Engee documentation

Simple Gear

Simple transmission of driving and driven gears with adjustable gear ratio, friction losses.

blockType: Engee1DMechanical.Transmission.Gears.Simple

simple gear

Path in the library:

/Physical Modeling/1D Mechanical/Gears/Simple Gear

Description

Block Simple Gear It is a gearbox in which the axes of the drive gear are connected ( ) and the driven gear ( ) rotate with a fixed gear ratio that you specify. You also choose whether the axis of the driven gear rotates in the same or opposite direction as the axis of the driving gear.

  • If they rotate in the same direction, the angular velocity of the driven gear ( ) and the angular velocity of the driving gear ( ) have the same sign.

  • If they rotate in opposite directions, and they have opposite signs.

You can add and remove backlash and thermal effects.

The thermal model

You can simulate the effects of heat flow and temperature changes by turning on an additional heat port H. To use the thermal port H, set the parameter Friction model meaning Temperature-dependent efficiency.

In addition, you can select an efficiency model that varies depending on load and temperature by setting the parameter Friction model meaning Temperature and load-dependent efficiency. Enabling the thermal model:

  • Opens the non-directional port H.

  • Includes the parameter Thermal mass, which allows you to specify the ability of a component to resist temperature changes.

  • Enables the Initial Temperature parameter, which allows you to set the initial temperature.

Ideal gears and transmission ratios

Block Simple Gear imposes two kinematic constraints on two connected axes:

where

  • — radius of the driven gear;

  • — angular velocity of the driven gear;

  • — radius of the driving gear;

  • — angular velocity of the driving gear.

The gear ratio for engagement of the driven and driving gears is:

where

  • — the number of teeth of the driving gear;

  • — the number of teeth of the driven gear.

Two degrees of freedom are reduced to one independent gear.

The transmission of torque is carried out as follows:

where

  • — input torque;

  • — output torque;

  • — losses during transmission of torque.

For the perfect occasion .

Imperfect limitations and losses in gears

In an imperfect case . For more information, see the article Modeling of mechanical gears with losses.

In an imperfect gear pair The angular velocity, gear ratio, and restrictions on the number of teeth remain unchanged. But the transmitted torque and power are reduced by:

  • Coulomb friction between tooth surfaces on gears and , which is determined by the efficiency, .

  • Viscous friction of the coupling of drive shafts with bearings, which is determined by the coefficients of viscous friction, .

Constant EFFICIENCY

In the case of constant efficiency, It is a constant value that does not depend on the load or the transmitted power.

Load-dependent efficiency

EFFICIENCY ( ) depends on the load or power transmitted through the gears. For any of the power streams:

where

  • — a torque that depends on Coulomb friction;

  • — proportionality coefficient;

  • — the torque acting on the input shaft in idle mode.

EFFICIENCY ( ) is associated with in the standard form, but becomes load-dependent:

The backlash effect

You can include the backlash effect in your model.

A luft is an excess space between a gear tooth and the teeth of another gear mated to it. The increased backlash compensates for the decrease in manufacturing tolerances and ensures the free movement of lubricants in the gears to prevent jamming. However, excessive backlash can lead to premature wear of system components and affect measurements that depend on the gear position. This block applies backlash to launch and reverse using the block implementation Translational Hard Stop.

If you enable the parameter Enable backlash, the block correlates gear rotation with linear backlash as:

where

  • — the relative linear speed of the gear tooth;

  • — corresponds to the value of the parameter Base (B) gear radius;

  • — the radius of the driven gear, where , and the parameter Follower (F) to base (B) teeth ratio (NF/NB) corresponds to the ratio ;

  • and — angular velocities of the driving and driven gears, respectively;

  • — the sign of the direction of rotation of the gear. If for the parameter Output shaft rotates the value is set:

    • In same direction as input shaft Then .

    • In opposite direction to input shaft Then .

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

If set for the parameter Hard stop model meaning Based on coefficient of restitution, then a hard stop can use a non-zero parameter value Coefficient of restitution, , in the equation of conservation of momentum. In case of collision:

where and — the time points before and after the collision, respectively. It is believed that is in the range of [0, 1].

