Chapter 4 - Linear and Angular Measurement

Basic Dimensional Metrology

Mani Maran Ratnam (2009)

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Chapter 4. Linear and angular measurement 4.1 Linear Measurement

Instruments used for linear measurement can be divided broadly into two categories: non-

precision and precision instruments. Non-precision instruments normally do not have a mechanism to amplify the reading. The measurement is taken directly from the scale reading. On the other hand, precision instruments use various types of mechanisms to amplify the input so that small readings can be detected. There a number of different types of precision and non-precision instruments. Some of these are shown in Figure 4.1.

Figure 4.1. Examples of non-precision and precision instruments.

4.1.1 Non-precision instruments (a) The steel rule

The steel is the simplest and most popular measuring instrument. It is made of a narrow high-quality hardened steel strip that has graduation marks etched on its surface. The distance between the graduations need not be the same throughout the length of the rule. On one part there could be 10 divisions for every centimeter, while on another part there could be 20 divisions for each centimeter. The part of the rule chosen for measurement depends on the accuracy required. For high accuracy the part that has the larger number of division, i.e. higher resolution or discrimination, is used. The scale of a steel rule should be read to the nearest graduation without interpolation for two reasons. Firstly, any interpolation between two divisions will only give an estimate and thus the reliability of the reading will decrease. Secondly, interpolation is usually not required because a finer scale can be used for higher accuracy. When a reading falls exactly between to divisions, the reading that gives a higher degree of safety or lower cost of manufacture is taken. Interpolation is done only if there is no other instrument capable of giving the measurement directly. There are four sources of errors that can occur when using a steel rule. These errors are:

(i) Instrument error (ii) Observational error (iii) Manipulative error (iv) Bias error

Non-precision instruments Precision instruments



26

Instrument error can be eliminated by using a high quality steel rule. A high quality steel rule usually has a maximum error of one-tenth the distance between each graduation. Some high

quality steel rules have accuracies of ±0.7 mm per 2 m, ±0.9 mm per 3 m and ±1.3 mm per 5 m. Parallax error is the most serious observational error. This type of error is cause by the

shift in the position of the object due to the position of the observer (Figure 4.1(a)). The error can be overcome simply by taking the reading normal to the rule. Alternatively, the error can be eliminated if the graduated side of the rule is placed on the measurement line as shown in Figure 4.1(b).

(a) (b)

Figure 4.1. (a) Parallax error occurs when reading is taken from position A or C,

(b) method of reducing parallax error. Manipulative error occurs because of the way the steel and workpiece are held when measurement is taken. For instance, when excessive pressure is applied to the rule, deflection can occur, thus changing the actual reading (Figure 4.2(a)). Error in reading can also occur if the steel rule is not placed along the line where measurement is to be made as shown in Figure 4.2(b).

Figure 4.2. Conditions that give rise to manipulative errors. Bias error is cause by the person taking the reading who unintentionally influences the measurement made. For instance, when a series of measurement is taken and the average value is determined, normally a reading that is far from the expected value is discarded. The reading that is discarded might be a correct reading. This causes bias error.

x

A B

C

x

A B

C

measured

measured

(a) (b)



27

(b) The combination set Figure 4.3 shows a combination set which is a very useful variation of the steel rule. This instrument is widely used in tool and die making, pattern making, model and prototype making and for setup of machine tools. It comprises four main components: Blade, square head, protractor head and center head.

The square head of the combination set is used to test right angles and measure height of an object as shown in Figure 4.4(a). The protractor head is used to measure angles on a workpiece as shown in Figure 4.4(b). The center head is used to center the scale on the blade so that the diameter of cylindrical workpiece can be measured (Figure 4.4(c)). The center head can also be used to simply mark location of the center on a solid cylindrical workpiece.

Figure 4.3. (a) Main components of a combination set, (b) the Mitutoyo metric combination set.

