Variable Ratio Transmission

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

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.

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.

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:

$ω_{B} = \pm g_{F B} (t) ω_{F}$

$τ_{B} = \pm g_{F B} (t) τ_{F}$

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:

$\frac{d ϕ}{d t} = \pm g_{F B} (t) ω_{F} - ω_{B},$

$τ_{B} = - k_{p} ϕ (t) - k_{v} \frac{d ϕ}{d t},$

$\pm g_{F B} (t) τ_{B} + τ_{F} - τ_{l o s s} = 0.$

τ_loss = 0 in the ideal case.

Estimating Compliance Parameters

You can estimate the base angular compliance k_p from the transmission time constant t_c and inertia J.
$k_{p} = J {(\frac{2 π}{t_{c}})}^{2}$
You can estimate the base angular velocity compliance k_v from the transmission time constant t_c, inertia J, and damping coefficient C.
$k_{v} = 2 C \frac{t_{c}}{2 π} = 2 C \sqrt{\frac{J}{k_{p}}}$

Nonideal Torque Transfer and Losses

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.

$\begin{array}{l} τ_{l o s s} = τ_{C o u l} \tanh (4 \frac{ω_{o u t}}{ω_{t h}}) + μ_{B} ω_{B} + μ_{F} ω_{F} \\ τ_{C o u l} = | τ_{F} | (1 - η) \end{array}$

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

Meshing Efficiency

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.

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

expand all

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

Physical signal port associated with the input to output shaft ratio.

Conserving

expand all

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

expand all

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
`3e5` `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
`0.05` `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.

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 angular velocity threshold — Point at which block applies full efficiency loss
`0.01` `rad/s` (default) | positive scalar

Absolute angular velocity threshold ω_th above which full efficiency loss is applied, for follower velocity ω_F.

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

Variable Ratio Transmission

Description

Ideal Motion and Torque Transfer

Nonideal Torque Transfer and Losses

Meshing Efficiency

Examples

Variable Ratio Gear

Vehicle with Manual Transmission

Ports

Input

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

Conserving

B — Base driveshaft mechanical rotational

F — Follower driveshaft mechanical rotational

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

Compliance

Transmission stiffness at base (B) — Base spring rate coefficient 3e5 N*m/rad (default) | positive scalar

Transmission damping at base (B) — Base damping coefficient 0.05 N*m/(rad/s) (default) | positive scalar

Initial input torque at base (B) — Starting torque at base 0 N*m (default) | scalar

Transmission Losses

Losses model — Optional loss modeling No losses — Suitable for HIL simulation (default) | Constant efficiency

Efficiency — Base-to-follower torque transfer efficiency 0.8 (default) | positive scalar

Dependencies

Follower angular velocity threshold — Point at which block applies full efficiency loss 0.01 rad/s (default) | positive scalar

Dependencies

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

Extended Capabilities

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

Version History

See Also

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

B — Base driveshaft
mechanical rotational

F — Follower driveshaft
mechanical rotational

Output shaft rotates — Output shaft direction versus input shaft direction
`In same direction as input shaft` (default) | `In opposite direction to input shaft`

Transmission stiffness at base (B) — Base spring rate coefficient
`3e5` `N*m/rad` (default) | positive scalar

Transmission damping at base (B) — Base damping coefficient
`0.05` `N*m/(rad/s)` (default) | positive scalar

Initial input torque at base (B) — Starting torque at base
`0` `N*m` (default) | scalar

Losses model — Optional loss modeling
`No losses — Suitable for HIL simulation` (default) | `Constant efficiency`

Efficiency — Base-to-follower torque transfer efficiency
`0.8` (default) | positive scalar

Follower angular velocity threshold — Point at which block applies full efficiency loss
`0.01` `rad/s` (default) | positive scalar

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

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