9.0 Taper And Angular Measurements
- Taper
Taper may be defined as a uniform increase or decrease in dia. of a piece of work measured along its length.
Taper elements
D = large dia. of taper in mm, 2α = full taper angle,
d= small dia. of taper in mm, α = angle of taper or half taper angle.
l = length of tapered part in mm,
9.2 Specification of taper
9.2.1 Conicity : The amount of taper in work piece is usually specified by the ratio of the difference in diameters of the taper to its length. This is termed as conicity and is designated by the letter K.
K = D - d
l
Standard tapers
Machine parts and tools having inside or outside taper are standardised to facilitate interchangeability of parts. Tapered surfaces, which follow standard dimensions, are called standard tapers. Standard tapers adapted by the Indian standard Institution for various tools and machine parts like drills, reamers, milling cutter shanks, arbors, lathe centers etc are Morse tapers.
Morse tapers are available in seven sizes numbered:
0,1,2,3,4,5, & 6. The amount of taper varies from number to number. The No. 0 (zero) Morse taper is the smallest while No. 6 is the largest in size. The non-uniformity of the angle of the taper for different Morse taper sizes is its greatest disadvantage.
Metric tapers are some times used as standard tapers. Metric tapers are made in seven sizes and designated by the number 4,6,80,100,120,160 &200. The taper number stands for the largest dia. of the taper in mm.
The advantage is that all metric tapers have the same angle of the taper.
9.4 Measurement of Angle
Angle is defined as the opening between two lines, which meet at a point. If one line is moved around a point a complete circle can be formed and it is from this circle that units of angle measurements are derived.
If a circle were divided into 360 parts each part is called a degree (o). So angle can be generated very easily without using any absolute standard and it is the precision with which the circle can be divided to get the correct measure of angle. Each degree is further divided into 60 minutes (‘) and each minute is further divided into 60 seconds (“)
Angle measuring tools
9.4.1 Bevel gauge
This is an instrument used for checking, comparing or transferring angles. It is also known as adjustable bevel. It consists of two adjustable blades, which may be adjusted to the required angle. From this direct reading is not obtained. The angle must be set or checked by some other angle measuring devices.
9.4.2 Bevel Try square
It is a common tool used for checking external and internal angles of 90o. These are usually made of tool steel, hardened tempered ground and lapped to a high degree of accuracy. Theses are divided into four classes i.e. 0, 1, 2, and 3. The 0 class is the most accurate one where as class 3 is used for rough work
9.4.3 Plane protractor
It consists of a semi circular steel disc on which degree graduations are marked. A pointer is fixed for setting the instrument to various angles. This is used for setting and checking of angles.
Universal Bevel Protractor
This is an instrument used for measuring and testing angles. Angles can be measured to an accuracy of 5'. 
1) Outer disc 2) Vernier scale
3) Knob 4) Blade
5) Blade locking screw 6) Base
7) Measuring faces 8) Main scale
8) Inner disc
Main parts are base, main scale, vernier scale, clamping screws and blade. The base is the integral part of the outer disc and marked with the main scale in degrees. Main scale consists of 4 quadrants and is graduated in degree from 0 - 90 on one quadrant and 90 - 0 on the next. Thus, the main scale is capable of reading 0o to 360o. The inner disc is attached with the vernier scale, which can be slid on the circular main scale in both clockwise and anticlockwise directions with the help of a locking screw.
Vernier scale is formed by dividing 23 main scale div. into 12 equal parts.
Least count = value of one main scale div. - value of one vernier scale div.
value of one main scale div = 2 ( To make the coinciding line more visible, 2main
scale div. is used to construct the vernier scale.)
value of one vernier scale div = 23/12
Least Count = 2 - 23/12 = 1/ 12
= 5'
For the purpose of reading clockwise and anticlockwise directions, two vernier scales are fitted to the inner disc. Vernier scale is numbered 15,30,45& 60 indicating minute.
9.5.1 Method of reading : Read from the scale the number of whole degrees between main scale zero and vernier scale zero. Read the vernier scale in the same direction and find the number of coinciding division. Multiply this number by the least count and the product will be the no of minutes to be added to the number of whole degrees.
