Spiral and Straight Bevel Gears for machine designers

The standard definition of a Bevel Gear is a cone-shaped gear which transmits power between 2 intersecting axels.

Looking at bevel gears from the differences in helix angles, they can be generally classified into straight bevel gears, which do not have helix angles, and spiral bevel gears (including zerol bevel gears), which do have helix angles. However, because of the fact that manufacture facilities for straight bevel gears are becoming rare and the fact that straight bevel gears teeth cannot be polished, making spiral bevel gears which can be polished superior in terms of noise reduction, spiral bevel gears are likely to become more common in the future.

Bevel gears can be generally classified by their manufacturing methods, namely the Gleason method and Klingelnberg method, which each have differing teeth shapes, and presently most gears use the Gleason method. Incidentally, all gears manufactured by KHK use the Gleason method.

Furthermore, there are also variations in gears in terms of teeth pitch (modules, etc.), whether polished or not, and materials used. For example in the case of materials, S45C of machine structural carbon steel, SCM415 of machine structual alloy steel and MC901 of engineering plastic, etc. are often used, and duracon, etc. are used for plastic molded parts.

button of gear selector

Related links :
Miter Gears – Main page of miter gears
Zerol Bevel Gears – Main page of zerol bevel gears
Hypoid Gears – Main page of hypoid gears
Equivalent tables of each standard relating to raw materials and precision grades of gears
锥齿轮 – 中文页

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Bevel Gears

This article is reproduced with the permission.
Masao Kubota, Haguruma Nyumon, Tokyo : Ohmsha, Ltd., 1963.


The gears used when two shafts intersect are based on two cones in rolling contact with apexes meeting at the point of intersection of the two axes and having teeth at the same distance from the apexes. These are called bevel gears. Above mentioned cones are called pitch cones and their half peak angles are called pitch cone angles.

Pitch Angles of Bevel Gears
Figure 8.1    Pitch Angles of Bevel Gears

In Figure 8.1, assume the shaft angle to be Σ, the respective numbers of teeth  zi (i = 1, 2 ), angular speed ωi, and pitch cone angle (or simply pitch angle) ɣ0i , then consider the rotating speed of a point on the common contact line of the cones at distance K from the apex :

ω1 K sin ɣ01= ω2 K sin ɣ02
ɣ01 + ɣ02 = Σ,     angular speed ratio ω12  = z2/z1

tan ɣ01 = sin Σ / [(z2/z1) + cos Σ ] ,       tan ɣ02 = sin Σ / [(z1/z2) + cos Σ ]

Normally, Σ = 90° so that :
tan ɣ01 = z1/z2,     tan ɣ02 = z2/z1   and   ɣ01 + ɣ02 = 90°

In particular, when ɣ01 + ɣ02 = 45°, the bevel gear is called a miter gear. Furthermore, when Σ ≠ 90°, as shown in Figure 8.2, it is called an angle gear.

Angle Gear

Figure 8.2    Angle Gear

When the large gear has a pitch angle of 90°, it is called a crown gear. It is equivalent to a rack in spur gear and becomes the base for tooth form and tooth cutting.

Bevel gears are divided into straight bevel gears and spiral bevel gears based on their tooth lines at the pitch cone. Conical gears and face gears can also be considered as belonging to the spiral bevel gear group. Because they are not based on a pitch cone and rely on a specialized tooth cutting method however, they are discussed separately from spiral bevel gears.

Efficiency of Bevel Gear

Because most bevel gears are intersecting shaft gears, their mesh is almost always rolling contact, therefore their general efficiency is high, typically 98-99%.

Straight Bevel Gears


Bevel gears are cone shaped gears which transmit motion between two intersecting shafts. Straight bevel gears are the simplest of these bevel gears with their teeth being straight and pointing toward the apex of the cone. They are easier than spiral bevel gears to make and do not produce inward thrust (in the minus direction), simplifying bearing construction. On the other hand, they have the disadvantage of not being able to grind teeth after heat treatment.

Straight bevel gears are divided into two groups: profile shifted Gleason type and non-profile shifted ones called standard type or Klingelnberg type. Over all, the Gleason system is presently the most widely used. In addition, the Gleason Company’s adoption of the tooth crowning method called Coniflex gears produces gears that tolerate slight assembly errors or shifting due to load and increases safety by eliminating stress concentration on the edges of the teeth.

Straight bevel gears are generally used in relatively slow speed applications (less than 2m/s circumferential speed). They are often not used when it is necessary to transmit large forces. Generally they are utilized in machine tool equipment, printing machines and differentials.

