 I welcome you for the series of lecture on gear measurement under module 7. The following topics are covered in this module, introduction to gears covering the various types of gears and then how the gears are manufactured and some details about the tooth profile, those things we will be studying and then we will study about the various terms used in association with the gear and then we will move on to the measurement of spur gear. Now, we will start the lecture 1 in module 7. So, in this lecture the following topics are covered, introduction to gears, the various types of gears, gear terminology and gear manufacturing processors, what are the sources of errors in gears and measurement of spur gear. Now, let us start the first topic that is introduction to gears. So, these are very important transmission elements to transmit the power, they are positive in action that means there is no slip page. In the case of belt ride there is a chance of slip page whereas, in gear drive it is positive transmission and exact speed ratio can be achieved by using the gears. Transmission efficiency in the case of gears is almost 99 percent, but it depends on the actual compliance with the specified dimensions of the gear. So, here we can see a tooth of a spur gear and some of the terms also we can see this is the face and the flank of tooth and the bottom land of tooth. Now, what we can observe here is a particular profile of the tooth. Different profiles are used normally cycloidal profile and even involute profile are used and here you can see an involute profile. So, we will discuss about these profiles in detail. The different forms or profiles of teeth normally used are cycloidal teeth and involute teeth. So, cycloidal profile is it is a curve traced by a point on the circumference of a circle which rolls without slipping on a fixed straight line. For example, say we have a fixed straight line and then we have a wheel which will rotate, which will roll on this fixed line without any slip page. Now, we consider a point on the circumference. So, when we roll this wheel how what is the path followed by this particular point. So, this profile is called cycloid. When a circle or a roller rolls over rolls without slipping on the outside of a fixed circle, the curve traced by a point on the circumference of the rolling circle is known as AP cycloid. On the other hand, if a circle rolls without slipping on the inside of a fixed circle, then the curve traced by a point on the circumference of a rolling circle is called hypo cycloid. Now, you can see let us learn how to construct a cycloidal path. You can see here we have a fixed circle known as pitch circle and then we have a rolling circle C which is which rolls on the outside of the pitch circle and we have another circle D which rolls inside the pitch circle. Now, the cycloidal teeth of a gear may be constructed as shown in the figure. The circle C is this circle is rolled without slipping on the outside of the pitch circle and the point P. So, we have a point P here on the circumference of circle C. The point P on the circle C traces AP cycloid PA. I can see here when it rolls like this in this the direction it traces a path PA. So, this PA represents the face of the cycloidal tooth and then the circle D the second circle which rolls inside the fixed circle. The circle D is rolled on the inside of a pitch circle and the point P again we consider the same point. The point P on the circle D traces hypo cycloid PB. So, when it rolls in this direction it the point P traces this path PB which represents the flank of the tooth profile. Now, in this diagram you can understand what is the face of the tooth and what is the flank of the tooth. Now, the profile BPA the profile BPA is one side of the cycloidal tooth the opposite side that is B dash P dash A dash the opposite side of the tooth is traced in a similar manner. Now, when we combine the two profiles B dash P dash A dash and B PA and when we join A and A dash with a curve then this becomes one tooth of the gear. Now, how do we construct an involute tooth? This is about construction of cycloidal profile. Now, we will try to understand how what is involute profile and how to construct involute profile. Now, we can see we have a base circle and then we have a tangent TT. So, in this involute profile the tangent is rolled over the base circle without any slip page and we consider a point A on the TT. When the tangent TT rolls over the base circle what is the path traced by this fixed point A. So, that becomes the involute profile and involute of a circle it is a plane curve generated by a point on a tangent which rolls on the circle base circle in the case of gear in the case of gear without slipping or it is a point generated by a it is a curve generated by a point on a stiff string which is unwrapped from a reel as shown in this picture. So, now let us study how to construct the involute profile. So, let A be the starting point we have the starting point A here. The base circle is divided into equal