Figure 4.4. Use of combination set for: (a) testing right angle, (b) angle measurement and (c)

measuring diameter of cylinder. (c) The spirit (bubble) level The spirit (or bubble) level is an instrument that is used to indicate whether a surface is horizontal or otherwise. There is a large variety of spirit levels used by different people, e.g. carpenters, masonry workers, machinist etc. Some spirit levels, such as one shown in Figure 4.5(a), can be mounted onto a steel rule blade. Figure 4.5(b) shows a machinist level with

Center head

Protractor head

Square head

Blade

(a) (b)

(a) (b) (c)



28

graduated vial. Some of these levels can be used to measure angle of inclination to a good degree of accuracy. Figure 4.5. (a) Spirit level mounted onto a rule blade, (b) machinist level (Source: http://www.fine-

tools.com/level.htm) 4.1.2 Precision instruments (a) Vernier instruments

The precision of a measuring instrument can be increased by adding a means of amplifying the input. One of the simplest methods of amplification is by the use of the vernier principle invented by Pierre Vernier in 1631. This principle is used in various instruments such as the vernier caliper, depth gage, height gage, gear tooth gage and the protractor. In a vernier caliper, the vernier scale is attached to the sliding jaw and moves along the main scale as shown in Figure 4.5.

Figure 4.5. (a) Parts of vernier caliper, (b) the Mitutoyo Series 530 vernier caliper (Source: www.mitutoyo.com.sg)

In one of the design metric vernier caliper, each centimeter of the main scale is divided

into 20 divisions as shown in Figure 4.6. Thus, each division of the main scale represents 1/20 cm, i.e. 0.05 cm or 0.5 mm. The vernier scale has 20 divisions in the same distance as 19

divisions of the main scale. Thus, each division of the vernier scale is equal to (1/20) × 19 × 0.5 mm, i.e. 0.475 mm.

At zero setting, i.e. when the jaws are fully closed, the first division (0 cm) of the main scale coincides with the first division of the vernier scale. When this occurs, the difference between the second divisions of the main and vernier scales is 0.5 mm – 0.475 mm = 0.025 mm.

Main scale

Vernier scale

Fixed jaw Movable jaw

(a)

(b)

(a)

(b)



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When the vernier scale is moved so that the second divisions of both scales coincide, the amount by which the vernier scale, i.e. movable jaw, moves is 0.025 mm. Thus, the smallest measurement that can be made using this instrument is 0.025 mm, i.e. the resolution of the caliper is 0.025 mm.

If the zero reading of the vernier scale falls between two divisions of the main scale, the fractional reading can be determined by seeing which of the vernier division coincides with the main scale division. For instance, the reading shown in Figure 4.7 is 2.5 mm + 6(0.025) mm = 2.650 mm. The same principle is also used for the inch scale.

Figure 4.6. Main and vernier scale of a vernier caliper

Figure 4.7. Example of measurement using a vernier caliper. The advantages of the vernier scale are:

(a) The amplification is achieved by design and does not depend on moving parts that can wear and affect accuracy of the instrument,

(b) interpolation is not necessary or possible when taking measurement, (c) the range of the scale has no physical limit; it is limited only by the length of

the scale. The disadvantages of the vernier scale are:

(a) The reliability of the measurement depends more on the person taking the reading rather than the instrument,

(b) the resolution is limited by the width of the graduation lines on the scale, (c) there are no other ways of adjusting the error but except by determining the

zero error and manually correcting the reading taken.

0 cm 1 cm

Main Scale

Vernier Scale

0.025 mm

0 10 mm

Coincide

2.5 mm



30

The use of vernier scale is not the only way of expanding the main scale. Two newer methods are by the use of the dial mechanism and digital electronics as shown in Figures 4.7(a)-(b).

