Sun-Planet Worm Gear

Planetary gear set of carrier, worm planet, and sun wheels with adjustable gear ratio, worm thread type, and friction losses

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Libraries:
Simscape / Driveline / Gears / Planetary Subcomponents

Description

The Sun-Planet Worm Gear block represents a two-degree-of-freedom planetary gear built from a carrier and sun and planet gears. By type, the sun and planet gears are crossed helical spur gears arranged as a worm drive, in which the planet gear is the worm. Such transmissions are used in the Torsen^® T-1 differential. When transmitting power, the sun gear can be independently rotated by the worm (planet) gear, by the carrier, or by both.

You specify a fixed gear ratio, which signifies the worm angular velocity divided by the sun gear angular velocity. You control the direction by setting the worm thread type to left-hand or right-hand. Rotation of the right-hand worm in the positive direction causes the sun gear to rotate in the positive direction. The positive directions of the sun gear and the carrier are the same.

Thermal Model

You can model the effects of heat flow and temperature change by enabling the optional thermal port. To enable the port, set Friction model to Temperature-dependent efficiency.

Equations

Variables

Equation variables are:

RWG	Gear ratio that signifies the worm angular velocity divided by the sun gear angular velocity: The ratio is positive for the right-hand worm and negative for the left-hand worm
ωS	Angular velocity of the sun gear
ωP	Angular velocity of the worm gear
ωC	Angular velocity of the carrier
ωSC	Angular velocity of the sun gear with respect to the carrier
α	Normal pressure angle
λ	Worm lead angle
L	Worm lead
d	Worm pitch diameter
τS	Torque applied to sun shaft
τP	Torque applied to planet shaft
τC	Torque applied to carrier shaft
τ	Torque due to meshing friction: The loss depends on the device efficiency and the power flow direction. To avoid an abrupt change of the friction torque at ωS = 0, the friction torque is introduced via the hyperbolic function.
τinstfr	Instantaneous value of the friction torque used to simulate friction losses
τfr	Torque due to friction in steady-state
k	Friction coefficient
ηWG	Efficiency of worm to gear power transfer
ηGW	Efficiency of gear to worm power transfer
pth	Power threshold
μSC	Viscous friction coefficient for the sun-carrier interface
μWC	Viscous friction coefficient for the worm-carrier interface

Ideal Gear Constraints and Gear Ratio

The Sun-Planet Worm Gear block imposes one kinematic constraint on the three connected axes:

$${\omega }_{\text{S}}=\frac{{\omega }_{\text{P}}}{R_{\text{WG}}}+{\omega }_{\text{C}}.$$

The gear has two independent degrees of freedom. The gear pair is (1,2) = (S,P).

The torque transfer is:

$$R_{\text{WG}}{\tau }_{\text{P}}+{\tau }_{\text{S}}–{\tau }_{\text{loss}}=0$$

$${\tau }_{\text{C}}=–{\tau }_{\text{S}}$$

In the ideal case where there is no torque loss, τ_loss = 0.

Nonideal Gear Constraints

In the nonideal case, τ_loss ≠ 0. For more information, see Model Gears with Losses.

In a nonideal gear, the angular velocity and geometric constraints are unchanged, but the transferred torque and power are reduced by:

• Coulomb friction due to worm-sun gear meshing, which is characterized by the friction coefficient k or constant efficiencies [η_WG, η_GW]

• Viscous couplings of driveshafts with bearings, which are parametrized by viscous friction coefficients μ_SC and μ_WC

Because the transmission incorporates a worm gear, the efficiencies are different for the direct and reverse power transfer. The table shows the value of the efficiency for all combinations of the power transfer.

Driving Shaft	Driven Shaft
	Planet	Sun	Carrier
Planet	N/A	ηWG	ηWG
Sun	ηGW	N/A	No loss
Carrier	ηGW	No loss	N/A

Geometric Surface Contact Friction

When you set Friction model to Constant efficiency and leave Friction parameterization set to Friction coefficient and geometrical parameters, the model considers geometric surface contact friction. In this case, η_WG and η_GW rely on:

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

• The surface contact friction coefficient k.

$${\eta }_{\text{WG}}=\frac{\left(\text{cos}\alpha –k·\text{tan}\lambda \right)}{\left(\text{cos}\alpha +\frac{k}{\text{tan}\lambda }\right)}$$

$${\eta }_{\text{GW}}=\frac{\left(\text{cos}\alpha –\frac{k}{\text{tan}\lambda }\right)}{\left(\text{cos}\alpha +k·\text{tan}\alpha \right)}$$

Constant Efficiencies

When you set Friction model to Constant efficiency and set Friction parameterization to Efficiencies, or when you set Friction model to Temperature-dependent efficiency, the model treats the efficiencies as constant. In this case, you specify η_WG and η_GW independently of geometric details.

Self-Locking and Negative Efficiency

You can enable self-locking behavior by making the efficiency negative. Power cannot be transmitted from the sun gear to the worm or from the carrier to the worm unless some torque is applied to the worm to release the train. In this case, the absolute value of the efficiency specifies the ratio at which the train is released. The smaller the train lead angle, the smaller the reverse efficiency.

Meshing Efficiency

The efficiencies, η, of meshing between the worm gear and planet 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.

Viscous Friction Force

The viscous friction coefficients of the worm-carrier and sun-carrier bearings control the viscous friction torque experienced by the carrier from lubricated, nonideal gear threads. For details, see Nonideal Gear Constraints.

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.

Assumptions and Limitations

• Gear inertia is assumed to be negligible.

• Gears are treated as rigid components.

• Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.