number of parts A P 1 P 1 P 2 P 2 P 3 like this. So, the base circle is divided into equal number of parts. Then the tangents at P 1 P 2 etcetera are drawn you can see here at point A we have a tangent TT and at P 1 point P 1 again we have another tangent A 1 P 1 at the point P 2 we have another tangent A 2 P 2 that is tangents at P 1 P 2 etcetera are drawn and lengths P 1 A 1 P 2 A 2 etcetera equal to the arcs A P 1 A P 2 etcetera are marked. Then by joining the points A A 1 A 2 A A 1 A 2 A 3 etcetera we get the involute curve. So, this is the involute curve. Now, what are the advantages of involute profiles or the gears with involute profiles? In actual practice, involute gears are more commonly used due to the following advantages. The most important advantage of the involute profile gear is that the center distance for a pair of involute gears can be varied within certain limits without changing the velocity ratio. This is not true with the cycloidal gears which requires exact center distance to be maintained. In the involute gears the pressure angle remains constant throughout the engagement of the teeth. That means from the beginning of engagement till the end of engagement, the pressure angle remains constant. We will discuss about the pressure angle later. The pressure angle remains constant throughout the engagement of the teeth which is necessary for smooth running and less wear of gears. But in the case of cycloidal gears, the pressure angle is maximum at the beginning of the engagement and it reduces to 0 the pitch point and again it becomes maximum at the end of the engagement. Hence, the running of gears with cycloidal profile is not smooth. Then the face and flank of involute teeth are generated by a single curve whereas in the case of cycloidal gears, double curves that is epicycloid and hypocycloid are required for the face and flank respectively. Thus, the involute teeth are easy to manufacture than the cycloidal teeth. In the involute system, the basic rack has a striped teeth and the same can be cut with simple tools. The only disadvantage of the involute teeth is that the interference occurs with the pinions having smaller number of teeth. This may be avoided by altering the heights of adendum and deadendum of the mating teeth or the angle of abliquity of the teeth. Now, let us move to the various types of gears. Different gears are available. So, first one is spur gear, very commonly used type of gear is spur gear. It is having radial teeth parallel to the axis. So, we have the axis, gear axis and the teeth are parallel to the gear axis and the teeth are striped and are mounted on parallel shafts. So, if we want to transmit the power, we have a shaft here over which a gear is mounted and then we have another shaft with another mating gear. It means the gears are mounted on parallel shafts. So, if the axis, the shafts are at some inclination, then spur gears cannot be used. Sometimes, many spur gears are used at once to create very large gear reductions. Each time, a gear tooth engages a tooth on the other gear. That means, when the gears are rotating, so we have the teeth of one gear and then we have the teeth of another gear. So, when it rotates like this, this tooth will impact on this tooth, mating tooth. That means, there is collision between the teeth, mating teeth. So, this impact makes a noise. It also increases the stress on the gear teeth. So, they are normally used in electric screwdriver, oscillating sprinkler, washing machine and clothes driver, clothes dryer. The other type of gear is helical gear. The teeth on helical gears are cut at an angle to the face of the gear. I can see here, the teeth, they are at some inclination, some angle is provided. When the tooth teeth on helical gear system engage, the contact starts at one end of the tooth and gradually spreads as the gears rotate until the tooth teeth are in full engagement. So, this gradual engagement makes helical gears operate much more smoothly and quietly than the spur gears. In the case of spur gears, there will be impact from one tooth to the other. Whereas, in the case of helical gears, since the engagement is gradual, there is no impact and noise is very, very less. For this reason, helical gears are used almost, they are used in car transmissions. Now, you can see here, this is a single helical gear. The bigger one is called gear and the smaller one is called pinion. And here, we have parallel configuration. That means, the two axis, they are parallel to each other. And here, crossed configuration, you can see, we have the axis, one axis here and the other one axis here. And here, we see double helical gears. You can see the angle in this fashion as well as in the other direction. So, this is double helical gear. Now, we have another kind called bevel gears. The bevel gears are used to connect shafts which intersect usually, but not necessarily at 90 degree. So, we have one axis here and another axis here. So, the angle is 90 degree. So, when the shafts are like this, then we can go for