Figure 4.7 (a) Dial caliper and (b) digital caliper. The advantage of the dial caliper is that less skilled users can easily take measurements because it is much easier to read compared to the vernier caliper. The main setback is that the dial mechanism can be easily damaged. The moving parts in the caliper such as gears can wear after a long time of usage and thus affect the accuracy of the measurement. The digital caliper is the easiest to read. This type of calipers operates on batteries. Besides giving digital readout they have built-in calculation functions that can perform a variety of statistical calculations and conversions between units. A special feature of electronic caliper is that its output can be transferred directly to a computer or other devices for further processing. The zero reading can also be set easily at any position of the sliding jaw (known as floating zero). (b) Micrometer instruments Micrometer instruments achieve a higher amplification compared to vernier instruments using a screw thread. Rotation of screw causes axial movement of a spindle as shown in Figure 4.8.

In one type of micrometer using metric thread, the screw has 4 threads per mm, i.e. a pitch of 0.25 mm. Thus, each rotation of the thimble moves the spindle by 0.25 mm. If the thimble is divided into 50 division, the resolution of the micrometer will be 0.25/50 = 0.005 mm. The number of divisions on the thimble can be increased by increasing its diameter, hence increasing the resolution. The ratio of diameter of thimble to displacement of the spindle gives the amplification of the micrometer.

Figure 4.8. Main parts of the micrometer.

Like the digital electronic caliper, digital micrometers are also available. Figure 4.9 shows a typical digital micrometer. Digital micrometers enable various statistical information relating to

(a) (b)

Thimble Main scale Spindle Anvil

Ratchet



31

the measurement to be made, such as maximum reading, minimum reading, average value, standard deviation and number of readings. This type of micrometers can be connected to a special printer. Micrometers, in general, have resolutions in the range 0.01 mm to 0.001 mm with

accuracies from ±0.002 mm to ±0.003 mm.

Figure 4.9. A typical digital micrometer (Source: www.tesa.com). (c) Block gages and length bars

Block gages are, strictly speaking, not measuring instruments but used with other instruments such as dial or digital indicators for measurement. Block gages are one type of end standards having lengths up to 125 mm (Figure 4.10). The smaller dimension gages are known as slip gages. End standards for dimensions from 125 mm to 1200 mm are called length bars.

Block gages are made from hardened steel that has very high resistance against wear. They are made in the form of rectangular blocks of high dimensional stability. They are made to close tolerance so that the distance between the working surfaces, i.e. length L of block, has accuracies up to 0.001 mm.

Figure 4.10. Block gages.

Both working faces of the gage block must be sufficiently flat and parallel. They must have the property of wringing, i.e. two gage blocks will stick together when pressed. This is illustrated in Figure 4.11. Wringing occurs because the thin layer of air in between the blocks are expelled with they are pressed, thus creating a vacuum. When left sufficiently long the block can become permanently bonded due to molecular attraction.

Working surfaces

Length, L

Face width

Face length



32

Figure 4.11. Wringing of gage blocks. (Source: www. mitutoyo.com)

According to BS4311:1968, there are three grades of block gages for general use, one special grade for calibration of other block gages and one additional grade for the highest accuracy. These grades are as shown in Table 4.1.

Table 4.1. Different grades of block gages.

Grade Description

2 For rough inspection in the workshop

1 Used in the making of tools, components and other gages in special inspection areas by skilled inspectors

0 Used for the inspection of other gages and high precision measurement in a controlled environment

Special grade Used only for calibration of other block gages

00 Highest grade; used only in specialized calibration laboratories

The definition of the length of block gage takes into account the thickness of the

wringing layer and is measured at room temperature of 20°C. These gages require close care. The working surfaces must be wiped clean carefully using chamois leather or lint-free cloth before and after use. The working surfaces must not be touched with finger in order to prevent scratches from appearing on these surfaces.

The use of lengths bars is similar to the use of block gages except that length bars are used for larger workpieces, typically (>150 mm). Length bars are cylindrical in shape with diameters of 22 mm. Figure 4.12 shows the diagram of a length bar.

Figure 4.12. Length standard.

