Worm gearing explained

Worm gearing is a type of mechanical transmission that allows for high reduction ratios in a compact form, using a worm(resembling a screw) and a worm wheel (a helical gear).

Worm gear mechanism featuring a steel worm and brass worm wheel, commonly used in high-torque, low-speed applications such as elevators, hoists, and conveyor systems.

These systems are common in hoists, elevators, and applications where back-driving is undesirable.

1. Basic Components and Geometry

A worm gear set consists of:

Worm: A shaft with one or more helical threads.
Worm Wheel: A gear that meshes with the worm at a 90° angle.

Terminology:

Lead (L): Axial distance advanced by the worm in one complete turn.
Pitch (p): Axial distance between corresponding points on adjacent threads.
Lead Angle (λ): Angle between the helix and a plane perpendicular to the worm axis.
Normal Pressure Angle (φₙ): The pressure angle measured in the plane normal to the tooth surfaces.
Number of Starts (Nₛ): The number of individual helical threads on the worm.

The lead angle is given by:

$\tan \lambda = \frac{L}{\pi d_w}$

Where:

$L$ = Lead of the worm
$d_w$ = Pitch diameter of the worm

If the worm has $N_s$ starts and a pitch $p$ , then:

$L = N_s \cdot p$

2. Velocity Ratio and Gear Ratio

The velocity ratio (VR) is defined as:

$\text{VR} = \frac{N_g}{N_s}$

Where:

$N_g$ = Number of teeth on the worm wheel
$N_s$ = Number of starts on the worm

Example:

If $N_s = 1$ and $N_g = 40$ , then:

$\text{VR} = \frac{40}{1} = 40:1$

3. Efficiency of Worm Gearing

Worm gear efficiency depends heavily on friction and the lead angle. The efficiency is approximated by:

$\eta = \frac{\tan \lambda (1 - \mu \tan \phi_n)}{\tan \lambda + \mu}$

Where:

$\eta$ = Efficiency
$\mu$ = Coefficient of friction (typically 0.05–0.15)
$\phi_n$ = Normal pressure angle
$\lambda$ = Lead angle

Note:

Efficiency increases with a larger lead angle.
If the efficiency is low enough, the gear set becomes self-locking, preventing the worm wheel from driving the worm in reverse.

4. Torque and Power Transmission

Let:

$T_1$ = Input torque on the worm
$T_2$ = Output torque on the wheel
$\omega_1, \omega_2$ = Angular velocities

Then the output torque is:

$T_2 = T_1 \cdot \text{VR} \cdot \eta$

And output power is:

$P_{\text{out}} = T_2 \cdot \omega_2 = \eta \cdot P_{\text{in}}$

5. Hertzian Contact Stress (Simplified)

The contact stress can be approximated using:

$\sigma_H = C \sqrt{ \frac{T_2}{d_w \cdot b} }$

Where:

$\sigma_H$ = Hertzian contact stress
$C$ = Geometry/material factor
$d_w$ = Pitch diameter of the worm
$b$ = Face width of the worm wheel

6. Example: Worm Gear Design Calculation

Given:

Single-start worm $(N_s = 1)$
Worm wheel teeth $N_g = 40$
Pitch of worm $p = 12 , \text{mm}$
Coefficient of friction $\mu = 0.10$
Normal pressure angle $\phi_n = 20^\circ$
Input torque $T_1 = 5 , \text{Nm}$
Worm pitch diameter $d_w = 40 , \text{mm}$
Face width of worm wheel $b = 20 , \text{mm}$

Step 1: Calculate Lead

$L = N_s \cdot p = 1 \cdot 12 = 12 , \text{mm}$

Step 2: Calculate Lead Angle

$\tan \lambda = \frac{L}{\pi d_w} = \frac{12}{\pi \cdot 40} \approx 0.0955$

$\Rightarrow \lambda = \tan^{-1}(0.0955) \approx 5.46^\circ$

Step 3: Efficiency

$\eta = \frac{\tan \lambda (1 - \mu \tan \phi_n)}{\tan \lambda + \mu}$

Compute each term:

$\tan \lambda \approx 0.0955$
$\tan \phi_n = \tan(20^\circ) \approx 0.3640$

Now:

$\eta = \frac{0.0955 (1 - 0.10 \cdot 0.3640)}{0.0955 + 0.10} = \frac{0.0955 \cdot 0.9636}{0.1955} \approx \frac{0.0921}{0.1955} \approx 0.471$

So:

$\eta \approx 47.1%$

Step 4: Output Torque

$\text{VR} = \frac{40}{1} = 40$

$T_2 = 5 \cdot 40 \cdot 0.471 = 94.2 , \text{Nm}$

Step 5: Contact Stress (Simplified)

Assume $C = 60 , \text{MPa} \cdot \sqrt{\text{mm/Nm}}$ (typical for bronze/steel pair):

$\sigma_H = 60 \cdot \sqrt{ \frac{94.2}{40 \cdot 20} } = 60 \cdot \sqrt{ \frac{94.2}{800} } = 60 \cdot \sqrt{0.11775} \approx 60 \cdot 0.343 = 20.6 , \text{MPa}$

7. Applications of Worm Gearing

Common uses include:

Hoists and lifts: Take advantage of self-locking.
Conveyor systems: Require large torque reduction.
Rotary tables: Smooth and precise indexing.
Musical tuning mechanisms
Valve actuators and gate mechanisms

Product Development Engineers Ltd

1. Basic Components and Geometry

Terminology:

2. Velocity Ratio and Gear Ratio

Example:

3. Efficiency of Worm Gearing

Note:

4. Torque and Power Transmission

5. Hertzian Contact Stress (Simplified)

6. Example: Worm Gear Design Calculation

Given:

Step 1: Calculate Lead

Step 2: Calculate Lead Angle

Step 3: Efficiency

Step 4: Output Torque

Step 5: Contact Stress (Simplified)

7. Applications of Worm Gearing

8. Summary – Advantages and Disadvantages

Advantages:

Disadvantages:

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Worm gearing explained

1. Basic Components and Geometry

Terminology:

2. Velocity Ratio and Gear Ratio

Example:

3. Efficiency of Worm Gearing

Note:

4. Torque and Power Transmission

5. Hertzian Contact Stress (Simplified)

6. Example: Worm Gear Design Calculation

Given:

Step 1: Calculate Lead

Step 2: Calculate Lead Angle

Step 3: Efficiency

Step 4: Output Torque

Step 5: Contact Stress (Simplified)

7. Applications of Worm Gearing

8. Summary – Advantages and Disadvantages

Advantages:

Disadvantages:

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Like this:

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