Harmonic Drive
High-ratio speed reducer based on elastic deformation of an elliptical gear
Libraries:
Simscape /
Driveline /
Gears
Description
The Harmonic Drive block represents a compact, high-ratio speed reduction mechanism that contains three key components:
Strain wave generator
Elliptical gear
Circular ring gear
A harmonic drive is backlash-free with a high-ratio of speed reduction by design. This is advantageous for systems that require precise gear positioning. The base shaft turns the strain wave generator, which is elliptical with bearings around the circumference. The bearings allow the strain wave generator to rotate within the elliptical gear. The elliptical gear is flexible, and the motion of the strain wave generator causes the vertices of the elliptical gear to move. This deformation of the ellipse causes the teeth of the elliptical gear to slowly climb the teeth of the circular ring gear. The ring gear has more teeth than the elliptical gear to allow for this motion. The elliptical gear transmits torque to the output shaft while rotating within the ring gear.
Meshing occurs concurrently at both vertices of the elliptical gear. This design doubles the teeth in mesh, thereby increasing the torque capacity of the drive system.
The internal meshing between the two gears causes the elliptical gear axis to spin counter to the elliptical strain wave generator.
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 number of teeth on the circular ring gear.
nE is the number of teeth on the elliptical gear.
The Simple Gear block provides the foundation for this block.
Ring Gear Rotation
You can enable rotation of the circular ring gear by setting Ring gear
rotation to On
. The block exposes port
R, which allows you to control the rotational motion of the
ring gear by using a mechanical rotational input signal.
Note
The Ring gear rotation parameter determines how you parameterize meshing losses.
When you set Ring gear rotation to
Off
, the block calculates the efficiencies using the
Simple Gear block implementation. In this case,
you parameterize the meshing losses the same way you would for the
Simple Gear block. However, when you select
On
, the block calculates the efficiencies by
implementing the Planetary Gear block. For this case,
you parameterize the meshing losses the same way you would for the
Planetary Gear block.
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
.
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.
Ports
Conserving
B — Base shaft
mechanical rotational
Mechanical rotational conserving port associated with the base shaft and the strain wave generator.
When you set Ring gear rotation to
On
, this port corresponds to port
S of the Planetary
Gear block. This is the sun gear.
F — Follower shaft
mechanical rotational
Mechanical rotational conserving port associated with the follower shaft and the flexible spline.
When you set Ring gear rotation to
On
, this port corresponds to port
R of the Planetary
Gear block. This is the ring gear.
C — Circular spline
mechanical rotational
Mechanical rotational conserving port associated with the circular spline.
When you set Ring gear rotation to
On
, this port corresponds to port
C of the Planetary
Gear block. This is the carrier gear.
Dependencies
To enable this port, set Ring gear rotation
to On
.
H — Heat transfer
thermal
Thermal conserving port associated with heat transfer.
Dependencies
To enable this port, set Friction model to
either Temperature-dependent efficiency
or Temperature and load-dependent
efficiency
.
Parameters
Main
Ring gear rotation — Option to simulate ring gear motion
Off
(default) | On
Option to simulate ring gear rotation. Set this parameter to
On
to expose port
R.
Number of teeth on elliptical gear — Elliptical gear teeth
100
(default) | positive scalar
Total number of teeth protruding outward from the elliptical gear perimeter. The value of this parameter should be slightly smaller than the number of teeth on the circular ring gear. The default difference of two teeth is common. 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. The value of this parameter should be slightly larger than the number of teeth on the elliptical gear. The default difference of two teeth is common. The ratio of the two gear tooth numbers defines the relative angular velocities of the base and follower shafts.
Meshing Losses
To enable these parameters, set Ring gear rotation to
Off
.
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
Option to include friction meshing losses:
No meshing losses - Suitable for HIL simulation
— The block ignores meshing losses.Constant efficiency
— You specify a component efficiency that remains constant throughout the simulation.Load-dependent efficiency
— The block reduces 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
— The block determines the torque transfer efficiency from the input at port H. This factor falls in the range 0 < η ≤ 1 and is independent from load.Temperature and load-dependent efficiency
— The block reduces torque transfer by a variable efficiency factor that depends on temperature and load. This factor falls in the range 0 < η < 1 and varies with the torque load.
Dependencies
To enable this parameter, set Ring gear
rotation to Off
.
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:
Temperature-dependent efficiency
Temperature and load-dependent efficiency
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:
Temperature-dependent efficiency
Temperature and load-dependent efficiency
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
.
To enable these parameters, set Ring gear rotation to
On
.
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.
Sun-planet and ring-planet ordinary efficiencies — Torque transfer efficiency vector
[.96, .98]
(default) | vector
Vector of torque transfer efficiencies where the first element represents the relationship between the strain wave generator and the elliptical gear, and the second element represents the relationship between the elliptical gear and the circular ring gear.
Dependencies
To enable this parameter, set Friction model
to Constant efficiency
.
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
.
Sun-planet efficiency — Sun gear to planet gear torque transfer efficiency
[.95, .9, .85]
(default) | vector
Vector of output-to-input power ratios that describe the power flow from the strain wave generator to the elliptical gear. The block uses the 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 Friction model
to Temperature-dependent
efficiency
.
Ring-planet efficiency — Torque transfer efficiency from the ring gear to the planet gear
[.95, .9, .85]
(default) | vector
Vector of output-to-input power ratios that describe the power flow from the elliptical gear to the flexible gear. The block uses the 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 Friction model
to Temperature-dependent
efficiency
.
Sun-carrier and planet-carrier power thresholds — Minimum efficiency power threshold for the sun-carrier and planet-carrier gear couplings
[.001, .001]
W
(default) | vector
Vector of power thresholds above which full efficiency factors apply. Enter the thresholds in the order strain wave generator-circular gear, elliptical gear-circular gear. Below these values, 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
block 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
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.
Inertia
To enable these parameters, set Ring gear rotation to
On
.
Inertia — Option to include inertia
Off
(default) | On
Option to include inertia due to ring gear rotation in your simulation.
Wave generator gear inertia — Inertia of the wave generator gear
0.001
kg*m^2
(default) | positive scalar
Inertia of the wave generator gear.
Dependencies
To enable this parameter, set Inertia to
On
.
Flexspline gear inertia — Inertia of the flexspline gear
0.0005
kg*m^2
(default) | positive scalar
Inertia of the flexspline gear.
Dependencies
To enable this parameter, set Inertia to
On
.
Ring gear inertia — Ring gear inertia
0.001
kg*m^2
(default) | positive scalar
Ring gear inertia.
Dependencies
To enable this parameter, set Inertia to
On
.
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™.
Version History
Introduced in R2014aR2024b: Inertia updates
You now can specify the inertia of all gear subcomponents including the sun, ring, and carrier gears. You also now can include the effects of planet procession in the planet inertia behavior.
See Also
Simscape Blocks
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