High-ratio speed reducer based on elastic deformation of an elliptical gear

Description

This block represents a compact, high-ratio, speed reduction mechanism that contains three key components:

Strain wave generator
Elliptical gear
Circular ring gear

The strain wave generator comprises an elliptical plug mated to a raced ball bearing. It sits inside an elastic metal gear, deforming it into a slight elliptical pattern. Rotation of the elliptical pattern in the body of the gear constitutes a strain wave.

The elliptically deformed gear engages the internal teeth of a fixed circular ring gear of only slightly larger diameter. Meshing occurs concurrently at the two elongated ends of the elliptical gear. This design doubles the teeth in mesh, boosting the torque capacity of the drive system.

During normal operation, the base shaft drives the strain wave generator. The elliptical plug spins freely inside the elastic metal gear, propagating the strain wave about the gear rotation axis. This strain wave causes the elliptical gear teeth to engage the internal teeth of the circular ring gear progressively.

The internal meshing between the two gears causes the elliptical gear axis to spin counter to the elliptical strain wave. For every clockwise rotation that the strain wave generator completes, the elliptical gear axis rotates counterclockwise by a small amount.

Large reduction ratios arise from the near-equal gear tooth numbers. The effective gear reduction ratio is:

$r = \frac{n_{E}}{n_{C} - n_{E}},$

where:

r is the gear reduction ratio.
n_C is the tooth number of the circular ring gear.
n_E is the tooth number of the deformable elliptical gear.

Optional parameters account for power losses due to gear meshing and viscous friction. The Simple Gear block provides the foundation for this block. For more information, see Simple Gear.

Thermal Modeling

You can model the effects of heat flow and temperature change through an optional thermal conserving port. By default, the thermal port is hidden. To expose the thermal port, right-click the block in your model and, from the context menu, select Simscape > Block choices. Select a variant that includes a thermal port. Specify the associated thermal parameters for the component.

Ports

Port	Description
B	Conserving rotational port representing the base shaft
F	Conserving rotational port representing the follower shaft

Parameters

Main

Number of teeth on elliptical gear: Total number of teeth protruding outward from the elliptical gear perimeter. This number should be slightly smaller than the number of teeth on the circular ring gear. The ratio of the two gear tooth numbers defines the relative angular velocities of the base and follower shafts. The default value is 100.
Number of teeth on circular gear: Number of teeth protruding inward from the circular ring gear perimeter. This number should be slightly larger than the number of teeth on the elliptical gear. The ratio of the two gear tooth numbers defines the relative angular velocities of the base and follower shafts. The default value is 102.

Meshing Losses

Parameters for meshing losses vary with the block variant chosen—one with a thermal port for thermal modeling and one without it.

Without Thermal Port

Friction model

List of friction models at various precision levels for estimating power losses due to meshing.

No meshing losses - Suitable for HIL simulation — Neglect friction between gear cogs. Meshing is ideal.
Constant efficiency — Reduce torque transfer by a constant efficiency factor. This factor falls in the range 0 < η ≤ 1 and is independent of load. Selecting this option exposes additional parameters.
Constant efficiency
Efficiency
Torque transfer efficiency (η) between base and follower shafts. This parameter is inversely proportional to the meshing power losses. Values can range from 0 to 1. The default value is 0.95.
Follower power threshold
Absolute value of the follower shaft power above which the full efficiency factor is in effect. A hyperbolic tangent function smooths the efficiency factor from zero when at rest to the full efficiency value at the power threshold.
As a guideline, the power threshold should be lower than the expected power transmitted during simulation. Higher values might cause the block to underestimate efficiency losses. Very low values might, however, raise the computational cost of simulation. The default value is 0.001 W.
Load-dependent efficiency — Reduce torque transfer by a variable efficiency factor. This factor falls in the range 0 < η < 1 and varies with the torque load. Selecting this option exposes additional parameters.
Load-Dependent Efficiency
Input shaft torque at no load
Net torque (τ_idle) acting on the input shaft in idle mode, e.g., when torque transfer to the output shaft equals zero. For nonzero values, the power input in idle mode completely dissipates due to meshing losses. The default value is 0.1 N*m.
Nominal output torque
Output torque (τ_F) at which to normalize the load-dependent efficiency. The default value is 5 N*m.
Efficiency at nominal output torque
Torque transfer efficiency (η) at the nominal output torque. Efficiency values must fall in the interval [0, 1]. Larger efficiency values correspond to greater torque transfer between the input and output shafts. The default value is 0.95.
Follower angular velocity threshold
Absolute value of the follower shaft angular velocity above which the full efficiency factor is in effect (ω_F). Below this value, a hyperbolic tangent function smooths the efficiency factor to one, lowering the efficiency losses to zero when at rest.
As a guideline, the angular velocity threshold should be lower than the expected angular velocity during simulation. Higher values might cause the block to underestimate efficiency losses. Very low values might, however, raise the computational cost of simulation.
The default value is 0.01 rad/s.

