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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:

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

You can incorporate the effects of backlash in your model. *Backlash* is
the excess space between a gear tooth and the mating gear teeth. Increasing the backlash
compensates for lowering manufacturing tolerances and allows the free motion of lubricants
in the gears to prevent jamming. However, excess backlash can cause premature wear on your
system components and can affect measurements that rely on gear position. The block applies
backlash for start-ups and reversals.

You can set **Backlash model** to:

`No backlash`

— The block ignores the effects of backlash.`Spring-damper`

— The block uses a Translational Hard Stop implementation to simulate backlash similar to a spring and damper in parallel.`Instantaneous impulse`

— The block locks the gears in contact while the relative backlash velocity remains below the**Static Contact Threshold**parameter. The block unlocks the gears when the contact force changes direction.This option is suitable for HIL simulation.

When you set **Backlash model** to `Spring-damper`

or `Instantaneous impulse`

, the block relates gear rotation to
linear backlash as:

$${v}_{Tooth}={r}_{B}{\omega}_{B}-\beta {r}_{F}{\omega}_{F},$$

where:

*v*is the relative linear velocity of the gear tooth._{Tooth}*r*is the_{B}**Base (B) gear radius**parameter associated with the base gear.*r*is the follower gear radius, where_{F}*r*_{F}= N_{F}/N_{B}·*r*, and the_{B}**Follower (F) to base (B) teeth ratio (NF/NB)**parameter represents*N*._{F}/N_{B}*ω*and_{B}*ω*are the angular velocities of the base and follower gears, respectively._{F}*β*is the gear direction sign. When you set:**Output shaft rotates**to`In same direction as input shaft`

,*β = 1*.**Output shaft rotates**to`In opposite direction as input shaft`

,*β = -1*.

The block treats the meshing gear tooth as a position,
*x _{Tooth}*, with respect to the linear backlash,

When you set **Backlash model** to ```
Instantaneous
impulse
```

, the hard stop can incorporate a nonzero value for the
**Coefficient of restitution** parameter, *e*, into
the momentum balance equation. During a collision,

$$e=\frac{{v}_{Backlash,t-}}{{v}_{Backlash,t+}},$$

where *t-* and *t+* are the instants
before and after the collision, respectively. The block asserts *0 < e <
1*. For more information, see State Reset Modeling. Simscape™ logs the mode state of the gear as the intermediate
**M**.

State | Value |
---|---|

M = 0 | Disengaged |

M = 1 | Forwards engaged with x_{tooth} =
1/2·Backlash |

M = -1 | Backwards engaged with x_{tooth} =
-1/2·Backlash |

The hard stop simulates static contact at the bounds. The gears lock when a collision occurs and *|v _{Tooth}| < v_{tol}*, such that

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 enabling the optional thermal port. To enable
the port, set **Friction model** to ```
Temperature-dependent
efficiency
```

.

Additionally, you can choose to model efficiency that varies with loading and
temperature by setting **Friction model** to
`Temperature and load-dependent efficiency`

. 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.

Use the **Variables** settings to set the priority and initial target
values for the block variables before simulating. For more information, see Set Priority and Initial Target for Block Variables.

Gear inertia is assumed to be negligible.

Gears are treated as rigid components.

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