Simple Gear

Simple drive and driven gear transmission with adjustable gear ratio, friction losses.

simple gear

Description

The Simple Gear unit is a gearbox in which the connected drive pinion ( ) and idler pinion ( ) axes rotate at a fixed gear ratio that you specify. You also select whether the idler pinion axis rotates in the same or opposite direction as the drive pinion axis.

If they rotate in the same direction, the angular speed of the idler pinion ( ) and the angular speed of the master pinion ( ) have the same sign.
If they rotate in opposite directions, and have opposite signs.

You can add and remove backlash and thermal effects.

Thermal model

You can model the effects of heat flow and temperature changes by enabling the optional H heat port. To use the H thermal port, set the Friction model parameter to `Temperature-dependent efficiency'.

Alternatively, you can select an efficiency model that varies with load and temperature by setting the Friction model parameter to Temperature and load-dependent efficiency. Enabling the thermal model:

Opens a non-directional H port.
Enables the Thermal mass parameter, which allows you to specify the component’s ability to resist temperature changes.
Enables the Initial Temperature parameter, which allows you to specify the initial temperature.

Ideal gears and gear ratios

The Simple Gear block imposes two kinematic constraints on two linked axes:

where

- is the radius of the idler gear;
- angular velocity of the driven pinion;
- radius of the drive pinion;
- angular velocity of the drive pinion.

The gear ratio for the meshing of the idler gear and the master gear is equal:

where

- is the number of teeth of the driving gear;
- the number of teeth of the idler gear.

The two degrees of freedom are reduced to one independent gear.

Torque transmission is realised as follows:

where

- inlet torque;
- output torque;
- torque transmission losses.

For the ideal case .

Non-ideal constraints and losses in gears

For the non-ideal case . For more details, see Modelling of mechanical gears with losses.

In a non-ideal gear pair angular velocity, gear ratio and tooth count limits remain unchanged. But the transmitted torque and power are reduced due to:

Coulomb friction between the tooth surfaces on the gears and , which is determined by the efficiency, .
Viscous friction of the drive shaft coupling with bearings, which is determined by the viscous friction coefficients, .

Constant efficiency

In the case of constant efficiency, is a constant value, independent of load or transmitted power.

Load-dependent efficiency

The efficiency ( ) depends on the load or power transmitted through the gears. For either power flow:

where

- is the torque depending on the Coulomb friction;
- is the proportionality coefficient;
- is the torque acting on the input shaft at idle speed.

The efficiency ( ) is related to in the standard form, but becomes load dependent:

Backlash effect

You can include a backlash effect in your model.

Backlash is the excess space between a gear tooth and the mating teeth of another gear. Increased backlash compensates for reduced manufacturing tolerances and allows free movement of lubricants in the gears to prevent jamming. However, excessive backlash can cause premature wear of system components and affect measurements that depend on gear position. This block applies backlash for starting and reversing using the block implementation Translational Hard Stop.

If you enable the Enable backlash parameter, the block relates pinion rotation to linear backlash as:

where

- is the relative linear velocity of the gear tooth;
- corresponds to the value of the parameter Base (B) gear radius;
- driven gear radius, where , and the parameter Follower (F) to base (B) teeth ratio (NF/NB) corresponds to the ratio ;
and are the angular velocities of the driving and driven gears, respectively;
- sign of the gear rotation direction. If Output shaft rotates is set to:
- In same direction as input shaft, then .
- In opposite direction as input shaft, then .

The block considers tooth engagement as a position, , in relation to linear backlash, , where . corresponds to the Linear backlash parameter. The initial value of the Backlash position variable corresponds to the initial position .

The hard stop simulates static contact at the boundaries. The gear locks on impact and when . Once the gear is locked, . Once the condition , the gear is unlocked, where:

- Static contact release force threshold parameter value.
- is the value of Static contact speed threshold.
- is the meshing force between the gear teeth, such that .