Measurement using the length bar is usually carried out vertically. When used horizontally, the bar must be supported at points located at the same distance from the ends and

the distance between the supports must be 0.577 × length of bar. In this condition the value of deflection must be minimum and thus the error in measurement can be reduced.

End surface



33

(d) Comparators Comparators are instruments used to compare with high accuracy the workpiece,

workshop gages and other instruments with standard block gages. There are several designs of comparators each with its own amplification mechanism. Comparators are classified according to the following operating principles:

(i) mechanical comparator (ii) electrical comparator (iii) optical comparator (iv) pneumatic comparator

Mechanical comparators use mechanical method to amplify the small displacement of a

measuring probe. The method of amplification used in all mechanical comparators is based on lever, gear or combination of both. Mechanical comparators has amplification ratio of 300 to 5000 for each unit movement of the probe. An example of this type of comparator is the dial gage whose amplification mechanism is shown in Figure 4.13.

Figure 4.13. Internal arrangement of a dial comparator.

In Figure 4.13, the movement of the rod is transferred through the toothed rack to a set of gears (Gear 2 and Gear 3). The movement of Gear 3 is transferred to the pointer needle. A dial indicator is used to measure the change in distance and not the absolute distance. The dimension measured is compared with the standard such as a block gage as shown in Figure 4.14.

Gear 1

Gear 2

Pointer

Gear 3

Probe

Reading of dial indicator

Measured dimension

Block gage

Reference surface

Figure 4.14. Dial gage used as a comparator.



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The electrical comparator comprises a transducer to change displacement to an equivalent current. The change in current is calibrated to show to displacement. Normally, an amplifier is used to obtain very high sensitivity. The working principle of an electrical comparator based on alternating current circuit is shown in Figure 4.15.

Figure 4.15. Operating principle of the electrical comparator.

In the electrical comparator shown in Figure 4.15, the movement of the plunger changes the position of the armature and hence causes changes in the inductance of a pair of coil the forms one arm of an alternating current bridge. This causes the impedance in the bridge to change. The effect of imbalance in the bridge is measured using a micro ammeter that is calibrated to give linear displacement. This type of comparator can give an amplification of up to 30 000 times.

Another design of a transducer popularly used in the electrical comparator is the Linear Variable Differential Transformer (LVDT). This transducer converts the linear displacement into voltage signal. The general arrangement of the LVDT is shown in Figure 4.15.

Figure 4.15. Layout of the linear variable differential transformer. The LVDT consists of a primary winding (P) and two secondary windings (S1 and S2).

Both secondary windings are connected in reverse. An iron core that is connected to the moving component is placed in between the primary and secondary windings. The location of the iron core is such that it is in neutral condition and the value of magnetic flux linking the primary and both secondary are equal. Thus the voltage output from S1 and S2 are equal but have opposite signs. This results in a net zero voltage at the output of the transducer. If the core moves to the right, then the magnetic flux linking S2 will increase while that linking S1 will decrease. Therefore, the net voltage will increase, e.g. to a positive value. If the core moves to the left, the voltage output of S1

exceeds that of S2 and the net voltage becomes negative. In this manner, the displacement is changed into voltage signal.

Coil

Spring

Armature Arm

Plunger

Core

P

S 1 S 2



35

The electrical comparator has the following advantages compared to the mechanical comparator:

(i) It can maintain the accuracy over a longer period of time because there are almost no moving components.

(ii) The accuracy can be easily adjusted. (iii) It has high amplification, i.e. as high as 30 000 times

In optical comparators, the amplification is obtained with the aid of a light beam. The

advantage of using light beam is that light travels in straight line and does not have mass. As such, this type of comparator has high degree of accuracy and sensitivity. There are two main types of optical comparators, i.e. the profile projector and the mechanical-optical comparator. The profile projector is used to magnify small and complex components such as screw thread and gear. The general arrangement of the profile projector is shown in Figure 4.16.

