Toothed gearing Notes

fundamentals of toothed gearing and advantages of toothed gearing, law of toothed gearing and gear tooth addendum modification pdf free download
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 382 Theory of Machines 12 F F F F Fea ea ea ea eatur tur tur tur tures es es es es 1. Introduction. 2. Friction Wheels. T Toothed oothed T T Toothed oothed oothed 3. Advantages and Disadvantages of Gear Drive. 4. Classification of Toothed Wheels. Gearing Gearing Gearing Gearing Gearing 5. Terms Used in Gears. 6. Gear Materials. 7. Law of Gearing. 8. Velocity of Sliding of Teeth. 12.1. 12.1. 12.1. 12.1. 12.1. Intr Intr Intr Intr Introduction oduction oduction oduction oduction 9. Forms of Teeth. We have discussed in the previous chapter, that the 10. Cycloidal Teeth. slipping of a belt or rope is a common phenomenon, in the 11. Involute Teeth. transmission of motion or power between two shafts. The 12. Effect of Altering the Centre effect of slipping is to reduce the velocity ratio of the system. Distance. 13. Comparison Between Involute In precision machines, in which a definite velocity ratio is of and Cycloidal Gears. importance (as in watch mechanism), the only positive drive 14. Systems of Gear Teeth. is by means of gears or toothed wheels. A gear drive is also 15. Standard Proportions of Gear provided, when the distance between the driver and the fol- Systems. lower is very small. 16. Length of Path of Contact. 17. Length of Arc of Contact. 12.2. 12.2. 12.2. 12.2. 12.2. Friction Wheels Friction Wheels Friction Wheels Friction Wheels Friction Wheels 18. Contact Ratio The motion and power transmitted by gears is kine- 19. Interference in Involute matically equivalent to that transmitted by friction wheels or Gears. discs. In order to 20. Minimum Number of Teeth on understand how the Pinion. the motion can be 21. Minimum Number of Teeth on the Wheel. transmitted by 22. Minimum Number of Teeth on two toothed a Pinion for Involute Rack in wheels, consider Order to Avoid Interference. two plain circular 23. Helical Gears. wheels A and B 24. Spiral Gears. mounted on 25. Centre Distance For a Pair of shafts, having sufficient rough surfaces and pressing against Spiral Gears. each other as shown in Fig. 12.1 (a). 26. Efficiency of Spiral Gears. 382 Chapter 12 : Toothed Gearing 383 Let the wheel A be keyed to the rotating shaft and the wheel B to the shaft, to be rotated. A little consideration will show, that when the wheel A is rotated by a rotating shaft, it will rotate the wheel B in the opposite direction as shown in Fig. 12.1 (a). The wheel B will be rotated (by the wheel A) so long as the tangential force exerted by the wheel A does not exceed the maximum frictional resistance between the two wheels. But when the tangential force (P) exceeds the frictional resistance (F), slipping will take place between the two wheels. Thus the friction drive is not a positive drive. (a) Friction wheels. (b) Toothed wheels. Fig. 12.1 In order to avoid the slipping, a number of projections (called teeth) as shown in Fig. 12.1 (b), are provided on the periphery of the wheel A, which will fit into the corresponding recesses on the periphery of the wheel B. A friction wheel with the teeth cut on it is known as toothed wheel or gear. The usual connection to show the toothed wheels is by their pitch circles. Note : Kinematically, the friction wheels running without slip and toothed gearing are identical. But due to the possibility of slipping of wheels, the friction wheels can only be used for transmission of small powers. 12.3. Advantages and Disadvantages of Gear Drive The following are the advantages and disadvantages of the gear drive as compared to belt, rope and chain drives : Advantages 1. It transmits exact velocity ratio. 2. It may be used to transmit large power. 3. It has high efficiency. 4. It has reliable service. 5. It has compact layout. Disadvantages 1. The manufacture of gears require special tools and equipment. 2. The error in cutting teeth may cause vibrations and noise during operation. The frictional force F is equal to µ. R , where µ = Coefficient of friction between the rubbing surface of N two wheels, and R = Normal reaction between the two rubbing surfaces. N For details, please refer to Art. 12.4. 384 Theory of Machines 12.4. Classification of Toothed Wheels The gears or toothed wheels may be classified as follows : 1. According to the position of axes of the shafts. The axes of the two shafts between which the motion is to be transmitted, may be (a) Parallel, (b) Intersecting, and (c) Non-intersecting and non-parallel. The two parallel and co-planar shafts connected by the gears is shown in Fig. 12.1. These gears are called spur gears and the arrangement is known as spur gearing. These gears have teeth parallel to the axis of the wheel as shown in Fig. 12.1. Another name given to the spur gearing is helical gearing, in which the teeth are inclined to the axis. The single and double helical gears con- necting parallel shafts are shown in Fig. 12.2 (a) and (b) respectively. The double helical gears are known as herringbone gears. A pair of spur gears are kinematically equivalent to a pair of cylindrical discs, keyed to parallel shafts and having a line contact. The two