1/2/2015 2:04 PM. Chapter 13. Gears General. Dr. Mohammad Suliman Abuhaiba, PE

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1 Chapter 13 Gears General 1

2 2 Chapter Outline 1. Types of Gears 2. Nomenclature 3. Conjugate Action 4. Involute Properties 5. Fundamentals 6. Contact Ratio 7. Interference 8. The Forming of Gear Teeth 9. Straight Bevel Gears 10. Parallel Helical Gears 11. Worm Gears 12. Tooth Systems 13. Gear Trains 14. Force Analysis Spur Gearing 15. Force Analysis Bevel Gearing 16. Force Analysis Helical Gearing 17. Force Analysis Worm Gearing

3 Example A gearset consists of a 16-tooth pinion driving a 40-tooth gear. The diametral pitch is 2, and the addendum and dedendum are 1/P and 1.25/P, respectively. The gears are cut using a pressure angle of 20. a. Compute the circular pitch, the center distance, and the radii of the base circles. b. In mounting these gears, the center distance was incorrectly made ¼ in larger. Compute the new values of the pressure angle and the pitch-circle diameters.

4 Straight Bevel Gears Figure Terminology of bevel gears Pitch of bevel gears is measured at the large end of tooth Both the circular pitch and the pitch diameter are calculated in the same manner as for spur gears

5 Straight Bevel Gears Shape of the teeth, when projected on back cone, is the same as in a spur gear having a radius equal to the back-cone distance r b. The number of teeth in this imaginary gear is

6 Parallel Helical Gears Used to transmit motion between parallel shafts Helix angle is the same on each gear, one gear must have a RH helix and the other a LH helix. The shape of the tooth is an involute helicoid (Fig ).

7 Parallel Helical Gears spur-gear line contact extending all the way across face of tooth line of contact is parallel to axis of rotation helical-gear Initial contact of teeth is a point that extends into a line as teeth come into more engagement. line is diagonal across face of tooth. gradual engagement of teeth and smooth transfer of load from one tooth to another that gives helical gears the ability to transmit heavy loads at high speeds.

8 Parallel Helical Gears Figure 13 22: a portion of top view of a helical rack ab & cd: centerlines of two adjacent helical teeth taken on the same pitch plane ψ = helix angle ac = transverse circular pitch p t in the plane of rotation (circular pitch) ae = normal circular pitch

9 13 10 Parallel Helical Gears 9 ad = axial pitch p x

10 13 10 Parallel Helical Gears Figure The oblique plane ab cuts out an arc having a radius of curvature of R. when ψ = 0, radius of curvature is R = D/2. If ψ slowly increases from zero to 90, R begins at a value of D/2 and increases until, when ψ = 90, R =. 10

11 Parallel Helical Gears R = apparent pitch radius of a helical gear tooth when viewed in direction of the tooth elements. A gear of same pitch and with radius R will have a greater number of teeth, because of increased radius. In helical-gear terminology this is called the virtual number of teeth.

12 12 Example 13 2 A stock helical gear has a normal pressure angle of 20, a helix angle of 25, and a transverse diametral pitch of 6 teeth/in, and has 18 teeth. Find: a. The pitch diameter b. Transverse, normal, and axial pitches c. The normal diametral pitch d. The transverse pressure angle

13 Parallel Helical Gears For a given gear ratio m G = N G /N P = m, the smallest pinion tooth count is The largest gear with a specified pinion is given by The smallest pinion that can be run with a rack is

14 13 11 Worm Gears 14 Helix angle on worm is large, and that on gear very small. For a 90 shaft angle, lead angle λ on worm = helix angle ψ G on gear

15 13 11 Worm Gears 15 Axial pitch p x of worm = transverse circular pitch p t of mating gear if shaft angle is 90 Pitch diameter of gear = diameter measured on a plane containing the worm axis, Pitch diameter of worm should be selected so as to fall into the range

16 13 11 Worm Gears 16 Lead L & lead angle λ of worm have the following relations:

17 Tooth Systems Table 13 1: Standard & Commonly Used Tooth Systems for Spur Gears

18 Tooth Systems Table 13 2: Tooth Sizes in General Uses

19 Tooth Systems Table 13 3: Tooth Proportions for 20 Straight Bevel-Gear Teeth

20 13 12 Tooth Systems Table 13 4: Standard Tooth Proportions for Helical Gears 20

21 13 12 Tooth Systems Table 13 5: Recommended Pressure Angles and Tooth Depths for Worm Gearing 21

22 Gear Trains Figure 13 26: Thrust, rotation, and hand relations for crossed helical gears

23 23

24 Gear Trains Figure 13 26: Thrust, rotation, and hand relations for crossed helical gears

25 Gear Trains Figure 13 26: Thrust, rotation, and hand relations for crossed helical gears

26 Gear Trains Figure 13 26: Thrust, rotation, and hand relations for crossed helical gears

27 27 Example 13 3 A gearbox is needed to provide a 30:1 (±1%) increase in speed, while minimizing the overall gearbox size. Specify appropriate teeth numbers

28 28 Example 13 4 A gearbox is needed to provide an exact 30:1 increase in speed, while minimizing the overall gearbox size. Specify appropriate teeth numbers

29 29 Example 13 5 A gearbox is needed to provide an exact 30:1 increase in speed, while minimizing the overall gearbox size. The input and output shafts should be in-line. Specify appropriate teeth numbers

30 Force Analysis Spur Gearing Figure 13 32

31 Force Analysis Spur Gearing Figure 13 33

32 Force Analysis Spur Gearing Figure V = pitch-line velocity, fpm d = gear diameter, in n = gear speed, rpm

33 Force Analysis Spur Gearing English SI W t = transmitted load lbf kn H = power hp kw V = pitch-line velocity ft/min d = gear diameter n = speed mm rpm

34 Example Pinion 2 in Fig a runs at 1750 rpm and transmits 2.5 kw to idler gear 3. The teeth are cut on the 20 full-depth system and have a module of m = 2.5 mm. Draw a FBD of gear 3 and show all the forces that act upon it.

35 Force Analysis Bevel Gearing r av = pitch radius at midpoint of tooth for the gear

36 Example The bevel pinion in Fig a rotates at 600 rpm in the direction shown and transmits 5 hp to the gear. The mounting distances, the location of all bearings, and the average pitch radii of the pinion and gear are shown in the figure. For simplicity, the teeth have been replaced by pitch cones. Bearings A and C should take the thrust loads. Find the bearing forces on the gearshaft.

37 Example

38 Example

39 Force Analysis Helical Gearing

40 Example In Fig a 1-hp electric motor runs at 1800 rpm in the cw direction, as viewed from the positive x axis. Keyed to the motor shaft is an 18-tooth helical pinion having a normal pressure angle of 20, a helix angle of 30, and a normal diametral pitch of 12 teeth/in. The hand of the helix is shown in the figure. Make a three-dimensional sketch of the motor shaft and pinion, and show the forces acting on the pinion and the bearing reactions at A and B. The thrust should be taken out at A.

41 Example

42 Example

12/6/2013 9:09 PM. Chapter 13. Gears General. Dr. Mohammad Suliman Abuhaiba, PE

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