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

Couplings & Drives

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.

**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`

.

**Transmission stiffness at base (B)**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`

).**Transmission damping at base (B)**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)`

).**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`

).

**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 friction coefficients at base (B) and follower (F)**Vector of viscous damping coefficients [

*μ*_{B}*μ*_{F}] 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)`

).

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}:

*d**ϕ*/*dt* =
±*g*_{FB}(*t*)·*ω*_{F} – *ω*_{B} ,

*τ*_{B} =
–*k*_{p}*ϕ* – *k*_{v}*d**ϕ*/*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*(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**t*_{c})/2*π*= 2*C*√(*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(4*ω*_{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.

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