Gear Geometry and Applied Theory Episode 2 Part 7 pps

P1: JTH

CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51

16.5 Design of Crossed Helical Gears 463

The new shortest center distance is

Eo = ro1 + ro2 = 116.1537 mm.

The new crossing angle is

γo = βo1 + βo2 = 91.0055◦

.

The new radii of addendum and dedendum cylinders:

roa1 = ro1 + mon = 40.2473 mm

roa2 = ro2 + mon = 83.9760 mm

rod1 = ro1 − 1.25mon = 31.1690 mm

rod2 = ro2 − 1.25mon = 74.8977 mm.

It is easy to verify that Eq. (16.B.10) is satisfied for the obtained parameters of non￾standard crossed helical gears.

Numerical Example 3: Approach 2 for Design of Nonstandard

Crossed Helical Gears

Numerical example 2 (Approach 1) of design of nonstandard gears has shown that the

crossing angle of the drive is slightly changed in comparison with the crossing angle of

a similar design of standard gears. The main goal of Approach 2 of design is to keep the

same crossing angle that is applied in a similar design of standard gears. The approach

is based on the following considerations:

(i) The assigned crossing angle γo = γp and the gear ratio m12 have to be observed.

(ii) Module mpn and normal pressure angle αpn of the common rack-cutter are given.

(iii) Settings of rack-cutter χ1 and χ2 for the pinion and the gear are applied respectively,

and the tooth thicknesses of the pinion and gear must fit each other.

The observation of the assigned crossing angle of the gear drive is satisfied by modifica￾tion of the skew angles of the rack-cutters. The computational procedure is an iterative

process accomplished as follows.

Step 1: Determination of parameters on the pitch cylinders as a function of βp1 and

βp2:

r pi = mpnNi

2 cos βpi

(i = 1, 2)

αpti = arctan

tan αpn

cos βpi

(i = 1, 2)

s pti = πmpn

2 cos βpi

+ 2χi mpn tan αpti (i = 1, 2).

P1: JTH

CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51

464 Involute Helical Gears with Crossed Axes

Step 2: Determination of parameters on the base cylinders:

rbi = r pi cos αpti (i = 1, 2)

λbi = arctan

1

tan βpi cos αpti

(i = 1, 2)

sbti = rbi s pti

r pi

+ 2invαpti

(i = 1, 2).

Step 3: Determination of parameters on the operating pitch cylinders:

cos αon = (cos2 λb1 ± 2 cos λb1 cos λb2 cos γo + cos2 λb2)

0.5

sin γo

roi = rbi sin λbi



cos2 αon − cos2 λbi

(i = 1, 2)

λoi = arctan

rbi tan λbi

roi

(i = 1, 2)

αoti = arccos

rbi

roi

(i = 1, 2)

soti = roi sbi

rbi

− 2invαoti

(i = 1, 2)

mon = 2ro1 sin λo1

N1

.

Step 4: Determination of the following functions:

f1 = rb2 sin λb2

rb1 sin λb1

− m12

f2 = sot1 sin λo1 + sot2 sin λo2 − πmon.

The iterative process for determination of βp1 and βp2 is applied as follows:

(i) Initially, in the first iteration, the applied magnitudes βp1 and βp2 are the same as in

standard design. Generally, the equations of Step 4 are not satisfied simultaneously.

(ii) In the process of iterations, βp1 and βp2 are changed and steps 1, 2, and 3 are

repeated until observation of Eqs. f1 = 0 and f2 = 0.

The computations have been applied for the following example. The settings of the

rack-cutters are χ1 = 0.3mpn, χ2 = 0.2mpn. The crossing angle γo = γp = 90◦. The

iterative process yields:

βp1 = 46.9860◦

, βp2 = 42.0010◦

.

Using the equations from Step 1 to Step 3 all the parameters of the gear drive can be

determined. The new center distance is

Eo = ro1 + ro2 = 115.1898 mm.

The assigned crossing angle γo = 90◦ is observed because

γo = 180◦ − λo1 − λo2 = 180◦ − 42.4631◦ − 47.5369◦ = 90.0000◦

.

P1: JTH

CB672-16 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:51

16.6 Stress Analysis 465

16.6 STRESS ANALYSIS

The goal of stress analysis presented in this section is determination of contact and

bending stresses and the investigation of formation of the bearing contact in a crossed

helical gear drive formed by an involute helical worm that is in mesh with an involute

helical gear. A similar approach may be applied for stress analysis in a gear drive formed

by mating helical gears. The performed stress analysis is based on the finite element

method [Zienkiewicz & Taylor, 2000] and application of a general purpose computer

program [Hibbit, Karlsson & Sirensen, Inc., 1998]. The developed approach for the

finite element models is described in Section 9.5.

Numerical Example

Finite element analysis has been performed for a gear drive formed by an involute worm

and an involute helical gear. The applied design parameters are the same as those shown

in Table 16.3.1, but an involute worm and not an Archimedes’ worm is considered

in this case. Therefore, transmission errors do not occur. The output from TCA [see

Figs. 16.6.1(a) and 16.6.1(b)] and the developed approach for the finite element models

automatically builds one model for every point of contact.

Figure 16.6.2 shows a three-tooth model of an involute worm. Figure 16.6.3 shows the

finite element model of the whole worm gear drive. A three-tooth model has been applied

for finite element analysis at each chosen point of the path of contact (Fig. 16.6.4). An

angle of 60◦ has been applied to delimit the worm gear body. Elements C3D8I of first

order (enhanced by incompatible modes to improve their bending behavior) [Hibbit,

Karlsson & Sirensen, Inc., 1998] have been used to form the finite element mesh. The

total number of elements is 59,866 with 74,561 nodes. The material is steel with the

properties of Young’s Modulus E = 2.068 × 105 MPa and Poisson’s ratio of 0.29. A

torque of 40 Nm has been applied to the worm.

Figure 16.6.1: Paths of contact on (a) the worm and (b) the gear.