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Dynamic gearbox with variable and controllable gear ratio, transmission compliance, and friction losses
The Variable Ratio Transmission block represents a gearbox that dynamically transfers motion and torque between the two connected driveshaft axes, base and follower.
Ignoring the dynamics of transmission compliance, the driveshafts are constrained 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 they rotate in the same direction, ω_{F} and ω_{B} have the same sign. If they 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 therefore does not act as a kinematic constraint. You can also control torque loss caused by transmission and viscous bearing losses. For model details, see Variable Ratio Transmission Model.
B and F are rotational conserving ports representing, respectively, the base and follower driveshafts.
You specify the unitless variable gear ratio g_{FB}(t) as a function of time at the physical signal input at port r. If the signal value becomes zero or negative, the simulation stops with an error.
From the drop-down list, choose how the output driveshaft rotates relative to the input driveshaft. The default is In same direction as input shaft.
Reciprocal of transmission angular compliance k_{p}, angular displacement per unit torque, measured at the base. The default is 30000.
From the drop-down list, choose units. The default is newton-meters/radian (N*m/rad).
Reciprocal of transmission angular compliance damping k_{v}, angular speed per unit torque, measured at the base. The default is 0.05.
From the drop-down list, choose units. The default is newton-meters/(radian/second) (N*m/(rad/s)).
Torque applied at the base driveshaft at the start of simulation (t = 0). The default is 0.
From the drop-down list, choose units. The default is newton-meters (N*m).
Select how to implement friction losses from nonideal torque transfer. The default is No losses.
No losses — Suitable for HIL simulation — Torque transfer is ideal.
Constant efficiency — Transfer of torque across gearbox is reduced by a constant efficiency η satisfying 0 < η ≤ 1. If you select this option, the panel changes from its default.
Variable Ratio Transmission is a dynamical mechanism for transferring motion and torque between base and follower.
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:
ω_{B} = ±g_{FB}(t)·ω_{F} , τ_{F} = ±g_{FB}(t)·τ_{B} .
However, Variable Ratio Transmission does include compliance between the axes. Dynamical motion and torque transfer replace the kinematic constraint, with a nonzero ϕ that dynamically responds through base compliance parameters k_{p} and k_{v}:
dϕ/dt = ±g_{FB}(t)·ω_{F} – ω_{B} ,
τ_{B} = –k_{p}ϕ – k_{v}dϕ/dt ,
±g_{FB}(t)·τ_{B} + τ_{F} – τ_{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(2π/t_{c})^{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} = (2Ct_{c})/2π = 2C√(J/k_{p}).
With nonideal torque transfer, τ_{loss} ≠ 0. Losses in the Variable Ratio Transmission are modeled similarly to how losses are modeled in nonideal gears. For general information on nonideal gear modeling, see Model Gears with Losses.
In a nonideal gearbox, the angular velocity and compliance dynamics are unchanged. The transferred torque and power are reduced by:
Coulomb friction (for example, 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.
τ_{loss} = τ_{Coul}·tanh(4ω_{out}/ω_{th}) + μ_{B}ω_{B} + μ_{F}ω_{F} , τ_{Coul} = |τ_{F}|·(1 – η) .
When the angular velocity changes sign, the hyperbolic tangent regularizes the sign 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 friction loss represented by efficiency η is fully applied only 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 actual efficiency is automatically regularized to one at zero velocity.
The sdl_variable_gearsdl_variable_gear example model provides a basic example of a simple gear.