bevel gears. It is not necessary that it should be 90. It can be lesser than that also. The teeth on a bevel gear are subjected to much the same action as spur gear teeth. Bevel gears are not interchangeable and in consequence, they are always designed in pairs. I can see here, we have striped bevel gears. The teeth are striped whereas, here spiral bevel gears, there is curve, the teeth are curved. Again, this type is called hypoid gears. We have rack and pinion type. I can see the picture here. The striped toothed bar is called the rack and the smaller wheel is called pinion. Rack and pinion gears are used to convert rotation into linear motion. So, when the pinion rotates, then the rack moves linearly. So, the pinion is a small gear. The teeth of which fit into those of larger gear are those of rack. Then, we have a schematic diagram of rack here. So, we have the pitch line and then the addendum, didendum, pressure angle, etc. The terminology we will see after some time. We have another type called worm gear. The picture shows the worm gear. A worm drive is a gearing arrangement in which a worm which looks like a screw. So, this is called worm. The worm measures with other wheel called worm wheel. So, this bigger one is called worm wheel. Worm drive reduces rotational speed and it allows higher torque to be transmitted. Now, we will move to anti-back clash gear. I can see the photograph here. We have split gears. So, we have this one part here and another part here. These are designed for precision applications. For example, radio tuning dial. The springs are used for tensioning the gears and then to eliminate the backlash. Plastic, brass, stainless steel and aluminum are the most normally used materials for manufacturing anti-backlash gear. They are available in spur, bevel and worm gears. So, these anti-backlash gear have very minimal backlash. So, wherever very precision transmission is needed, for example, CNC machine tools, etc., these anti-backlash gears are used. So, this schematic diagram of anti-backlash gear. We have two parts of the gear. Two hulls of a split scissor gear. They rotate slightly in opposite directions to increase the tooth thickness. One is spring loaded. Two hulls of a split scissor gear rotate slightly in opposite directions to increase tooth thickness. I can see the diagram here. It is a split. So, there is slight relative motion between one part and the other part. So, thereby increasing the tooth thickness. So, this is the tooth thickness. So, when the tooth thickness increases, it reduces the backlash. So, we have composite gears, a composite gear containing a plastic center. At the center, we have plastic material and on the other side, we have metallic material. Plastic center with a slightly thicker profile than the metal teeth. So, the thickness of plastic teeth is slightly more when compared to the tooth thickness of metal part. So, this also will help in reducing the backlash. So, here we can see different kinds of gears. The combination of gears, compound gears and then gears with key ways. So, different types are used for depending upon the application. Now, we will move to the gear nomenclature. What are the various terms used to discuss the gear? You can see we have two teeth here and this is the profile of the tooth. Either emollute profile or cyclical profile is used to make the gears. So, this portion, this is the face width. So, this is called face of tooth and this is called flank and this is the bottom land. So, the distance from here to here is called face width. So, this is called top land and then we have a curve passing through the top land here. So, this is called the addendum circle. This circle is addendum circle and then we have another imaginary circle called pitch circle. So, pitch circle is it will on the pitch circle see if you observe here we have tooth thickness. That means this is the material thickness is equal to the width of space. So, such a way we have to write the pitch circle and the distance from this point to the corresponding point on the next tooth is called circular pitch and the radial distance from the pitch circle to the addendum circle is called addendum and we have a didendum circle which will pass through the root. So, this is the didendum circle and the radial distance between the didendum circle and the pitch circle is known as didendum. And then we have a clearance circle here and then a small fillet radius and this is the clearance. So, it is a gap between clearance circle and didendum circle. Now, pitch circle is a theoretical circle upon which all calculations are based. Pitch circles of mating gears are tangential to each other and then pinion is a smaller of the two mating gears. Here you can say a mating gear bigger one is called gear and the smaller one is called pinion. The circular pitch is equal to the sum of two thickness this is the circular pitch is equal to sum of two thickness and width of space measured along the pitch circle. Diameter pitch is the ratio of number of teeth to the