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

Gears

The Simple Gear block represents a gearbox that constrains the
two connected driveline axes, base (B) and follower (F), to corotate
with a fixed ratio that you specify. You can choose whether the follower
axis rotates in the same or opposite direction as the base axis. If
they rotate in the same direction, *ω*_{F} and *ω*_{B} have
the same sign. If they rotate in opposite directions, *ω*_{F} and *ω*_{B} have
opposite signs. For model details, see Simple Gear Model.

The block models the effects of heat flow and temperature change
through an optional thermal port. To expose the thermal port, right-click
the block and select **Simscape** > **Block choices** > **Show thermal
port**. Exposing the thermal port causes
new parameters specific to thermal modeling to appear in the block
dialog box.

**Follower (F) to base (B) teeth ratio (NF/NB)**Fixed ratio

*g*_{FB}of the follower axis to the base axis. The gear ratio must be strictly positive. The default is`2`

.**Output shaft rotates**Direction of motion of the follower (output) driveshaft relative to the motion of the base (input) driveshaft. The default is

`In opposite direction to input shaft`

.

Parameters for meshing losses vary with the block variant chosen—one with a thermal port for thermal modeling and one without it.

**Viscous friction coefficients at base (B) and follower (F)**Two-element array with the viscous friction coefficients in effect at the base and follower shafts. The default array,

`[0 0]`

, corresponds to zero viscous losses.

**Thermal mass**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. The default value is

`50`

J/K.**Initial temperature**Component temperature at the start of simulation. The initial temperature influences the starting meshing or friction losses by altering the component efficiency according to an efficiency vector that you specify. The default value is

`300`

K.

Simple Gear imposes one kinematic constraint on the two connected axes:

*r*_{F}*ω*_{F} = *r*_{B}*ω*_{B} .

The follower-base gear ratio *g*_{FB} = *r*_{F}/*r*_{B} = *N*_{F}/*N*_{B}. *N* is
the number of teeth on each gear. The two degrees of freedom reduce
to one independent degree of freedom.

The torque transfer is:

*g*_{FB}*τ*_{B} + *τ*_{F} – *τ*_{loss} =
0 ,

with *τ*_{loss} =
0 in the ideal case.

In the nonideal case, *τ*_{loss} ≠
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*

*τ*_{loss} = *τ*_{Coul}·tanh(4*ω*_{out}/*ω*_{th})
+ *μ**ω*_{out} ,
*τ*_{Coul} = |*τ*_{F}|·(1
– *η*) .

The hyperbolic tangent regularizes the sign change in the Coulomb friction torque when the angular velocity changes sign.

Power Flow | Power Loss Condition | Output Driveshaft ω_{out} |
---|---|---|

Forward | ω_{B}τ_{B} > ω_{F}τ_{F} | Follower, ω_{F} |

Reverse | ω_{B}τ_{B} < ω_{F}τ_{F} | Base, ω_{B} |

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, *τ*_{Coul} = *g*_{FB}*τ*_{idle} + *k**τ*_{F}. *k* is
a proportionality constant. *η* is related
to *τ*_{Coul} in the standard,
preceding form but becomes dependent on load:

*η* = *τ*_{F}/[*g*_{FB}*τ*_{idle} +
(*k* + 1)*τ*_{F}]
.

Gear inertia is assumed negligible.

Gears are treated as rigid components.

Coulomb friction slows down simulation. See Adjust Model Fidelity.

Port | Description |
---|---|

B | Rotational conserving port representing the base shaft |

F | Rotational conserving port representing the follower shaft |

H | Thermal conserving port for thermal modeling |

The sdl_simple_gearsdl_simple_gear example model gives a basic example of a simple gear.

The sdl_backlashsdl_backlash example model illustrates a gear with backlash.

The sdl_gearbox_efficiencysdl_gearbox_efficiency example model measures the efficiency of a nonideal simple gear by comparing output to input power.

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