Variable Ratio Transmission
Dynamic gearbox with variable and controllable gear ratio, transmission compliance, and friction losses
Simscape / Driveline / Couplings & Drives
The Variable Ratio Transmission block represents a gearbox that dynamically transfers motion and torque between the two connected driveshaft axes, the base and the follower.
When you choose to have the block ignore the dynamics of transmission compliance, it constrains the driveshafts to corotate with a variable gear ratio that you control. You can choose whether the follower axis rotates in the same or opposite direction as the base axis. If the follower and base axis rotate in the same direction, ωF and ωB have the same sign. If the follower and base axis rotate in opposite directions, ωF and ωB have opposite signs.
Transmission compliance introduces internal time delay between the axis motions. Unlike a gear, a variable ratio transmission does not act as a kinematic constraint. You can also control the torque loss caused by transmission and viscous losses.
Ideal Motion and Torque Transfer
The Variable Ratio Transmission block dynamically transfers motion and torque between the base shaft and the follower shaft.
If the relative compliance ϕ between the axes is absent, the block is equivalent to a gear with a variable ratio gFB(t). Such a gear imposes a time-dependent kinematic constraint on the motions of the two driveshafts:
However, the Variable Ratio Transmission does include compliance between the axes. Dynamic motion and torque transfer replace the kinematic constraint, with a nonzero ϕ that dynamically responds through the base compliance parameters kp and kv:
τloss = 0 in the ideal case.
You can estimate the base angular compliance kp from the transmission time constant tc and inertia J.
You can estimate the base angular velocity compliance kv from the transmission time constant tc, inertia J, and damping coefficient C.
Nonideal Torque Transfer and Losses
With nonideal torque transfer, τloss ≠ 0. The Variable Ratio Transmission block models losses similarly to nonideal gears. For general information about nonideal gear modeling, see Model Gears with Losses.
In a nonideal gearbox, the angular velocity and compliance dynamics remain the same as in the ideal case. The transferred torque and power are reduced by:
Coulomb friction, such as the friction between belt and wheel, or internal belt losses due to stretching, characterized by an efficiency η.
Viscous coupling of driveshafts with bearings, parametrized by viscous friction coefficients μ.
When the angular velocity changes sign, the hyperbolic tangent function smooths the change in the Coulomb friction torque.
|Power Flow||Power Loss Condition||Output Driveshaft ωout|
|Forward||ω B τ B > ω F τ F||Follower, ωF|
|Reverse||ω B τ B < ω F τ F||Base, ωB|
The block only fully applies the friction loss represented by efficiency η if the absolute value of the follower angular velocity ωF is greater than a velocity threshold ωth.
If this absolute velocity is less than ωth, the block smooths the efficiency to one at zero velocity.
r — Input-to-output shaft velocity ratio, unitless
Physical signal port associated with the input to output shaft ratio.
B — Base driveshaft
Mechanical rotational conserving port associated with the base driveshaft.
F — Follower driveshaft
Mechanical rotational conserving port associated with the follower driveshaft.
Output shaft rotates — Output shaft direction versus input shaft direction
In same direction as input
shaft (default) |
In opposite direction to input
Output driveshaft rotation relative to the input driveshaft.
Transmission stiffness at base (B) — Base spring rate coefficient
N*m/rad (default) | positive scalar
Reciprocal of the transmission angular compliance kp measured at the base.
Transmission damping at base (B) — Base damping coefficient
N*m/(rad/s) (default) | positive scalar
Reciprocal of the transmission angular compliance damping kv measured at the base.
Initial input torque at base (B) — Starting torque at base
N*m (default) | scalar
Torque applied at the base driveshaft at the start of simulation.
Losses model — Optional loss modeling
No losses — Suitable for HIL
simulation (default) |
Implementation of friction losses from nonideal torque transfer.
No losses — Suitable for HIL simulation— Torque transfer is ideal.
Constant efficiency— The block reduces gear box torque transfer by a constant efficiency η satisfying 0 < η ≤ 1.
Efficiency — Base-to-follower torque transfer efficiency
0.8 (default) | positive scalar
Effective torque transfer efficiency η between the base and follower driveshafts.
To enable this parameter, set Losses model to
Follower angular velocity threshold — Point at which block applies full efficiency loss
rad/s (default) | positive scalar
Absolute angular velocity threshold ωth above which full efficiency loss is applied, for follower velocity ωF.
To enable this parameter, set Losses model to
Viscous friction coefficients at base (B) and follower (F) — Base and follower \viscous friction vector
N*m/(rad/s) (default) | vector of nonnegative elements
Vector of viscous damping coefficients [μB, μF] applied at the base and follower driveshafts, respectively.