pitch diameter. Also known as pitch circle diameter. The module is a very common term used with the gears. The module is the ratio of pitch diameter to the number of teeth and then we have addendum is the radial distance from the top land to the pitch circle and didendum is the radial distance from the bottom land to the pitch circle. Whole depth is the sum of addendum and didendum. So, this is depth of tooth profile depth of tooth. Clearance circle we have the clearance circle here clearance circle is a circle that is tangent to the addendum it is tangent to the addendum circle of mating gear. Clearance the amount by which the didendum in a given gear exceeds the addendum of its mating gear and most commonly we use another term that is back lash. It is the amount by which the width of tooth space width of tooth space exceeds the thickness of the engaging tooth thickness there will be another engaging tooth between these two there will be a small gap which is known as back lash. It is the amount by which the width of tooth space exceeds the amount of exceeds the thickness of engaging tooth measured on the pitch circle. Now, we will discuss about the pressure angle is also known as angle of obliquity. It is the angle between the common normal to the two gear teeth at the point of contact and the common tangent at the pitch point. It is usually denoted by this particular symbol and the standard pressure angles used in the manufacture is 14.5 degree or 20 degree. So, pitch circle it is an imaginary circle which by pure rolling action would give the same motion as the actual gear. Then we have a pitch circle diameter it is the diameter of the pitch circle. The size of the gear is usually specified by the pitch circle diameter which is also known as pitch diameter. Then pitch point it is a common point of contact between two pitch circles. Now, you can see some more terminologies here we have a gear and a pinion mating the together and the distance from the centre of pinion to the distance of to the centre of gear. So, this distance is called centre distance and you can see here the is the pressure angle and line of action base circle. This is the root diameter and then we have circular tooth thickness. So, it is measured on the pitch line. So, you can observe in the case of the gear tooth. So, this is the gear tooth. So, in the case of gear tooth the thickness will be varying the thickness will be varying from tooth to tooth. So, we will write the peak. So, here you can see this is the thickness. So, as we move down from top line to bottom line the thickness of the tooth is varying. So, when we want to measure the tooth thickness we should note what depth we want to measure the thickness at the pitch circle. So, we have the thickness which is called circular tooth thickness or chordal tooth thickness. Now, we will move to the gear manufacturing processes. We can make the gears by forming operation. So, which is a chipless operation or we can machine the gears. In the forming process different methods are available casting methods if the gear is very big then we can cast the gears and then by sintering also we can make the gears, ingestion molding process, extruding process, cold drawing process and if we want very thin gears we can stamp the gears and free forming, forging. So, these are different forming operations by which we can make the gears. Coming to the machining of gears we have roughing operations and finishing operations. In rough machining operation we can form we can make the gears by form milling, rack generation, gear shaping, shaping operation, hobbing operation. So, like this we can make the gears by rough machining and then we can finish them by different operations called gear shaving, gear grinding, burnishing, lapping and honing process. Now, under machining we have two methods of machining gears. The first one is reproducing method and we have another type generating method. Reproducing method in which the cutting tool is a form cutter. That means the form that is available in the cutter is reproduced on the blank. So, which forms the gear T profiles by reproducing the shape of the cutter itself. So, in this method each tooth space is cut independently of the other tooth spaces. Schematically you can see here the cutter is rotating as well as it is moving up and down and so at a time one tooth will be cut. You can see the photograph of reproduction method. This is the cutter and this is the gear blank. Now, the second method of machining is generating method in which the cutting tool forms the profiles of several teeth simultaneously. So, in this case several teeth are cut simultaneously during constant relative motion of tooth tool and blank. So, this is the gear blank on which we want to make the gear and this is the hob, the gear cutter. The direction of hob feed also you can see and the gear blank is rotated simultaneously. So, there are various sources of errors in gear manufacturing. This lists the sources of errors in the reproducing method or incorrect profile