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Active Differential

Spur or planetary active differential gear

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  • Active Differential block

Description

The Active Differential block implements an active differential to account for the power transfer from the transmission to the axles. The block models the active differential as an open differential coupled to either a spur or planetary differential gear set. The block uses external pressure signals to regulate the clutch pressure to either speed up or slow down each axle rotation.

Use the block in hardware-in-the-loop (HIL) and optimization workflows to dynamically couple the driveshaft to the wheel axles when you want to direct the transmission torque to a specific axle. For detailed front wheel driving studies, use the block to couple the driveshaft to universal joints. The block is suitable to use in system-level closed-loop control studies, for example, yaw stability and torque vectoring. All the parameters are tunable.

To specify the active differential, open the Active Differential parameters and specify Active differential type.

Setting

Block Implementation
Spur gears, superposition clutches

Clutches are in superposition through a three-gang gear system and a differential case

Double planetary gears, stationary clutches

Clutches are fixed to the carrier and axles through double planetary gear sets

Use the Open Differential parameter Crown wheel (ring gear) located to specify the open differential location, either to the left or right of the center-line.

Depending on the available data, to specify the method to couple the different torques applied to the axles, use the Slip Coupling parameter Coupling type.

Setting

Block Implementation
Pre-loaded ideal clutch

Torque modeled as a dry clutch with constant friction coefficients

Slip speed dependent torque data

Torque determined from a lookup table that is a function of slip-speed and clutch pressure

The Active Differential block does not include a controller or external clutch actuator dynamics. Use this information to control the input clutch pressure. The info bus contains the slip speeds at clutch 1, Δωcl1, and clutch 2, Δωcl2.

Input Axle Torque

Δωcl1

Δωcl2

Input Clutch Pressure

Positive axle 1 torque

> 0

N/A

Increase clutch 1 pressure

Positive axle 1 torque

< 0

N/A

Disengage clutch 1 and 2

Positive axle 2 torque

N/A

> 0

Increase clutch 1 pressure

Positive axle 2 torque

N/A

< 0

Disengage clutch 1 and 2

Differentials

The Active Differential block implements these equations to represent the mechanical dynamic response for the superposition and stationary clutch configurations. To determine the gear ratios, the block uses the clutch speed and the number of teeth for each gear pair. The allowable wheel speed difference (AWSD) limits the wheel speed difference for positive torque.

Mechanical Dynamic Response

Equations

Superposition Clutches and Spur Gearing

Stationary Clutches and Planetary Gearing

Crown gear

ω˙d(Jd+Jgs)=TdωdbdTi

ω˙d( Jd+Js1+Js2)=TdωdbdTi

Axle 1

ω˙1J1=ηT1-ω1b1-Ti1

ω˙1(J1+Jr1)=T1ω1b1Ti1

Axle 2

ω˙2J2=ηT2-ω2b2-Ti2

ω˙2(Jaxle2+Jr1)=T2ω2b2Ti2

Gear ratios

ωcl1ωd=Ns1=z1z6z4z3ωcl2ωd=Ns2=z1z5z4z2

ωcl1ωd=Np1=z1z6z4z3ωcl2ωd=Np2=z1z5z4z2

Rigid Coupling Constraints

T1= NTi2Ns22Tcl2+Ns12Tcl1T2= NTi2+(1Ns22)Tcl2(1Ns12)Tcl1ωd==N2(ω1+ω2)

T1= NTi2Np2(Np21)2Tcl2+(2Np1)(Np11)2Tcl1T2= NTi2+(2Np2)(Np21)2Tcl2Np1(Np11)2Tcl1ωd==N2(ω1+ω2)

Allowable wheel speed difference (AWSD)

Δω¯max=(Ns2Ns1)100%

Δω¯max=(Np1,21)100%

Superposition Clutches and Spur Gearing

These superposition clutch illustrations show the clutch configuration and schematic for torque transfer to the left wheel.

Detailed illustration of torque transfer to the left wheel

Schematic of clutch torque transfer to the left wheel

Stationary Clutches and Planetary Gearing

The illustrations show the stationary clutch configuration and schematic.

