This is the second of a three-part series of instructions to assist engineering designers and detailers with the process of correctly laying out bevel and hypoid gear teeth. Each guide assumes that the basic gear tooth design information is already at hand (shaft angle, pinion and gear pitch diameters, outer cone distance, face width, pinion and gear pitch angles, face angles, root angles and the outer pinion, and gear addendums and dedendums). See the appendix at the end, and in Part 1, for the definitions of these items. This basic information is usually available in the form of a gear “dimension sheet,” which may have been calculated by your own gear engineer. As a gear design and manufacturing company, Nissei provides gear dimension sheet information to our customers as part of our service. Publications on the subject are also readily available from the AGMA (American Gear Manufacturers Association) or The Gleason Works.
To create a dimension sheet, certain information is usually required. This information includes the applied loads, speed, shaft angle, offset, gear materials, heat treatment, lubrication method, operating temperature, required life, and operating conditions affecting the driving and driven loads. An initial size with pitch, face width, tooth pressure angle, and spiral angle is chosen, and the bending stresses and contact stresses for the resulting design are calculated. The resulting estimated life is compared to the requirement. This can be an iterative procedure involving several sizing trials before a dimension sheet is finalized.
Taking the data listed on a gear dimension sheet and converting it into a drawing is the key next step. It is the connecting step between the gear tooth design and the creation of the rest of the structure — the gear blank, its bearing arrangement, input or output features and, ultimately, the housing itself.
Spiral bevel gears feature curved teeth that are set at an angle, generally 35 degrees, to the axis of the pinion and of the gear. A comparison can be made to helical gears, which also have their teeth set at an angle to their axes. Whereas helical gears are characterized by straight teeth set at an angle on an imaginary pitch cylinder, spiral bevel gears are characterized by curved teeth set at an angle on an imaginary pitch cone. Helical gears can be theoretically represented by two tangent cylinders rolling together without slipping. In a similar way, spiral bevel gears are represented by two tangent cones rolling on each other without slipping. ZerolÆ bevel gears are a special case of spiral bevel gears. The curved teeth are set at an angle of zero to 10 degrees.
Step one in drawing any bevel gear set is to lay out the pinion and gear axes at the desired shaft angle. The vast majority of bevel gears are designed for a shaft angle of 90 degrees, which we will use for our examples. The intersection point of the axes is the “pitch apex” (Figure 1).
The next step is to locate the “pitch line,” drawn as an extension of the common tangent cone element, for the gear set. The angle between the pinion axis and the pitch line is the “pinion pitch angle”; the angle between the gear axis and the pitch line is the “gear pitch angle.” The sum of the pinion pitch angle and the gear pitch angle is the “shaft angle” (Figure 2).
Next, locate the pitch point along the pitch line at the dimension given for the outer cone distance. This point simultaneously identifies the pitch diameter for the pinion and for the gear (Figure 3). Construct a normal to the pitch line through the pitch point. This locates the heel end of the teeth (Figure 4). Construct another normal at a distance along the pitch line equal to the face width to find the toe end of the teeth (Figure 5). On the normal at the outside, measure off the gear and pinion outer addendums and also the gear and pinion outer dedendums (Figure 6).
From the pinion outer addendum, and at the given pinion face angle, draw a line connecting the normal at the heel to the normal at the toe. You now have the pinion face surface for the blank. Likewise draw a line, at the gear face angle, from the gear outer addendum connecting the heel and toe normals to establish the gear face surface. In the same way draw lines at the pinion and gear root angles, from the pinion and gear outer dedendum points, connecting the normals to create the pinion and gear tooth root lines (Figure 7).
The teeth may taper in depth from the heel to the toe, or not, as is the case for “parallel depth” tooth designs. The root lines, when extended, generally do not pass through the pitch apex. The points where they cross the axes may be found on the dimension sheet as the “root apex beyond crossing point” dimension. The construction should also demonstrate another characteristic of bevel and hypoid gears in general: parallel clearance. The face line of one part should lie parallel to the root line of the mating part.
With the standardized theoretical form of the teeth now correctly identified, it is possible to start detailing the rest of the pinion and gear blank. Note that the traditional (non-FEM) gear rating calculations are generally always based on this standardized theoretical form. Consequently, blanks that vary widely from this standardized form may have significantly different stresses than predicted by the calculations. Depending on the modifications, the parts may be weaker or stronger than estimated (Figure 8).
It is not uncommon with fine pitch parts to see blanks with design changes to the standardized form such as those shown in the following graphics. Many times the gear member outside diameter is trimmed off at the pitch diameter. In the following example, the pinion outside diameter is shown trimmed at the theoretical outside diameter of the teeth. It is undesirable on a high ratio to trim the pinion outside diameter to the pitch diameter dimension, as is sometimes done to pass through a small diameter bearing. In such a case, a large amount of the working area of the pinion tooth can be removed, compromising the design (Figure 9). Trimming the blanks as shown increases the root line face width. This can provide some additional bending strength.
A caution when making blank form changes is related to the practice of drawing the pinion and gear independently. When introducing changes to the theoretical form, the pinion and gear teeth should be drawn together to judge the effect on the mutual tooth contact position. In the following example, the blanks are trimmed and the mating of the teeth results in a contact pattern placement dilemma. The pattern is in the middle of the pinion face width, but on the toe end of the gear face width. Achieving the usually desired “central toe” contact on both members is not possible. The contact placement issue for this gear set was not discovered until the parts were manufactured and run together in the test machine (Figure 10).
The potential for the occurrence of this mismatch is not readily evident if the gear and pinion are drawn separately. The mismatch results in poor positioning of the load on one member. Usually the localized tooth contact is placed “central toe” on both pinion and gear. If the localized tooth contact was placed central toe on this example pinion member, the contact would run off the ends of the gear member teeth. This is generally undesirable for reasons of noise, vibration, and load concentration.
We hope this article has been helpful for the design of your bevel gear sets. As stated in last month’s installment, hopefully you can now move forward with the rest of your application design and communicate your needs more effectively with your gear suppliers. The final layout guide to be presented in this series, on hypoid gearing, will appear in the March issue of Gear Solutions magazine.
Appendix
Bevel Gear Terminology: Hand of Spiral
A left-hand gear always mates with a right-hand pinion. A right-hand gear always mates with a left-hand pinion. The “rule of thumb curvature” shown here can be applied to either the pinion or the gear and works for Zerol bevel, spiral bevels, and hypoids. The gear or pinion is positioned so you are looking down the axis with the teeth toward you and with your hand facing palm-down. (More in Part 1.)