Figure 4.16. General arrangement of the profile projector.

In the profile projector the shadow of the object is projected onto the screen. The projection lens magnifies the image up to 100 times. The object is placed onto a translation stage so that its position can be changed. Using this method, the shadow on the screen can be moved and measurement can be made at different locations of the object shadow. The scale on the screen can be used to change the readings taken to the actual dimensions of the object. The weakness of the profile projector is that it is capable of making measurements in two dimensions only. In one arrangement of the mechanical-optical comparator, as shown in Figure 4.17, a small displacement at the plunger is initially amplified using a mechanical system that comprises levers and then by an optical system. The mechanical system causes the mirror to rotate through

a small angleδθ. This causes the beam reflected from the mirror to rotate through angle of 2δθ (see Figure 4.18).

Screen

Object

Light source

Projection lens

Mirror



36

Figure 4.17. The mechanical-optical comparator.

In Figure 4.18, AO is the incident beam, OB and OB’ are the reflected beams and angle

BOB’ = θ +δθ − (θ +δθ) = 2δθ . For the system shown in Figure 4.18, mechanical amplification

= 1 × 20 = 20 units and optical amplification = 50 × 2 = 100 units. Notice that the factor 2 in

the optical amplification is caused by the angle 2δθ . The total amplification is therefore 20 × 100 = 20000 units. Thus, a movement of 1 unit of the plunger is equivalent to a movement of 2000 units on the screen.

Figure 4.18. Operating mechanism of the mechanical-optical comparators. In the pneumatic comparator, a small displacement is changed to change in pressure or air flow velocity. High amplification can be obtained and contact does not exist between the gage and the object being measured. A popular design is the back pressure type shown in Figure 4.19.

20

Light source cahaya

1

50

1

Lens

Indicator

Mirror

Object

Plunger

Screen with scale

O

B'

B

normal 2

normal 1

A



37

Figure 4.19. Pneumatic comparator.

Air at pressure Ps is directed into the probe head. After going through an orifice, the air enters into a space in which the pressure changes to Pb, known as the back pressure. When the distance d between the object surface and the end of the probe changes, the back pressure also changes. The relationship between Pb and d is shown in the figure. Pressure Pb can be measured to determine the distance d. An important condition for the proper operation of this type of pneumatic comparator is that the supply pressure Ps must be constant. If Ps varies then Pb also varies and this variation may be misinterpreted as variation in dimension d. 4.1.3 Characteristics of linear measuring instruments Linear measuring instruments must have the following characteristics:

(i) high degree of accuracy (ii) high degree of sensitivity (iii) high precision (iv) minimum inertia (v) freedom of variance

In addition to these characteristics, the instrument should obey the principle of alignment.

This principle states that the line of measurement and line of dimension must be collinear. An example of an instrument that obeys this principle is the micrometer.

Figure 4.20. Instrument that obeys the principle of alignment.

This principle is not obeyed by the vernier caliper because the measurement line is not coincident with the dimension line as shown in Figure 4.21. The slight curvature in the vernier beam can occur due to pressure applied at the caliper jaw. This makes the measurement B

To pressure measurement instrument

Supply Ps

Object

d

Pb

Pb

Ps

d

A Measurement line and dimension line



38

smaller than the dimension A being measured. Therefore, the vernier caliper has lower reliability than the micrometer.

Figure 4.21. Instrument that does not satisfy alignment principle.

The terms sensitivity, precision and accuracy were explained in Chapter 1. Sensitivity can also be stated as the ratio of displacement of stylus to the quantity being measured. If the ratio is high this means that the measurement instrument has high sensitivity. Meanwhile, precision refers to the closeness in the readings that can be obtained using the instrument. It makes no comparison with the actual value. An instrument that has high precision also has high sensitivity.