Detailed illustration of a stationary clutch

Schematic of stationary clutch

Slip Coupling

For both the ideal clutch and slip-speed configurations, the slip coupling is a function of the slip-speed and clutch pressure. The slip-speed depends on the slip velocity at each of the clutch interfaces.

ϖ=[Δωc1,Δωc2]

Ideal Clutch

The ideal clutch coupling model uses the axle slip speed, clutch pressure, and friction to calculate the clutch torque. The friction coefficient is a function of the slip speed.

TC=FTNdμ(|ω¯|)Refftanh(4ω¯)

To calculate the total clutch force, the block uses the effective radius, clutch pressure, and clutch preload force.

FT= FC+P1,2Aeff, FT0

The disc radii determine the effective clutch radius over which the clutch force acts.

Reff=2(Ro3-Ri3)3(Ro2-Ri2)

Slip-Speed

To calculate the clutch torque, the slip speed coupling model uses torque data that is a function of slip speed and clutch pressure. The angular velocities of the axles determine the slip speed.

TC=TC(ϖ, P1,2)

The equations use these variables.

AeffEffective clutch pressure area
bd

Crown gear linear viscous damping

b1, b2

Axle 1 and 2 linear viscous damping, respectively

Fc, FT

Clutch preload force and total force, respectively

Jd

Carrier rotational inertia

Jgc

Three-gang gear rotational inertia

Jc1, Jc2

Planetary carrier 1 and 2 rotational inertia, respectively

Jr1, Jr2

Planetary ring gear 1 and 2 rotational inertia, respectively

Js1, Js2

Planetary sun gear 1 and 2 rotational inertia, respectively

J1, J2

Axle 1 and 2 rotational inertia, respectively

N

Carrier-to-drive shaft gear ratio

Nd

Number of disks

Ns1, Ns2

Clutch 1 and 2 carrier-to-spur gear ratio, respectively

Np1, Np2

Planetary 1 and 2 carrier-to-axle gear ratio, respectively

P1, P2

Clutch 1 and 2 pressure, respectively

Reff

Effective clutch radius

Ri, Ro

Annular disk inner and outer radius, respectively

Tc

Clutch torque

Tcl1, Tcl2

Clutch 1 and 2 coupling torque, respectively

Td

Driveshaft torque

T1, T2

Axle 1 and 2 torque, respectively

Ti

Axle internal resistance torque

Ti1, Ti2

Axle 1 and 2 internal resistance torque

ωd

Driveshaft angular velocity

ϖ

Slip speed

ω1, ω2

Axle 1 and 2 angular velocity, respectively

Δωcl1, Δωcl2

Clutch 1 and 2 slip speed at interface, respectively

ωcl1, ωcl2

Clutch 1 and 2 angular velocity, respectively

μ

Clutch coefficient of friction

zi

Number of teeth on gear i

Ports

Inputs

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Clutch 1 pressure, P1, in Pa.

Clutch 2 pressure, P2, in Pa.

Applied input torque, Td, typically from the engine driveshaft, in N·m.

Axle 1 torque, T1, in N·m.

Axle 2 torque, T2, in N·m.

Output

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Bus signal containing these block calculations.

SignalDescriptionUnits

Driveshft

DriveshftTrq

Drive shaft torque

N·m

DriveshftSpd

Drive shaft angular velocity

rad/s

Axl1

Axl1Trq

Axle 1 torque

N·m

Axl1Spd

Axle 1 angular velocity

rad/s

Axl2

Axl2Trq

Axle 2 torque

N·m

Axl2Spd

Axle 2 angular velocity

rad/s

Cplng

CplngTrq1

Clutch 1 coupling torque

N·m

CplngTrq2

Clutch 2 coupling torque N·m
CplngSlipSpd1

Clutch 1 slip speed

rad/s
CplngSlipSpd2Clutch 2 slip speedrad/s
CplngPrs1

Clutch 1 input pressure

Pa
CplngPrs2Clutch 2 input pressurePa

Driveshaft angular velocity, ωd, in rad/s.

Axle 1 angular velocity, ω1, in rad/s.

Axle 2 angular velocity, ω2, in rad/s.

Parameters

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Active Differential

Specify the type of active differential.