Ports

Conserving

C — Planet gear carrier

rotational mechanical

Rotational mechanical conserving port associated with the planet gear carrier.

W — Worm gear

rotational mechanical

Rotational mechanical conserving port associated with the worm gear.

S — Sun gear

rotational mechanical

Rotational mechanical conserving port associated with the sun gear.

H — Heat flow

thermal

Thermal conserving port associated with heat flow. Heat flow affects the power transmission efficiency by altering the gear temperatures.

Dependencies

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

Parameters

Main

Gear ratio — Gear ratio

25 (default) | positive scalar

Gear or transmission ratio, R_WG, which signifies the worm gear angular velocity divided by the sun gear angular velocity. This gear ratio must be strictly positive.

Worm thread type — Thread rotation direction

Right-hand (default) | Left-hand

Direction of positive worm rotation. If you select Left-hand, positive worm rotation results in negative gear rotation.

Meshing Losses

The table shows how the options that you choose for Friction model affect the visibility of other parameters in the Meshing Losses tab.

Meshing Losses Parameter Dependencies

Meshing Losses Setting Parameters and Values
Friction Model
No meshing losses - Suitable for HIL simulation		Constant efficiency		Temperature-dependent efficiency
		Friction parameterization		Temperature
		Friction coefficient and geometrical parameters	Efficiencies
		Normal pressure angle	Worm-gear efficiency	Worm-gear efficiency
		Lead angle	Gear-worm efficiency	Gear-worm efficiency
		Friction coefficient
		Power threshold	Power threshold	Power threshold

Friction model — Friction model

No meshing losses - Suitable for HIL simulation (default) | Constant efficiency | Temperature-dependent efficiency

Friction model for the block:

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

• Constant efficiency — Transfer of torque between the gear wheel pairs is reduced by a constant efficiency, η, such that 0 < η ≤ 1.

• Temperature-dependent efficiency — Transfer of torque between the gear wheel pairs is defined by the table lookup based on the temperature.

Friction parameterization — Friction parameterization method

Friction coefficient and geometrical parameters (default) | Efficiencies

Characterization of the friction between gear threads:

• Friction coefficient and geometrical parameters — Friction is determined by geometric surface contact friction.

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

Dependencies

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

Normal pressure angle — Thread pressure angle

17.5 deg (default) | scalar

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.

Lead angle — Thread helix angle

20 deg (default) | positive scalar

Thread helix angle, where λ = arctan[L/(πd)]. L is the worm lead and d is the worm pitch diameter. This 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.

Friction coefficient — Thread friction

0.08 (default) | positive scalar

Dimensionless coefficient of normal friction in the thread. 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.

Temperature — Temperature

[280, 300, 320] K (default) | vector

Vector of temperatures used to construct a 1-D temperature-efficiency lookup table. The vector elements must increase from left to right.

Dependencies

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

Worm-gear efficiency — Torque transfer efficiency from the worm gear to the sun gear

[.75, .65, .6] or 0.74 (default) | vector or scalar

Vector of output-to-input power ratios that describe the power flow from the worm gear to the sun gear, η_WG. When you set Friction model to Constant efficiency, specify the value as a scalar. When you set Friction model to Temperature-dependent efficiency, specify the value as a vector. The block uses the vector values to construct a 1-D temperature-efficiency lookup table.

Each element is an efficiency that relates to a temperature in the Temperature vector. The length of the vector must be equal to the length of the Temperature vector. Each element in the vector must be in the range (0,1].

Dependencies

To enable this parameter, set either:

• Friction model to Constant efficiency and Friction parameterization to Efficiencies — In this case, specify the value as a scalar.

• Friction model to Temperature-dependent efficiency — In this case, specify the value as a vector.

Gear-worm efficiency — Torque transfer efficiency from the sun gear to the worm gear

[.5, .45, .4] or 0.65 (default) | vector or scalar

Vector of output-to-input power ratios that describe the power flow from the sun gear to the worm gear, η_GW. When you set Friction model to Constant efficiency, specify the value as a scalar. When you set Friction model to Temperature-dependent efficiency, specify the value as a vector. The block uses the vector values to construct a 1-D temperature-efficiency lookup table.

Each element is an efficiency that relates to a temperature in the Temperature vector. The length of the vector must be equal to the length of the Temperature vector. Each element in the vector must be in the range (0,1].

Dependencies

To enable this parameter, set either:

• Friction model to Constant efficiency and Friction parameterization to Efficiencies — In this case, specify the value as a scalar.

• Friction model to Temperature-dependent efficiency — In this case, specify the value as a vector.

Power threshold — Minimum efficiency power threshold

0.001 W (default) | scalar

Power threshold, p_th, above which full efficiency is in effect. Below this value, a hyperbolic tangent function smooths the efficiency factor.

When you set Friction model to Constant efficiency, the block lowers the efficiency losses to zero when no power is transmitted. When you set Friction model to Temperature-dependent efficiency, the function smooths the efficiency factors between zero when at rest and the values provided by the temperature-efficiency lookup tables at the power thresholds.

Dependencies

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

Viscous Losses

Worm-carrier and sun-carrier viscous friction coefficients — Gear viscous friction

[0, 0] N*m/(rad/s) (default) | vector

Vector of viscous friction coefficients [μ_WC μ_SC], for the worm-carrier and sun-carrier shafts, respectively.

Thermal Port

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

Thermal mass — Thermal mass

50 J/K (default) | scalar

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

Dependencies

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

More About

Hardware-in-the-Loop Simulation

For optimal simulation performance, set Friction model to the default value, No meshing losses - Suitable for HIL simulation.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2011a