Design of Worm Gears

In this tutorial, you will learn about the basics of how worm gear is designed and how they are used in an assembly. In short, you will learn about the parameters involved in designing the worm gear from scratch and check whether the gear can handle the applied stress.

Contents:

1. What are Worm Gears?
2. Where are Worm Gears used?
3. Terminology of Worm Gears
4. What Forces act on Worm Gears?
5. Friction in Worm Gears
6. Strength Rating of Worm Gears
7. Wear Rating of Worm Gears
8. Thermal Consideration of Worm Gears

What are Worm Gears?

Worm gear drives find application when there is a need to transmit power between two non-intersecting shafts placed at right angles to one another. The drive consists of the worm, which is a threaded screw, and the worm wheel, which is a toothed gear. The teeth on the wheel cover the worm and give line contact between mating points.

• Worm Gears are most widely used when there is a requirement for high-speed reduction. They can achieve a reduction as high as 100:1 with a single pair.
• These gear drives are very compact when compared in application to an equivalent spur or helical drive. The entire operation is smooth and silent.
• Due to the worm wheel, the gear drives can be made to be self-locking in action, where the motion can only be transmitted from the worm to the worm wheel.
• In application, a large quantity of heat is generated which needs to be dissipated using a lubricating oil placed in the housing walls and the surrounding.
• The power transmission capacity is low for worm gear drives. They are only used for up to around 100 kW of power transmission.
• Single-threaded worm gears give large speed reduction but at very low efficiency. Multi-thread worm gears, on the other hand, have higher efficiency, but at the cost of a lower speed reduction.

Where are Worm Gears used?

Based on the special characteristics of the worm gear drives, they find application in four major aspects of engineering. They are as follows:

• In manually operated intermittent mechanisms, a large mechanical advantage is required, and efficiency is of minor importance. Some examples of these mechanisms include gate valves which are opened and closed manually.
• In motorized operated intermittent mechanisms, a small capacity low-cost motor drive is used to drive the mechanism and efficiency is of minor importance. Some examples of these mechanisms include drives for small hoists and large gate valves operated by electric motors.
• In motorized continuous operations, worm gear drives are used in place of conventional drives due to space limitations and require a silent operation. Here efficiency is important. Hence, multi-thread worms are used in these applications. An example of this would be machine tools and elevators.
• Motorized speed-increasing applications are where worm gear drives are preferred due to high velocity and silent operation. Speed increasing applications include drives for the automotive supercharger and centrifugal cream charger. In superchargers, a six-threaded worm with a lead angle of 45° is used.

Terminology of Worm Gears

Worm gears are designated uniquely with a combination of four quantities. The designation is as follows z₁/z₂/q/m, where

z₁ = number of starts on the worm
z₂ = number of teeth on the worm wheel
q = diametral quotient
m = module (mm)

Here the diametrical quotient is given by
q = d₁/m
With d₁ as the pitch circle diameter of the worm. The worm gear is similar to a screw with single-start or multi-start threads.

• The axial pitch of the worm is the distance measured from one point on the thread to the corresponding point on the adjacent thread measured along the axis of the worm. It is represented by px.
• The lead of the worm gear is defined as the distance any point on the helical profile will cover when the worm gear is one complete revolution. In simple terms, it is the thread advance in one complete turn. For single start threads, the axial pitch and lead are equal. The Lead is represented by l.
  l=p_x z₁
  The pitch circle diameter of the worm wheel is given by d₂ = mz₂.
• For successful power transfer, the axial pitch of the worm should be equal to the circular pitch of the worm wheel.
  p₂ = \(\frac{πd_2}{z_2} = \frac{π(mz_2)}{z_2}\) = πm
  Therefore the lead can also be given by the following relation.
  l= πmz₁
• The lead angle is the angle between the tangent to the thread at the pitch diameter and a plane normal to the worm axis. It is represented by γ.
  tan γ = \(\frac{l}{πd_1}=\frac{πmz_1}{πqm}=\frac{z_1}{q}\)
• The helix angle () is defined as the angle between a tangent to the thread at the pitch diameter and the axis of the worm. The worm helix angle is the complement of the worm lead angle. Ideally, the angle must be limited to 6° per thread. Hence if = 30°, then the worm should have at least five threads.
• The tooth pressure angle () is measured in the plane containing the axis of the worm and is equal to one-half of the thread angle.