With Thermal Port

Friction model

Choice of model for representing friction. The model can incorporate the temperature and load dependences of friction or just the temperature dependence. The default setting is Temperature-dependent efficiency.

Temperature

Array of temperatures used to construct an efficiency lookup table. The array values must increase from left to right. The temperature array must be the same size as the efficiency array in temperature-dependent models. It must be the same size as a single row of the efficiency matrix in temperature-and-load-dependent models. The default array is [280 300 320] K.

Load at elliptical gear

Array of elliptical-gear loads used to construct a 2-D temperature-load-efficiency lookup table for temperature-and-load-dependent efficiency models. The array values must increase from left to right. The load array must be the same size as a single column of the efficiency matrix. The default array is [1 5 10] N*m.

Efficiency

Array of efficiencies used to construct a 1-D temperature-efficiency lookup table for temperature-dependent efficiency models. The array elements are the efficiencies at the temperatures in the Temperature array. The two arrays must be the same size.

This parameter appears only when you select a temperature-dependent friction model. The default array is [0.95 0.9 0.85].

Efficiency matrix

Matrix of component efficiencies used to construct a 2-D temperature-load-efficiency lookup table. The matrix elements are the efficiencies at the temperatures in the Temperature array and at the loads in the Load at elliptical gear array.

The number of rows must be the same as the number of elements in the Temperature array. The number of columns must be the same as the number of elements in the Load at elliptical gear array.

This parameter appears only when you select a temperature-and-load-dependent efficiency model. The default matrix is [ 0.85 0.8 0.75; 0.95 0.9 0.85; 0.85 0.8 0.7 ].

Follower power threshold

Visible only if you select Temperature-dependent efficiency as the friction model. Absolute value of the follower shaft power above which the full efficiency factor is in effect. A hyperbolic tangent function smooths the efficiency factor between zero when at rest and the value provided by the temperature-efficiency lookup table when at the power threshold. The default value is 0.001 W.

Follower angular velocity threshold

Visible only if you select Temperature and load-dependent efficiency as the friction model. Absolute value of the follower shaft angular velocity above which the full efficiency factor is in effect. Below this value, a hyperbolic tangent function smooths the efficiency factor to one, lowering the efficiency losses to zero when at rest. The default value is 0.01 rad/s.

Viscous Losses

Viscous friction coefficients at base (B) and follower (F): Two-element array with the viscous friction coefficients in effect at the base and follower shafts. The default array, [0 0] N*m/(rad/s), corresponds to zero viscous losses.

Thermal Port

Thermal mass: 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. The default value is 50 J/K.
Initial temperature: 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. The default value is 300 K.

Real-Time Simulation

Hardware-in-the-Loop Simulation

For optimal simulation performance, use the Meshing Losses > Friction model parameter default setting, No meshing losses - Suitable for HIL simulation.

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Harmonic Drive

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