non-parallel or intersecting, but coplanar shafts connected by gears is shown in Fig. 12.2 (c). These gears are called bevel gears and the arrangement is known as bevel gearing. The bevel gears, like spur gears, may also have their teeth inclined to the face of the bevel, in which case they are known as helical bevel gears. The two non-intersecting and non-parallel i.e. non-coplanar shaft connected by gears is shown in Fig. 12.2 (d). These gears are called skew bevel gears or spiral gears and the arrangement is known as skew bevel gearing or spiral gearing. This type of gearing also have a line contact, the rotation of which about the axes generates the two pitch surfaces known as hyperboloids. Notes : (a) When equal bevel gears (having equal teeth) connect two shafts whose axes are mutually perpen- dicular, then the bevel gears are known as mitres. (b) A hyperboloid is the solid formed by revolving a straight line about an axis (not in the same plane), such that every point on the line remains at a constant distance from the axis. (c) The worm gearing is essentially a form of spiral gearing in which the shafts are usually at right angles. (a) Single helical gear. (b) Double helical gear. (c) Bevel gear. (d) Spiral gear. Fig. 12.2 2. According to the peripheral velocity of the gears. The gears, according to the peripheral velocity of the gears may be classified as : (a) Low velocity, (b) Medium velocity, and (c) High velocity. The gears having velocity less than 3 m/s are termed as low velocity gears and gears having velocity between 3 and 15 m/s are known as medium velocity gears. If the velocity of gears is more than 15 m/s, then these are called high speed gears. Chapter 12 : Toothed Gearing 385 Helical Gears Spiral Gears Double helical gears 3. According to the type of gearing. The gears, according to the type of gearing may be classified as : (a) External gearing, (b) Internal gearing, and (c) Rack and pinion. In external gearing, the gears of the two shafts mesh externally with each other as shown in Fig. 12.3 (a). The larger of these two wheels is called spur wheel and the smaller wheel is called pinion. In an external gearing, the motion of the two wheels is always unlike, as shown in Fig. 12.3 (a). (a) External gearing. (b) Internal gearing. Fig. 12.3 Fig. 12.4. Rack and pinion. In internal gearing, the gears of the two shafts mesh internally with each other as shown in Fig. 12.3 (b). The larger of these two wheels is called annular wheel and the smaller wheel is called pinion. In an internal gearing, the motion of the two wheels is always like, as shown in Fig. 12.3 (b). 386 Theory of Machines Sometimes, the gear of a shaft meshes externally and internally with the gears in a straight line, as shown in Fig. 12.4. Such type of gear is called rack and pinion. The straight line gear is called rack and the circular wheel is called pinion. A little consideration will show that with the help of a rack and pinion, we can convert linear motion into rotary motion and vice-versa as shown in Fig. 12.4. 4. According to position of teeth on the gear surface. The teeth on the gear surface may be (a) straight, (b) inclined, and (c) curved. We have discussed earlier that the spur gears have straight teeth where as helical gears have their teeth inclined to the wheel rim. In case of spiral gears, the teeth are curved over the rim surface. Rack and pinion Internal gears 12.5. Terms Used in Gears The following terms, which will be mostly used in this chapter, should be clearly understood at this stage. These terms are illustrated in Fig. 12.5. Fig. 12.5. Terms used in gears. 1. Pitch circle. It is an imaginary circle which by pure rolling action, would give the same motion as the actual gear. A straight line may also be defined as a wheel of infinite radius. Chapter 12 : Toothed Gearing 387 2. Pitch circle diameter. It is the diameter of the pitch circle. The size of the gear is usually specified by the pitch circle diameter. It is also known as pitch diameter. 3. Pitch point. It is a common point of contact between two pitch circles. 4. Pitch surface. It is the surface of the rolling discs which the meshing gears have replaced at the pitch circle. 5. Pressure angle or angle of obliquity. It is the angle between the common normal to two gear teeth at the point of contact and the common tangent at the pitch point. It is usually denoted by φ. 1 The standard pressure angles are 14 ° and 20°. 2 6. Addendum. It is the radial distance of a tooth from the pitch circle to the top of the tooth. 7. Dedendum. It is the radial distance of a tooth from the pitch circle to the bottom of the tooth. 8. Addendum circle. It is the circle drawn through the top of the teeth and is concentric with the pitch circle. 9. Dedendum circle. It is the circle drawn through the bottom of the teeth. It is also called root circle. Note : Root circle diameter = Pitch circle diameter × cos φ, where φ is the pressure angle. 