Simple gear of base and follower wheels with adjustable gear ratio, friction losses, and triggered faults

**Library:**Simscape / Driveline / Gears

The Simple Gear block represents a gearbox that constrains
the connected driveline axes of the base gear, *B*, and the follower
gear, *F*, to corotate with a fixed ratio that you specify. You choose
whether the follower axis rotates in the same or opposite direction as the base axis. If
they rotate in the same direction, the angular velocity of the follower,
*ω _{F}*, and the angular velocity of the
base,

The kinematic constraint that the Simple Gear block imposes on the two connected axes is

$${r}_{F}{\omega}_{F}={r}_{B}{\omega}_{B}$$

where:

*r*is the radius of the follower gear._{F}*ω*is the angular velocity of the follower gear._{F}*r*is the radius of the base gear._{B}*ω*is the angular velocity of the base gear._{B}

The follower-base gear ratio is

$${g}_{FB}=\frac{{r}_{F}}{{r}_{B}}=\frac{{N}_{F}}{{N}_{B}}$$

where:

*N*is the number of teeth in the base gear._{B}*N*is the number of teeth in the follower gear._{BF}

Reducing the two degrees of freedom to one independent degree of freedom yields the torque transfer equation

$${g}_{FB}{\tau}_{B}+{\tau}_{F}-{\tau}_{loss}=0$$

where:

*τ*is the input torque._{B}*τ*is the output torque._{F}*τ*is the torque loss due to friction._{loss}

For the ideal case, $${\tau}_{loss}=0$$.

In the nonideal case, $${\tau}_{loss}\ne 0$$. For general considerations on nonideal gear modeling, see Model Gears with Losses.

In a nonideal gear pair (B,F), the angular velocity, gear radii, and gear teeth constraints are unchanged. But the transferred torque and power are reduced by:

Coulomb friction between teeth surfaces on gears

*B*and*F*, characterized by efficiency,*η*Viscous coupling of driveshafts with bearings, parametrized by viscous friction coefficients,

*μ*

In the constant efficiency case, *η* is constant, independent
of load or power transferred.

In the load-dependent efficiency case, *η* depends on the load or power
transferred across the gears. For either power flow,

$${\tau}_{Coul}={g}_{FB}{\tau}_{idle}+k{\tau}_{F}$$

where:

*τ*is the Coulomb friction dependent torque._{Coul}*k*is a proportionality constant.*τ*is the net torque acting on the input shaft in idle mode._{idle}

Efficiency, *η*, is related to
*τ _{Coul}* in the standard,
preceding form but becomes dependent on load:

$$\eta =\frac{{\tau}_{F}}{{g}_{FB}{\tau}_{idle}+(k+1){\tau}_{F}}$$

If you enable faults for the block, the efficiency changes in response to one or both of these triggers:

Simulation time — A fault occurs at a specified time.

Simulation behavior — A fault occurs in response to an external trigger. Enabling an external fault trigger exposes port

**T**.

If a fault trigger occurs, for the remainder of the simulation, the block uses the faulted efficiency in one of these ways:

Throughout rotation

When the rotation angle is within a faulted range that you specify

You can program the block to issue a fault report as a warning or error message.

You can model the effects of heat flow and temperature change by selecting a thermal block variant. Selecting a thermal variant:

Exposes port

**H**, a conserving port in the thermal domain.Enables the

**Thermal mass**parameter, which allows you to specify the ability of the component to resist changes in temperature.Enables the

**Initial Temperature**parameter, which allows you to set the initial temperature.

To select a thermal variant, right-click the block in your model and, from the
context menu, select **Simscape** > **Block choices**. Select a variant that includes a thermal port.

Gear inertia is assumed negligible.

Gears are treated as rigid components.

Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.