Planetary Gear

Gear train with sun, planet, and ring gears

Libraries:
Simscape / Driveline / Gears

Description

The Planetary Gear block models a gear train with sun, planet, and ring gears. Planetary gears are common in transmission systems, where they provide high gear ratios in compact geometries. A carrier connected to a drive shaft holds the planet gears. Ports C, R, and S represent the shafts connected to the planet gear carrier, ring gear, and sun gear.

The block models the planetary gear as a structural component based on Sun-Planet and Ring-Planet Simscape™ Driveline™ blocks. The figure shows the block diagram of this structural component.

To increase the fidelity of the gear model, you can specify properties such as gear inertia, meshing losses, and viscous losses. By default, gear inertia and viscous losses are assumed to be negligible. The block enables you to specify the inertias of the internal planet gears. To model the inertias of the carrier, sun, and ring gears, connect Simscape Inertia blocks to ports C, S, and R.

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.

Equations

Ideal Gear Constraints and Gear Ratios

The Planetary Gear block imposes two kinematic and two geometric constraints,

$r_{C} ω_{C} = r_{S} ω_{S} + r_{P} ω_{P}$

$r_{R} ω_{R} = r_{C} ω_{C} + r_{P} ω_{P}$

$r_{C} = r_{S} + r_{P}$

$r_{R} = r_{C} + r_{P}$

where:

r_C is the radius of the carrier gear.
ω_C is the angular velocity of the carrier gear.
r_S is the radius of the sun gear.
ω_S is the angular velocity of the sun gear.
r_P is the radius of planet gear.
ω_p is the angular velocity of the planet gears.
r_R is the radius of the ring gear.

The ring-sun gear ratio is

$g_{R S} = r_{R} / r_{S} = N_{R} / N_{S},$

where N is the number of teeth on each gear.

In terms of this ratio, the key kinematic constraint is:

$(1 + g_{R S}) ω_{C} = ω_{S} + g_{R S} ω_{R} .$

The four degrees of freedom reduce to two independent degrees of freedom. The gear pairs are (1, 2) = (S, P) and (P, R).

Warning

The gear ratio g_RS must be strictly greater than one.

The torque transfer is

$g_{R S} τ_{S} + τ_{R} - τ_{l o s s} = 0,$

where:

τ_S is torque transfer for the sun gear.
τ_R is torque transfer for the ring gear.
τ_loss is torque transfer loss.

In the ideal case where there is no torque loss, τ_loss = 0.

Nonideal Gear Constraints and Losses

In the nonideal case, τ_loss ≠ 0. For more information, see Model Gears with Losses.

Assumptions and Limitations

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

expand all

C — Planet gear carrier
rotational mechanical

Rotational mechanical conserving port associated with the planet gear carrier.

R — Ring gear
rotational mechanical

Rotational mechanical conserving port associated with the ring gear.

S — Sun gear
rotational mechanical

Rotational mechanical conserving port associated with the sun gear.

H — Heat flow
thermal

Thermal conserving port associated with heat flow. Heat flow affects the power transmission efficiency by altering the gear temperatures.

Dependencies

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

Parameters

expand all

Main

Ring (R) to sun (S) teeth ratio (NR/NS) — Ring gear to sun gear rotation ratio
`2` (default) | scalar integer

Fixed ratio, g_RS, of the ring gear to the sun gear rotations as defined by the number of ring gear teeth divided by the number of sun gear teeth. The gear ratio must be strictly greater than 1.

Meshing Losses

Friction model — Friction model
`No meshing losses - Suitable for HIL simulation` (default) | `Constant efficiency` | `Temperature-dependent efficiency`

Friction model for the block:

No meshing losses - Suitable for HIL simulation — Gear meshing is ideal.
Constant efficiency — Transfer of torque between the gear wheel pairs is reduced by a constant efficiency, η, such that 0 < η ≤ 1.
Temperature-dependent efficiency — Transfer of torque between the gear wheel pairs is defined by the table lookup based on the temperature.

Sun-planet and ring-planet ordinary efficiencies — Torque transfer efficiencies from the sun gear to the planet gear and from the ring gear to the planet gear
`[.96, .98]` (default) | vector

Vector of torque transfer efficiencies, [η_SP η_RP], for sun-planet and ring-carrier gear wheel pair meshings, respectively.

Dependencies

To enable this parameter, set Friction model to Constant efficiency.

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

Vector of temperatures used to construct a 1-D temperature-efficiency lookup table. The vector elements must increase from left to right.

Dependencies

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

Wave generator flexspline efficiency — Wave generator to flexspline torque transfer efficiency
`[.95, .9, .85]` (default) | vector

Vector of output-to-input power ratios that describe the power flow from the wave generator gear to the flexspline gear, η_SP. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Each element is an efficiency that relates to a temperature in the Temperature vector. The length of the vector must be equal to the length of the Temperature vector. Each element in the vector must be in the range (0,1].

Dependencies

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

Ring-flexspline efficiency — Torque transfer efficiency from the ring gear to the flexspline gear
`[.95, .9, .85]` (default) | vector

Vector of output-to-input power ratios that describe the power flow from the ring gear to the flexspline gear, η_RP. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Dependencies

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

Sun-carrier and planet-carrier power thresholds — Minimum efficiency power threshold for the sun-carrier and planet-carrier gear couplings
`[.001, .001]` `W` (default) | vector

Vector of power thresholds above which full efficiency factors apply. Enter the thresholds in the order sun-carrier, planet-carrier. Below these values, a hyperbolic tangent function smooths the efficiency factor.

When you set Friction model to Constant efficiency, the block lowers the efficiency losses to zero when no power is transmitted. When you set Friction model to Temperature-dependent efficiency, the block smooths the efficiency factors between zero when at rest and the values provided by the temperature-efficiency lookup tables at the power thresholds.

Dependencies

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

Viscous Losses

Sun-carrier and planet-carrier viscous friction coefficients — Gear viscous friction
`[0, 0]` `N*m/(rad/s)` (default) | vector

Vector of viscous friction coefficients, [μ_S, μ_P], for the sun-carrier and planet-carrier gear motions, respectively.

Inertia

Inertia — Inertia model
`off` (default) | `on`

Inertia model for the block:

Off — Model gear inertia.
On — Neglect gear inertia.