Differential

Gear mechanism that allows driven shafts to spin at different speeds

Libraries:
Simscape / Driveline / Gears

Description

The Differential block represents a gear mechanism that allows the driven shafts to spin at different speeds. Differentials are common in automobiles, where they enable the various wheels to spin at different speeds while cornering. Ports D, S1, and S2 represent the longitudinal driveshaft and the sun gear shafts of the differential, respectively. Any one of the shafts can drive the other two.

The block models the differential mechanism as a structural component based on the Simple Gear and Sun-Planet Bevel Simscape™ Driveline™ blocks. The figure demonstrates the equivalent block diagram for the Differential block.

To increase the fidelity of the gear model, 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 gear carrier and internal planet gears. To model the inertias of the outer gears, connect Simscape Inertia blocks to ports D, S1, and S2.

Thermal Modeling

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 differential imposes one kinematic constraint on the three connected axes, such that

$ω_{S 1} - ω_{S 2},$

where:

ω_S1 is the velocity of sun gear shaft 1.
ω_S2 is the velocity of sun gear shaft 2.

Negative values imply that the differential is left of centerline. The three degrees of freedom reduce to two independent degrees of freedom. The gear pairs are (1,2) = (S, S) and (C, D). C is the carrier.

The sum of the lateral motions is the transformed longitudinal motion. The difference of side motions, $ω_{S 1} - ω_{S 2}$ , is independent of the longitudinal motion. The general motion of the lateral shafts is a superposition of these two independent degrees of freedom, which have this physical significance:

The longitudinal degree of freedom is equivalent to the two lateral shafts rotating at the same angular velocity, $ω_{S 1} = ω_{S 2}$ , and at a fixed ratio with respect to the longitudinal shaft.
The differential degree of freedom is equivalent to keeping the longitudinal driving shaft locked, $ω_{D} = 0$ , where ω_D is the velocity of the driving shaft, while the lateral shafts rotate with respect to each other in opposite directions, $ω_{S 1} = - ω_{S 2}$ .

The lateral axis torques are constrained by the longitudinal axis torque such that the net power flow sums to zero:

$ω_{S 1} τ_{S 1} + ω_{S 2} τ_{S 2} + ω_{D} τ_{D} - P_{l o s s} = 0,$

where:

τ_S1 and τ_S2 are the torques along the lateral axes.
τ_D is the longitudinal torque.
P_loss is the power loss.

When the kinematic and power constraints are combined, the ideal case yields

$g_{D} τ_{D} = 2 \frac{(ω_{S 1} τ_{S 1} + ω_{S 2} τ_{S 2})}{ω_{S 1} + ω_{S 2}},$

where g_D is the gear ratio for the longitudinal driveshaft.

Ideal Fundamental Constraints

The effective Differential block constraint is composed of two sun-planet bevel gear subconstraints.

The first subconstraint is due to the coupling of the two sun-planet bevel gears to the carrier:
$\frac{ω_{S 1} - ω_{C}}{ω_{S 2} - ω_{C}} = - \frac{g_{S P 2}}{g_{S P 1}},$
where g_SP1 and g_SP2 are the gear ratios for the sun-planet gears.
The second subconstraint is due to the coupling of the carrier to the longitudinal driveshaft:

$ω_{D} = - g_{D} ω_{C} .$

The sun-planet gear ratios of the underlying sun-planet bevel gears, in terms of the radii, r, of the sun-planet gears are:

$g_{S P 1} = \frac{r_{S 1}}{r_{P 1}}$

$g_{S P 2} = \frac{r_{S 2}}{r_{P 2}}$

The Differential block is implemented with $g_{S P 1} = g_{S P 2} = 1$ , leaving g_D free to adjust.

Nonideal Gear Constraints and Losses

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

Assumptions and Limitations

The gears are assumed to be rigid.
Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.

Ports

Conserving

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D — Driveshaft
rotational mechanical

Rotational mechanical conserving port associated with the longitudinal driveshaft.

S1 — Sun gear 1
rotational mechanical

Rotational conserving port associated with sun gear 1.