In the figure, Main scale reading = 12o
Vernier scale reading = 50’ (10X5’)
Total reading = 12o 50’
Applications of the vernier bevel protractor
Checking of angle by keeping the base directly supported on the surface plate
Checking of angle by using sine plate, height gauge and surface plate
9.6 Sine bar
Sine bar is used for the accurate setting and checking of angles. It consists of a bar of square or rectangular section, which is stepped at both the ends with rollers secured into each step. Sine bar is specified by the distance between the centers of the two rollers that is 100 mm, 200 mm etc.
For accurate measurements the following points in construction are important.
1. Rollers must be of the same diameter
2. The distance between the roller centers must be accurately maintained
3. The plane of the roller centers must be parallel to the top and bottom surfaces of the
sine bar
For setting sine bar to a particular angle, first the height of slip gauge to be used is calculated. Keep sine bar on a surface plate in such a way that one roller is supported on the slip gauges of calculated height kept over the surface plate and the roller directly on the surface plate
In the figure,
Sin α = h/ l
where h = height of slip gauge used
l = length of sine bar
α = Angle set
The height of slip gauges, h = l sin α
Problems
1) A sine bar of length 200 mm used to be set to an angle of 200 30’. Find the height of slip gauge to be used.
We have h = l sin α
L = 200 mm, α = 200 30’, h=?
h = 200 x 200 30’ = 70.04 mm.
Slip gauges to a height of 70.04 mm have to be used.
2) To set a sine bar of 200 mm length, slip gauges of 13.26 mm height were used. Find the angle
h = l sin α i.e. α = sin (h/l)
We have h =13.26 mm, l = 200 mm
α = sin -1 ( 13.26/ 200)
α = 30 48’ 54”.
3) A Sin bar of length 250 mm is to be set to an angle of 240 40’.Find the height of slip gauges to be used. Select the slip 2gauge from the set, which contains the following blocks,
1.005, 1No
1.01-1.09, 9Nos.
1,1 -1.9, 9Nos.
1-10, 10Nos.
20, 30 and 60 mm. 1No each
h = l sin α
We have l = 250 mm, α =240 40’
h = 250 x sin 240 40’,
= 104.335.
i.e. 60+30+10+1.3+1.03+1.005+1 =104.335 mm.
The slip gauges, which are to be selected to make a height of 104.335 mm, are 60,30,10,1.3,1.03,1.005 and 1 mm blocks.
Various types and sizes of sine bars
9.6.1 Sine plate
The purpose of sine plate is to set the work piece on machines to any workable angles and also to check angular surfaces. It works on the principles of sine bar. It consists of two gray cast iron plates hinged at one end. Top plate is provided with tapered holes for clamping the work piece. Two rollers are provided at the ends of the top plate and one roller acts as a hinge pin .The distance between the centers of rollers is of a known length. Bottom plate is provided with slots for clamping it into the bed of the machine. On the bottom plate a steel plate hardened and ground to a close accuracy is fixed. On this plate slip gauges of calculated height are kept for adjusting the required angle. Plates are clamped in position by using the clamping nuts.
Height of slip gauges to be used can be calculated by the formula h = l sin. Where l = length of sine plate, α = angle which is set. The maximum angle, which can be set on the sine plate, is 450.
9.6.2 Sine center
It is a very useful device for testing the conical work centered at each end. The principle of setting is same as that of sine table.
9.7 Spirit Level
Spirit level is used for measuring small angles or inclination. It consists of a sealed glass tube, the inside surface of which is made to a convex form with a large radius of curvature. A scale is engraved on the glass at the top of the tube. The tube is nearly filled with ether and a small volume contains ether vapor in the form of a bubble. A glass tube is set in a base is adjusted so that the base is horizontal; the bubble sets the center of the scale. When the base is moved out of horizontal, the bubble tries to remains at the highest point of the tube and thus moves along the scale.
In the figure C is the top of the tube radius and the position of the bubble when the based at OB’. If the base is tilted through an angle α and to position OB, the bubble will move a distance d to D where angle COD = α in radian
If R is the radius of the tube, then d = α in radian
R
Also if L is the length of the level and h the difference in height between it’s ends, then for small values of h
h = α hence h = d and d = Rh
L L R L
i.e. the sensitivity of the level increases as R increases.