Spiral Bevel Gears


Bevel gears are cone shaped gears which transmit motion between two intersecting shafts. Spiral bevel gears are one type in which the teeth are curved spirally. Unlike straight bevel gears, these teeth contact each other gradually and smoothly from one end to the other. The meshing of teeth are, as in straight bevel gears, rolling contacts on the pitch cone surface.

With regard to design and gear cutting, just as in straight bevel gears, the Gleason type is most widely used in spiral bevel gears. However, in Germany, the Klingelnberg type with equal toe and heel tooth depth is still deeply rooted in use.

Spiral bevel gears have the advantage of being able to grind teeth after heat treatment, making it possible to produce high precision gears. Also, because the teeth contact ratio is higher than with straight bevel gears, noise and vibration are reduced and they are better suited for high speed applications. For example, noise and vibration are markedly reduced at high operating speed (more than 10m/s). They are also stronger and more durable than straight bevel gears allowing for higher load operations. On the other hand, it is more difficult to manufacture
spiral bevel gears and needs attention regarding change in thrust directions depending on the rotation and twist angle. These are some of the disadvantages.

In use, the right-hand spiral is mated with the left-hand spiral. As for their applications, they are frequently used in automotive speed reducers and machine tools.

Tooth Shape of Spiral Bevel Gears

This article is reproduced with the permission.
Masao Kubota, Haguruma Nyumon, Tokyo : Ohmsha, Ltd., 1963.

Spiral bevel gears are gears that have the teeth arranged on a pitch cone along curved lines which produces a quiet operation even at high speed.  Especially when the peripheral velocity exceeds 5 m/s, it is difficult to achieve a quiet operation and use of spiral bevel gears are considered desirable.

Types of Spiral Bevel Gears (Tooth Lines of Crown Gear)

(a) Straight / (b) Circular Arc / (c) Involute
Figure 8.13  Types of Spiral Bevel Gears (Tooth Lines of Crown Gear)

The tooth form line is determined based on the standard crown gear tooth form (the intersection of the crown tooth surface and pitch surface). If this is considered as the logarithmic swirling line, the tilt angle of the tooth form is constant regardless of the radius which is most desirable from a tooth meshing consideration. However, for tooth cutting, it is not convenient and several curves more suitable for cutting are in actual use. As shown in Figure 8.13, these are tilted straight line (Reinecker form) [these are sometimes called helical bevel gears], circular arc (Gleason form), and involute (Klingelnberg form). Besides these, there are trochoid (Oerlikon form, Fiat form) and Archimedes spiral, etc.

In particular, as shown in Figure 8.14, when the tooth form is a circular arc and at the midpoint of the tooth form, the tilt angle is 0 is called Zerol gear. While the loading of the Zerol tooth is similar to the straight tooth, the meshing is smoother. All the circular arc gears other than Zerol gears are sometimes called helical bevel gears.

Zerol Bevel Gear
Figure 8.14  Zerol Bevel Gear

At the point on the tooth line where it intersects the pitch cone generating line at angle β, if the perpendicular cross section is drawn, then the equivalent spur gear’s number of teeth zvi is

z vi  = zi / cos ϓ 0i cos 3 β

and the normal to tooth surface pressure angle αn relates to the spherical surface pressure angle αs as

tan αs = tan αn / cos β

The twist direction of the teeth is, when looking from the small end of the teeth, if the teeth curve clockwise, it is a right spiral and if the teeth curve counterclockwise, it is a left spiral. For mating curved bevel gears, if one gear is right spiral, the opposite gear is left spiral.

Conical gear

This article is reproduced with the permission.
Masao Kubota, Haguruma Nyumon, Tokyo : Ohmsha, Ltd., 1963.

The combination of helical racks which move straightly using crown gears, or cone shape gears guided through helical rack are called conical gear. Each gear is thought to be similar to helical gear in pic 8.27, whose addendum modification changes are oriented in the axial direction. It makes point-contact where contact lines of the mediating rack and each gear cross. Conical gearing is sometimes used instead of bevel gearing when the load is small because they can be cut by modified hob machinery or gear shavers, or by attaching auxiliary equipment. In case of parallel axes, it becomes a  conical shifted gear (see p.93) and makes point-contact.

Conical gear
Pic 8.27 Conical gear
1) Cut edge of tooth by pitch cylinder

Manufacturing of Bevel Gears