ME 452: Machine Design II

Fall Semester 2004

Solution to Homework Set 9

Problem 14-15.

Since the pinion and the gear of the given spur gearset have the same material properties then the

power that the gearset can transmit is governed by the failure of the pinion. The problem is to

determine: (i) the power that can be transmitted considering pinion tooth bending failure, and (ii) the

power that can be transmitted considering pinion tooth wear. The power rating of the gearset will then

be the minimum of these two power ratings (for the specified design factor of 2.25).

Consider bending of the pinion tooth.

The diameter of the pinion, see Equation (13-1), page 666, can be written as

where is the diametral pitch; i.e., number of teeth per inch. Therefore, the diameter of the pinion is

The pitch line velocity, see Example (14-1), page 728, can be written as

Therefore, the pitch line velocity is given as

The uncorrected endurance strength for , see Equation (7-8), page 325, can be written as

Therefore, the uncorrected endurance strength is given as

1

The surface factor, see Equation (7-18), page 329 can be written as

For milled (machined) teeth, see Table (7-4), page 329, the factor and the exponent are given as

Therefore, the surface factor is given as

Assuming full depth teeth, the sum of the addendum and dedendum, see Figure (14-1), page 725, and

Table (13-1), page 686, can be written as

The tooth thickness, see Equation (b), page 725 and Figure (14-1), page 725, can be written as

where the value of the distance , see Equation (14-3), page 725, can be written as

The Lewis form factor for the pinion with 17 teeth, see Table (14-2), page 726, is

Therefore, the distance for the pinion is

Therefore, the tooth thickness for the pinion is

The effective diameter for bending, see Equation (7-24), page 330, can be written as

where and . Therefore, the effective

diameter is

2

The size factor, see Equation (7-19), page 329, can be written as

The loading factor for bending, see Equation (7-25), page 331, is

The temperature factor, assuming operation at room temperature, see Table (7-6), page 332, is

The reliability factor, assuming 50 % reliability, see Table (7-7), page 334, is

Two effects are used to evaluate the miscellaneous-effects factor (see Example 14-2, page 728).

The first of these is the effect of one-way bending. The second effect to be accounted for in using the

miscellaneous-effects factor is stress concentration. The miscellaneous effects factor for one-way

bending, see Example 14-2, page 728, is

The miscellaneous effects factor due to stress concentration, see Example 14-2, page 728, is

where is the fatigue stress concentration factor. The tooth fillet radius for a full-depth tooth, see

Figure (14-1), page 725 and Example 14-2, page 728, is

The ratio in Figure A-15-6, page 983, is

The ratio in Figure A-15-6, page 983, is equal to infinity for a gear tooth. Since Figure A-15-6

does not have a curve for then approximate . Therefore, the theoretical stress

concentration factor from Figure A-15-6 is

3

The fatigue stress concentration factor, see Equation (7-35), page 337, can be written as

where the value of the Neuber constant for a shoulder, see Table (7-8), page 337, is

Therefore, the fatigue stress concentration factor is

Finally, the miscellaneous-effects factor due to stress concentration is

The final value of the miscellaneous-effects factor, see Example 14-2, page 728, is

The fully corrected endurance strength, see Equation (7-17), page 328, is

Therefore, the fully corrected endurance strength is

The allowable bending stress for a design factor of 2.25, see Example 14-2, page 728, can be written

as

The tangential component of the load, see Equation (14-7), page 727 and Example 14-2, page 728, can

be written as

4

The dynamic (or velocity) factor for milled teeth, see Equation (14-4b), page 727, can be written as

Therefore, the dynamic factor is

Therefore, the tangential component of the load is

The transmitted power (with bending of pinion teeth as the design criterion), see Example 14-2, page

728, can be written as

Therefore, the transmitted power (with bending of pinion teeth as the design criterion) is

Pinion Tooth Wear.

The surface stress (i.e., Hertzian contact stress), see Equation (14-14), page 732, can be written as

1/ 2

,1 2

1 1

cos

tv

C all p

K WC

F r r

The elastic coefficient , for a steel pinion in mesh with a steel gear, see Table 14-8, page 745, is

The radii of curvature of the tooth profiles at the pitch point, see Equation (14-12), page 732, are

Therefore the radii of curvature of the tooth profiles at the pitch point are

5

and

The surface endurance strength, see Equation (7-68), page 372, can be written as

Therefore, the surface endurance strength is

The allowable contact stress, see Example 14-3, page 732, can be written as

Therefore, the allowable contact stress is

Note that the negative sign is because is a compressive stress.

The tangential component of the load, see Equation (14-14), page 732, can be written as

Therefore, the tangential component of the load is

The transmitted power, see Example 14-2, page 728, can be written as

Therefore, the transmitted power (with wear of pinion teeth as the design criterion) is

Conclusion: The power rating considering pinion tooth wear is lower than the power rating

considering pinion tooth bending. Therefore, the power rating of the spur gearset is

6

Problem 14-19.

