# Variable Ratio Transmission

Dynamic gearbox with variable and controllable gear ratio, transmission compliance, and friction losses

## Library

Simscape / Driveline / Couplings & Drives

## Description

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, ωF and ωB have the same sign. If they rotate in opposite directions, ωF and ωB have opposite signs.

Transmission compliance introduces internal time delay between the axis motions. Therefore, unlike a gear, a variable ratio transmission does not act as a kinematic constraint. You can also control torque loss caused by transmission and viscous losses. For model details, see Variable Ratio Transmission Model.

### Ports

B and F are rotational conserving ports representing, respectively, the base and follower driveshafts.

You specify the unitless variable gear ratio gFB(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.

## Variable Ratio Transmission Model

### Ideal Motion and Torque Transfer

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 gFB(t). Such a gear imposes a time-dependent kinematic constraint on the motions of the two driveshafts:

ωB = ±gFB(tωF , τF = ±gFB(tτB .

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 kp and kv:

dϕ/dt = ±gFB(tωFωB ,

τB = –kpϕkvdϕ/dt ,

±gFB(tτB + τFτloss = 0 .

τloss = 0 in the ideal case.

Estimating Compliance Parameters
• You can estimate the base angular compliance kp from the transmission time constant tc and inertia J: kp = J(2π/tc)2.

• You can estimate the base angular velocity compliance kv from the transmission time constant tc, inertia J, and damping coefficient C: kv = (2Ctc)/2π = 2C√(J/kp).

### Nonideal Torque Transfer and Losses

With nonideal torque transfer, τloss ≠ 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.

τloss = τCoul·tanh(4ωout/ωth) + μBωB + μFωF , τCoul = |τF|·(1 – η) .

When the angular velocity changes sign, the hyperbolic tangent regularizes the sign change in the Coulomb friction torque.

Power FlowPower Loss ConditionOutput Driveshaft ωout
ForwardωBτB > ωFτFFollower, ωF
ReverseωBτB < ωFτFBase, ωB

### Meshing Efficiency

The friction loss represented by efficiency η is fully applied only 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 actual efficiency is automatically regularized to one at zero velocity.

## Parameters

### Main

Output shaft rotates

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```.

### Compliance

Transmission stiffness at base (B)

Reciprocal of transmission angular compliance kp, 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`).

Transmission damping at base (B)

Reciprocal of transmission angular compliance damping kv, 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)`).

Initial input torque at base (B)

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`).

### Transmission Losses

Losses model

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.

### Viscous Losses

Viscous friction coefficients at base (B) and follower (F)

Vector of viscous damping coefficients [μB μF] applied at the base and follower driveshafts, respectively. The default is ```[ 0 0 ]```.

From the drop-down list, choose units. The default is newton-meters/(radian/second) (`N*m/(rad/s)`).

## Real-Time Simulation

### Hardware-in-the-Loop Simulation

For optimal simulation performance, use the Transmission Losses > Losses model parameter default setting, ```No losses - Suitable for HIL simulation```.