Gears and Gear Train for the students of Mechanical

Gears and Gear Train

1. KINEMATICS & DYNAMICS OF MACHINES (KDM) Chapter-– Gears

2. Wheel-1 Wheel-2 Q O 1 O 2 ω 2 ω 1 T T M N Common normal C V 1 D V 2 α β α β V 1 *cos α =V 2 * cos β (ω 1 *O 1 Q ) *cos α = (ω 2 *O 2 Q ) * cos β (ω 1 *O 1 Q ) * = (ω 2 *O 2 Q ) * P LAW OF GEARING Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi The common normal at the point of contact between a pair of teeth must always pass through the pitch point . E

3. Wheel-1 Wheel-2 Q O 1 O 2 ω 2 ω 1 T T M N Common normal C V 1 D V 2 α β α β P Velocity of Sliding of Teeth Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi From similar triangles QEC and O 1 MQ, E   From similar triangles QED and O 2 NQ,   Let Vs= Velocity of sliding at Q.               The velocity of sliding is the velocity of one tooth relative to its mating tooth along the common tangent at the point of contact .

4. P C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 1 2 3 4 5 6 7 C D CYCLOID Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

5. C C 1 C 2 C 3 C 4 C 5 C 8 C 6 C 7 EPI CYCLOID : P O R r = CP + r R 360 = 1 2 3 4 5 6 7 Generating/ Rolling Circle Directing Circle Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

6. HYPO CYCLOID C P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 1 2 3 6 5 7 4 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 O OC = R ( Radius of Directing Circle) CP = r (Radius of Generating Circle) + r R 360 = Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

7. C C 1 C 2 CYCLOIDAL : P O R 1 2 3 4 5 6 7 D c b a h g f e d D 2 D 1 Face Flank P1 Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

8. Base circle of Pinion Base circle of Wheel Pitch circle of Pinion & Wheel P O 2 Wheel O 1 Pinion Addandum circles Tooth of Pinion Tooth of wheel Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

9. 1 2 3 4 5 6 7 8 P P 8 1 2 3 4 5 6 7 8 P 3 3 to p P 4 4 to p P 5 5 to p P 7 7 to p P 6 6 to p P 2 2 to p P 1 1 to p D A Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

10. 4 3 2 1 8 7 6 5 A Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

11. Base circle of Pinion Base circle of Wheel Pitch circles of Pinion & Wheel P O 2 Wheel O 1 Pinion Addendum circles A B A’ B’ Q N M ϕ Common Tangent of Pitch Circle Pressure angle ϕ ϕ Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

12. Base circle of Pinion Base circle of Wheel Pitch circle of Pinion & Wheel P O 2 Wheel O 1 Pinion Addendum circle of Wheel N M ϕ Pressure angle ϕ ϕ K L Addendum circle of Pinion R R A r A r r A = O 1 L R A = O 2 K r = O 1 P R = O 2 P LENGTH OF PATH OF CONTACT Path of approach Path of recess Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

13. P O 2 Wheel O 1 Pinion N M ϕ ϕ K L R A r A O 1 M =O 1 P cos ϕ = r cos ϕ O 2 N =O 2 P cos ϕ = R cos ϕ From right angled triangle O 2 KN KN = ((O 2 K) 2 - (O 2 N) 2 ) 1/2 KN = { (R A ) 2 - (R 2 cos 2 ϕ ) } 1/2 PN = O 2 P sin ϕ = R sin ϕ Length of the part of the path of contact, or the path of approach KP = KN –PN = {(R A ) 2 - (R 2 cos 2 ϕ ) } 1/2 - R sin ϕ From right angled triangle O 1 ML ML = {(O 1 L) 2 - (O 1 M) 2 } 1/2 ML = {(r A ) 2 - (r 2 cos 2 ϕ ) } 1/2 MP = O 1 P sin ϕ = r sin ϕ Length of the part of the path of contact, or the path of recess PL = ML – MP = {(r A ) 2 - (r 2 cos 2 ϕ ) } 1/2 - r sin ϕ Length of the part of contact, KL = LP +PK = {(R A ) 2 - (R 2 cos 2 ϕ ) } 1/2 +{(r A ) 2 - (r 2 cos 2 ϕ ) } 1/2 – ( R + r) sin ϕ

14. LENGTH OF ARC OF CONTACT Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

15. P O 2 Wheel O 1 Pinion N M ϕ ϕ K L r cos ϕ G H Arc of Approach Arc of Recess θ LENGTH OF ARC OF CONTACT θ C D CO 1 D = GO 1 H = θ Arc (CD) = r b θ = r cos ϕ θ … (2) Arc (GH) = r θ ……….(1) From Eq. (1) and (2) Arc (GH)/ Arc (CD) = r θ /r b θ =r/ r cos ϕ Arc (GH) = Arc (CD) /cos ϕ Arc (GH) = Length of path of contact /cos ϕ Arc (GH) = KL/cos ϕ Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi

16. 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 path of contact /P c Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi The contact ratio is usually a real number and it’s value lies between 1 to 2 When contact ratio is 1, it means that only one pair of teeth of two mating gears will occupy the entire arc GPH In case the contact ratio is more than 1, say its valve 1.5. It means that when one pair of teeth of two meshing gear will just entering in to engagement, the preceding pair of teeth is still in contact.