Block Simple Gear registers the mode state of the transmission as the intermediate state M.

Condition Meaning

Turned off

Direct running with

Reverse gear from

Instant mode transition between forward and reverse running

Instant mode transition between reverse and forward running

Instant impact mode

The rigid stop simulates static contact at the borders. The gear is blocked in the event of a collision and when . As soon as the gear is locked: . As soon as the condition is met , the gear is unlocked.

In these formulas:

  • — parameter value Static contact release force threshold;

  • — parameter value Static contact speed threshold;

  • — this is the engagement force between the teeth of the gear, such that .

The thermal model

You can simulate the effects of heat flow and temperature changes by turning on an additional heat port. To use the thermal port, set the parameter Friction model meaning Temperature-dependent efficiency or Temperature and load-dependent efficiency.

When choosing a thermal model:

  • A non-directional thermal port H is used.

  • The parameter is used Thermal mass, which allows you to determine the ability of a component to withstand temperature changes.

Assumptions and limitations

  • The inertia of the gears is negligible.

  • Gears are treated as solids.

  • Coulomb friction slows down the simulation. (for more information, see here)

Ports

Conserving

# B — driving gear
rotational mechanics

Details

A non-directional port connected to the drive gear.

Program usage name

base_flange

# F — driven gear
rotational mechanics

Details

A non-directional port connected to a driven gear.

Program usage name

follower_flange

# H — heat flow
warm

Details

A non-directional port connected to the heat flow.

The thermal port allows you to simulate the heat flow between the unit and the connected network.

Dependencies

To use this port, set the parameter Friction model meaning 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 the driven gear to the driving gear

Details

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

Default value

2.0
Program usage name

ratio
Evaluatable

Yes

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

Details

The direction of movement of the driven gear relative to the movement of the driving 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 — the friction model
No meshing losses - Suitable for HIL simulation | Constant efficiency | Load-dependent efficiency | Temperature-dependent efficiency | Temperature and load-dependent efficiency

Details

The model of friction in transmission. Set as:

  • No meshing losses - Suitable for HIL simulation – gear engagement is considered ideal.

  • Constant efficiency – the transmission of torque between the gear pairs is reduced by a constant amount of efficiency, , such that .

  • Load-dependent efficiency – reduction of torque transmission by means of variable efficiency. This coefficient is in the range of and it depends on the load.

  • Temperature-dependent efficiency – the transmission of torque between pairs of gears is determined by the interpolation table of the correspondence of temperature and torque efficiency.

  • Temperature and load-dependent efficiency – reducing the transmission of torque by the amount of efficiency, depending on temperature and load. This coefficient is in the range of and it varies depending on the load. The efficiency of torque transmission is determined based on user-provided 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 — idle torque
N*m | uN*m | mN*m | kN*m | MN*m | GN*m | kgf*m | lbf*in | lbf*ft

Details

The torque, acting on the drive gear in idle mode, i.e. when the transmission of torque to the driven gear is zero. At non-zero values, the input power in idle mode is completely dissipated due to gearing losses.

Dependencies

To use this parameter, set for the parameter Friction model meaning Load-dependent efficiency.

Units

N*m | uN*m | mN*m | kN*m | MN*m | GN*m | kgf*m | lbf*in | lbf*ft
Default value

0.1 N*m
Program usage name

T_no_load
Evaluatable

Yes

# Nominal output torque — Rated torque
N*m | uN*m | mN*m | kN*m | MN*m | GN*m | kgf*m | lbf*in | lbf*ft

Details

Torque on the driven gear, , at which the efficiency is normalized depending on the load.

Dependencies

To use this parameter, set for the parameter Friction model meaning Load-dependent efficiency.

Units

N*m | uN*m | mN*m | kN*m | MN*m | GN*m | kgf*m | lbf*in | lbf*ft
Default value

5.0 N*m
Program usage name

T_nominal
Evaluatable

Yes

# Efficiency at nominal output torque — rated EFFICIENCY

Details

Efficiency of torque transmission, , at the nominal torque on the driven gear. High efficiency values correspond to a greater transmission of torque between the driving and driven gears.

