# 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, ωF, and the angular velocity of the base, ωB, have the same sign. If they rotate in opposite directions, ωF and ωB have opposite signs. You can easily add and remove backlash, faults, and thermal effects.

### 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:

• rF is the radius of the follower gear.

• ωF is the angular velocity of the follower gear.

• rB is the radius of the base gear.

• ωB is the angular velocity of the base gear.

The follower-base gear ratio is

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

where:

• NB is the number of teeth in the base gear.

• NBF is the number of teeth in the follower gear.

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:

• τB is the input torque.

• τF is the output torque.

• τloss is the torque loss due to friction.

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.

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:

• τCoul is the Coulomb friction dependent torque.

• k is a proportionality constant.

• τidle is the net torque acting on the input shaft in idle mode.

Efficiency, η, is related to τCoul in the standard, preceding form but becomes dependent on load:

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

### 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:

• vTooth is the relative linear velocity of the gear tooth.

• rB is the Base (B) gear radius parameter.

• rF is the follower gear radius, where rF = NF/NB·rB, and the Follower (F) to base (B) teeth ratio (NF/NB) parameter represents NF/NB.

• ωB and ωF are the angular velocities of the base and follower gears, respectively.

• 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, xTooth, with respect to the linear backlash, Backlash, where -1/2·Backlash < xTooth < 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 xTooth.

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, coeffrest, 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 coeffrest 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.

StateValue
M = 0Disengaged
M = 1Forwards engaged with xtooth = 1/2·Backlash
M = -1Backwards engaged with xtooth = -1/2·Backlash
M = 2Instantaneous mode transition between forward engaged and forward disengaged
M = -2Instantaneous mode transition between backward engaged and backward disengaged
M = 3Instantaneous impact mode

The hard stop simulates static contact at the bounds. The gear locks when a collision occurs and |vTooth| < vtol. Once the gear locks, vTooth = 0. Once fTooth > fTol, the gear unlocks, where

• fTol is the Static contact release force threshold parameter.

• vtol is the Static contact speed threshold parameter.

• fTooth is the meshing force between the gear teeth such that fTooth = TB/rB = TF/rF.

### Faults

To model a fault in the Simple Gear block, in the Faults section, click the Add fault hyperlink next to the fault that you want to model. When the Add Fault window opens, you can to specify the fault properties. For more information about fault modeling, see Fault Behavior Modeling and Fault Triggering.

When you trigger a fault, the block applies the value of the Faulted efficiency parameter to the range of the gear specified in the Faulted angle range parameter.

### 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.

## Ports

### Conserving

expand all

Mechanical rotational conserving port associated with the base shaft.

Mechanical rotational conserving port associated with the follower shaft.

Thermal conserving port associated with heat flow. Heat flow affects gear temperature, and therefore, power transmission efficiency.

#### Dependencies

To enable this port, set Friction model to either:

• ```Temperature-dependent efficiency```

• ```Temperature and load-dependent efficiency```

## Parameters

expand all

### Parameter Dependencies Table

The table shows how the visibility of some Meshing Losses parameters and Faults parameters depend on the thermal model and the option that you choose for other parameters.

Default Model — For nonthermal models, thermal port H is not visible.Thermal Model — For thermal models, thermal port H is visible.
Meshing LossesMeshing Losses

Friction model — Choose ```No meshing losses - Suitable for HIL simulation```, ```Constant efficiency```, or ```Load-dependent efficiency```

Friction model — Choose ```Temperature-dependent efficiency``` or ```Temperature and load-dependent efficiency```

No meshing losses - Suitable for HIL simulationConstant efficiencyLoad-dependent efficiencyTemperature-dependent efficiencyTemperature and load-dependent efficiency

Efficiency

Input shaft torque at no load

Temperature

Temperature

Follower power threshold

Nominal output torque

Efficiency

Efficiency at nominal output torque

Follower power threshold

Efficiency matrix

Follower angular velocity threshold

Follower angular velocity threshold

### Main

Fixed ratio gFB of the follower axis to the base axis. The gear ratio must be strictly positive.

Direction of motion of the follower (output) driveshaft relative to the motion of the base (input) driveshaft.

### Meshing Losses

Meshing losses parameters depend on the thermal model. For more information, see Parameter Dependencies Table.

Friction models at various precision levels for estimating power losses due to meshing.

• ```No meshing losses - Suitable for HIL simulation``` — Neglect friction between gear cogs. Meshing is ideal.

• `Constant efficiency` — Reduce torque transfer by a constant efficiency factor. This factor falls in the range 0 < η ≤ 1 and is independent from load.

