Main Content

Variable Ratio Transmission

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

  • Variable Ratio Transmission block

Libraries:
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, the base and the follower.

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. When the input ratio is positive, the rotating direction of the output shaft is the one you specify. When the ratio is negative, the output shaft rotates in the opposite direction of the input shaft. When the ratio is zero, the two shafts become disengaged and do not transfer torque.

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:

ωB=±gFB(t)ωF

τF=±gFB(t)τB

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:

dϕdt=(±gFB(t)ωFωB )αkpkv ϕ(1α)τB=(kpϕ(t)kv(±gFB(t)ωFωB))α±gFB(t)τB+τFτloss=0

where ɑ = 3u2-2u3, and u=|gFB(t)|gFB,thr. The function, α, enables smooth transitions between engaged and disengaged states, and the ratio threshold gFB,thr determines the magnitude of the ratio to initiate a transition. When you set Losses model to No losses — Suitable for HIL simulation, τloss = 0.

Estimating Compliance Parameters
  • You can estimate the base transmission stiffness, kp, from the transmission time constant tc and inertia J.

    kp=J(2πtc)2

  • You can estimate the base transmission damping kv from the transmission time constant tc, inertia J, and damping coefficient C.

    kv=2Ctc2π=2CJkp

Nonideal Torque Transfer and Losses

When you set Losses model to Constant efficiency, τloss ≠ 0. The 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, μ1 and μ2.

The torque loss in the torque balance equation is due to Coulomb friction, such that

τloss=(1η)(1+tanh4ppthr2(±gFB(t)τB)+1tanh4ppthr2τF)

where pthr is the Follower power threshold parameter. The block calculates the viscous shaft frictions as internal damping torques attached to ports B and F, such that

τdamper,B=μBωBτdamper,F=μFωF

where the total torque is the sum of τB or τF and its respective damping torque.

Examples

Ports

Input

expand all

Physical signal port associated with the ratio of base to follower torques, which equals the ratio of follower to base angular velocities.

Conserving

expand all

Mechanical rotational conserving port associated with the base driveshaft.

Mechanical rotational conserving port associated with the follower driveshaft.

Parameters

expand all

Main

Output driveshaft rotation relative to the input driveshaft.

Compliance

Reciprocal of the transmission angular compliance kp measured at the base.

Reciprocal of the transmission angular compliance damping kv measured at the base.

Torque applied at the base driveshaft at the start of simulation.

Transmission ratio below which the block uses smoothing. The block smooths the transmission ratio when the ratio is between zero and the value of the Ratio threshold parameter.

Transmission Losses

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.

Effective torque transfer efficiency η between the base and follower driveshafts.

Dependencies

To enable this parameter, set Losses model to Constant efficiency.

Absolute value of the follower shaft power above which the full efficiency factor is in effect. Below this value, a hyperbolic tangent function smooths the efficiency factor to 1, lowering the efficiency losses to 0 when no power is transmitted.

As a guideline, the power threshold should be lower than the expected power transmitted during simulation. Higher values might cause the block to underestimate efficiency losses. Very low values tend to raise the computational cost of simulation.

Dependencies

To enable this parameter, set Losses model to Constant efficiency.

Viscous Losses

Vector of viscous damping coefficients [μB, μF] applied at the base and follower driveshafts, respectively.

Extended Capabilities

expand all

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2011a

expand all