17. P O 2 Wheel O 1 Pinion N M ϕ Pressure angle ϕ ϕ K L L’ Interference in Involutes Gears Under cut Prepared by Prof. D.B.Patel Mechanical Engg. Dept LE. College, Morbi At every instance the portion of the tooth profile which are in contact must be involute so that line of action does not deviate. If any portion of the tooth profile in contact is not involute, the surfaces in contact will not touch each other tangentially causing the improper transmission of motion and power. In such case, the relative motion between tooth surface will not be proper sliding motion and thus the transmission of motion will be rough. Mating of two non-conjugate or non involute tooth profiles is known as interference.

18. P O 2 Wheel O 1 Pinion N M ϕ ϕ K L Minimum Number of Teeth on the Pinion in Order to Avoid Interference From triangle O 1 NP, R R A r A r ( O 1 N ) 2 = ( O 1 P ) 2 +( PN ) 2 -2 ( O 1 P) (PN) cos O 1 PN ( O 1 N ) 2 = r 2 +(Rsin φ ) 2 - 2 ( r) ( Rsin φ ) cos (90 + φ ) φ ( O 1 N ) 2 = r 2 +R 2 sin φ 2 + 2 ( r) (R)( sin 2 φ ) cos (90 + φ ) = - sin φ φ + 90 ( O 1 N ) 2 = r 2 { 1+(R 2 sin φ 2 )/r 2 + 2 (R)( sin 2 φ )/r } ( O 1 N ) 2 = r 2 { 1+R/r [ R/r + 2 ] ( sin 2 φ ) } ( O 1 N ) = r ( { 1+R/r [ R/r + 2 ] ( sin 2 φ ) }) 1/2 ( O 1 N ) = mt/2 ( { 1+T/t [ T/t + 2 ] ( sin 2 φ ) }) 1/2 t = No. of teeth on Pinion T = No. of teeth on Wheel m = 2r/t = 2R/T , r = mt/2 , R = mT/2 ( O 1 N ) = mt/2 ( { 1+G [ T/t + 2 ] ( sin 2 φ ) }) 1/2

19. A P .m = Addendum of the pinion, where A P is a fraction by which the standard addendum of one module for the pinion should be multiplied in order to avoid interference. We know that the addendum of the pinion = O 1 N – O 1 P A P .m = mt/2 ( {1+T/t [T/t + 2 ] ( sin φ ) }) 1/2 - mt/2 A P .m = mt/2[ ( {1+T/t [T/t + 2 ] ( sin φ ) }) 1/2 - 1] A P . = t/2[ ( {1+T/t [T/t + 2 ] ( sin φ ) }) 1/2 - 1] t = 2A P /[ ( {1+T/t [T/t + 2 ] ( sin φ ) }) 1/2 - 1] t = 2A P /[ ( {1+G [G + 2 ] ( sin φ ) }) 1/2 - 1] O 1 P = r =mt/2 Gear ratio G= T/t If the pinion and wheel have equal teeth, t = 2A P /[ ( {1+3 ( sin φ ) }) 1/2 - 1] then G = 1.

20. P O 2 Wheel O 1 Pinion N M ϕ ϕ K L Minimum Number of Teeth on the Wheel in Order to Avoid Interference From triangle O 2 MP, R R A r A r ( O 2 M ) 2 = ( O 2 P ) 2 +( PM ) 2 -2 ( O 2 P) (PM) cos O 2 PM ( O 2 M ) 2 = R 2 +(r × sin φ ) 2 - 2 ( R) ( r ×sin φ ) cos (90 + φ ) φ ( O 2 M ) 2 = R 2 +r 2 sin 2 φ + 2 ( r) (R)( sin 2 φ ) cos (90 + φ ) = - sin φ φ + 90 ( O 2 M ) 2 = R 2 { 1+(r 2 sin φ 2 )/R 2 + 2 (r)( sin 2 φ )/R } ( O 2 M ) 2 = R 2 { 1+r/R [ r/R + 2 ] ( sin 2 φ ) } ( O 2 M ) = R ( { 1+r/R [ r/R + 2 ] ( sin 2 φ ) } ) 1/2 ( O 2 M ) = mT/2 ( { 1+t/T [ t/T+ 2 ] ( sin 2 φ ) }) 1/2 t = No. of teeth on Pinion T = No. of teeth on Wheel m = 2r/t = 2R/T , r = mt/2 , R = mT/2 ( O 2 M ) = mT/2 ( { 1+1/G [ 1/G+ 2 ] ( sin 2 φ ) }) 1/2

21. A W .m = Addendum of the wheel, where A W is a fraction by which the standard addendum of one module for the wheel should be multiplied in order to avoid interference. We know that the addendum of the wheel = O 2 M – O 2 P A W .m = mT/2 ( { 1+t/T [ t/T+ 2 ] ( sin 2 φ ) }) 1/2 - mT/2 A W .m = mT/2 [( { 1+t/T [ t/T+ 2 ] ( sin 2 φ ) }) 1/2 - 1] A W . = T/2 [( { 1+t/T [ t/T+ 2 ] ( sin 2 φ ) }) 1/2 - 1] T = 2A W / [( { 1+t/T [ t/T+ 2 ] ( sin 2 φ ) }) 1/2 - 1] T = 2A W /[ ( { 1+1/G [ 1/G+ 2 ] ( sin 2 φ ) }) 1/2 - 1] O 2 P =R= mT/2 Gear ratio G= T/t If the pinion and wheel have equal teeth, t = 2A W /[ ( {1+3 ( sin φ ) }) 1/2 - 1] then G = 1.

22. Thank you