on the cutting tool. That means we have selected a cutting tool which is having incorrect profile. So, that incorrect profile will be reproduced on the blank and hence we get wrong or some error in the gear. Incorrect positioning of the tool in relation to the work. That means the proper angles, proper orientations, proper depths are not given during the machining operation and incorrect indexing of the blank. So, improper indexing arrangement also leads to the error in gear. In the case of generating method, errors in the manufacture of cutting tool, again the cutting tool is improper or it is having some error. So, we get error in gear that is cut. Errors in positioning of the tool in relation to the work that means improper orientation of tool and work and errors in the relative motion of tool and blank. The relative motions between blank and tool are not correct. They are not rotating at the required orientation or speed or whatever it is. Hence, we get the error in the gear. Now, let us understand why, what is the need for gear measurement? Why do we want to measure the various gear elements? So, to compare the capability of different suppliers, say we are procuring gears from different suppliers and we want to assess which supplier is supplying the better gears. So, when we want to assess the suppliers, it is necessary that we should go for gear measurement. To identify why gears are failing, 90% of the premature failures have excessive geometrical errors. So, to identify what is the reason for failure, we need to measure the gear and to identify the causes of manufacturing the errors. Why the error is occurring? So, to identify that we have to measure the gears. To minimize the noise from a gear set, poor specification or manufacturing of gear causes noise. So, when the gears are made to the specification and you can always reduce the noise. To increase the potential power density, accurate gears transmit more power. To achieve accurate transmission, good transmission, we need to check whether the gears are proper or not. To benchmark suppliers, manufacturing capability. So, whenever we are in the process of benchmarking the suppliers, we need to measure the gears. To minimize the product through life cost, to increase the competitiveness. So, when if we are manufacturer of gears, we want to increase the competitiveness. So, we have to produce good gears. So, it is very essential that we should assess the quality of the gears produced. To prove that gears are supplied to the specification. So, to check whether all the gears are made to specification or not or if there are any errors, if there are errors how to eliminate. To study those things, we need to measure the gears. To reduce premature failure risk also, we need to measure gears. Now, we will move to the gear tooth measurement. Different methods are available to measure the tooth thickness. So, one very commonly measured parameter in the case of gear is gear tooth thickness and the thickness, gear tooth thickness can be measured by various methods. Gear tooth by using the gear tooth varnier caliper by constant chord method and use of base tangent method and by measuring the dimensions over pins. So, these are some of the methods available for measurement of gear tooth. Now, we will study the first method that is measurement of tooth by gear tooth varnier caliper. So, we have seen the conventional varnier caliper wherein there will be only one main scale and one varnier scale. Whereas in the gear tooth varnier caliper, you can see here, we have one horizontal scale and then we have one vertical scale. So, two scales, horizontal scales and two vertical scales are available. We have one set of jaws. So, you can see the jaws, we have one set of jaws and here you can see there is a blade or sometimes it is called tongue. So, we have a blade here. So, which will move up and down. So, this blade can be moved up and down and the jaw opening can be adjusted by moving the horizontal part of the varnier caliper. Now, we should know at what depth we want to measure the thickness. So, this is thickness, tooth thickness is varying at different depths. So, we should know at what depth we want to measure. So, that depth, that depth we should fix here by moving the blade. The tongue should be moved up and down and the depth should be adjusted and then using the, after adjusting the tongue, the gear tooth caliper is placed on the gear tooth which is to be inspected and then the jaw should be moved and the thickness should be measured. If the corridor tooth thickness is to be measured, then we should define what is the value of this corridor height and then that value we should fix here and then we can measure the thickness, corridor thickness. So, this photo shows a gear tooth varnier caliper. You can see the jaws here. We have two jaws and then we have the horizontal scale with varnier and then we have vertical scale with varnier here and this is the blade or tongue which will move up and down. Now you can see this geometry of the