S2 — Sun gear 2
rotational mechanical

Rotational conserving port associated with sun gear 2.

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

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Main

Crown gear located — Crown gear location
`To the right of centerline` (default) | `To the left of centerline`

Location of the bevel crown gear relative to the centerline of the gear assembly.

Carrier (C) to driveshaft (D) teeth ratio (NC/ND) — Carrier to driveshaft gear rotation ratio
`4` (default) | positive scalar

Fixed ratio, g_D, of the carrier gear to the longitudinal driveshaft gear rotations as defined by the number of carrier gear teeth divided by the number of driveshaft gear teeth. This gear ratio must be strictly greater than 0.

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-sun and carrier-driveshaft ordinary efficiencies — Torque transfer efficiencies
`[.85, .92]` (default) | vector

Vector of torque transfer efficiencies, [η_SS, η_CD], from driving to driven sun gear and from the carrier to the longitudinal driveshaft, respectively. The vector elements must be in the range (0,1].

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.

Sun-sun efficiency — Driving sun gear to driven sun gear torque transfer efficiency
`[.95, .9, .85]` (default) | vector

Vector of output-to-input power ratios that describe the power flow from the driving sun gear to the driven sun gear, η_SS. 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.

Carrier-driveshaft efficiency — Carrier to driveshaft gear torque transfer efficiency
`[.95, .9, .85]` (default) | vector

Vector of output-to-input power ratios that describe the power flow from the carrier to the driveshaft, η_CD. 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 driveshaft-casing power thresholds — Minimum efficiency power threshold for the sun-carrier and driveshaft-casing gear couplings
`[0.001, 0.001]` `W` (default) | vector

Vector of power thresholds, p_th, for sun-carrier and longitudinal driveshaft-casing [p_S, p_D], respectively. The full efficiency loss applies above these values. 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 driveshaft-casing viscous friction coefficients — Viscous friction coefficients
`[0, 0]` `N*m/(rad/s)` (default) | vector

Vector of viscous friction coefficients, [μ_S μ_D ], for the sun gear-to-carrier gears and longitudinal driveshaft-to-casing gear motions, respectively.

Inertia

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

Inertia model for the block:

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

Sun gear inertia — Sun gear inertia
`0.001` `kg*m^2` (default) | positive scalar

Sun gear inertia.

Dependencies

To enable this parameter, set Inertia to On.

Sun gear radius — Sun gear radius
`0.01` `m` (default) | positive scalar

Sun gear radius.

Dependencies

To enable this parameter, set Inertia to On.

Carrier inertia — Carrier gear inertia
`0.001` `kg*m^2` (default) | positive scalar

Moment of inertia of the planet gear carrier.

Dependencies

To enable this parameter, set Inertia to On.

Planet gear spin inertia — Planet gear spin inertia
`0.001` `kg*m^2` (default) | positive scalar

Planetary gear inertia about its own axis.

Dependencies

To enable this parameter, set Inertia to On.

Planet gear mass — Planet gear mass
`0.001` `kg*m^2` (default) | positive scalar

Planet gear mass.

Dependencies

To enable this parameter, set Inertia to On.

Thermal Port

To enable these settings, set Friction model to Temperature-dependent efficiency.

Thermal mass — Thermal mass
`50` `J/K` (default) | scalar

Thermal energy required to change the component temperature by a single temperature unit. The greater the thermal mass, the more resistant the component is to temperature change.

Dependencies

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

Initial temperature — Initial temperature
`300` `K` (default) | scalar

Block temperature at the start of simulation. The initial temperature sets the initial component efficiencies according to their respective efficiency vectors.

Dependencies

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

More About

expand all

Hardware-in-the-Loop Simulation

For optimal simulation performance, set Friction model to the default value, No meshing losses - Suitable for HIL simulation.

Extended Capabilities

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

Version History

Introduced in R2011a

expand all

R2024b: Inertia updates

You now can specify the inertia of all gear subcomponents including the sun, ring, and carrier gears. You also now can include the effects of planet procession in the planet inertia behavior.