Dependencies

To use this parameter, set for the parameter Friction model meaning 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 driving and driven gears. The efficiency value is inversely proportional to the power loss in the engagement.

Dependencies

To use this parameter, set for the parameter Friction model meaning Constant efficiency.

Default value

0.8
Program usage name

efficiency_const
Evaluatable

Yes

# Temperature — vector of temperature values
K | degC | degF | degR | deltaK | deltadegC | deltadegF | deltadegR

Details

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

Dependencies

To use this parameter, set for the parameter Friction model meaning 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 | uN*m | mN*m | kN*m | MN*m | GN*m | kgf*m | lbf*in | lbf*ft

Details

The load vector on the base gear used to build a two-dimensional interpolation table of efficiency matching depending on the temperature and load values. The elements of the vector should increase monotonously. The load vector must be the same size as one column of the efficiency matrix.

Dependencies

To use this parameter, set for the parameter Friction model meaning Temperature and load-dependent efficiency.

Units

N*m | uN*m | mN*m | kN*m | MN*m | GN*m | kgf*m | lbf*in | lbf*ft
Default value

[1.0, 5.0, 10.0] N*m
Program usage name

load_vector
Evaluatable

Yes

# Efficiency — vector of values of efficiency of torque transmission

Details

The vector of the values of the efficiency of torque transmission for the gearing of the driving and driven gears.

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

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

Dependencies

To use this parameter, set for the parameter Friction model meaning 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

A matrix of torque transmission efficiency values for the gearing of the driving and driven gears.

The block uses these values to build a two-dimensional interpolation table of efficiency matching depending on temperature and load values.

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

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

Dependencies

To use this parameter, set for the parameter Friction model meaning 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 — the angular velocity of the driven gear at which the maximum efficiency value is applied
rad/s | deg/s | rad/min | deg/min | rpm | rps

Details

The absolute value of the angular velocity of the driven gear, at which the maximum value of the torque transmission efficiency is achieved, . At values below the specified value, the efficiency is smoothed using the hyperbolic tangent function to 1, reducing losses to 0.

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

Dependencies

To use this parameter, set for the parameter Friction model meaning Temperature and load-dependent efficiency.

Units

rad/s | deg/s | rad/min | deg/min | rpm | rps
Default value

0.01 rad/s
Program usage name

w_threshold
Evaluatable

Yes

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

Details

The absolute value of the driven gear power, above which the full values of the torque transmission efficiency are applied, . At values below those indicated, the efficiency is smoothed using the hyperbolic tangent function to 1, reducing losses to 0.

The power threshold must 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 modeling.

Dependencies

To use this parameter, set for the parameter Friction model meaning Constant efficiency.

Units

W | uW | mW | kW | MW | GW | V*A | HP_DIN
Default value

0.001 W
Program usage name

power_threshold
Evaluatable

Yes

Backlash

# Enable backlash — enabling backlash

Details

Check this box to account for backlash.

Default value

false (switched off)
Program usage name

enable_backlash
Evaluatable

No

# Hard stop model — behavior during 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 | Based on coefficient of restitution

Details

The stiffness and rebound parameter for the rigid limiter model. Set 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;

  • Based on coefficient of restitution

For more information, see Translational Hard Stop.

Dependencies

To use this option, enable the option checkbox. Enable backlash.

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 | Based on coefficient of restitution
Default value

Stiffness and damping applied smoothly through transition region, damped rebound
Program usage name

hardstop_model
Evaluatable

No

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

Details

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

Dependencies

To use this option, check the box for the option Enable backlash.

Units

m | um | mm | cm | km | in | ft | yd | mi | nmi
Default value

1e-3 mm
Program usage name

backlash_distance
Evaluatable

Yes

# Base (B) gear radius — radius of the driving gear
m | um | mm | cm | km | in | ft | yd | mi | nmi

Details

The distance from the center of the gear to the point of engagement of the teeth.

