Simple Gear

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

Library:
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_B, have the same sign. If they rotate in opposite directions, ω_F and ω_B have opposite signs.

Ideal Gear Constraint and Gear Ratio

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

$r_{F} ω_{F} = r_{B} ω_{B}$

where:

r_F is the radius of the follower gear.
ω_F is the angular velocity of the follower gear.
r_B is the radius of the base gear.
ω_B is the angular velocity of the base gear.

The follower-base gear ratio is

$g_{F B} = \frac{r_{F}}{r_{B}} = \frac{N_{F}}{N_{B}}$

where:

N_B is the number of teeth in the base gear.
N_BF 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

$g_{F B} τ_{B} + τ_{F} - τ_{l o s s} = 0$

where:

τ_B is the input torque.
τ_F is the output torque.
τ_loss is the torque loss due to friction.

For the ideal case, $τ_{l o s s} = 0$ .

Nonideal Gear Constraint and Losses

In the nonideal case, $τ_{l o s s} \neq 0$ . 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,

$τ_{C o u l} = g_{F B} τ_{i d l e} + k τ_{F}$

where:

τ_Coul is the Coulomb friction dependent torque.
k is a proportionality constant.
τ_idle is the net torque acting on the input shaft in idle mode.

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

$η = \frac{τ_{F}}{g_{F B} τ_{i d l e} + (k + 1) τ_{F}}$

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 exposing an optional thermal port. To expose the port, in the Meshing Losses tab, set the Friction model parameter 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.

Dependencies

Variable settings are exposed only when, in the Meshing Losses settings, the Friction model parameter is set to Temperature-dependent efficiency.

Assumptions

Gear inertia is assumed negligible.
Gears are treated as rigid components.
Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.

Ports

Input

expand all

`T` — External fault trigger
physical signal

Physical signal input port for an external fault trigger.

Dependencies

To expose the T port:

On the Meshing Losses tab, set Friction model toConstant efficiency, Load-dependent efficiency, Temperature-dependent efficiency, or Temperature and load-dependent efficiency.
On the Faults tab:
- Set Enable faults to On.
- Set Enable external fault trigger to On.
Click OK or Apply.

For information on related dependencies, see Parameter Dependencies Table.

Conserving

expand all

`B` — Base
rotational mechanical

Rotational mechanical conserving port associated with the base, or input, shaft.

`F` — Follower
rotational mechanical

Rotational mechanical conserving port associated with the follower, or output, shaft.

`H` — Heat flow
thermal

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

Dependencies

To enable this port, on the Meshing Losses tab 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. To learn how to read the table, see Parameter Dependencies.

Default Model — For nonthermal models, thermal port H is not visible.					Thermal Model — For thermal models, thermal port H is visible.
Meshing Losses					Meshing 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 simulation	Constant efficiency			Load-dependent efficiency	Temperature-dependent efficiency	Temperature and load-dependent efficiency
	Efficiency			Input shaft torque at no load	Temperature	Temperature
	Follower power threshold			Nominal output torque	Efficiency	Load at base gear
				Efficiency at nominal output torque	Follower power threshold	Efficiency matrix
				Follower angular velocity threshold	Follower power threshold	Follower angular velocity threshold
Faults	Faults
	Enable faults — Choose `Off` or `On`
	Off	On
		Faulted efficiency
		Enable external fault trigger — Choose `Off` or `On`. Selecting `On` makes thermal port T visible.
		Enable temporal fault trigger — Choose `Off` or `On`
		Off	On
		Off	Simulation time for fault event
		Faulted angle range
		Reporting when fault occurs — Choose `None`, or `Warning`, or `Error`

Main

`Follower (F) to base (B) teeth ratio (NF/NB)` — Gear ratio
`2` (default) | positive scalar

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

`Output shaft rotates` — Motion direction
`In opposite direction to input shaft` (default) | `In same direction as input shaft`

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 model` — Default friction model
`No meshing losses - Suitable for HIL simulation` (default) | `Constant efficiency` | `Load-dependent efficiency` | `Temperature-dependent efficiency` | `Temperature and load-dependent efficiency`

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.

`Efficiency` — Torque transfer efficiency, η
`0.8` (default) | `0`≤ η < `1`

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.

For more information, see Parameter Dependencies Table.

`Follower power threshold` — Efficiency factor power basis
`0.001` `W` (default) | positive scalar

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.

For more information, see Parameter Dependencies Table.

`Input shaft torque at no load` — No-load torque
`0.1` `N*m` (default) | positive scalar

Net torque,τ_idle, 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.

For more information, see Parameter Dependencies Table.

`Nominal output torque` — Nominal torque
`5` `N*m` (default) | positive scalar

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

Dependencies

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

For more information, see Parameter Dependencies Table.

`Efficiency at nominal output torque` — Nominal efficiency, η
`0.8` (default) | `0` ≤ η < `1` | scalar

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.

For more information, see Parameter Dependencies Table.

`Follower angular velocity threshold` — Efficiency factor angular velocity basis
`0.01` `rad/s` (default) | positive scalar

Absolute value of the follower shaft angular velocity above which the full efficiency factor is in effect, ω_F. 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.

For more information, see Parameter Dependencies Table.

`Temperature` — Temperature
`[280, 300, 320]` `K` (default) | positive array

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

For more information, see Parameter Dependencies Table.

`Efficiency` — Efficiency, η
`[.95, .9, .85]` (default) | `0` ≤ η < `1` | array

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.

For more information, see Parameter Dependencies Table.

`Follower power threshold` — Efficiency factor power basis
`0.001` `W` (default) | positive array

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

Dependencies

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

For more information, see Parameter Dependencies Table.

`Load at base gear` — Base gear load
`[1, 5, 10]` `N*m` (default) | positive array

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.

For more information, see Parameter Dependencies Table.

`Efficiency matrix` — Efficiency
`[.85, .8, .75; .95, .9, .85; .85, .8, .7]` (default) | numerical array with positive elements in the range of [0,1)

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.

For more information, see Parameter Dependencies Table.

`Follower angular velocity threshold` — Efficiency factor angular velocity basis
`0.01` `rad/s` (default) | positive scalar

Dependencies

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

For more information, see Parameter Dependencies Table.

Viscous Losses

`Viscous friction coefficients at base (B) and follower (F)` — Coefficients of viscous friction
`[0, 0]` `N*m/(rad/s)` (default) | positive array

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

The Faults tab is not visible when you set Friction model to No meshing losses - Suitable for HIL simulation.

`Enable faults` — Fault option
`on` (default) | `off`

Enable externally or temporally triggered faults.

Dependencies

This parameter is not visible when you set Friction model to No meshing losses - Suitable for HIL simulation on the Meshing Losses tab. This parameter affects the visibility of other Faults parameters.

For more information, see Parameter Dependencies Table.