Differential

Description

Thermal Modeling

Equations

Assumptions and Limitations

Ports

Conserving

D — Driveshaft rotational mechanical

S1 — Sun gear 1 rotational mechanical

S2 — Sun gear 2 rotational mechanical

H — Heat flow thermal

Dependencies

Parameters

Main

Crown gear located — Crown gear location To the right of centerline (default) | To the left of centerline

Carrier (C) to driveshaft (D) teeth ratio (NC/ND) — Carrier to driveshaft gear rotation ratio 4 (default) | positive scalar

Meshing Losses

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

Sun-sun and carrier-driveshaft ordinary efficiencies — Torque transfer efficiencies [.85, .92] (default) | vector

Dependencies

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

Dependencies

Sun-sun efficiency — Driving sun gear to driven sun gear torque transfer efficiency [.95, .9, .85] (default) | vector

Dependencies

Carrier-driveshaft efficiency — Carrier to driveshaft gear torque transfer efficiency [.95, .9, .85] (default) | vector

Dependencies

Sun-carrier and driveshaft-casing power thresholds — Minimum efficiency power threshold for the sun-carrier and driveshaft-casing gear couplings [0.001, 0.001] W (default) | vector

Dependencies

Viscous Losses

Sun-carrier and driveshaft-casing viscous friction coefficients — Viscous friction coefficients [0, 0] N*m/(rad/s) (default) | vector

Inertia

Inertia — Inertia model off (default) | on

Sun gear inertia — Sun gear inertia 0.001 kg*m^2 (default) | positive scalar

Dependencies

Sun gear radius — Sun gear radius 0.01 m (default) | positive scalar

Dependencies

Carrier inertia — Carrier gear inertia 0.001 kg*m^2 (default) | positive scalar

Dependencies

Planet gear spin inertia — Planet gear spin inertia 0.001 kg*m^2 (default) | positive scalar

Dependencies

Planet gear mass — Planet gear mass 0.001 kg*m^2 (default) | positive scalar

Dependencies

Thermal Port

Thermal mass — Thermal mass 50 J/K (default) | scalar

Dependencies

Initial temperature — Initial temperature 300 K (default) | scalar

Dependencies

More About

Hardware-in-the-Loop Simulation

Extended Capabilities

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

Version History

R2024b: Inertia updates

See Also

Topics

D — Driveshaft
rotational mechanical

S1 — Sun gear 1
rotational mechanical

S2 — Sun gear 2
rotational mechanical

H — Heat flow
thermal

Crown gear located — Crown gear location
`To the right of centerline` (default) | `To the left of centerline`

Carrier (C) to driveshaft (D) teeth ratio (NC/ND) — Carrier to driveshaft gear rotation ratio
`4` (default) | positive scalar

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

Sun-sun and carrier-driveshaft ordinary efficiencies — Torque transfer efficiencies
`[.85, .92]` (default) | vector

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

Sun-sun efficiency — Driving sun gear to driven sun gear torque transfer efficiency
`[.95, .9, .85]` (default) | vector

Carrier-driveshaft efficiency — Carrier to driveshaft gear torque transfer efficiency
`[.95, .9, .85]` (default) | vector

Sun-carrier and driveshaft-casing power thresholds — Minimum efficiency power threshold for the sun-carrier and driveshaft-casing gear couplings
`[0.001, 0.001]` `W` (default) | vector

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

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

Sun gear inertia — Sun gear inertia
`0.001` `kg*m^2` (default) | positive scalar

Sun gear radius — Sun gear radius
`0.01` `m` (default) | positive scalar

Carrier inertia — Carrier gear inertia
`0.001` `kg*m^2` (default) | positive scalar

Planet gear spin inertia — Planet gear spin inertia
`0.001` `kg*m^2` (default) | positive scalar

Planet gear mass — Planet gear mass
`0.001` `kg*m^2` (default) | positive scalar

Thermal mass — Thermal mass
`50` `J/K` (default) | scalar

Initial temperature — Initial temperature
`300` `K` (default) | scalar

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