10. Circular pitch. It is the distance measured on the circumference of the pitch circle from a point of one tooth to the corresponding point on the next tooth. It is usually denoted by p . c Mathematically, Circular pitch, p = π D/T c where D = Diameter of the pitch circle, and T = Number of teeth on the wheel. A little consideration will show that the two gears will mesh together correctly, if the two wheels have the same circular pitch. Note : If D and D are the diameters of the two meshing gears having the teeth T and T respectively, then for 1 2 1 2 them to mesh correctly, ππ DD D T 12 1 1 == or = p c TT D T 12 2 2 11. Diametral pitch. It is the ratio of number of teeth to the pitch circle diameter in millimetres. It is denoted by p . Mathematically, d T π D  π p== ...  p = Diametral pitch, d  c D p T  c where T = Number of teeth, and D = Pitch circle diameter. 12. Module. It is the ratio of the pitch circle diameter in millimeters to the number of teeth. It is usually denoted by m. Mathematically, Module, m = D /T Note : The recommended series of modules in Indian Standard are 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, and 20. The modules 1.125, 1.375, 1.75, 2.25, 2.75, 3.5, 4.5, 5.5, 7, 9, 11, 14 and 18 are of second choice. 13. Clearance. It is the radial distance from the top of the tooth to the bottom of the tooth, in a meshing gear. A circle passing through the top of the meshing gear is known as clearance circle. 14. Total depth. It is the radial distance between the addendum and the dedendum circles of a gear. It is equal to the sum of the addendum and dedendum. 388 Theory of Machines 15. Working depth. It is the radial distance from the addendum circle to the clearance circle. It is equal to the sum of the addendum of the two meshing gears. 16. Tooth thickness. It is the width of the tooth measured along the pitch circle. 17. Tooth space . It is the width of space between the two adjacent teeth measured along the pitch circle. 18. Backlash. It is the difference between the tooth space and the tooth thickness, as mea- sured along the pitch circle. Theoretically, the backlash should be zero, but in actual practice some backlash must be allowed to prevent jamming of the teeth due to tooth errors and thermal expansion. 19. Face of tooth. It is the surface of the gear tooth above the pitch surface. 20. Flank of tooth. It is the surface of the gear tooth below the pitch surface. 21. Top land. It is the surface of the top of the tooth. 22. Face width. It is the width of the gear tooth measured parallel to its axis. 23. Profile. It is the curve formed by the face and flank of the tooth. 24. Fillet radius. It is the radius that connects the root circle to the profile of the tooth. 25. Path of contact. It is the path traced by the point of contact of two teeth from the beginning to the end of engagement. 26. Length of the path of contact. It is the length of the common normal cut-off by the addendum circles of the wheel and pinion. 27. Arc of contact. It is the path traced by a point on the pitch circle from the beginning to the end of engagement of a given pair of teeth. The arc of contact consists of two parts, i.e. (a) Arc of approach. It is the portion of the path of contact from the beginning of the engagement to the pitch point. (b) Arc of recess. It is the portion of the path of contact from the pitch point to the end of the engagement of a pair of teeth. Note : The ratio of the length of arc of contact to the circular pitch is known as contact ratio i.e. number of pairs of teeth in contact. 12.6. Gear Materials The material used for the manufacture of gears depends upon the strength and service condi- tions like wear, noise etc. The gears may be manufactured from metallic or non-metallic materials. The metallic gears with cut teeth are commercially obtainable in cast iron, steel and bronze. The non- metallic materials like wood, raw hide, compressed paper and synthetic resins like nylon are used for gears, especially for reducing noise. The cast iron is widely used for the manufacture of gears due to its good wearing properties, excellent machinability and ease of producing complicated shapes by casting method. The cast iron gears with cut teeth may be employed, where smooth action is not important. The steel is used for high strength gears and steel may be plain carbon steel or alloy steel. The steel gears are usually heat treated in order to combine properly the toughness and tooth hardness. The phosphor bronze is widely used for worm gears in order to reduce wear of the worms which will be excessive with cast iron or steel. 12.7. Condition for Constant Velocity Ratio of Toothed Wheels–Law of Gearing Consider the portions of the two teeth, one on the wheel 1 (or pinion) and the other on the For details, see Art. 12.16. For details, see Art. 12.17. Chapter 12 : Toothed Gearing 389 wheel 2, as shown by thick line curves in Fig. 12.6. Let the two teeth come in contact at point Q, and the wheels rotate in the directions as shown in the figure. Let T T be the common tangent and MN be the common normal to the curves at the point of contact Q. From the centres O and O , draw O M and O N perpendicular to MN. A 1 2 1 2 little consideration will show that the point Q moves in the direction QC, when considered as a point on wheel 1, and in the direction QD when considered as a point on wheel 2. Let v and v be the velocities of the point Q on the wheels 1 2 1 and 2 respectively. If the teeth are to remain in contact, then the components of these velocities along the common normal MN must be equal. ∴ vv cos α= cosβ Fig. 12.6. Law of gearing. 