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, ω_{F} and ω_{B} have the same sign. If the follower and base axis rotate in opposite directions, ω_{F} and ω_{B} 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.
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 g_{FB}(t). Such a gear imposes a time-dependent kinematic constraint on the motions of the two driveshafts:
$${\omega}_{B}=\pm {g}_{FB}(t){\omega}_{F}$$
$${\tau}_{B}=\pm {g}_{FB}(t){\tau}_{F}$$
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 k_{p} and k_{v}:
$$\frac{d\varphi}{dt}=\pm {g}_{FB}(t){\omega}_{F}-{\omega}_{B},$$
$${\tau}_{B}=-{k}_{p}\varphi (t)-{k}_{v}\frac{d\varphi}{dt},$$
$$\pm {g}_{FB}(t){\tau}_{B}+{\tau}_{F}-{\tau}_{loss}=0.$$
τ_{loss} = 0 in the ideal case.
You can estimate the base angular compliance k_{p} from the transmission time constant t_{c} and inertia J.
$${k}_{p}=J{\left(\frac{2\pi}{{t}_{c}}\right)}^{2}$$
You can estimate the base angular velocity compliance k_{v} from the transmission time constant t_{c}, inertia J, and damping coefficient C.
$${k}_{v}=2C\frac{{t}_{c}}{2\pi}=2C\sqrt{\raisebox{1ex}{$J$}\!\left/ \!\raisebox{-1ex}{${k}_{p}$}\right.}$$
With nonideal torque transfer, τ_{loss} ≠ 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.
$$\begin{array}{l}{\tau}_{loss}={\tau}_{Coul}\mathrm{tanh}\left(4\raisebox{1ex}{${\omega}_{out}$}\!\left/ \!\raisebox{-1ex}{${\omega}_{th}$}\right.\right)+{\mu}_{B}{\omega}_{B}+{\mu}_{F}{\omega}_{F}\\ {\tau}_{Coul}=\left|{\tau}_{F}\right|(1-\eta )\end{array}$$
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 ω_{out} |
---|---|---|
Forward | ω _{B} τ _{B} > ω _{F} τ _{F} | Follower, ω_{F} |
Reverse | ω _{B} τ _{B} < ω _{F} τ _{F} | Base, ω_{B} |
The block only fully applies the friction loss represented by efficiency η if the absolute value of the follower angular velocity ω_{F} is greater than a velocity threshold ω_{th}.
If this absolute velocity is less than ω_{th}, the block smooths the efficiency to one at zero velocity.