• `Load-dependent efficiency` — Reduce torque transfer by a variable efficiency factor. This factor falls in the range 0 < η < 1 and varies with the torque load.

• ```Temperature-dependent efficiency``` — Reduce torque transfer by a constant efficiency factor that is dependent on temperature but does not consider the gear load. This factor falls in the range 0 < η ≤ 1 and is independent from load. Torque transfer is determined from user-supplied data for gear efficiency and temperature.

• ```Temperature and load-dependent efficiency``` — Reduce torque transfer by a variable efficiency factor that is dependent on temperature and load. This factor falls in the range 0 < η < 1 and varies with the torque load. Torque transfer efficiency is determined from user-supplied data for gear loading and temperature.

Torque transfer efficiency, η, between base and follower shafts. Efficiency is inversely proportional to the meshing power losses.

#### Dependencies

To enable this parameter, set Friction model to `Constant efficiency`.

Absolute value of the follower shaft power above which the full efficiency factor is in effect. Below this value, a hyperbolic tangent function smooths the efficiency factor to 1, lowering the efficiency losses to 0 when no power is transmitted.

As a guideline, the power threshold should be lower than the expected power transmitted during simulation. Higher values might cause the block to underestimate efficiency losses. Very low values tend to raise the computational cost of simulation.

#### Dependencies

To enable this parameter, set Friction model to `Constant efficiency`.

Net torque,τidle, acting on the input shaft in idle mode, that is, when torque transfer to the output shaft equals zero. For nonzero values, the power input in idle mode completely dissipates due to meshing losses.

#### Dependencies

To enable this parameter, set Friction model to `Load-dependent efficiency`.

Output torque, τF, at which to normalize the load-dependent efficiency.

#### Dependencies

To enable this parameter, set Friction model to `Load-dependent efficiency`.

Torque transfer efficiency, η, at the nominal output torque. Larger efficiency values correspond to greater torque transfer between the input and output shafts.

#### Dependencies

To enable this parameter, set Friction model to `Load-dependent efficiency`.

Absolute value of the follower shaft angular velocity above which the full efficiency factor is in effect, ωF. Below this value, a hyperbolic tangent function smooths the efficiency factor to one, lowering the efficiency losses to zero when at rest.

As a guideline, the angular velocity threshold should be lower than the expected angular velocity during simulation. Higher values might cause the block to underestimate efficiency losses. Very low values tend to raise the computational cost of simulation.

#### Dependencies

To enable this parameter, set Friction model to `Load-dependent efficiency`.

Array of temperatures used to construct an efficiency lookup table. The array values must increase from left to right. The temperature array must be the same size as the efficiency array in temperature-dependent models. The array must be the same size as a single row of the efficiency matrix in temperature and load dependent models.

#### Dependencies

To enable this parameter, set Friction model to either:

• ```Temperature-dependent efficiency```

• ```Temperature and load-dependent efficiency```

Array of efficiencies used to construct a 1-D temperature-efficiency lookup table for temperature-dependent efficiency models. The array values are the efficiencies at the temperatures in the Temperature array. The number of elements must be the same as the number of elements in the Temperature array.

#### Dependencies

To enable this parameter, set Friction model to ```Temperature-dependent efficiency```.

Absolute value of the follower shaft power above which the full efficiency factor is in effect, pF. Below this value, a hyperbolic tangent function smooths the efficiency factor to 1, lowering the efficiency losses to 0 when no power is transmitted.

As a guideline, the power threshold should be lower than the expected power transmitted during simulation. Higher values might cause the block to underestimate efficiency losses. Very low values tend to raise the computational cost of simulation.

#### Dependencies

To enable this parameter, set Friction model to ```Temperature-dependent efficiency```.

Array of base-gear loads used to construct a 2-D temperature load efficiency lookup table for temperature and load dependent efficiency models. The array values must increase left to right. The load array must be the same size as a single column of the efficiency matrix.

#### Dependencies

To enable this parameter, set Friction model to ```Temperature and load-dependent efficiency```.

Matrix of component efficiencies used to construct a 2-D temperature load efficiency lookup table. The matrix elements are the efficiencies at the temperatures given by the Temperature array and at the loads given by the Load at base gear array.

The number of rows must be the same as the number of elements in the Temperature array. The number of columns must be the same as the number of elements in the Load at base gear array.

#### Dependencies

To enable this parameter, set Friction model to ```Temperature and load-dependent efficiency```.

Absolute value of the follower shaft angular velocity above which the full efficiency factor is in effect, ωF. Below this value, a hyperbolic tangent function smooths the efficiency factor to one, lowering the efficiency losses to zero when at rest.

