Documentation |
High-ratio speed reducer based on cycloidal disk motion
This block represents a compact, high-ratio, speed-reduction mechanism that contains four key components:
Eccentric cam
Cycloidal disk
Ring-gear housing
Pin rollers
The eccentric cam, which extends from the base shaft, sits inside the cycloidal disk. This disk meshes with the ring-gear housing. The pin rollers, which extend from the follower shaft, sit in matching holes on the cycloidal disk.
During normal operation, the base shaft drives the eccentric cam. The cam spins inside the cycloidal disk, causing it to rotate in an eccentric pattern about an offset axis. As it moves, the cycloidal disk engages the internal teeth of the ring-gear housing. The internal meshing reverses the rotational velocity direction.
Pin rollers extending from cycloidal disk holes transmit rotational motion to the follower shaft. This shaft spins counter to the base shaft at a highly reduced speed. The large reduction ratio results from the near-equal cycloidal disk and ring gear tooth numbers. The effective gear reduction ratio is
$$r=\frac{{n}_{R}-{n}_{C}}{{n}_{C}},$$
where:
r is the gear reduction ratio.
n_{R} is the number of teeth on the ring gear.
n_{C} is the number of teeth on the cycloidal disk.
The gear reduction ratio constrains the angular velocities of the base and follower shafts according to the expression
$${\omega}_{F}=-r{\omega}_{B},$$
where:
ω_{F} is the angular velocity of the follower shaft.
ω_{C} is the angular velocity of the base shaft.
The gear reduction ratio also constrains the torques acting on the base and follower shafts, according to the expression
$${T}_{B}=r{T}_{F}+{T}_{f},$$
where:
T_{B} is the net torque at the base shaft.
T_{F} is the net torque at the follower shaft.
T_{f} is the torque loss due to friction.
The magnitude of the frictional torque depends on the torque transfer efficiency and power flow direction according to the expression
$${T}_{f}=\{\begin{array}{cc}f\xb7\text{\hspace{0.17em}}abs\left({T}_{B}\right)\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{{\eta}_{BF}}-1\right),& {T}_{B}\xb7{\omega}_{B}\le 0\\ f\xb7\text{\hspace{0.17em}}abs\left({T}_{B}\right)\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(1-{\eta}_{FB}\right),& {T}_{B}\xb7\text{\hspace{0.17em}}{\omega}_{B}>0\end{array},$$
where:
f is a blending factor that smooths the frictional torque function near zero (e.g., during reversal of the rotational velocity direction).
η_{BF} is the torque transfer efficiency from the base shaft to the follower shaft.
η_{FB} is the torque transfer efficiency from the follower shaft to the base shaft.
The blending factor f is a hyperbolic function of the cycloidal disk angular velocity:
$$f=\mathrm{tanh}\left(\frac{4{\omega}_{B}}{{\omega}_{Th}}\right),$$
where:
ω_{Th} is the angular velocity threshold at the base shaft above which full efficiency losses are in effect.
The figure shows the cycloidal drive in front and side views. The kinematics of the drive system cause a reversal of the base and follower shaft angular velocities so that the two shafts spin in opposite directions.
The cycloidal drive can operate in reverse mode, e.g., with power flowing from the follower shaft to the base shaft. In this mode, torque transfer efficiencies are typically negligible. You can adjust the efficiency value in the block dialog box using the Efficiency from follower shaft to base shaft parameter.
Total number of teeth projecting outward from the cycloidal disk perimeter. This number should be slightly smaller than the number of teeth or pins on the ring gear. The ratio of the gear tooth numbers defines the relative angular velocities of the base and follower shafts. The default value is 20.
Total number of teeth or pins projecting inward from the ring gear housing. This number should be slightly larger than the number of teeth on the cycloidal disk. The ratio of the two gear tooth numbers defines the relative angular velocities of the base and follower shafts. The default value is 24.
Torque transfer efficiency in normal operation mode, e.g., with the base shaft driving the follower shaft. Efficiency values must fall in the interval [0, 1]. Larger efficiency values correspond to greater torque transfer between the base and follower shafts. Values approaching unity are typical. The default value is 0.90.
Torque transfer efficiency in reverse operation mode, e.g., with the follower shaft driving the base shaft. Efficiency values must fall in the interval [0, 1]. Larger efficiency values correspond to greater torque transfer between base and follower shafts. Values approaching zero are typical. The default value is 0.05.
Absolute value of the cycloidal disk 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.
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.