Simple Gear

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

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

• 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

$$g_{FB}=\frac{r_F}{r_B}=\frac{N_F}{N_B}$$

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

$$g_{FB}{\tau }_B+{\tau }_F-{\tau }_{loss}=0$$

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

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_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:

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

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_Tooth is the relative linear velocity of the gear tooth.

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

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

• ω_B and ω_F 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, 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 xtooth = 1/2·Backlash
M = -1	Backwards engaged with xtooth = -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_Tooth = 0. Once f_Tooth > f_Tol, the gear unlocks, where

• f_Tol is the Static contact release force threshold parameter.

• v_tol is the Static contact speed threshold parameter.

• f_Tooth is the meshing force between the gear teeth such that f_Tooth = T_B/r_B = T_F/r_F.

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. 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.

Examples

Simple Gear

A simple gear coupling two inertias (shafts). The gear ratio between the follower (F) and base (B) is 2:1. Thus the angular velocity of the follower shaft is half the angular velocity of the base shaft. The follower shaft torque is twice the base shaft torque.

• Open Model

Gearbox Efficiency Measurement

A test for measuring gear box efficiency. The input shaft is driven at a fixed speed and a variable torque is applied to the output shaft. Power is measured on both the input and output shafts, and efficiency is calculated as the ratio of output to input power. The simulation results match the parameters specified for the efficiency of the gear.

• Open Model

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

Conserving

B — Base

rotational mechanical

Mechanical rotational conserving port associated with the base shaft.

F — Follower

mechanical rotational

Mechanical rotational conserving port associated with the follower 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, set Friction model to either:

• Temperature-dependent efficiency

• Temperature and load-dependent efficiency

Parameters

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 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 angular velocity threshold

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 — Shaft rotation 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.

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.

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

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 Temperature and load-dependent efficiency.

For more information, see Parameter Dependencies Table.

Backlash

Enable backlash — Backlash option

off (default) | on

Whether to enable backlash.

Hard stop model — Hard stop transition behavior

Stiffness and damping applied smoothly through transition region, damped rebound (default) | ...

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

For more information, see Translational Hard Stop.

Dependencies

To enable this parameter, select Enable backlash.

Linear backlash — Translational backlash distance

1e-3 mm (default) | positive scalar

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

Dependencies

To enable this parameter, select Enable backlash.

Base (B) gear radius — Base gear pitch radius

0.1 m (default) | positive scalar

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

Dependencies

To enable this parameter, select Enable backlash.

Transition region — Region of gradual hard stop effects

1e-4 mm (default) | positive scalar

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.

Linear stiffness — Resistance to translational deformation

1e6 N/m (default) | positive scalar

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

Linear damping — Tendency to dissipate translational energy

1e3 N/(m/s) (default) | positive scalar

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

Coefficient of restitution — Collision energy loss coefficient

0.7 (default) | positive scalar

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.

Static contact speed threshold — Threshold relative speed between gear teeth before collision

0.001 m/s (default) | nonnegative scalar

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

Dependencies

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

Static contact release force threshold — Threshold force needed to transition from contact mode to free mode

0.001 N (default) | positive scalar

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

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

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.

Gear fault — Option to model gear fault

Add fault

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

Faulted efficiency — Faulted efficiency

0.05 (default) | scalar in the range [0,1]

Efficiency when a fault triggers.

Dependencies

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

Faulted angle range — Fault angle range

pi / 12 rad (default) | nonnegative scalar

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 Add fault hyperlink.

Thermal Port

Thermal mass — Thermal mass

50 J/K (default) | positive scalar

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

Initial temperature — Temperature

300 K (default) | positive scalar

Temperature at simulation start.

Dependencies

To enable this parameter, set Friction model to either:

• Temperature-dependent efficiency

• Temperature and load-dependent efficiency

More About

Hardware-in-the-Loop Simulation

For optimal performance of your real-time simulation, set the Friction model to No meshing losses - Suitable for HIL simulation on the Meshing Losses tab.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2011a

R2024a: Model faults with the Simscape faults interface

The Simple Gear block now supports the Simscape faults interface. To learn more, see Introduction to Simscape Faults.

R2023a: Backlash updates

Due to updates in the Translational Hard Stop block, the Simple Gear block provides new options when modeling backlash. Previously, the block used the Backlash model drop down parameter. Now, you can enable backlash when you select Enable backlash.

The updated Hard stop model parameter provides the same options as the Translational Hard Stop block. The parameter names align with new options introduced in theTranslational Hard Stop block.