12 or () ω×OQcosα=(ω ×O Q)cosβ 11 2 2 OM O N 12 () ω×OQ =(ω ×O Q) or ω ×OM =ω ×O N 11 2 2 11 2 2 OQ O Q 12 ω ON 12 ∴ …(i) = ω OM 21 Also from similar triangles O MP and O NP, 1 2 ON OP 22 = ...(ii) OM O P 11 Combining equations (i) and (ii), we have ω ON O P 12 2 == ...(iii) ω OM OP 21 1 From above, we see that the angular velocity ratio is inversely proportional to the ratio of the distances of the point P from the centres O and O , or the common normal to the two surfaces at the 1 2 point of contact Q intersects the line of centres at point P which divides the centre distance inversely as the ratio of angular velocities. Therefore in order to have a constant angular velocity ratio for all positions of the wheels, the point P must be the fixed point (called pitch point) for the two wheels. In other words, the common normal at the point of contact between a pair of teeth must always pass through the pitch point. This is the fundamental condition which must be satisfied while designing the profiles for the teeth of gear wheels. It is also known as law of gearing. Notes : 1. The above condition is fulfilled by teeth of involute form, provided that the root circles from which the profiles are generated are tangential to the common normal. 2. If the shape of one tooth profile is arbitrarily chosen and another tooth is designed to satisfy the above condition, then the second tooth is said to be conjugate to the first. The conjugate teeth are not in common use because of difficulty in manufacture, and cost of production. 3. If D and D are pitch circle diameters of wheels 1 and 2 having teeth T and T respectively, then 1 2 1 2 velocity ratio, ω OP D T 12 2 2 == = ω OP D T 21 1 1 390 Theory of Machines 12.8. Velocity of Sliding of Teeth The sliding between a pair of teeth in contact at Q occurs along the common tangent T T to the tooth curves as shown in Fig. 12.6. The velocity of sliding is the velocity of one tooth relative to its mating tooth along the common tangent at the point of contact. The velocity of point Q, considered as a point on wheel 1, along the common tangent T T is represented by EC. From similar triangles QEC and O MQ, 1 EC v ==ω or EC=ω .MQ 11 MQ O Q 1 Similarly, the velocity of point Q, considered as a point on wheel 2, along the common tan- gent T T is represented by ED. From similar triangles QCD and O NQ, 2 ED v 2 ==ω or ED=ω .QN 22 QN O Q 2 Let vQ = Velocity of sliding at . S ∴ vE=−D EC=ω.. QN−ωMQ S2 1 =ω() QP + PN − ω(MP − QP) 21 =() ω +ω QP +ω. PN −ω.MP ...(i) 12 2 1 ω OP PN 12 Since == or ω . MP=ω .PN , therefore equation (i) becomes 12 ω OP MP 21 vQ =ω() +ωP ...(ii) S1 2 Notes : 1. We see from equation (ii), that the velocity of sliding is proportional to the distance of the point of contact from the pitch point. 2. Since the angular velocity of wheel 2 relative to wheel 1 is (ω + ω ) and P is the instantaneous 1 2 centre for this relative motion, therefore the value of v may directly be written as v (ω + ω ) QP, without the s s 1 2 above analysis. 12.9. Forms of Teeth We have discussed in Art. 12.7 (Note 2) that conjugate teeth are not in common use. Therefore, in actual practice following are the two types of teeth commonly used : 1. Cycloidal teeth ; and 2. Involute teeth. We shall discuss both the above mentioned types of teeth in the following articles. Both these forms of teeth satisfy the conditions as discussed in Art. 12.7. 12.10. Cycloidal Teeth A cycloid is the curve traced by a point on the circumference of a circle which rolls without slipping on a fixed straight line. When a circle rolls without slipping on the outside of a fixed circle, the curve traced by a point on the circumference of a circle is known as epi-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 circle is called hypo-cycloid. Chapter 12 : Toothed Gearing 391 In Fig. 12.7 (a), the fixed line or pitch line of a rack is shown. When the circle C rolls without slipping above the pitch line in the direction as indicated in Fig. 12.7 (a), then the point P on the circle traces epi-cycloid PA. This represents the face of the cycloidal tooth profile. When the circle D rolls without slipping below the pitch line, then the point P on the circle D traces hypo-cycloid PB, which represents the flank of the cycloidal tooth. The profile BPA is one side of the cycloidal rack tooth. Similarly, the two curves P' A' and P'B' forming the opposite side of the tooth profile are traced by the point P' when the circles C and D roll in the opposite directions. (a)(b) Fig. 12.7. Construction of cycloidal teeth of a gear. In the similar way, the cycloidal teeth of a gear may be constructed as shown in Fig. 12.7 (b). The circle C is rolled without slipping on the outside of the pitch circle and the point P on the circle C traces epi-cycloid PA , which represents the face of the cycloidal tooth. The circle D is rolled on the inside of pitch circle and the point P on the circle D traces hypo-cycloid PB, which represents the flank of the tooth profile. The profile BPA is one side of the cycloidal tooth. The opposite side of the tooth is traced as explained above. The construction of the two mating cycloidal teeth is shown in Fig. 12.8. A point on the circle D will trace the flank of the tooth T when circle D rolls without slipping on the inside of pitch circle 1 of wheel 1 and face of tooth T when the circle D rolls without slipping on the outside of pitch circle 2 of wheel 2. Similarly, a point on the circle C will trace the face of tooth T and flank of tooth T . The 1 2 rolling circles C and D may have unequal diameters, but if several wheels are to be interchangeable, they must have rolling circles of equal diameters. Fig. 12.8. Construction of two mating cycloidal teeth. A little consideration will show, that the common normal XX at the point of contact between two cycloidal teeth always passes through the pitch point, which is the fundamental condition for a constant velocity ratio. 