Accuracy is the relationship between the instrument reading and the true value. Accuracy is represented by the amount of correction to be made to the instrument reading to obtain the actual value. If this correction is small this means that the instrument has high accuracy. Usually proper calibration of the instrument can overcome problems with low accuracy during manufacture of the instrument. Yet, the driving mechanism in the measurement instrument can deteriorate over time and reduce its accuracy. Variance is defined as the range of change in instrument reading obtained from repeated measurement of a certain quantity. Variance usually exists in most instruments. Its effect on the readings depends on the manufacturing quality and operating principle of the instrument. Variance in simple instruments such as dial gage can be determined as follows. The dial gage is fixed onto a flat surface and measurement is carried out on block gages in increasing values. Then, the measurement is repeated for decreasing values. If the error in reading is plotted against the quantity being measured we can obtain the result shown in Figure 4.22. The readings will lie on two curves. The boundary of the curves represents the maximum deviation in reading for a particular measurement.

Figure 4.22. Graph showing variance in an instrument.

All instruments based on mechanical and pneumatic systems have the disadvantage of inertia. The inertia effect can be determined by observing the change in the quantity being measured that causes a change in the reading. If the instrument readings take some time to

Dimension A

Measurement line

Measurement A

Increasing values

-

+

Error

Maximum difference

Measured quantity

Decreasing values



39

change when the input quantity is changed, then the instrument has high inertia. This effect is also causes by improper elastic behavior in a component such as a spring. 4.2 Angular measurement 4.2.1 Sine bar

Sine bars are used to measure angles indirectly. The angle measured is obtained from the sine bar and length standard by using a sine formula that relates the two. Sine bar is made from hardened steel and has a pair of rollers placed at center distances of known dimension, usually 100 mm or 250 mm (Figure 4.23).

Figure 4.23 The sine bar.

The use of the sine bar is shown in Figure 4.24. If l is the linear distance between the roller centers and h is the height of the block gages, then

= −

l

h1sinθ (4.1)

Figure 4.24. Use of sine bar.

For high precision measurement, the following conditions must be followed during manufacture of the sine bar:

(i) The rollers must have the same diameter and must be perfectly cylindrical in shape (ii) The distance between rollers must be known to a high degree of accuracy (iii) The straight line between the roller centers must be parallel with the upper and lower

surfaces of the sine bar.

Block gages Base plate

l

h

Sine bar

Rollers

, h



40

The sine bar can be used to measure angle using two different methods as shown in Figure 4.25(a) and (b).

(a) (b) Figure 4.25. Method of measuring angle using sine bar.

In the method shown in Figure 4.25(a) the angle of the workpiece is first estimated using

another instrument such as a combination set. Based on this estimate the height h of block gages is calculated using equation (4.1). Suitable gages are then selected to build the height h and the combination is placed under one of the rollers as shown. The workpiece is placed below the sine bar and the assembly is observed from the side. If the lower part of the sine bar is not parallel with the surface of the workpiece then a gap can be seen. The height of block gages is then adjusted so that the gap is minimized. The angle is then calculated from the equation.

The foregoing method is difficult to use in practice because it is based on trial and error and the resulting error is high. In the second method, the workpiece is placed on top of the sine bar as shown in Figure 4.25(b). A dial gage is moved along the surface of the workpiece. The difference in the dial gage reading is recorded. The value of height h is increased or decreased until the difference in reading is zero. The value of height of block gage that needs to be increased or decreased is equal to the difference in dial gage reading multiplied by the ratio of length of sine bar to the distance of travel of dial gage. For instance, if the end of workpiece closer to the block gage is 0.01 mm lower than the other end, the length of sine bar is 250 mm and the distance of travel of dial gage is 100 mm, then the block gage must be increased by 0.01

× 250/100 = 0.025 mm.

The sine bar should not be used when the angle to be measured exceed 45° because error caused by distances between the rollers and the block gages will increase. 4.2.2 Optical instruments for angle measurement

Many of the optical instruments used for angle measurement are based on the

collimation of light beam. If a light source is placed at focal point of a lens, the light will be transmitted as a parallel beam as shown in Figure 4.26.

