Worm gear with adjustable gear ratio and friction losses
Simscape / Driveline / Gears
The block represents a rotational gear that constrains the two connected driveline axes, worm (W) and gear (G), to rotate together in a fixed ratio that you specify. You can choose whether the gear rotates in a positive or negative direction. Right-hand rotation is the positive direction. If the worm thread is right-hand, ω_{W} and ω_{G} have the same sign. If the worm thread is left-hand, ω_{W} and ω_{G} have opposite signs.
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
.
R_{WG} | Gear ratio |
ω_{W} | Worm angular velocity |
ω_{G} | Gear angular velocity |
α | Normal pressure angle |
λ | Worm lead angle |
L | Worm lead |
d | Worm pitch diameter |
τ_{G} | Gear torque |
τ_{W} | Torque on the worm |
τ_{loss} | Torque loss due to meshing friction. The loss depends on the device efficiency and the power flow direction. To avoid abrupt change of the friction torque at ω_{G} = 0, the friction torque is introduced via the hyperbolic function. |
τ_{fr} | Steady-state value of the friction torque at ω_{G} → ∞. |
k | Friction coefficient |
η_{WG} | Torque transfer efficiency from worm to gear |
η_{GW} | Torque transfer efficiency from gear to worm |
p_{th} | Power threshold |
[μ_{W} μ_{G}] | Vector of viscous friction coefficients for the worm and gear |
Worm gear imposes one kinematic constraint on the two connected axes:
ω_{W} = R_{WG}ω_{G} . | (1) |
The two degrees of freedom are reduced to one independent degree of freedom. The forward-transfer gear pair convention is (1,2) = (W,G).
The torque transfer is:
R_{WG}τ_{W} – τ_{G} – τ_{loss} = 0 , | (2) |
with τ_{loss} = 0 in the ideal case.
In the nonideal case, τ_{loss} ≠ 0. For general considerations on nonideal gear modeling, see Model Gears with Losses.
In the contact friction case, η_{WG} and η_{GW} are determined by:
The worm-gear threading geometry, specified by lead angle λ and normal pressure angle α.
The surface contact friction coefficient k.
η_{WG} = (cosα – k·tanλ)/(cosα + k/tanλ) , | (3) |
η_{GW} = (cosα – k/tanλ)/(cosα + k·tanλ) . | (4) |
In the constant friction case, you specify η_{WG} and η_{GW}, independently of geometric details.
η_{GW} has two distinct regimes, depending on lead angle λ, separated by the self-locking point at which η_{GW} = 0 and cosα = k/tanλ.
In the overhauling regime, η_{GW} > 0, and the force acting on the nut can rotate the screw.
In the self-locking regime, η_{GW} < 0, and an external torque must be applied to the screw to release an otherwise locked mechanism. The more negative is η_{GW}, the larger the torque must be to release the mechanism.
η_{WG} is conventionally positive.
The efficiencies η of meshing between worm and 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.
You can set the meshing losses friction model to:
No meshing losses - suitable for HIL
simulation
.
Constant efficiency
, which is the default
friction setting for block versions prior to R2020b.
Temperature-dependent efficiency
, which
models variability in the base-shaft efficiencies calculated in the
Constant efficiency
setting according to
a user-supplied look-up table. The temperature-dependency setting
enables a thermal conserving port H. This port
receives the heat flow into the block, which is translated into the
block temperature according to the gear Thermal
mass.
The viscous friction coefficient μ_{W} controls the viscous friction torque experienced by the worm from lubricated, nonideal gear threads and viscous bearing losses. The viscous friction torque on a worm driveline axis is –μ_{W}ω_{W}. ω_{W} is the angular velocity of the worm with respect to its mounting.
The viscous friction coefficient μ_{G} controls the viscous friction torque experienced by the gear, mainly from viscous bearing losses. The viscous friction torque on a gear driveline axis is –μ_{G}ω_{G}. ω_{G} is the angular velocity of the gear with respect to its mounting.
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.
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.
Variable settings are exposed only when, in the Meshing Losses
settings, the Friction model parameter is set to
Temperature-dependent efficiency
.
Gear inertia is assumed negligible.
Gears are treated as rigid components.
Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.