Active Differential

Spur or planetary active differential gear

Libraries:
Vehicle Dynamics Blockset / Powertrain / Drivetrain / Final Drive Unit

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, Δω_cl1, and clutch 2, Δω_cl2.

Input Axle Torque	Δω_cl1	Δω_cl2	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
Crown gear	${\dot{ω}}_{d} (J_{d} + J_{g s}) = T_{d} - ω_{d} b_{d} - T_{i}$	${\dot{ω}}_{d} (J_{d} + J_{s 1} + J_{s 2}) = T_{d} - ω_{d} b_{d} - T_{i}$
Axle 1	${\dot{ω}}_{1} J_{1} = η T_{1} - ω_{1} b_{1} - T_{i 1}$	${\dot{ω}}_{1} (J_{1} + J_{r 1}) = T_{1} - ω_{1} b_{1} - T_{i 1}$
Axle 2	${\dot{ω}}_{2} J_{2} = η T_{2} - ω_{2} b_{2} - T_{i 2}$	${\dot{ω}}_{2} (J_{a x l e 2} + J_{r 1}) = T_{2} - ω_{2} b_{2} - T_{i 2}$
Gear ratios	$\begin{array}{l} \frac{ω_{c l 1}}{ω_{d}} = N_{s 1} = \frac{z_{1} z_{6}}{z_{4} z_{3}} \\ \frac{ω_{c l 2}}{ω_{d}} = N_{s 2} = \frac{z_{1} z_{5}}{z_{4} z_{2}} \end{array}$	$\begin{array}{l} \frac{ω_{c l 1}}{ω_{d}} = N_{p 1} = \frac{z_{1} z_{6}}{z_{4} z_{3}} \\ \frac{ω_{c l 2}}{ω_{d}} = N_{p 2} = \frac{z_{1} z_{5}}{z_{4} z_{2}} \end{array}$
Rigid Coupling Constraints	$\begin{array}{l} T_{1} = \frac{N T_{i}}{2} - \frac{N_{s 2}}{2} T_{c l 2} + \frac{N_{s 1}}{2} T_{c l 1} \\ T_{2} = \frac{N T_{i}}{2} + (1 - \frac{N_{s 2}}{2}) T_{c l 2} - (1 - \frac{N_{s 1}}{2}) T_{c l 1} \\ ω_{d =} = \frac{N}{2} (ω_{1} + ω_{2}) \end{array}$	$\begin{array}{l} T_{1} = \frac{N T_{i}}{2} - \frac{N_{p 2}}{(N_{p 2} - 1) 2} T_{c l 2} + \frac{(2 - N_{p 1})}{(N_{p 1} - 1) 2} T_{c l 1} \\ T_{2} = \frac{N T_{i}}{2} + \frac{(2 - N_{p 2})}{(N_{p 2} - 1) 2} T_{c l 2} - \frac{N_{p 1}}{(N_{p 1} - 1) 2} T_{c l 1} \\ ω_{d =} = \frac{N}{2} (ω_{1} + ω_{2}) \end{array}$
Allowable wheel speed difference (AWSD)	${\bar{Δ ω}}_{m a x} = (N_{s 2} - N_{s 1}) \cdot 100 %$	${\bar{Δ ω}}_{m a x} = (N_{p 1, 2} - 1) \cdot 100 %$

Mechanical Dynamic Response

Equations

Superposition Clutches and Spur Gearing

Stationary Clutches and Planetary Gearing

Crown gear

${\dot{ω}}_{d} (J_{d} + J_{g s}) = T_{d} - ω_{d} b_{d} - T_{i}$

${\dot{ω}}_{d} (J_{d} + J_{s 1} + J_{s 2}) = T_{d} - ω_{d} b_{d} - T_{i}$

Axle 1

${\dot{ω}}_{1} J_{1} = η T_{1} - ω_{1} b_{1} - T_{i 1}$

${\dot{ω}}_{1} (J_{1} + J_{r 1}) = T_{1} - ω_{1} b_{1} - T_{i 1}$

Axle 2

${\dot{ω}}_{2} J_{2} = η T_{2} - ω_{2} b_{2} - T_{i 2}$

${\dot{ω}}_{2} (J_{a x l e 2} + J_{r 1}) = T_{2} - ω_{2} b_{2} - T_{i 2}$

Gear ratios

$\begin{array}{l} \frac{ω_{c l 1}}{ω_{d}} = N_{s 1} = \frac{z_{1} z_{6}}{z_{4} z_{3}} \\ \frac{ω_{c l 2}}{ω_{d}} = N_{s 2} = \frac{z_{1} z_{5}}{z_{4} z_{2}} \end{array}$

$\begin{array}{l} \frac{ω_{c l 1}}{ω_{d}} = N_{p 1} = \frac{z_{1} z_{6}}{z_{4} z_{3}} \\ \frac{ω_{c l 2}}{ω_{d}} = N_{p 2} = \frac{z_{1} z_{5}}{z_{4} z_{2}} \end{array}$

Rigid Coupling Constraints

$\begin{array}{l} T_{1} = \frac{N T_{i}}{2} - \frac{N_{s 2}}{2} T_{c l 2} + \frac{N_{s 1}}{2} T_{c l 1} \\ T_{2} = \frac{N T_{i}}{2} + (1 - \frac{N_{s 2}}{2}) T_{c l 2} - (1 - \frac{N_{s 1}}{2}) T_{c l 1} \\ ω_{d =} = \frac{N}{2} (ω_{1} + ω_{2}) \end{array}$

