# Worm Gear

Worm gear with adjustable gear ratio and friction losses

• Library:
• Simscape / Driveline / Gears

## Description

The block represents a rotational gear that constrains the two connected driveline axes, worm (W) and gear (G), to rotate together in a fixed ratio that you specify. You can choose whether the gear rotates in a positive or negative direction. Right-hand rotation is the positive direction. If the worm thread is right-hand, ωW and ωG have the same sign. If the worm thread is left-hand, ωW and ωG have opposite signs.

### Thermal Model

You can model the effects of heat flow and temperature change by exposing an optional thermal port. To expose the port, in the Meshing Losses tab, set the Friction model parameter to Temperature-dependent efficiency.

### Model Variables

 RWG Gear ratio ωW Worm angular velocity ωG Gear angular velocity α Normal pressure angle λ Worm lead angle L Worm lead d Worm pitch diameter τG Gear torque τW Torque on the worm τloss Torque loss due to meshing friction. The loss depends on the device efficiency and the power flow direction. To avoid abrupt change of the friction torque at ωG = 0, the friction torque is introduced via the hyperbolic function. τfr Steady-state value of the friction torque at ωG → ∞. k Friction coefficient ηWG Torque transfer efficiency from worm to gear ηGW Torque transfer efficiency from gear to worm pth Power threshold [μW μG] Vector of viscous friction coefficients for the worm and gear

### Ideal Gear Constraint and Gear Ratio

Worm gear imposes one kinematic constraint on the two connected axes:

 ωW = RWGωG . (1)

The two degrees of freedom are reduced to one independent degree of freedom. The forward-transfer gear pair convention is (1,2) = (W,G).

The torque transfer is:

 RWGτW – τG – τloss = 0 , (2)

with τloss = 0 in the ideal case.

### Nonideal Gear Constraint

In the nonideal case, τloss ≠ 0. For general considerations on nonideal gear modeling, see Model Gears with Losses.

Geometric Surface Contact Friction

In the contact friction case, ηWG and ηGW are determined by:

• The worm-gear threading geometry, specified by lead angle λ and normal pressure angle α.

• The surface contact friction coefficient k.

 ηWG = (cosα – k·tanλ)/(cosα + k/tanλ) , (3)
 ηGW = (cosα – k/tanλ)/(cosα + k·tanλ) . (4)
Constant Efficiencies

In the constant friction case, you specify ηWG and ηGW, independently of geometric details.

Self-Locking and Negative Efficiency

ηGW has two distinct regimes, depending on lead angle λ, separated by the self-locking point at which ηGW = 0 and cosα = k/tanλ.

• In the overhauling regime, ηGW > 0, and the force acting on the nut can rotate the screw.

• In the self-locking regime, ηGW < 0, and an external torque must be applied to the screw to release an otherwise locked mechanism. The more negative is ηGW, the larger the torque must be to release the mechanism.

ηWG is conventionally positive.

### Meshing Efficiency

The efficiencies η of meshing between worm and gear are fully active only if the transmitted power is greater than the power threshold.

If the power is less than the threshold, the actual efficiency is automatically regularized to unity at zero velocity.

You can set the meshing losses friction model to:

• No meshing losses - suitable for HIL simulation.

• Constant efficiency, which is the default friction setting for block versions prior to R2020b.

• Temperature-dependent efficiency, which models variability in the base-shaft efficiencies calculated in the Constant efficiency setting according to a user-supplied look-up table. The temperature-dependency setting enables a thermal conserving port H. This port receives the heat flow into the block, which is translated into the block temperature according to the gear Thermal mass.

### Viscous Friction Force

The viscous friction coefficient μW controls the viscous friction torque experienced by the worm from lubricated, nonideal gear threads and viscous bearing losses. The viscous friction torque on a worm driveline axis is –μWωW. ωW is the angular velocity of the worm with respect to its mounting.

The viscous friction coefficient μG controls the viscous friction torque experienced by the gear, mainly from viscous bearing losses. The viscous friction torque on a gear driveline axis is –μGωG. ωG is the angular velocity of the gear with respect to its mounting.

