Bevel, Helical and Worm Gears(3)

Worm gearing

The worm gearing is classified as non-interchangeable, because a worm wheel cut with a hob of one diameter will not operate satisfactorily with a worm of given different measured diameter even if the thread pitch is same.

Types of Worms

The following are two types of worms

• Cylindrical or straight worm and
• Cone or double enveloping worm

Fig: Types of worm

The cylindrical or straight worm, as shown in fig (a) is most commonly used. The shape of the thread is involute helicoid of pressure angle of 14 ½° for given single and double threaded worms and 20° for that triple and quadruple threaded worms. The worm threads are cut by straight sided milling cutter which having its diameter not less than the outside diameter of worm or greater than that 1.25 times of the outside diameter of given worm. The cone or double enveloping worm as shown in fig(b) which is used to some extent but it need extremely accurate alignment.

Types of Worm Gears

The following are three types of worm gears are most important from the subject point of view.

1. Straight face worm gear as shown in fig (a),
2. Hobbed straight face worm gear as shown in fig (b) and
3. Concave face worm gear as shown in fig (c)

Fig. Types of worms gears.

The straight face worm gear is like a helical gear in which the straight teeth are cut with that of form cutter. Since it has only one point contact with the worm thread. Therefore it is used for light service parts. The hobbed straight face worm gear also used for light service but its teeth are cut with a hob after which the outer surface is turned. The concave face worm gear is the accepted standard form and is used for all heavy service and normal industrial uses. The teeth of this type of gear are cut with a hob of the same pitch diameter as the mating worm to increase the contact area.

Terms used in Worm Gearing

Fig: Terms used in Worm Gearing

1. Axial pitch: It is also known as linear pitch of a worm. It is the distance measured axially i.e. parallel to the axis of worm from a point on one thread to the corresponding point on adjacent thread on worm, as shown in fig. It is noted that the axial pitch (pa) of a worm is equal to the circular pitch (Pc) of mating worm gear type when the shafts are at right angles.
2. Lead: It is linear distance through which a point on the thread moves ahead in one revolution of the worm. For single start threads lead is equal to axial pitch but for multiple start threads the lead is equal to product of axial pitch and number of starts.

Mathematically,

Lead (l) = P_a.n

Where P_a = Axial pitch

And n = Number of starts.

3. Lead angle: It is angle between the tangent to thread helix on pitch cylinder and plane normal to the axis of worm. It is represented by λ. A little supposition will show that if one complete turn of a worm thread be imagined to be unwound from body of worm. It will form an inclined plane whose base is equal to the pitch circumference of worm and altitude equal to lead of worm that shown in fig.

Fig: Development of a helix thread

From the geometry of the figure, we find that

\[\begin{array}{l}
\tan \lambda = \frac{{{\rm{Lead of the worm}}}}{{{\rm{Pitch circumference of the worm}}}} = \frac{l}{{\pi {D_w}}}\\
{\rm{ }} = \frac{{{P_a}.n}}{{\pi {D_w}}} = \frac{{{P_c}.n}}{{\pi {D_w}}} = \frac{{\pi m.n}}{{\pi {D_w}}} = \frac{{m.n}}{{{D_w}}}\\
where{\rm{ }}m = Module\\
{\rm{ }}{D_w} = {\rm{Pitch circle diameter of worm}}
\end{array}\]

4. Tooth pressure angle: It is measured in a plane containing the axis of worm and is equal to one-half thread profile angle. For automotive applications the pressure angle of 30° is appreaciated to obtain a high efficiency and to permit overhauling.
5. Normal pitch: It is the distance measured along the normal to threads between the two corresponding points on two adjacent threads of worm.

Mathematically,

Normal pitch (p_N) = p_a.cos λ

6. Helix angle: It is the angle between tangent to thread helix on pitch cylinder and axis of worm. It is represented by αW as in fig. The worm helix angle is complement of worm lead angle i.e. α_W + λ = 90°

It is noted that helix angle on worm is generally quite large and that on worm gear is very small. So that it is usual to specify lead angle (λ) on worm and helix angle (α_G) on the worm gear. These two angles are equal for a 90° shaft angle.

7. Velocity ratio: It is the ratio of the speed of worm (NW) in r.p.m. to the speed of the worm gear (NG) in r.p.m.

Mathematically,

Velocity ratio (V.R) = N_w/N_G

Proportions for Worms

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.