Assumptions and limitations

The inertia of the gears is negligible.
Gears are treated as solids.
Coulomb friction slows down the simulation. (See here for more details)

Ports

Conserving

# B — drive pinion
`rotational mechanics

Details

A non-directional port associated with the drive gear.

Program usage name

base_flange

# F — idler gear
`rotational mechanics

Details

A non-directional port associated with the driven gear.

Program usage name

follower_flange

# H — heat flux
`heat

Details

A non-directional port associated with heat flow.

The heat port allows modelling the heat flow between the unit and the connected network.

Dependencies

To use this port, set Friction model to `Temperature-dependent efficiency' or `Temperature and load-dependent efficiency'.

Program usage name

thermal_port

Parameters

Main

# Follower (F) to base (B) teeth ratio (NF/NB) — gear ratio from driven pinion to driving pinion

Details

Constant gear ratio, , revolutions of the driven pinion to revolutions of the driving pinion. The gear ratio must be strictly >0.

Default value	`2.0`
Program usage name	`ratio`
Evaluatable	Yes

# Output shaft rotates — direction of rotation of the drive pinion
In same direction as input shaft | In opposite direction to input shaft

Details

The direction of movement of the idler gear in relation to the movement of the drive gear.

Values	`In same direction as input shaft` \| `In opposite direction to input shaft`
Default value	`In opposite direction to input shaft`
Program usage name	`rotation_direction_type`
Evaluatable	No

Meshing Losses

# Friction model — friction model
No meshing losses - Suitable for HIL simulation | Constant efficiency | Load-dependent efficiency | Temperature-dependent efficiency | Temperature and load-dependent efficiency

Details

A model of friction in a transmission. Defined as:

No meshing losses - Suitable for HIL simulation - the gearing is assumed to be perfect.
Constant efficiency - torque transmission between gear pairs is reduced by a constant efficiency value, , such that .
Load-dependent efficiency - torque transmission is reduced by a variable efficiency factor. This coefficient is in the range of and is load-dependent.
Temperature-dependent efficiency - torque transmission between gear pairs is determined from an interpolation table of temperature and torque efficiency correspondence.
Temperature and load-dependent efficiency - the reduction in torque transmission by the temperature and load-dependent efficiency. This coefficient is in the range of and varies with load. The torque transmission efficiency is determined based on user supplied data on gearbox load and temperature.

Values	`No meshing losses - Suitable for HIL simulation` \| `Constant efficiency` \| `Load-dependent efficiency` \| `Temperature-dependent efficiency` \| `Temperature and load-dependent efficiency`
Default value	`No meshing losses - Suitable for HIL simulation`
Program usage name	`friction_model`
Evaluatable	No

# Input shaft torque at no load — idling torque
N*m | mN*m | lbf*ft

Details

Torque, , acting on the drive pinion at idle speed, i.e. when the torque transmission to the driven pinion is zero. At non-zero values, the input power in the idle mode is completely dissipated due to losses in the meshing.

Dependencies

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

Units	`Nm` \| `mNm` \| `lbf*ft`
Default value	`0.1 N*m`
Program usage name	`T_no_load`
Evaluatable	Yes

# Nominal output torque — nominal torque
N*m | mN*m | lbf*ft

Details

Torque at the driven pinion, , at which the efficiency is normalised depending on the load.

Dependencies

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

Units	`Nm` \| `mNm` \| `lbf*ft`
Default value	`5.0 N*m`
Program usage name	`T_nominal`
Evaluatable	Yes

# Efficiency at nominal output torque — nominal efficiency

Details

The efficiency of torque transmission, , at nominal torque on the driven pinion. Higher efficiency values correspond to greater torque transmission between the drive and idler gears.

Dependencies

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

Default value	`0.8`
Program usage name	`nominal_efficiency`
Evaluatable	Yes

# Efficiency — Torque transmission efficiency

Details

The efficiency of torque transmission between the drive and driven gears. The efficiency value is inversely proportional to the power loss in the meshing.

Dependencies

To use this parameter, set the Friction model parameter to `Constant efficiency'.