Figure 4.26. Schematic of collimated beam.

Workpiece

Dial gage

h

Sine bar

Workpiece

h

Sine bar

Light source Optical axis

Lens Flat reflector



41

If the parallel beam hits a reflecting surface, the beam will be reflected back along its original path and focus at the source O (Figure 4.27). When the reflecting surface is rotated

through a small angle δθ, the reflected beam will rotate through an angle of 2δθ and focus at point O’.

Figure 4.27. Schematic showing rotation of reflected beam.

For a small angle δθ, x = f tan2δθ ≈ f 2δθ , where f is the lens focal length. This principle is used in an autocollimator. Figure 4.28 shows the general arrangement of the autocollimator.

Figure 4.28. General arrangement of the autocollimator. In this instrument the cross-hair target is projected onto the reflecting surface and the

image of the reflected cross-hair is focused onto the plane of the target. The original cross-hair target and its image are observed through a microscope lens. The microscope is also equipped with a scale that gives readings in angles (Figure 4.29). Angular measurement up to 0.5 seconds for a length of 10 m can be obtained using this instrument.

Figure 4.29. Location of cross-hair and its image.

x

O

O’

f

δθ

2δθ

Partial refelctor

Microscope objective

Light source

Cross-hair target Reflecting surface

Image of cross-hair

Collimating lens

Eye

Cross-hair target

Image of cross-hair



42

4.2.3. Combination angle gages Combination angle gages are the basic standard for angle measurements. The gages are

made of hardened steel blocks that are polished on both surfaces. The flatness of the polished

surface is within 0.2µm and the angle on each gage has an accuracy of ±2 seconds from the nominal size. The angle gages can be wrung to obtain angles of various sized. Each set of block gage comprises 13 gages in the following decimations:

Degrees: 1, 3, 9, 27, 41, 90 Minutes: 1, 3, 9, 27 Seconds: 3, 9, 27

The angles can be added or subtracted as shown in Figure 4.30. Using this method,

angles from 0° to 90° can be built in steps of 3 seconds. For instance, the angle 34° 9` can be obtained from the combination of angles 27° + 9° - 3° + 1° + 9`.

Figure 4.30. Angle built-up from combination angle gages. Revision questions Question 4.1

A sine bar was used to measure the angle θ on a specimen block as shown in Figure Q4.1. The length of the sine bar, i.e. distance between rollers, is 250 mm. The total length of the block gages is 52.9 mm. If the reading on the dial indicator was set to zero at point A and the

reading at point B is 0.20 mm, determine the angle θ to the nearest second. The distance between points A and B is 50 mm.

Total angle = 18° Total angle = 36°

36°

27°

9° 18° 27°

9°

A B

h

θ



43

Question 4.2 Two sets of angle gages having the following denominations are available:

Degrees : 1, 3, 9, 27, 41, 90 Minutes : 1, 3, 9, 27 Seconds : 3, 9, 27

Determine the combination of gages that will build the following angles:

(a) 52° 33` 12`` (b) 39° 51` 21`` (c) 68° 19` 18``

Question 4.3

A dial comparator was used to determine the dimensions of a specimen block. Each

dimension was compared with the length of a combination of gage blocks wrung together. The readings obtained are as follows:

Dimension Dimension of gage blocks (mm) Comparator reading(mm) Length 50.000, 2.500, 1.080 - 0.052 Width 20.000, 1.800, 1.120 - 0.205 Height 15.000, 2.500, 1.020 + 1.150

The error in the comparator readings for each reading can be determined from the

calibration graph shown in Figure Q4.3. By taking into account the errors determine the dimensions of the specimen block. Assume that the reading of the comparator was set to zero using the gage blocks.

Dial comparator reading

Error

-0.052

-0.205

0.002

-0.003

1.150

Chapter 4 - Linear and Angular Measurement

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