gear tooth. So, we have the pitch circle and this is the top land and this O is the center of gear and O A is the radius and then C to D. So, this is the point D. C to D is the depth and then A to B. A to B is the corridor thickness which is termed as W. So, we should calculate what is the value of D and then we should fix move the tongue and then we can measure the value of W, that is corridor thickness. Now the tooth thickness is generally measured at the pitch circle. Since the gear tooth thickness varies from tip to the base circle, the tooth thickness is measured at a specified depth on the tooth. Normally at pitch circle it is measured. The gear tooth varnier has two varnier scales as already discussed. The vertical varnier scale is used to set the depth from the top surface of the tooth that is top land at which the width is to be measured. After fixing D we can measure the corridor thickness. The horizontal varnier scale is used to measure the width of the teeth. Now we can see we have considered one tooth here. The theoretical values of W and D can be found out which then the values are verified by the instrument. From the figure W is A D B, W is equal to A D B but the tooth thickness this W is a chord A D B is a chord but tooth thickness is specified as an arc distance A E B. So we have A we have E here we have B. A E B is arc on the pitch circle. The tooth thickness is specified as an arc distance A E B. Also the depth D adjusted on the instrument is slightly greater than the addendum CE. So this distance from top land to the point E on the pitch circle this is the addendum whereas the D is slightly greater than the addendum. So the D is more than the addendum by an amount ED. Width W is therefore called a chordal thickness and D is called a chordal addendum. So the accuracy of measurement is limited by the least count of the instrument normally it is 0.05 millimeter and the wear is normally concentrated on the two jaws. Two jaws they get worn out more quickly. Now from this geometry from the triangle A D O we have W is equal to twice AD that is AD plus DB this is twice AD which is equal to 2 times A O sine theta. So we can using this geometry we can derive this equation for W that is chordal thickness. So it is equal to N M sine 90 by N where N is the number of teeth. N is number of teeth and R is pitch circle radius. So this is OA is equal to R which is pitch circle radius and M is the module. N is number of teeth and M is module. Now also from this geometry so here we have the D chordal addendum this is equal to OC minus OD. So chordal depth is equal to OC minus OD and then we can derive the equation for D using this relationship. So D that is chordal depth or chordal addendum is equal to NM by 2 times 1 plus 2 by N minus cos 90 by N where M is module and N is number of teeth. So we can understand from this derivation that the W and D they depend on number of teeth and module. So if the number of teeth varies then again we have to calculate what is the W and D. So we will solve a numerical problem. Calculate the gear tooth caliper settings to measure the chordal thickness of a gear having 45 teeth having a module of 4. So the data that is given is module M is equal to 4 and number of teeth N is equal to 45. We have to calculate the gear tooth caliper setting that means what is the depth. So this is the gear tooth profile and we want to measure this is the say pitch line we want to measure chordal thickness. So this is the chordal thickness W. So we should find out what is the value of D which can be set in the Wernier caliper and then we can measure the W. So chordal depth is equal to NM by 2 times 1 plus 2 upon N minus cos 90 by N so which we derived here. Now I can see D is equal to NM by 2 times 1 plus 2 upon N minus cos 90 by N. So we have to feed the values of N and M. N is 45 times M is 4 divided by 2 plus 1 upon 2 divided by 45 minus cos 90 by 45. This will give a value of 4.086 millimeter. So this value we have to set in the Wernier caliper. So the D this is the pitch circle and then chordal thickness. So this is D. D is equal to 4.086. So we have to move the tongue of the Wernier caliper and we have to adjust this 4.086 millimeter in the instrument and then we can measure the thickness. So theoretical value of thickness to thickness chordal to thickness is equal to NM sin 90 by N. So N is 45 number teeth and M is 4 sin 90 by 45 this will give a value of 6.28 millimeter. So chordal thickness is 6.28 millimeter. So this we can verify by using the Wernier gear tooth Wernier. So with this we will conclude this session. In this session we discussed about the introduction to the gears, what are the different types of gears manufactured, how they are manufactured and what are the various terms used in relation to gear and how the gear tooth thickness can be measured using the gear tooth Wernier. And we also saw a numerical problem we will stop at this point. In the next class we will continue the discussion on measurement of tooth thickness by other methods. Thank you.