$\begin{array}{l} T_{1} = \frac{N T_{i}}{2} - \frac{N_{p 2}}{(N_{p 2} - 1) 2} T_{c l 2} + \frac{(2 - N_{p 1})}{(N_{p 1} - 1) 2} T_{c l 1} \\ T_{2} = \frac{N T_{i}}{2} + \frac{(2 - N_{p 2})}{(N_{p 2} - 1) 2} T_{c l 2} - \frac{N_{p 1}}{(N_{p 1} - 1) 2} T_{c l 1} \\ ω_{d =} = \frac{N}{2} (ω_{1} + ω_{2}) \end{array}$

Allowable wheel speed difference (AWSD)

${\bar{Δ ω}}_{m a x} = (N_{s 2} - N_{s 1}) \cdot 100 %$

${\bar{Δ ω}}_{m a x} = (N_{p 1, 2} - 1) \cdot 100 %$

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.

$ϖ = [Δ ω_{c 1}, Δ ω_{c 2}]$

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.

$T_{C} = F_{T} N_{d} μ (| \bar{ω} |) R_{e f f} \tanh (4 \bar{ω})$

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

$F_{T} = F_{C} + P_{1, 2} A_{e f f}, F_{T} \geq 0$

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

$R_{e f f} = \frac{2 (R_{o}^{3} - R_{i}^{3})}{3 (R_{o}^{2} - R_{i}^{2})}$

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.

$T_{C} = T_{C} (ϖ, P_{1, 2})$

The equations use these variables.

A_eff	Effective clutch pressure area
b_d	Crown gear linear viscous damping
b₁, b₂	Axle 1 and 2 linear viscous damping, respectively
F_c, F_T	Clutch preload force and total force, respectively
J_d	Carrier rotational inertia
J_gc	Three-gang gear rotational inertia
J_c1, J_c2	Planetary carrier 1 and 2 rotational inertia, respectively
J_r1, J_r2	Planetary ring gear 1 and 2 rotational inertia, respectively
J_s1, J_s2	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
N_d	Number of disks
N_s1, N_s2	Clutch 1 and 2 carrier-to-spur gear ratio, respectively
N_p1, N_p2	Planetary 1 and 2 carrier-to-axle gear ratio, respectively
P₁, P₂	Clutch 1 and 2 pressure, respectively
R_eff	Effective clutch radius
R_i, R_o	Annular disk inner and outer radius, respectively
T_c	Clutch torque
T_cl1, T_cl2	Clutch 1 and 2 coupling torque, respectively
T_d	Driveshaft torque
T₁, T₂	Axle 1 and 2 torque, respectively
T_i	Axle internal resistance torque
T_i1, T_i2	Axle 1 and 2 internal resistance torque
ω_d	Driveshaft angular velocity
ϖ	Slip speed
ω₁, ω₂	Axle 1 and 2 angular velocity, respectively
Δω_cl1, Δω_cl2	Clutch 1 and 2 slip speed at interface, respectively
ω_cl1, ω_cl2	Clutch 1 and 2 angular velocity, respectively
μ	Clutch coefficient of friction
z_i	Number of teeth on gear i

Examples

Double Lane Change Reference Application

Simulate a full vehicle dynamics model undergoing a double lane change maneuver standard ISO 3888-2. Use for vehicle dynamics ride and handling analysis and chassis controls development, including yaw stability and lateral acceleration limits.

Open Script

Ports

Inputs

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Prs1 — Clutch 1 pressure
`scalar`

Clutch 1 pressure, P₁, in Pa.

Prs2 — Clutch 2 pressure
`scalar`

Clutch 2 pressure, P₂, in Pa.

DriveshftTrq — Driveshaft torque
`scalar`

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

Axl1Trq — Torque
`scalar`

Axle 1 torque, T₁, in N·m.

Axl2Trq — Torque
`scalar`

Axle 2 torque, T₂, in N·m.

Output

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Info — Bus signal
bus

Bus signal containing these block calculations.

Signal		Description	Units
`Driveshft`	`DriveshftTrq`	Drive shaft torque	N·m
`Driveshft`	`DriveshftSpd`	Drive shaft angular velocity	rad/s
`Axl1`	`Axl1Trq`	Axle 1 torque	N·m
`Axl1`	`Axl1Spd`	Axle 1 angular velocity	rad/s
`Axl2`	`Axl2Trq`	Axle 2 torque	N·m
`Axl2`	`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
	`CplngSlipSpd2`	Clutch 2 slip speed	rad/s
	`CplngPrs1`	Clutch 1 input pressure	Pa
	`CplngPrs2`	Clutch 2 input pressure	Pa

DriveshftSpd — Angular velocity
`scalar`

Driveshaft angular velocity, ω_d, in rad/s.

Axl1Spd — Angular velocity
`scalar`

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

Axl2Spd — Angular velocity
`scalar`

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

Parameters

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Active Differential

Active differential type — Differential
`Spur gears, superposition clutches` (default) | `Double planetary gears, stationary clutches`

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 differential case gear ratio, Ns1 — Clutch 1-spur gear ratio
`.875` (default) | `scalar`

Clutch 1-to-carrier spur gear ratio, N_s1, dimensionless.

Dependencies

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

Clutch 2 to differential case gear ratio, Ns2 — Clutch 2-spur gear ratio
`1.125` (default) | `scalar`

Clutch 2-to-carrier spur gear ratio, N_s2, dimensionless.

Dependencies

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

Three-gang gear inertia, Jgc — Rotational inertia
`.003` (default) | `scalar`

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

Dependencies

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