392 Theory of Machines 12.11. Involute Teeth An involute of a circle is a plane curve generated by a point on a tangent, which rolls on the circle without slipping or by a point on a taut string which in unwrapped from a reel as shown in Fig. 12.9. In connection with toothed wheels, the circle is known as base circle. The involute is traced as follows : Let A be the starting point of the involute. The base circle is divided into equal number of parts e.g. AP , P P , 1 1 2 P P etc. The tangents at P , P , P etc. are drawn and the 2 3 1 2 3 length P A , P A , P A equal to the arcs AP , AP and AP are 1 1 2 2 3 3 1 2 3 set off. Joining the points A, A , A , A etc. we obtain the involute 1 2 3 Fig. 12.9. Construction of involute. curve AR. A little consideration will show that at any instant A , the tangent A T to the involute is perpendicular to P A and P A is the normal to the involute. In 3 3 3 3 3 3 other words, normal at any point of an involute is a tangent to the circle. Now, let O and O be the fixed centres of the two base circles as shown in Fig. 12.10 (a). Let 1 2 the corresponding involutes AB and A B be in contact at point Q. MQ and NQ are normals to the 1 1 involutes at Q and are tangents to base circles. Since the normal of an involute at a given point is the tangent drawn from that point to the base circle, therefore the common normal MN at Q is also the common tangent to the two base circles. We see that the common normal MN intersects the line of centres O O at the fixed point P (called pitch point). Therefore the involute teeth satisfy the 1 2 fundamental condition of constant velocity ratio. (a)(b) Fig. 12.10. Involute teeth. From similar triangles O NP and O MP, 2 1 OM O P ω 11 2 == ... (i) ON OP ω 22 1 which determines the ratio of the radii of the two base circles. The radii of the base circles is given by OM=φ OP cos , and O N= O P cosφ 11 2 2 Also the centre distance between the base circles, OM O N OM + O N 12 1 2 OO=+ O P O P= + = 12 1 2 cosφφ cos cosφ Chapter 12 : Toothed Gearing 393 where φ is the pressure angle or the angle of obliquity. It is the angle which the common normal to the base circles (i.e. MN) makes with the common tangent to the pitch circles. When the power is being transmitted, the maximum tooth pressure (neglecting friction at the teeth) is exerted along the common normal through the pitch point. This force may be resolved into tangential and radial or normal components. These components act along and at right angles to the common tangent to the pitch circles. If F is the maximum tooth pressure as shown in Fig. 12.10 (b), then Tangential force, F = F cos φ T and radial or normal force, F = F sin φ. R ∴ Torque exerted on the gear shaft = F × r, where r is the pitch circle radius of the gear. T Note : The tangential force provides the driving torque and the radial or normal force produces radial deflection of the rim and bending of the shafts. 12.12.Effect of Altering the Centre Distance on the Velocity Ratio for Involute Teeth Gears In the previous article, we have seen that the velocity ratio for the involute teeth gears is given by OM O P ω 11 2 == ...(i) ON O P ω 22 1 Let, in Fig. 12.10 (a), the centre of rotation of one of the gears (say wheel 1) is shifted from O to O ' . Consequently the contact point shifts from Q to Q '. The common normal to the teeth at the 1 1 point of contact Q ' is the tangent to the base circle, because it has a contact between two involute curves and they are generated from the base circle. Let the tangent M' N' to the base circles intersects ′ O at the pitch point P' . As a result of this, the wheel continues to work correctly. O 1 2 Now from similar triangles O NP and O MP, 2 1 OM O P 11 = ...(ii) ON O P 22 and from similar triangles O N'P' and O 'M'P', 2 1 ′′ ′ OM ′ O P 11 = ...(iii) ′′ ON O P 22 But O N = O N', and O M = O ' M'. Therefore from equations (ii) and (iii), 2 2 1 1 OP O ′P′ 11 = ...Same as equation (i) ′ OP O P 22 Thus we see that if the centre distance is changed within limits, the velocity ratio remains unchanged. However, the pressure angle increases (from φ to φ′) with the increase in the centre distance. Example 12.1. A single reduction gear of 120 kW with a pinion 250 mm pitch circle diameter and speed 650 r.p.m. is supported in bearings on either side. Calculate the total load due to the power transmitted, the pressure angle being 20°. 3 Solution. Given : P = 120 kW = 120 × 10 W ; d = 250 mm or r = 125 mm = 0.125 m ; N = 650 r.p.m. or ω = 2π × 650/60 = 68 rad/s ; φ = 20° It is not the case with cycloidal teeth. 