Bevel, Helical and Worm Gears(1)

Straight Bevel gears

The most familiar kinds of bevel gears have pitch angles of less than 90 degrees. Therefore it is cone-shaped. This type of bevel gear is called external because the gear teeth point is outward. The pitch surfaces of the meshed external bevel gears are coaxial with the gear shafts; the apexes of that two surfaces are at point of intersection of given shaft axes.

Bevel gears that have pitch angles of greater than 90 degrees have teeth that it point inward and called internal bevel gears. Bevel gears that have pitch angles of exactly 90 degrees have teeth that point outward parallel with the axis and resemble the points on a crown. So this type of bevel gear is called a crown gear. Straight bevel gears have conical pitch surface and teeth are straight and tapering towards apex.

Straight bevel gears are the simplest type of bevel gears that transfer power between intersecting axes. They are widely used in low-speed applications or static-loading conditions. Differential gears are one of such application where speed is very low and the load type is mainly static. There are still several traditional applications of the straight bevel gears in aerospace, marine, agriculture and construction field. These have a wide range of shaft angles i.e. in marine applications. However right angle is the most common.

Terms used in Bevel Gears

Fig. Terms used in bevel gears.

A sectional view of two bevel gears in mesh is shown in fig. The following terms in connection with the bevel gears are important.

Bevel Gears Terms

• Pitch cone: It is a cone containing pitch elements of teeth.
• Cone centre: It is the apex of pitch cone. It may be defined as that point where axes of two mating types of gears intersect each other.
• Pitch angle: It is the angle made by the pitch line with the axis of shaft. It is denoted by ‘θ_P’.
• Cone distance: It is the length of pitch cone element. It is also called a pitch cone radius. It is denoted by ‘O_P’.

Mathematically,

cone distance or pitch cone radius,

\[OP = \frac{{{\rm{Pitch radius}}}}{{\sin {\theta _P}}} = \frac{{\frac{{{D_P}}}{2}}}{{\sin {\theta _P}_1}} = \frac{{\frac{{{D_G}}}{2}}}{{\sin {\theta _P}_2}}\]

• Addendum angle: It is the angle subtended by the addendum of the tooth at the cone centre. It is denoted by ‘α’.

Mathematically,

addendum angle,\[\alpha = {\tan ^{ - 1}}\left( {\frac{a}{{OP}}} \right)\]

where a = Addendum, and OP = Cone distance.

• Dedendum angle: It is angle subtended by dedendum of the tooth at cone centre. It is denoted by ‘β’. Mathematically,

dedendum angle,\[\beta = {\tan ^{ - 1}}\left( {\frac{d}{{OP}}} \right)\]

where d = Dedendum, and OP = Cone distance.

• Face angle: It is angle subtended by face of the tooth at cone centre. It is denoted by ‘φ’. The face angle is equal to pitch angle plus addendum angle.
• Root angle: It is angle subtended by the root of tooth at cone centre. It is denoted by ‘θR’. It is equal to pitch angle minus dedendum angle.
• Back (or normal) cone: It is an imaginary cone normal to pitch cone at end of the tooth.
• Back cone distance: It is the length of back cone. It is denoted by ‘R_B’. It is also called back cone radius.
• Backing: It is the distance of pitch point (P) from back of boss parallel to the pitch point of the gear. It is denoted by ‘B’.
• Crown height:It is the distance of crown point (C) from cone centre (O) parallel to the axis of gear. It is denoted by ‘H_C’.
• Mounting height: It is the distance of back of the boss from cone centre. It is denoted by ‘H_M’.
• Pitch diameter: It is the diameter of largest pitch circle.
• Outside or addendum cone diameter: It is the maximum diameter of teeth of gear. It is equal to diameter of the blank from which gear can be cut.

Mathematically,

outside diameter,\[{D_0} = {D_P} + 2a\cos {\theta _P}\]

where D_P = Pitch circle diameter,

a = Addendum, and

θ_P= Pitch angle.

• Inside or dedendum cone diameter: The inside or the dedendum cone diameter is given by

\[{D_d} = {D_P} - 2d\cos {\theta _P}\]

where Dd = Inside diameter, and

d = Dedendum.

Forces Acting on a Bevel Gear

Fig: Forces acting in bevel gear

Consider a bevel gear and pinion in mesh as shown in fig. The normal force (W_N) on the tooth is perpendicular to the tooth profile and so that makes an angle equal to the pressure angle (φ) to pitch circle. Thus normal force can be resolved into two components i.e. one is tangential component (W_T) and the other is the radial component (W_R). The tangential component i.e. the tangential tooth load produces the bearing reactions while the radial component produces end thrust in the shafts. The magnitude of tangential and radial components are as follows :

W_T = W_N cos φ, and W_R = W_N sin φ = W_T tan φ ...(i)

These forces are considered to act at mean radius (R_m). From the geometry of the fig,

We find that

\[{R_m} = \left( {L - \frac{b}{2}} \right)\sin {\theta _{P1}} = \left( {L - \frac{b}{2}} \right)\frac{{{D_P}}}{{2L}}\]

Now radial force (W_R) acting at the mean radius may be further resolved into two components, WRH and WRV, in the axial and radial directions as shown in fig. Therefore the axial force acting on pinion shaft,

\[{{\rm{W}}_{RH}}{\rm{ = }}{{\rm{W}}_R}\sin {\theta _{P1}} = {W_T}\tan \varphi .\sin {\theta _{P1}}\]

and the radial force acting on the pinion shaft,

\[{{\rm{W}}_{RV}}{\rm{ = }}{{\rm{W}}_R}\cos {\theta _{P1}} = {W_T}\tan \varphi .\cos {\theta _{P1}}\]

A little supposition will show that the axial force on pinion shaft is equal to the radial force on gear shaft but their directions are opposite. Similarly the radial force on pinion shaft is equal to the axial force on gear shaft but act in opposite directions.

Bevel Gear tooth proportions

Table: Bevel gear tooth proportion

References:
1. H.H. Mabie and C. F. Reinholtz, “Mechanism and Dynamics of Machinery”, Wiley.
2. J.S. Rao & R.V. Dukkipati Mechanisms and Machine Theory, New Age International (P) Limited..
3. J.E. Shigley and J.J. Uicker, Jr., “ Theory of Machines and Mechanisms”, McGraw Hill.
4. B. Paul, “Kinematics and Dynamics of Planar Machinery”, Prentice Hall.
5. C. E. Wilson, J.P. Sadler and W.J. Michels, “Kinematics and Dynamics of Machinery”, Harper Row.