Harmonic Drive
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:
where:
r is the gear reduction ratio.
nC is the tooth number of the
circular ring gear.
nE 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 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.
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.
DependenciesVariable settings are exposed only when, in the Meshing Losses
settings, the Friction model parameter is set to
Temperature-dependent efficiency.
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.
Ports
Conserving
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B — Base shaft
mechanical rotational
Conserving rotational port representing the base shaft torque and
angular velocity.
F — Follower shaft
mechanical rotational
Conserving rotational port representing the follower shaft torque and
angular velocity.
H — Optional heat transfer port
thermal
Thermal conserving port for thermal modeling.
Dependencies
To enable this port, set Friction model to
either:
.
Parameters
expand all
Main
Number of teeth on elliptical gear — Elliptical gear teeth
100 (default) | positive scalar
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.
Number of teeth on circular gear — Circular gear teeth
102 (default) | positive scalar
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.
Meshing Losses
Friction model — Gear friction model
No meshing losses — Suitable for
HIL simulation (default) | Constant efficiency | Load-dependent efficiency | Temperature-dependent efficiency | Temperature and load-dependent
efficiency
No meshing losses — Suitable for HIL
simulation — Gear meshing is
ideal.
Constant efficiency —
Transfer of torque between worm and gear is reduced by
friction.
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.
Temperature-dependent
efficiency — Reduce torque transfer
by a constant efficiency factor that is dependent on
temperature but does not consider the gear load. This factor
falls in the range 0 < η ≤ 1 and is independent from load. Torque
transfer is determined from user-supplied data for gear
efficiency and temperature.
Temperature and load-dependent
efficiency — Reduce torque transfer
by a variable efficiency factor that is dependent on
temperature and load. This factor falls in the range 0 < η <
1 and varies with the torque load. Torque
transfer efficiency is determined from user-supplied data
for gear loading and temperature.
Efficiency — Torque transfer efficiency
0.95 (default) | positive scalar in the range of (0, 1]
Torque transfer efficiency (η) between base and
follower shafts. This parameter is inversely proportional to the meshing
power losses.
Dependencies
To enable this parameter, set Friction model
to Constant efficiency.
Follower power threshold — Value above which the full efficiency factor is applied
0.001 W (default) | positive scalar
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. However, very low values may raise
the computational cost of simulation.
Dependencies
To enable this parameter, set Friction model
to Constant efficiency.
Input shaft torque at no load — Net torque on idle shaft
0.1 N*m (default) | positive scalar
Net torque (τidle) 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.
Dependencies
To enable this parameter, set Friction model
to Load-dependent efficiency.
Nominal output torque — Value of torque at which efficiency is normalized
5 N*m (default) | positive scalar
Output torque (τF) at which
to normalize the load-dependent efficiency.
Dependencies
To enable this parameter, set Friction model
to Load-dependent efficiency.
Efficiency at nominal output torque — Transfer efficiency
0.95 (default) | positive scalar in the range of (0,1]
Torque transfer efficiency (η) at the nominal
output torque. Larger efficiency values correspond to greater torque
transfer between the input and output shafts.
Dependencies
To enable this parameter, set Friction model
to Load-dependent efficiency.
Follower angular velocity threshold — Value above which the full efficiency factor is applied
0.01 rad/s (default) | positive scalar
Absolute value of the follower shaft angular velocity above which the
full efficiency factor is in effect
(ωF). 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. However, very low
values may raise the computational cost of simulation.
Dependencies
To enable this parameter, set Friction model
to Load-dependent efficiency.
Temperature — Array for tabular parameterization of efficiency
[280 300 320] K (default) | vector
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. The array must be the same size as the
Efficiency array.
Dependencies
To enable this parameter, set Friction model
to either:
Efficiency — Array of gear efficiencies
[0.95 0.9 0.85] (default) | vector
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.
Dependencies
To enable this parameter, set Friction model
to either:
Follower power threshold — Power below which numerical smoothing is applied
0.001 W (default) | positive scalar
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.
Dependencies
To enable this parameter, set Friction model
to Temperature-dependent
efficiency.
Load at elliptical gear — Elliptical-gear loads for tabular parameterization of efficiency
[1 5 10] N*m (default) | numerical array
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.
Dependencies
To enable this parameter, set Friction model
to Temperature and load-dependent
efficiency.
Efficiency matrix — Efficiency array for tabular parameterization of efficiency
[ 0.85 0.8 0.75; 0.95 0.9 0.85; 0.85 0.8 0.7
] (default) | numerical array
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.
Dependencies
To enable this parameter, set Friction model
to Temperature and load-dependent
efficiency.
Follower angular velocity threshold — Angular velocity below which numerical smoothing is applied
0.01 rad/s (default) | positive scalar
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.
Dependencies
To enable this parameter, set Friction model
to Temperature and load-dependent
efficiency.
Viscous Losses
Viscous friction coefficients at base (B) and follower (F) — Fluid dynamic friction coefficients
[0 0]
N*m/(rad/s) (default) | positive two-element vector
Two-element array with the viscous friction coefficients in effect at the base
and follower shafts. The default array corresponds to zero viscous
losses.
Thermal Port
Thermal mass — Thermal mass
50 J/K (default) | positive scalar
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.
Initial temperature — Initial temperature
300 K (default) | positive scalar
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.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Introduced in R2014a