394 Theory of Machines Let T = Torque transmitted in N-m. We know that power transmitted (P), 3 3 120 × 10 = T.ω = T × 68 or T = 120 × 10 /68 = 1765 N-m and tangential load on the pinion, F = T /r = 1765 / 0.125 = 14 120 N T ∴ Total load due to power transmitted, F = F / cos φ = 14 120 / cos 20° = 15 026 N = 15.026 kN Ans. T 12.13. Comparison Between Involute and Cycloidal Gears In actual practice, the involute gears are more commonly used as compared to cycloidal gears, due to the following advantages : Advantages of involute gears Following are the advantages of involute gears : 1. The most important advantage of the involute gears is that the centre distance for a pair of involute gears can be varied within limits without changing the velocity ratio. This is not true for cycloidal gears which requires exact centre distance to be maintained. 2. In involute gears, the pressure angle, from the start of the engagement of teeth to the end of the engagement, remains constant. It is necessary for smooth running and less wear of gears. But in cycloidal gears, the pressure angle is maximum at the beginning of engagement, reduces to zero at pitch point, starts decreasing and again becomes maximum at the end of engagement. This results in less smooth running of gears. 3. The face and flank of involute teeth are generated by a single curve where as in cycloidal gears, double curves (i.e. epi-cycloid and hypo-cycloid) are required for the face and flank respec- tively. Thus the involute teeth are easy to manufacture than cycloidal teeth. In involute system, the basic rack has straight teeth and the same can be cut with simple tools. Note : The only disadvantage of the involute teeth is that the interference occurs (Refer Art. 12.19) with pinions having smaller number of teeth. This may be avoided by altering the heights of addendum and dedendum of the mating teeth or the angle of obliquity of the teeth. Advantages of cycloidal gears Following are the advantages of cycloidal gears : 1. Since the cycloidal teeth have wider flanks, therefore the cycloidal gears are stronger than the involute gears, for the same pitch. Due to this reason, the cycloidal teeth are preferred specially for cast teeth. 2. In cycloidal gears, the contact takes place between a convex flank and concave surface, whereas in involute gears, the convex surfaces are in contact. This condition results in less wear in cycloidal gears as compared to involute gears. However the difference in wear is negligible. 3. In cycloidal gears, the interference does not occur at all. Though there are advantages of cycloidal gears but they are outweighed by the greater simplicity and flexibility of the involute gears. 12.14. Systems of Gear Teeth The following four systems of gear teeth are commonly used in practice : 1 1 ° ° 1. 14 Composite system, 2. 14 Full depth involute system, 3. 20° Full depth involute 2 2 system, and 4. 20° Stub involute system. 1 ° The 14 composite system is used for general purpose gears. It is stronger but has no inter- 2 Chapter 12 : Toothed Gearing 395 changeability. The tooth profile of this system has cycloidal curves at the top and bottom and involute curve at the middle portion. The teeth are produced by formed milling cutters or hobs. The tooth 1 ° 14 profile of the full depth involute system was developed for use with gear hobs for spur and 2 helical gears. The tooth profile of the 20° full depth involute system may be cut by hobs. The increase of 1 ° the pressure angle from 14 to 20° results in a stronger tooth, because the tooth acting as a beam is 2 wider at the base. The 20° stub involute system has a strong tooth to take heavy loads. 12.15. Standard Proportions of Gear Systems The following table shows the standard proportions in module (m) for the four gear systems as discussed in the previous article. Table 12.1. Standard proportions of gear systems. 1 ° 14 S. No. Particulars composite or full 20° full depth 20° stub involute 2 depth involute system involute system system 1. Addenddm 1 m 1 m 0.8 m 2. Dedendum 1.25 m 1.25 m 1 m 3. Working depth 2 m 2 m 1.60 m 4. Minimum total depth 2.25 m 2.25 m 1.80 m 5. Tooth thickness 1.5708 m 1.5708 m 1.5708 m 6. Minimum clearance 0.25 m 0.25 m 0.2 m 7. Fillet radius at root 0.4 m 0.4 m 0.4 m 12.16. Length of Path of Contact Consider a pinion driving the wheel as shown in Fig. 12.11. When the pinion rotates in clockwise direction, the contact between a pair of involute teeth begins at K (on the flank near the base circle of pinion or the outer end of the tooth face on the wheel) and ends at L (outer end of the tooth face on the pinion or on the flank near the base circle of wheel). MN is the common normal at the point of contacts and the common tangent to the base circles. The point K is the intersection of the addendum circle of wheel and the common tangent. The point L is the intersection of the addendum circle of pinion and common tangent. Fig. 12.11. Length of path of contact. If the wheel is made to act as a driver and the directions of motion are reversed, then the contact between a pair of teeth begins at L and ends at K. 396 Theory of Machines We have discussed in Art. 12.4 that the length of path of contact is the length of common normal cut- off