As a guideline, the angular velocity threshold should be lower than the expected angular velocity during simulation. Higher values might cause the block to underestimate efficiency losses. Very low values tend to raise the computational cost of simulation.

#### Dependencies

To enable this parameter, set Friction model to ```Temperature and load-dependent efficiency```.

### Backlash

Whether to enable backlash.

Stiffness and rebound options for the hard stop model. You can choose from these options:

• ```Stiffness and damping applied smoothly through transition region, damped rebound```

• ```Full stiffness and damping applied at bounds, undamped rebound```

• ```Full stiffness and damping applied at bounds, damped rebound```

• ```Based on coefficient of restitution```

#### Dependencies

To enable this parameter, select Enable backlash.

Translational distance that a gear tooth can travel between meshing teeth.

#### Dependencies

To enable this parameter, select Enable backlash.

Base gear distance from the center to the meshing point on the gear tooth.

#### Dependencies

To enable this parameter, select Enable backlash.

Distance where the block gradually applies effects of stiffness and damping. When you set Hard stop model to ```Stiffness and damping applied smoothly through transition region, damped rebound```, the block smoothly transitions the onset of stiffness and damping as the hard stop approaches full stiffness.

#### Dependencies

To enable this parameter, select Enable backlash and set Hard stop model to ```Stiffness and damping applied smoothly through transition region, damped rebound```.

Effective translational spring stiffness of the gear collision.

#### Dependencies

To enable this parameter, select Enable backlash and set Hard stop model to one of these options:

• ```Stiffness and damping applied smoothly through transition region, damped rebound```

• ```Full stiffness and damping applied at bounds, undamped rebound```

• ```Full stiffness and damping applied at bounds, damped rebound```

Effective translational damping of the gear collision.

#### Dependencies

To enable this parameter, select Enable backlash and set Hard stop model to one of these options:

• ```Stiffness and damping applied smoothly through transition region, damped rebound```

• ```Full stiffness and damping applied at bounds, undamped rebound```

• ```Full stiffness and damping applied at bounds, damped rebound```

Loss of translational kinetic energy during collisions. A value of `0` represents an inelastic collision, and a value of `1` represents a perfectly elastic collision where the gear retains all kinetic energy. The default of `0` is equivalent to simulating gear torque when the gears are in contact, and removing torque when the gear changes direction and the tooth travels the backlash distance.

#### Dependencies

To enable this parameter, select Enable backlash and set Hard stop model to ```Based on coefficient of restitution```.

Velocity, vtol, below which the gear tooth becomes locked with the meshing teeth. The block sets vTooth = 0 when |vTooth| < vtol.

#### Dependencies

To enable this parameter, select Enable backlash and set Hard stop model to ```Based on coefficient of restitution```.

Minimum force needed to release the gear from a static contact mode.

#### Dependencies

To enable this parameter, select Enable backlash and set Hard stop model to ```Based on coefficient of restitution```.

### Viscous Losses

Two-element array with the viscous friction coefficients in effect at the base and follower shafts. To neglect viscous losses, use the default setting, `[0, 0]`.

### Faults

To enable the Faults tab, set Friction model to

• `Constant efficiency`

• `Load-dependent efficiency`

• `Temperature-dependent efficiency`

• ```Temperature and load-dependent efficiency```

To modify the faults, create a fault and, in the block dialog, click Open fault properties. In the Property Inspector, click the Fault behavior link to open the faults.

Whether to model a fault in the block. When you trigger a fault, the block applies the value of the Faulted efficiency parameter to the range of the gear specified in the Faulted angle range parameter. To add a fault, click the Add fault hyperlink.

#### Dependencies

To enable this parameter, set Meshing Losses to:

• `Constant efficiency`

• ```Load-dependent efficiency```

• ```Temperature-dependent efficiency```

• ```Temperature and load-dependent efficiency```

Efficiency when a fault triggers.

#### Dependencies

To enable this parameter, enable faults for the block by clicking the hyperlink.

Rotational angle range for the faulted efficiency. For multiples of ```2π rad```, the block applies the value of the Faulted efficiency parameter throughout rotation.

#### Dependencies

To enable this parameter, enable faults for the block by clicking the hyperlink.

### Thermal Port

Thermal energy required to change the component temperature by a single degree. The greater the thermal mass, the more resistant the component is to temperature change.

#### Dependencies

To enable this parameter, set Friction model to either:

• ```Temperature-dependent efficiency```

• ```Temperature and load-dependent efficiency```

Temperature at simulation start.

#### Dependencies

To enable this parameter, set Friction model to either:

• ```Temperature-dependent efficiency```

• ```Temperature and load-dependent efficiency```

expand all

## Version History

Introduced in R2011a

expand all