# Simple Gear

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

**Libraries:**

Simscape /
Driveline /
Gears

## Description

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,

*ω*, have the same sign. If they rotate in opposite directions,

_{B}*ω*and

_{F}*ω*have opposite signs. You can easily add and remove backlash, faults, and thermal effects.

_{B}### Ideal Gear Constraint and Gear Ratio

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, $${\tau}_{loss}=0$$.

### Nonideal Gear Constraint and Losses

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*

**Constant Efficiency**

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

**Load-Dependent Efficiency**

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:

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

### Backlash

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 using an implementation of the Translational Hard Stop block.

When you select **Enable backlash**, 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.*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,

*Backlash*, where

*-1/2·Backlash < x*.

_{Tooth}< 1/2·Backlash*Backlash*is equivalent to the

**Linear backlash**parameter. The initial value of the

**Backlash position**variable is equivalent to the initial position of

*x*.

_{Tooth}When you set **Hard stop model** to ```
Based on coefficient of
restitution
```

, the hard stop can incorporate a nonzero value for the
**Coefficient of restitution** parameter,
*coeff _{rest}*, into the momentum balance
equation. During a collision,

$$coef{f}_{rest}=\frac{{v}_{Backlash,t-}}{{v}_{Backlash,t+}},$$

where *t-* and *t+* are the instants
before and after the collision, respectively. The block asserts
*coeff _{rest}* is in the range [0, 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 |

M = 2 | Instantaneous mode transition between forward engaged and forward disengaged |

M = -2 | Instantaneous mode transition between backward engaged and backward disengaged |

M = 3 | Instantaneous impact mode |

The hard stop simulates static contact at the bounds. The gear locks when a collision occurs
and *|v _{Tooth}| < v_{tol}*.
Once the gear locks,

*v*= 0. Once

_{Tooth}*f*>

_{Tooth}*f*, the gear unlocks, where

_{Tol}*f*is the_{Tol}**Static contact release force threshold**parameter.*v*is the_{tol}**Static contact speed threshold**parameter.*f*is the meshing force between the gear teeth such that_{Tooth}*f*=_{Tooth}*T*=_{B}/r_{B}*T*._{F}/r_{F}

### Faults

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.

### Thermal Model

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.

### Variables

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.

## Assumptions

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.

## Ports

### Input

### Conserving

## Parameters

## More About

## Extended Capabilities

## Version History

**Introduced in R2011a**