperature on a block gage as a function of time after being handled is shown in Figure 2.2. The figure shows that although the time has exceeded 30 minutes, the dimension of the block gage has not returned to its original value. 2.0

0.25

30

15

Time (min.)

Figure 2.2. Cooling curve for a block gage.

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2.3.5 Error due to elastic deformation When an elastic body is subjected to a force it will undergo deformation. The amount of deformation will depend mainly on the magnitude of the force and mechanical property of the material. When an instrument that uses a stylus for measurement is brought into contact with the part being measured, the pressure due to the stylus will cause the part to deform. This deformation will introduce an error in the reading. Consider, for example, a hollow cylinder whose diameter is being measured using a dial indicator as shown in Figure 2.3. Pressure at the tip of the stylus will cause deformation of the surface of the cylinder and a small deformation at the stylus itself.

Stylus

Deformation of cylinder Deformation of stylus

Cylinder

Figure 2.3 Error due to elastic deformation. The total deformation δ is given by sum of deformation of cylinder and deformation of stylus, and is given by the following equation, 1

2 3

δ = 1.774W (k1 + k

2 2 3

)

 1 1 3  +   R1 R 2 

(2.3)

where, R1 = radius of stylus, R2 = radius cylinder, W = pressure at the stylus, (1 −ν 12 ) k1 = E1 k2 =

(1 −ν ) 2 2

E2

where E1 and E2 are the modulus of Young for the stylus and cylinder material and ν1 and ν2 are the corresponding Poisson’s ratios. Another form of deformation occurs when a body deforms due to its own weight, known as dead weight. For a beam the error caused by this type of deformation can be minimized by placing supports at position shown in Figure 2.4. Under this condition, the deflection of the beam, and hence the error, is minimum.

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L

Beam

Support 0.577 L

Figure 2.4. Deflection of beam due to dead weight.

2.3.6 Parallax error Parallax errors occur in the use of instruments like dial gages where there is a gap between the scale and the pointer and the reading is not taken normal to the scale. An example of parallax error when reading a dial gage is shown in Figure 2.5(a). Figure 2.5(b) shows an example of parallax error that can occur when measuring distances using a simple scale rule. Parallax errors can be overcome by taking the readings normal to the scale or by using an instrument whose pointer needle is at the same plane as the scale. A Direction reading is taken

Normal direction

B

θ

Pointer

L

Scale Reading taken

Parallax error

Actual reading (b) (a)

Figure 2.5. Examples of parallax errors. 2.4 Random errors The random fluctuation of readings during repeated measurement of the same quantity is due to an error known as random error. The presence of random errors can only be detected by repeating the measurement a number of times. Statistical parameters, such as mean and standard deviation, are used to assess the random errors. For a series of reading x1, x2, x3 ..... xn, the arithmetic mean x is defined as: x=

x 1 + x 2 + x 3 + ....... + x n n

(2.4)

and the standard deviation σ is given by

12



(x − x )

2

σ =±

(2.5)

n −1

If the measurement of a value x is repeated a large number of times, we will get slightly different values due to the random errors. A frequency distribution of the measurement can be obtained by plotting the measured value against the number of times each measurement occurs or number of readings that fall within a range. The distribution will take the familiar bell-shaped (Gaussian) profile shown in Figure 2.6 where the variable z is given by z=

x −µ

(2.6)

σ

and the Gaussian function is given by f (x ) =

2 1 e −z σ 2π

2

(2.7)

The function in eq.(2.7) and shown plotted in Figure 2.6 using σ=2 is also known as the probability density function. An important property of the curve shown in Figure 2.6 is that 99.8% of the data lies within ±3σ of the mean value. For instance, if a sample of 2000 parts, say nails, is taken and their diameters measured, we can say with 99.8% level of confidence that the diameter will lie within ±3σ of the mean diameter, where σ is the standard deviation of the 2000 samples. Similarly, 95% of the data lie within ±2σ of the mean. f(z) 0.25

0.2

0.15

0.1

0.05

z

0 -5

-4

-3

-2

-1

0

1

2

3

4

5

Figure 2.6. Gaussion curve for σ=2. The uncertainty in measurement caused by random errors must be related to the manufacturing tolerance of a product. As a rule of thumb, the accuracy of the measuring instrument must be within 10% of the manufacturing tolerance of the part. For instance, if the manufacturing tolerance on the diameter of a product is ±0.1 mm, then the measuring instrument used to measure the diameter must have a minimum accuracy of ±0.01 mm. If σp dan σm are the standard deviation of the manufacturing process and standard deviation of the measurement process respectively, then the relationship between σp dan σm will determine number of products that will be rejected or accepted by mistake at both ends of the tolerance limits. Assume that the nominal diameter is 20.0 mm, then the maximum and minimum diameters acceptable are 20.1 mm and 19.9 mm. At the 13



upper limit of 20.1 mm if the measuring instruments shows a reading of 20.12 mm, the product will be rejected although the reading may be due to the deviation of the measurement process. If the standard deviation of the measurement process (σm) is large compared to the standard deviation of the manufacturing process (σp) more products will be rejected by mistake at the upper limit and more will be accepted at the lower limit.

Manufacturing process distribution

Measurement process distribution

(6σm)

Manufacturing tolerance (6σp)

Figure 2.7. Relationship between manufacturing process distribution and measurement process distribution.

2.5 Compound errors There are many situations where the measured quantities are used to determine another variable. For instance the diameter and length of a rod are measured to determine its volume. The errors in the measurement of diameter and length will contribute to error in the volume. In general, if a quantity M is a function of several quantities, say a, b, c etc. and the errors in the measurement of each of these quantities are δa, δb, δc etc., then the error δM in M is given by

δM =

where

∂M ∂M ∂M δa + δb + δc ∂a ∂b ∂c

∂M the partial derivative of M with respect to a ∂a ∂M the partial derivative of M with respect to b and so on. ∂b

The error δM is known as compound error. Exercise 2.3 illustrates a problem where the compound error is to be determined.

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Revision Exercises Exercise 2.1 Figure below shows an arrangement for measuring the dimension L on a block using a dial gage. If the dial gage gives a reading of 5.212 mm, calculate the error in the measurement. Given that angle θ = 5°. Dial gage

θ L block

Exercise 2.2 The diameter of an aluminium rod is measured using a micrometer in a laboratory where the temperature is 25°C. The reading shown by the micrometer is 12.54 mm. Given that the coefficient of thermal expansions of aluminium is 23×10-6/°C and that of the micrometer material is 11.7 ×10-6/°C determine the error in the measurement and the true diameter of the rod.

Exercise 2.3 The volume V of a cylinder is given by the expression: V = πr 2 h , where r is the radius and h is the height of the cylinder. If r = 50 mm and h = 200 mm, and the errors in the measurement of r and h are, respectively, ± 0.5 mm and ± 1.0 mm, determine the error in the volume V.

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Chapter 2 - Measurement Errors

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