Dependencies

To use this option, check the box for the option Enable backlash.

Units

m | um | mm | cm | km | in | ft | yd | mi | nmi
Default value

0.1 m
Program usage name

base_tooth_radius
Evaluatable

Yes

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

Details

The distance at which the block gradually applies the effects of stiffness and damping.

If set for the parameter Hard stop model meaning Stiffness and damping applied smoothly through transition region, damped rebound the block smoothly transitions from one stiffness to another as the rigid stop approaches full rigidity.

Dependencies

To use this option, select the option checkbox. Enable backlash and for the parameter Hard stop model meaning Stiffness and damping applied smoothly through transition region, damped rebound.

Units

m | um | mm | cm | km | in | ft | yd | mi | nmi
Default value

1e-4 mm
Program usage name

transition_region
Evaluatable

Yes

# Linear stiffness — translational rigidity
N/m | mN/m | kN/m | MN/m | GN/m | kgf/m | lbf/ft | lbf/in

Details

Translational stiffness of the spring when gears collide.

Dependencies

To use this option, select the option checkbox. Enable backlash and for the parameter 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 | mN/m | kN/m | MN/m | GN/m | kgf/m | lbf/ft | lbf/in
Default value

1e6 N/m
Program usage name

k_backlash
Evaluatable

Yes

# Linear damping — translational damping
N*s/m | kgf*s/m | lbf*s/ft | lbf*s/in

Details

Damping of translational energy during gear collisions.

Dependencies

To use this option, select the option checkbox. Enable backlash and for the parameter 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*s/m | kgf*s/m | lbf*s/ft | lbf*s/in
Default value

1e3 N*s/m
Program usage name

C_backlash
Evaluatable

Yes

# Coefficient of restitution — the coefficient of energy loss in a collision

Details

Loss of translational kinetic energy in collisions. Meaning 0 means inelastic collision, and the value 1 – a perfectly elastic collision in which the gear retains all kinetic energy.

Default value 0 It is equivalent to simulating the torque when the gears are in contact and eliminating the torque when the gear changes direction and the tooth travels the distance to the backlash.

Dependencies

To use this option, select the option checkbox. Enable backlash and for the parameter Hard stop model meaning Based on coefficient of restitution.

Default value

0.7
Program usage name

restitution_coefficient
Evaluatable

Yes

# Static contact speed threshold — threshold relative velocity between gear teeth before collision
m/s | mm/s | cm/s | km/s | m/hr | km/hr | in/s | ft/s | mi/s | ft/min | mi/hr | kn

Details

Speed, below which the gear tooth becomes locked with the engaging teeth. The block installs When .

Dependencies

To use this option, select the option checkbox. Enable backlash and for the parameter Hard stop model meaning Based on coefficient of restitution.

Units

m/s | mm/s | cm/s | km/s | m/hr | km/hr | in/s | ft/s | mi/s | ft/min | mi/hr | kn
Default value

0.001 m/s
Program usage name

v_static_contact_threshold
Evaluatable

Yes

# Static contact release force threshold — threshold force required to switch from contact mode to free mode
N | nN | uN | mN | kN | MN | GN | dyn | lbf | kgf

Details

The minimum force required to exit the gear from the static contact mode.

Dependencies

To use this option, select the option checkbox. Enable backlash and for the parameter Hard stop model meaning Based on coefficient of restitution.

Units

N | nN | uN | mN | kN | MN | GN | dyn | lbf | kgf
Default value

0.001 N
Program usage name

F_static_contact_release_threshold
Evaluatable

Yes

Viscous Losses

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

Details

The vector of values of viscous friction coefficients for the movement of the driving and driven gears, respectively. To ignore viscous losses, use the default value.

Units

N*m/(rad/s) | ft*lbf/(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 thermal energy required to change the temperature of a component by one degree. The higher the heat capacity, the more resistant the component is to temperature changes.

Dependencies

To use this parameter, set for the parameter Friction model 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