Setting

Block Implementation
Spur gears, superposition clutches

Clutches are in superposition through a three-gang gear system and a differential case

Double planetary gears, stationary clutches

Clutches are fixed to the carrier and axles through double planetary gear sets

Clutch 1-to-carrier spur gear ratio, Ns1, dimensionless.

Dependencies

To enable the spur gear parameters, select Spur gears, superposition clutches for the Active differential type parameter.

Clutch 2-to-carrier spur gear ratio, Ns2, dimensionless.

Dependencies

To enable the spur gear parameters, select Spur gears, superposition clutches for the Active differential type parameter.

Three-gang gear rotational inertia, Jgc, in kg·m^2.

Dependencies

To enable the spur gear parameters, select Spur gears, superposition clutches for the Active differential type parameter.

Planetary 1 carrier-to-axle gear ratio, Np1, dimensionless.

Dependencies

To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.

Planetary 1 sun gear inertia, Js1, in kg·m^2.

Dependencies

To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.

Planetary 1 carrier inertia, Jc1, in kg·m^2.

Dependencies

To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.

Planetary 1 ring gear inertia, Jr1, kg·m^2.

Dependencies

To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.

Planetary 2 carrier-to-axle gear ratio, Np2, dimensionless.

Dependencies

To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.

Planetary 2 sun gear inertia, Js2, in kg·m^2.

Dependencies

To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.

Planetary 2 carrier inertia, Jc2, in kg·m^2.

Dependencies

To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.

Planetary 2 ring gear inertia, Jr2, in kg·m^2.

Dependencies

To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.

Open Differential

Specify the crown wheel connection to the drive shaft.

Carrier-to-drive shaft gear ratio, N.

Rotational inertia of the crown gear assembly, Jd, in kg·m^2. You can include the drive shaft inertia.

Crown gear linear viscous damping, bd, in N·m·s/rad.

Axle 1 rotational inertia, J1, in kg·m^2.

Axle 1 linear viscous damping, b1, in N·m·s/rad.

Axle 2 rotational inertia, J2, in kg·m^2.

Axle 2 linear viscous damping, b2, in N·m·s/rad.

Axle 1 initial velocity, ωo1, in rad/s.

Axle 2 initial velocity, ωo2, in rad/s.

Slip Coupling

Specify the type of torque coupling.

Setting

Block Implementation
Pre-loaded ideal clutch

Torque modeled as a wet clutch with a constant velocity

Slip speed dependent torque data

Torque determined from a lookup table that is a function of slip-speed and clutch pressure

Effective applied pressure area, in N/m^2.

Dependencies

To enable the clutch parameters, select Ideal pre-loaded clutch for the Coupling type parameter.

Number of disks.

Dependencies

To enable the clutch parameters, select Ideal pre-loaded clutch for the Coupling type parameter.

The effective radius, Reff, used with the applied clutch friction force to determine the friction force. The effective radius is defined as:

Reff=2(Ro3-Ri3)3(Ro2-Ri2)

The equation uses these variables.

Ro

Annular disk outer radius

Ri

Annular disk inner radius

Dependencies

To enable the clutch parameters, select Ideal pre-loaded clutch for the Coupling type parameter.

Nominal preload force, in N.

Dependencies

To enable the clutch parameters, select Ideal pre-loaded clutch for the Coupling type parameter.

Friction coefficient vector.

Dependencies

To enable the clutch parameters, select Ideal pre-loaded clutch for the Coupling type parameter.

Slip speed vector, in rad/s.

To enable the clutch parameters, select Ideal pre-loaded clutch for the Coupling type parameter.

Torque matrix, Tc, in N·m.

Dependencies

To enable the slip speed parameters, select Slip speed dependent torque data for the Coupling type parameter.

Clutch pressure breakpoints vector, P1,2, in Pa.

Dependencies

To enable the slip speed parameters, select Slip speed dependent torque data for the Coupling type parameter.

Slip speed breakpoints vector, ω, in rad/s.

Dependencies

To enable the slip speed parameters, select Slip speed dependent torque data for the Coupling type parameter.

Coupling time constant, in s.

References

[1] Deur, J., Ivanović, V., Hancock, M., and Assadian, F. "Modeling of Active Differential Dynamics." In ASME proceedings. Transportation Systems. Vol. 17, pp: 427-436.

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Introduced in R2018b