### Hardware-in-the-Loop Simulation

For optimal performance of your real-time simulation, set the Friction model to No meshing losses - Suitable for HIL simulation on the Meshing Losses tab.

### Variables

Use the Variables settings to set the priority and initial target values for the block variables before simulating. For more information, see Set Priority and Initial Target for Block Variables.

Dependencies

Variable settings are exposed only when, in the Meshing Losses settings, the Friction model parameter is set to Temperature-dependent efficiency.

### Limitations

• Gear inertia is assumed negligible.

• Gears are treated as rigid components.

## Ports

### Conserving

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Rotational conserving port representing the worm torque and angular velocity.

Rotational conserving port representing the gear torque and angular velocity.

Thermal conserving port for thermal modeling.

#### Dependencies

To enable this port, set Friction model to either:

• Temperature-dependent efficiency

.

## Parameters

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### Main

Gear or transmission ratio RWG determined as the ratio of the worm angular velocity to the gear angular velocity.

Choose the directional sense of gear rotation corresponding to positive worm rotation. If you select Left-hand, rotation of the worm in the generally-assigned positive direction results in the gear rotation in negative direction.

### Meshing Losses

• No meshing losses — Suitable for HIL simulation — Gear meshing is ideal.

• Constant efficiency — Transfer of torque between worm and gear is reduced by friction.

• Temperature-dependent efficiency — Torque transfer is determined from user-supplied data for worm-gear efficiency, gear-worm efficiency, and temperature.

• Friction coefficient and geometrical parameters — Friction is determined by contact friction between surfaces.

• Efficiencies — Friction is determined by constant efficiencies 0 < η < 1.

#### Dependencies

To enable this parameter, set Friction model to Constant efficiency.

Thread pressure angle α in the normal plane. The value must be greater than zero and less than 90 degrees.

#### Dependencies

To enable this parameter, set Friction model to Constant efficiency and Friction parameterization to Friction coefficient and geometrical parameters.

Thread helix angle λ = arctan[L/(πd)], where:

• L is the worm lead.

• d is the worm pitch diameter.

The value must be greater than zero.

#### Dependencies

To enable this parameter, set Friction model to Constant efficiency and Friction parameterization to Friction coefficient and geometrical parameters.

Dimensionless coefficient of normal friction in the thread. The value must be greater than zero.

#### Dependencies

To enable this parameter, set Friction model to Constant efficiency and Friction parameterization to Friction coefficient and geometrical parameters.

Efficiency ηWG of the power transfer from worm to gear.

#### Dependencies

To enable this parameter, set Friction model to Constant efficiency and Friction parameterization to Efficiencies.

Efficiency ηGW of the power transfer from gear to worm.

#### Dependencies

To enable this parameter, set Friction model to Constant efficiency and Friction parameterization to Efficiencies.

Array of temperatures used to construct a 1-D temperature-efficiency lookup table. The array values must increase left to right. The temperature array must be the same size as the Worm-gear efficiency and Gear-worm efficiency arrays.

#### Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Array of component efficiencies with the worm as the driver— that is, with power flowing from the worm to the gear. The array values are the efficiencies at the temperatures in the Temperature array. The two arrays must be the same size.

#### Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Array of component efficiencies with the gear as the driver— that is, with power flowing from the gear to the worm. The array values are the efficiencies at the temperatures in the Temperature array. The two arrays must be the same size.

#### Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Power threshold above which full efficiency factor is in effect. A hyperbolic tangent function smooths the efficiency factor between zero at rest and the current efficiency set point.

### Viscous Losses

Vector of viscous friction coefficients [μW μG], for the worm and gear, respectively.

### Thermal Port

Thermal energy required to change the component temperature by a single degree. The greater the thermal mass, the more resistant the component is to temperature change.

Component temperature at the start of simulation. The initial temperature alters the component efficiency according to an efficiency vector that you specify, affecting the starting meshing or friction losses.