Variable Ratio Transmission

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

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

$${\omega }_B=\pm g_{FB}(t){\omega }_F$$

$${\tau }_F=\pm g_{FB}(t){\tau }_B$$

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:

$$\begin{array}{l}\frac{d\varphi }{dt}=\left(\pm g_{FB}\left(t\right){\omega }_F-{\omega }_B\right)\alpha -\frac{k_p}{k_v} \varphi \left(1-\alpha \right)\\ {\tau }_B=\left(-k_p\varphi (t)-k_v\left(\pm g_{FB}(t){\omega }_F-{\omega }_B\right)\right)\alpha \\ \pm g_{FB}(t){\tau }_B+{\tau }_F-{\tau }_{loss}=0\end{array}$$

where ɑ = 3u²-2u³, and $$\text{u}=\frac{\left|{\text{g}}_{FB}(\text{t})\right|}{g_{FB,thr}}$$. The function, α, enables smooth transitions between engaged and disengaged states, and the ratio threshold g_FB,thr determines the magnitude of the ratio to initiate a transition. When you set Losses model to No losses — Suitable for HIL simulation, τ_loss = 0.

Estimating Compliance Parameters

• You can estimate the base transmission stiffness, 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 transmission damping k_v from the transmission time constant t_c, inertia J, and damping coefficient C.

  $$k_v=2C\frac{t_c}{2\pi }=2C\sqrt{\raisebox{1ex}{$J$}\!\left/ \!\raisebox{-1ex}{$k_p$}\right.}$$

Nonideal Torque Transfer and Losses

When you set Losses model to Constant efficiency, τ_loss ≠ 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

$${\tau }_{loss}=(1-\eta )\left(\frac{1+\tanh \frac{4p}{p_{thr}}}{2}\left(\pm g_{FB}(t){\tau }_B\right)+\frac{1-\tanh \frac{4p}{p_{thr}}}{2}{\tau }_F\right)$$

where p_thr 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

$$\begin{array}{l}{\tau }_{damper,B}={\mu }_B{\omega }_B\\ {\tau }_{damper,F}={\mu }_F{\omega }_F\end{array}$$

where the total torque is the sum of τ_B or τ_F and its respective damping torque.

Examples

Variable Ratio Gear

A variable ratio gear coupling two inertias (shafts). The gear ratio between the follower (F) and the base (B) is steadily increased from 1:1 to 2:1. The ratio of the follower to base angular speed decreases with gear ratio, and conversely the ratio of the follower to base torque increases with gear ratio. The variable gear is implemented using the Variable Ratio Transmission block which includes compliance. Appropriate stiffness and damping values for the compliance will depend upon the levels of torque transmitted by the variable gear. If the compliance damping is unrealistically low, the system can develop high frequency oscillations that require small simulation time steps.

• Open Model

Vehicle with Manual Transmission

A vehicle that has a four-speed manual transmission. The key elements of the transmission are four synchronizers. By engaging or disengaging these synchronizers and associated dog clutches, the transmission provides four ratios 3.581, 2.022, 1.384, and 1, respectively. The synchronizers are modeled using the Cone Clutch and Dog Clutch blocks.

• Open Model

Ports

Input

r — Input-to-output shaft velocity ratio, unitless

physical signal

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

Conserving

B — Base driveshaft

mechanical rotational

Mechanical rotational conserving port associated with the base driveshaft.

F — Follower driveshaft

mechanical rotational

Mechanical rotational conserving port associated with the follower driveshaft.

Parameters

Main

Output shaft rotates — Output shaft direction versus input shaft direction

In same direction as input shaft (default) | In opposite direction to input shaft

Output driveshaft rotation relative to the input driveshaft.

Compliance

Transmission stiffness at base (B) — Base spring rate coefficient

5e5 N*m/rad (default) | positive scalar

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

Transmission damping at base (B) — Base damping coefficient

5e3 N*m/(rad/s) (default) | positive scalar

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

Initial input torque at base (B) — Starting torque at base

0 N*m (default) | scalar

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

Ratio threshold — Zero transition smoothing

0.01 (default) | scalar

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

Losses model — Optional loss modeling

No losses — Suitable for HIL simulation (default) | Constant efficiency

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.

Dependencies

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

Follower power threshold — Point at which block applies full efficiency losses

0.001 W (default) | positive scalar

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

Viscous friction coefficients at base (B) and follower (F) — Base and follower viscous friction vector

[0, 0] N*m/(rad/s) (default) | vector of nonnegative elements

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

Extended Capabilities

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

Version History

Introduced in R2011a

R2025a: Specify a ratio of zero

You can now specify a zero-ratio or a continuously variable ratio that changes sign in the block. When the ratio is zero, the block does not transfer power between ports B and F.