If we assume R =100 m then the movement of the bubble for a height difference of 0.01mm between the ends of a 300 mm level
d = Rh = 100 x 1000 x 0.01 = 3.3mm
L 300
The sensitivity of spirit level is expressed as the angle of tilt in seconds for which bubble will move by one division on the tube
Hence the sensitivity of the level depends upon the curvature of the glass tube. More the radius of curvature more sensitive is the spirit level.
Spirit level
9.7.1 Clinometers
This instrument is really a spirit level mounted on a rotating member whose angle of inclination relative to its base can be measured by a circular scale. It consists of a rigid frame with two flat surfaces accurately machined at 90o to each other. An arm, which carries a sensitive spirit level, is hinged at one end. The other end of the arm can be moved along a scale on which graduations are made form 0 to 45o
For checking the inclination of a surface frame is kept on the surface to be checked and the spirit level is adjusted to zero by moving the arm. When the spirit level shows zero the division coinciding with the datum line on the arm will give the inclination of the surface.
Clinometers are used for checking angular faces and relief angles on large cutting tools and milling cutter inserts. These can also be used for setting inclinable table on jig boring machine and angular work on grinding machines etc. Average clinometers read to one or more minutes, while instruments of the precision may be obtained which read to 0.1 minute.
Digital clinometer
Modern digital clinometers output electronically and are accurate to .001 degrees with a measuring range of +/- 45°. These have large digital display with the advantage to set all the measuring units commonly used.
9.8 Angle gauges
These are used for setting and checking of angular surfaces. The measuring faces of gauges are lapped and polished to a high degree of flatness and accuracy like slip gauges.13 grades are available in a set and can be divided into three series.
First series contains 10 30 90 270 410
Second series contains 1' 3' 9' 27'
The series contains 3" 6" 18" 30"
All this gauges in combination can be added or subtracted thus making a large number of combinations possible. It is possible to set angles to the nearest 3".
Each angle gauges is marked with a symbol "<" which indicates the direction included angle. When angles of individual gauges are to be added up, symbol of all gauges should be in line.
Direct combination enables computation of any angle up to 81o 40’ 9” and angles larger than this can be made up with the help of the square block. However an additional gauge of 9o can also be supplied with the set to obtain a full 90 angle without the use of the square.
Problems: -
1) Find the combination of angle gauges to build an angle of 260 56’ 30"
The slip gauges used are : 270 -3'-30"
2) Find the combination of angle gauges to build an angle of 570 34' 9".
The slip gauges used are : 410+270 -10 - 90 -27' +1'+6"+3".
Auto – Collimator
The auto-collimator or auto-collimating telescope is an optical instrument used for the measurement of small angular differences, changes or deflections. It is also used to determine straightness, flatness, alignment etc. the principle on which the instrument works is given below.
‘o’ is point source of light placed at the focus of a collimating lens. The rays of light from 0 incident on the lens will travel as a parallel beam of light. If this beam strikes a plain reflector, which is normal to the optical axis, it will be reflected back along its own path and re focused at the same point 0. If the plain deflector is tilted through a small angle then parallel beam will be deflected through twice this angel and will be brought to focus at another point 0 in the same plane.
In an auto-collimator pair of target wires is kept in the focal plane of the collimating lens. The cross wires are illuminated form the back and their image travels out through the lens as a parallel beam of light. On striking a reflecting surface reflection of the image is brought to a focus in the plane of target wires. Through the eyepiece the wires and their images are viewed simultaneously.
Setting wires are placed in the microscope unit and are adjustable by a micrometer mechanism. The displacement of the image can be measured by setting the wires to the original cross lines and then moving over those of the image. A scale provided on the eyepiece can read to the nearest half minute. The micrometer drum moves the setting wires at half minute per revolution and is divided into 60equal parts.
Therefore with the aid of the micrometer it is possible to read up to ½ second of arc. The instrument has generally a range of reading of 10 minutes of arc.
Auto collimator
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