by the addendum circles of the wheel and the pinion. Thus the length of path of contact is KL which is the sum of the parts of the path of contacts KP and PL. The part of the path of contact KP is known as path of approach and the part of the path of contact PL is known as path of recess. Let r = O L = Radius of addendum A 1 circle of pinion, R = O K = Radius of addendum A 2 circle of wheel, r = O P = Radius of pitch circle of Bevel gear 1 pinion, and R = O P = Radius of pitch circle of 2 wheel. From Fig. 12.11, we find that radius of the base circle of pinion, O M = O P cos φ = r cos φ 1 1 and radius of the base circle of wheel, O N = O P cos φ = R cos φ 2 2 Now from right angled triangle O KN, 2 2 22 22 KN=− () O K (O N)= () R− Rcosφ 22 A and PN=φ O Psin= R sinφ 2 ∴ Length of the part of the path of contact, or the path of approach, 2 22 KP=− KN PN= () R − R cosφ− R sinφ A Similarly from right angled triangle O ML, 1 22 222 and ML=− () O L (O M)= (r)− rcosφ 11 A MP=φ O P sin= r sinφ 1 ∴ Length of the part of the path of contact, or path of recess, 222 PL=− ML MP= () r − r cosφ− rsinφ A ∴ Length of the path of contact, 22 2 2 2 2 KL=+ KP PL= () R − R cosφ+ (r)− r cosφ−(R+ r)sinφ AA 12.17. Length of Arc of Contact We have already defined that the arc of contact is the path traced by a point on the pitch circle from the beginning to the end of engagement of a given pair of teeth. In Fig. 12.11, the arc of contact is EPF or GPH. Considering the arc of contact GPH, it is divided into two parts i.e. arc GP and arc PH. The arc GP is known as arc of approach and the arc PH is called arc of recess. The angles subtended by these arcs at O are called angle of approach and angle of recess respectively. 1 Chapter 12 : Toothed Gearing 397 We know that the length of the arc of approach (arc GP) Length of path of approach KP == cosφφ cos and the length of the arc of recess (arc PH) Lengthofpathofrecess PL == cosφφ cos Since the length of the arc of contact GPH is equal to the sum of the length of arc of approach and arc of recess, therefore, Length of the arc of contact KP PL KL = arc GP+ arc PH=+= cosφφφ cos cos Length of path of contact = cosφ 12.18. Contact Ratio (or Number of Pairs of Teeth in Contact) The contact ratio or the number of pairs of teeth in contact is defined as the ratio of the length of the arc of contact to the circular pitch. Mathematically, Contact ratio or number of pairs of teeth in contact Length of the arc of contact = p c where p== Circular pitchπm, and c m = Module. Notes : 1. The contact ratio, usually, is not a whole number. For example, if the contact ratio is 1.6, it does not mean that there are 1.6 pairs of teeth in contact. It means that there are alternately one pair and two pairs of teeth in contact and on a time basis the average is 1.6. 2. The theoretical minimum value for the contact ratio is one, that is there must always be at least one pair of teeth in contact for continuous action. 3. Larger the contact ratio, more quietly the gears will operate. Example 12.2. The number of teeth on each of the two equal spur gears in mesh are 40. The teeth have 20° involute profile and the module is 6 mm. If the arc of contact is 1.75 times the circular pitch, find the addendum. Solution. Given : T = t = 40 ; φ = 20° ; m = 6 mm We know that the circular pitch, p = π m = π × 6 = 18.85 mm c ∴ Length of arc of contact = 1.75 p = 1.75 × 18.85 = 33 mm c and length of path of contact = Length of arc of contact × cos φ = 33 cos 20° = 31 mm Let R = r = Radius of the addendum circle of each wheel. A A We know that pitch circle radii of each wheel, R = r = m.T / 2 = 6 × 40/2 = 120 mm 398 Theory of Machines and length of path of contact 2 2 2 222 31=− (RR ) cosφ+ (r )−r cosφ− (R+r) sinφ AA 22 2  = 2 ...(∵ R = r, and R = r ) ()RR−φ cos−Rsinφ A A  A 31 222 =− (R ) (120) cos 20°− 120 sin 20° A 2 2 15.5=− (R ) 12 715− 41 A 22 (15.5+= 41) (R )− 12 715 A 2 3192+= 12 715 (RR ) or = 126.12 mm AA We know that the addendum of the wheel, =RR−= 126.12− 120= 6.12 mm Ans. A Example 12.3. A pinion having 30 teeth drives a gear having 80 teeth. The profile of the gears is involute with 20° pressure angle, 12 mm module and 10 mm addendum. Find the length of path of contact, arc of contact and the contact ratio. Solution. Given : t = 30 ; T = 80 ; φ = 20° ; m = 12 mm ; Addendum = 10 mm Length of path of contact We know that pitch circle radius of pinion, Worm. r = m.t / 2 = 12 × 30 / 2 = 180 mm and pitch circle radius of gear, R = m.T / 2 = 12 × 80 / 2 = 480 mm ∴ Radius of addendum circle of pinion, r = r + Addendum = 180 + 10 = 190 mm A and radius of addendum circle of gear, R = R + Addendum = 480 + 10 = 490 mm A We know that length of the path of approach, 22 2 ...