Default value	`0.8`
Program usage name	`efficiency_const`
Evaluatable	Yes

Details

A vector of temperature values used to construct an interpolation table of temperature and torque transmission efficiency. The elements of the vector should be monotonically increasing.

Dependencies

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

Units	`K` \| `degC` \| `degF` \| `degR` \| `deltaK` \| `deltadegC` \| `deltadegF` \| `deltadegR`
Default value	`[280.0, 300.0, 320.0] K`
Program usage name	`temperature_vector`
Evaluatable	Yes

# Load at base gear — vector of loads on the base gear
N*m | mN*m | lbf*ft

Details

A vector of base gear loads used to construct a two-dimensional interpolation table of efficiency versus temperature and load values. The elements of the vector should be monotonically increasing. The load vector must be the same size as one column of the efficiency matrix.

Dependencies

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

Units	`Nm` \| `mNm` \| `lbf*ft`
Default value	`[1.0, 5.0, 10.0] N*m`
Program usage name	`load_vector`
Evaluatable	Yes

# Efficiency — vector of torque transmission efficiency values

Details

Vector of torque transmission efficiency values for the gear meshing of the driving and driven gears.

The unit uses these values to build an interpolation table of temperature and efficiency correspondence.

Each element is the efficiency related to the corresponding temperature value in the vector of Temperature parameter values. The length of the vector must be equal to the length of the Temperature parameter vector.

Dependencies

To use this parameter, set the Friction model parameter to `Temperature-dependent efficiency'.

Default value	`[0.95, 0.90, 0.85]`
Program usage name	`efficiency_vector`
Evaluatable	Yes

# Efficiency matrix — matrix of torque transmission efficiency values

Details

Matrix of torque transmission efficiency values for the gear meshing of the driving and driven gears.

The unit uses these values to build a two-dimensional interpolation table of the efficiency versus temperature and load values.

Each element is the efficiency related to the corresponding temperature value in the Temperature parameter value vector and at the loads specified in the Load at base gear parameter value vector.

The number of rows shall be the same as the number of elements in the Temperature parameter vector. The number of columns must be equal to the number of elements in the Load at base gear vector.

Dependencies

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

Default value	`[0.85 0.80 0.75; 0.95 0.90 0.85; 0.85 0.80 0.70]`
Program usage name	`efficiency_matrix`
Evaluatable	Yes

# Follower angular velocity threshold — angular speed of the driven gear at which the maximum efficiency value is applied
rpm | deg/s | rad/s

Details

The absolute value of the idler angular speed at which the maximum torque transmission efficiency value is achieved is . For values below this value, the efficiency is smoothed using a hyperbolic tangent function to 1, reducing the loss to 0.

The angular velocity threshold value should be lower than the expected angular velocity during simulation. Higher values may cause the block to underestimate the efficiency loss. Very low values increase the computational cost of the simulation.

Dependencies

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

Units	`rpm` \| `deg/s` \| `rad/s`
Default value	`0.01 rad/s`
Program usage name	`w_threshold`
Evaluatable	Yes

# Follower power threshold — minimum threshold power value
W | GW | MW | kW | mW | uW | HP_DIN

Details

The absolute value of the driven pinion power above which full torque transmission efficiency values apply, . For values below these values, the efficiency is smoothed using a hyperbolic tangent function to 1, reducing losses to 0.

The power threshold value should be lower than the expected power transmitted during the simulation. Higher values may cause the block to underestimate the efficiency loss. Very low values increase the computational cost of the simulation.

Dependencies

To use this parameter, set the Friction model parameter to `Constant efficiency'.

Units	`W` \| `GW` \| `MW` \| `kW` \| `mW` \| `uW` \| `HP_DIN`
Default value	`0.001 W`
Program usage name	`power_threshold`
Evaluatable	Yes

Backlash

# Enable backlash — backlash activation

Details

Select this checkbox to take backlash into account.