What Forces act on Worm Gears?

The analysis of three components of the resultant tooth force between the meshing teeth of worm and worm wheel is based on the following assumptions (i) The worm is the driving element, while the worm wheel is the driven element, (ii) The worm has right-handed threads, (iii) The worm rotates in anti-clockwise directions

Like the helical gear, the force acting on the worm gear will have three components as shown in the figure below.

P_t = tangential component on the worm (N). For the worm gear, it is the driving element.
P_a = axial component on the worm (N)
P_r = radial component on the worm (N). It acts towards the center of the gear.

The components have forces in P₁ and P₂, with P₁ acting on the worm and P₂ acting on the worm wheel. The force acting on the worm wheel is the equal and
opposite reaction of the force acting on the worm.
(P₂)_t = (P₁)a
(P₂)_a=(P₁)_t
(P₂)_r=(P₁)_r

Friction in Worm Gears

The coefficient of friction in worm gear drives is large and mainly depends on the rubbing speed. The rubbing velocity, also called the sliding velocity, is defined as the relative velocity between the worm and the wheel. The below figure shows the sliding velocity on a worm gear.

• The pitch line velocity is given by
  V₁ = \(\frac{πd_1 n_1}{60×1000}\)
• From the figure,
  V_s=\(\frac{V_1}{cos⁡γ} = \frac{πd_1 n_1}{60×1000×cos⁡γ}\)
• Compared to spur or helical gears whose efficiency is around 98-99%, the efficiency of worm gears is low and varies considerably in the range of 50-98% depending on the speed ratio.
• Worm gear drives can be designed to be self-locking in nature if the coefficient of friction is greater than the tangent of the lead angle. When the opposite is true, the drive is called reversible or back-driving.

Strength Rating of Worm Gears

The teeth of the worm wheel are considerably weaker than the threads of worms, hence the design for strength is based on the worm wheel teeth. The maximum permissible torque that the worm wheel can withstand without bending failure can be given by the following equation
M_t=17.65X_bS_bml_rd₂ cos γ

Here,
M_t = permissible torque on the worm wheel (N-mm)
X_b = speed factors for the strength of worm and worm wheel
S_b = bending stress factors of worm and worm wheel
m = module (mm)
l_r = length of the root of worm wheel teeth (mm) [Eq. (20.21)] d₂ = pitch circle diameter of worm wheel (mm)
= lead angle of the worm

Wear Rating of Worm Gears

Wear of the worm gears includes failure by pitting on the gear tooth. The maximum permissible torque that the wheel can withstand without pitting failure is given by the following equation.
M_t = 18.64 X_c S_c m Y_z (d₂)^1.8
Here,
M_t = permissible torque on the worm wheel (N-mm)
X_c = speed factors for wear of worm and worm wheel
S_c = surface stress factors of worm and worm wheel
Y_z = zone factor

Thermal Consideration of Worm Gears

As discussed earlier, the efficiency of a worm gear drive is low and in operation, the work done by friction is converted into heat. In continuous operation, a considerable amount of heat is generated. As such, the rate of heat generated is as follows
H_g = 1000(1-η) kW
Where,
H_g = rate of heat generation
η = efficiency of worm gears
kW = power transmitted by gears

• The heat generated by the worm gear is dissipated through the housing wall and then to the surrounding air. The rate of heat dissipation is given by the following equation
  H_d = k(t-t_o)A
  Where,
  H_d = rate of heat dissipation
  k = overall heat transfer coefficient of housing
  t = temperature of the lubricating oil
  t_o = temperature of the surrounding air
  A = effective surface area of housing
• Equating the two equations for complete heat transfer, we get the following relation.
  t = \(t_o+\frac{1000(1-η) kW}{kA}\)
• The overall heat transfer coefficient under normal conditions is with natural air circulation and is around 18 W/m² ℃. This value can be increased by using a fan on the worm shaft and arranging fins horizontally along the air stream.
• One can increase the coefficient to as high as 20 to 28 W/m² ℃. The maximum temperature for lubricating oils is 95 ℃, above which it loses its properties and there is a danger of gear tooth failures due to seizures.