(Refer Fig.12.11) KP=−φ () R R cos− Rsinφ A 222 =− 191.5 164.2= 27.3 mm =− (490) (480) cos 20°− 480 sin 20° and length of the path of recess, 222 PL=− () r r cosφ− rsinφ A 222 =−= 86.6 61.6 25 mm =− (190) (180) cos 20°− 180 sin 20° We know that length of path of contact, KL = KP + PL = 27.3 + 25 = 52.3 mm Ans. Chapter 12 : Toothed Gearing 399 Length of arc of contact We know that length of arc of contact Length of path of contact 52.3 Ans. === 55.66 mm cosφ° cos 20 Contact ratio We know that circular pitch, p = π.m = π × 12 = 37.7 mm c Length of arc of contact 55.66 ∴ Ans. Contact ratio = == 1.5 say 2 p 37.7 c Example 12.4. Two involute gears of 20° pressure angle are in mesh. The number of teeth on pinion is 20 and the gear ratio is 2. If the pitch expressed in module is 5 mm and the pitch line speed is 1.2 m/s, assuming addendum as standard and equal to one module, find : 1. The angle turned through by pinion when one pair of teeth is in mesh ; and 2. The maximum velocity of sliding. Solution. Given : φ = 20° ; t = 20; G = T/t = 2; m = 5 mm ; v = 1.2 m/s ; addendum = 1 module = 5 mm 1. Angle turned through by pinion when one pair of teeth is in mesh We know that pitch circle radius of pinion, r = m.t / 2 = 5 × 20 / 2 = 50 mm and pitch circle radius of wheel, R = m.T /2 = m.G.t / 2 = 2 × 20 × 5 / 2 = 100 mm ...(. TG = t) ∴ Radius of addendum circle of pinion, r = r + Addendum = 50 + 5 = 55 mm A and radius of addendum circle of wheel, R = R + Addendum = 100 + 5 = 105 mm A We know that length of the path of approach (i.e. the path of contact when engagement occurs), 22 2 ...(Refer Fig.12.11) KP=− () R R cosφ− Rsinφ A 222 = (105)−° (100) cos 20− 100 sin 20° =− 46.85 34.2= 12.65 mm and the length of path of recess (i.e. the path of contact when disengagement occurs), 222 PL=− () r r cosφ− rsinφ A 22 2 =− (55) (50) cos 20°− 50 sin 20°= 28.6− 17.1= 11.5 mm ∴ Length of the path of contact, KL = KP + PL = 12.65 + 11.5 = 24.15 mm 400 Theory of Machines and length of the arc of contact Length of path of contact 24.15 === 25.7 mm cosφ° cos 20 We know that angle turned through by pinion Length of arc of contact × 360° 25.7×° 360 === 29.45° Ans. Circumference of pinion 2π× 50 2. Maximum velocity of sliding Let ω = Angular speed of pinion, and 1 ω = Angular speed of wheel. 2 We know that pitch line speed, v = ω .r = ω .R 1 2 ∴ω = v/r = 120/5 = 24 rad/s 1 and ω = v/R = 120/10 = 12 rad/s 2 ∴Maximum velocity of sliding, = (ω + ω ) KP ...( KP PL) v S 1 2 = (24 + 12) 12.65 = 455.4 mm/s Ans. Example 12.5. A pair of gears, having 40 and 20 teeth respectively, are rotating in mesh, the speed of the smaller being 2000 r.p.m. Determine the velocity of sliding between the gear teeth faces at the point of engagement, at the pitch point, and at the point of disengagement if the smaller gear is the driver. Assume that the gear teeth are 20° involute form, addendum length is 5 mm and the module is 5 mm. Also find the angle through which the pinion turns while any pairs of teeth are in contact. Solution. Given : T = 40 ; t = 20 ; N = 2000 r.p.m. ; φ = 20° ; addendum = 5 mm ; m = 5 mm 1 We know that angular velocity of the smaller gear, 2 π N 22 π×000 1 ω= = = 209.5 rad/s 1 60 60 and angular velocity of the larger gear, ω t t 20  2  = ... ω=ω × = 209.5× = 104.75 rad/s  21 ω T T 40 1  Pitch circle radius of the smaller gear, r = m.t / 2 = 5 × 20/2 = 50 mm and pitch circle radius of the larger gear, R = m.t / 2 = 5 × 40/2 = 100 mm ∴ Radius of addendum circle of smaller gear, r = r + Addendum = 50 + 5 = 55 mm A and radius of addendum circle of larger gear, R = R + Addendum = 100 + 5 = 105 mm A The engagement and disengagement of the gear teeth is shown in Fig. 12.11. The point K is the point of engagement, P is the pitch point and L is the point of disengagement. MN is the common tangent at the points of contact. Chapter 12 : Toothed Gearing 401 We know that the distance of point of engagement K from the pitch point P or the length of the path of approach, 22 2 () cos sin KP=− R R φ− Rφ A 222 =− (105) (100) cos 20°− 100 sin 20° =− 46.85 34.2= 12.65 mm and the distance of the pitch point P from the point of disengagement L or the length of the path of recess, 22 2 PL=− () r r cosφ− rsinφ A 22 2 =− (55) (50) cos 20°− 50 sin 20°= 28.6− 17.1= 11.5 mm Velocity of sliding at the point of engagement We know that velocity of sliding at the point of engagement K, vK =ω ( +ω )P= (209.5+ 104.75) 12.65= 3975 mm/s Ans. SK 1 2 Velocity of sliding at the pitch point Since the velocity of sliding is proportional to the distance of the contact point from the pitch point, therefore the velocity of sliding at the pitch point is zero. Ans. Velocity of sliding at the point of disengagement We know that velocity of sliding at the point of disengagement L, vP =ω ( +ω )L= (209.5+ 104.75) 11.5= 3614 mm/s Ans. SL 1 2 Angle through which the pinion turns We know that length of the path of contact, KL = KP + PL = 12.65 + 11.5 = 24.15 mm KL 24.15 and length of arc of contact == = 25.7 mm cosφ° cos 20 Circumference of the smaller gear or pinion = 2 π r = 2π × 50 = 314.2 mm ∴ Angle through which the pinion turns 360° =× Length of arc of contact Circumference of pinion 360° =× 25.7 = 29.45° Ans. 314.2 Example 12.6. The following data relate to a pair of 20° involute gears in mesh : Module = 6 mm, Number of teeth on pinion = 17, Number of teeth on gear = 49 ; Addenda on pinion and gear wheel = 1 module. Find : 1. The number of pairs of teeth in contact ; 2. The angle turned through by the pinion and the gear wheel when one pair of teeth is in contact, and 3. The ratio of sliding to rolling motion when the tip of a tooth on the larger wheel (i) is just making contact, (ii) is just leaving contact with its mating tooth, and (iii) is at the pitch point.

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