The diameter of the pinion, see Equation (13-1), page 666, can be written as

Therefore, the diameter of the pinion is

Similarly, the diameter of the gear, see Equation (13-1), page 666, can be written as

Therefore, the diameter of the gear is

The pitch line velocity, see Example (14-1), page 728, can be written as

Therefore, the pitch line velocity is

The transmitted power, see Example 14-2, page 728, can be written as

Therefore, the tangential component of the load is given as

The AGMA bending stress equation, see Equation (14-15), page 734, can be written as

The overload factor, assuming uniform loading, see page 746, is

7

The dynamic factor, see Equation (14-27), page 744 can be written as

where the exponent , see Equation (14-28), page 744 is

Therefore, for the specified transmission accuracy-level number, of 6, the exponent is

The factor , see Equation (14-28), page 744, can be written as

Therefore, the factor is

Therefore, the dynamic factor is

The size factor, see Equation (a), page 747, can be written as

The Lewis form factor for the pinion with 16 teeth, see Table (14-2), page 726, is

The Lewis form factor for the gear with 48 teeth, see Table (14-2), page 726, using linear interpolation

is

Therefore, the size factor for the pinion is

Similarly, the size factor for the gear is

8

0.0535 0.0535

2 0.40561.192 1.192 1.097

6G

S G

F YK

P

The load distribution factor, see Equation (14-30), page 747, can be written as

The load correction factor for uncrowned teeth, see Equation (14-31), page 748, is

The pinion proportion factor for a face width of , see Equation (14-32), page 748, is

Therefore, the pinion proportion factor is

Assuming in Figure (14-10), page 748, then from Equation (14-33), page 748, the pinion

proportion modifier is

The mesh alignment factor , see Equation (14-34), page 748, can be written as

For commercial enclosed unit, see Table (14-9), page 748, the factors are

Therefore, the mesh alignment factor is

The mesh alignment correction factor , see Equation (14-35), page 748, is

Therefore, the load distribution factor is

9

Assuming the backup ratio , the rim thickness factor, see Equation (14-40), page 752 is

The geometry factor for bending strength for the pinion with 16 teeth from Figure (14-6), page 741,

assuming that the load is applied at the highest point of single-tooth contact is

The geometry factor for bending strength for the gear with 48 teeth from Figure (14-6), page 741,

assuming that the load is applied at the highest point of single-tooth contact is

The AGMA bending stress for the pinion can then be written as

Therefore, the AGMA bending stress for the pinion is

The AGMA bending stress for the gear can be written as

Therefore, the AGMA bending stress for the gear is

The safety factor guarding against bending fatigue failure, see Equation (14-41), page 753, can be

written as

From Figure (14-2), page 735, the AGMA bending strength for the pinion and the gear made of Grade

1 steel can be written as

10

Therefore, the AGMA bending strength for the pinion and the gear are

The pinion life is specified to be cycles. The gear ratio of the given gearset is

Therefore the life of the gear in cycles is cycles.

The stress cycle factor for bending strength for the pinion, see Figure (14-14), page 751, with

cycles and considering the upper curve in the shaded region is

The stress cycle factor for bending strength for the gear, see Figure (14-14), page 751, with

cycles and considering the upper curve in the shaded region is

The reliability factor, see Table (14-10) page 752, for a reliability of 0.9 is

The temperature factor, assuming oil and gear-blank temperatures less than , see page 752, is

Then the safety factor guarding against bending fatigue failure for the pinion can be written as

Therefore, the safety factor guarding against bending fatigue failure for the pinion is

The safety factor guarding against bending fatigue failure for the gear can be written as

Therefore, the safety factor guarding against bending fatigue failure for the gear is

11

The AGMA pitting resistance (contact stress) equation, see Equation (14-16), page 734, can be written

as

The elastic coefficient , for a steel pinion in mesh with a steel gear, see Table (14-8), page 745, is

The surface condition factor, , see page 746 is assumed to be equal to 1.

The geometry factor for pitting resistance, see Equation (14-23), page 743, for an external gearset can

be written as

For spur gears, see page 743, the load-sharing ratio and .

Therefore, for the given spur gearset the geometry factor for pitting resistance is

Then, the AGMA contact stress for the pinion is

Therefore, the AGMA contact stress for the pinion is

The AGMA contact stress for the gear is

12

Therefore, the AGMA contact stress for the gear is

The safety factor guarding against pitting failure, see Equation (14-42), page 753, can be written as

From Figure (14-5), page 738, the AGMA surface endurance strength for the pinion and the gear made

of Grade 1 steel can be written as

Therefore, the AGMA surface endurance strength for the pinion and the gear are

The stress cycle factor for pitting resistance for the pinion, see Figure (14-15), page 751, with

cycles and considering the upper curve in the shaded region is

The stress cycle factor for pitting resistance for the gear, see Figure (14-15), page 751, with

cycles and considering the upper curve in the shaded region is

Then the safety factor guarding against pitting failure for the pinion is

Therefore, the safety factor guarding against pitting failure for the pinion is

13

The safety factor guarding against pitting failure for the gear is

Therefore, the safety factor guarding against pitting failure for the gear is

14