Default value	`false (switched off)`
Program usage name	`enable_backlash`
Evaluatable	No

# Hard stop model — behaviour during the transition to a hard stop
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

Details

Stiffness and rebound parameter for the rigid stop model. Defined as:

`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'.

For more information, see Translational Hard Stop.

Dependencies

To use this option, enable the Enable backlash checkbox.

Values	`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`
Default value	`Stiffness and damping applied smoothly through transition region, damped rebound`
Program usage name	`hardstop_model`
Evaluatable	No

# Linear backlash — tooth free distance
m | cm | ft | in | km | mi | mm | um | yd

Details

The distance that a gear tooth can travel between the meshing teeth.

Dependencies

To use this parameter, select the Enable backlash checkbox.

Units	`m` \| `cm` \| `ft` \| `in` \| `km` \| `mi` \| `mm` \| `um` \| `yd`
Default value	`1e-3 mm`
Program usage name	`backlash_distance`
Evaluatable	Yes

# Base (B) gear radius — drive pinion radius
m | cm | ft | in | km | mi | mm | um | yd

Details

Distance from the centre of the pinion to the point of tooth engagement.

Dependencies

To use this parameter, select the Enable backlash check box.

Units	`m` \| `cm` \| `ft` \| `in` \| `km` \| `mi` \| `mm` \| `um` \| `yd`
Default value	`0.1 m`
Program usage name	`base_tooth_radius`
Evaluatable	Yes

# Transition region — the area of gradual impact of hard stop
m | cm | ft | in | km | mi | mm | um | yd

Details

The distance over which the unit gradually applies stiffness and damping effects.

If you set the Hard stop model to `Stiffness and damping applied smoothly through transition region, damped rebound', the block transitions smoothly from one stiffness to another as the hard stop approaches full stiffness.

Dependencies

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

Units	`m` \| `cm` \| `ft` \| `in` \| `km` \| `mi` \| `mm` \| `um` \| `yd`
Default value	`1e-4 mm`
Program usage name	`transition_region`
Evaluatable	Yes

# Linear stiffness — translational rigidity
N/m | lbf/ft | lbf/in

Details

The reciprocating stiffness of the spring when the gears collide.

Dependencies

To use this parameter, check the Enable backlash checkbox and Hard stop model one of the following values:

`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'.

Units	`N/m` \| `lbf/ft` \| `lbf/in`
Default value	`1e6 N/m`
Program usage name	`k_backlash`
Evaluatable	Yes

# Linear damping — progressive damping
kg/s | N*s/m | N/(m/s) | lbf/(ft/s) | lbf/(in/s)

Details

Damping of translational energy when gears collide.

Dependencies

To use this parameter, select the Enable backlash check box and one of the following values for Hard stop model:

`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'.

Units	`kg/s` \| `N*s/m` \| `N/(m/s)` \| `lbf/(ft/s)` \| `lbf/(in/s)`
Default value	`1e3 N*s/m`
Program usage name	`C_backlash`
Evaluatable	Yes

Viscous Losses

# Viscous friction coefficients at base (B) and follower (F) — viscous friction coefficients between gears
N*m/(rad/s) | ft*lbf/(rad/s)

Details

Vector of values of viscous friction coefficients for the movement of the drive and driven gears respectively. To neglect viscous losses, use the default value [0.0, 0.0].

Units	`Nm/(rad/s)` \| `ftlbf/(rad/s)`
Default value	`[0.0, 0.0] N*m/(rad/s)`
Program usage name	`viscous_coefficient_vector`
Evaluatable	Yes

Thermal Port

# Thermal mass — heat capacity
J/K | kJ/K

Details

The heat energy required to change the temperature of a component by one degree. The greater the heat capacity, the more resistant the component is to temperature change.

Dependencies

To use this parameter, set Friction model to one of the following values:

Temperature-dependent efficiency;
`Temperature and load-dependent efficiency'.

Units	`J/K` \| `kJ/K`
Default value	`50.0 J/K`
Program usage name	`thermal_mass`
Evaluatable	Yes