Limited Slip Differential

Limited differential as a planetary bevel gear

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

Description

The Limited Slip Differential block implements a differential as a planetary bevel gear train. The block matches the driveshaft bevel gear to the crown (ring) bevel gear. You can specify:

• Carrier-to-driveshaft ratio

• Crown wheel location

• Viscous and damping coefficients for the axles and carrier

• Type of slip coupling

Use the block in system-level driveline analysis to account for the power transfer from the transmission to the wheels. The block is suitable for use in hardware-in-the-loop (HIL) and optimization workflows. All the parameters are tunable.

In a limited slip differential, to prevent one of the wheels from slipping, the differential splits the torque applied to the left and right axles. With different torque applied to the axles, the wheels can move at different angular velocities, preventing slip. The block implements three methods for coupling the different torques applied to the axes:

• Pre-loaded ideal clutch

• Slip speed-dependent torque data

• Input torque dependent torque data

The block uses a coordinate system that produces positive tire and vehicle motion for standard engine, transmission, and differential configurations. The arrows indicate positive motion.

Efficiency

To account for the block efficiency, use the Efficiency factors parameter. This table summarizes the block implementation for each setting.

Setting	Implementation
Constant	Constant efficiency that you can set with the Constant efficiency factor, eta parameter.
Driveshaft torque, temperature and speed	Efficiency as a function of base gear input torque, air temperature, and driveshaft speed. Use these parameters to specify the lookup table and breakpoints: Efficiency lookup table, eta_tbl Efficiency torque breakpoints, Trq_bpts Efficiency speed breakpoints, omega_bpts Efficiency temperature breakpoints, Temp_bpts For the air temperature, you can either: Select Input ambient temperature to create an input port. Set a Ambient temperature, Tamb parameter value. To select the interpolation method, use the Interpolation method parameter. For more information, see Interpolation Methods.

Power Accounting

For the power accounting, the block implements these equations.

Bus Signal			Description	Equations
PwrInfo	PwrTrnsfrd — Power transferred between blocks Positive signals indicate flow into block Negative signals indicate flow out of block	PwrDriveshft	Mechanical power from driveshaft	$$\eta T_d{\omega }_d$$
		PwrAxl1	Mechanical power from axle 1	$$\eta T_1{\omega }_1$$
		PwrAxl2	Mechanical power from axle 2	$$\eta T_2{\omega }_2$$
	PwrNotTrnsfrd — Power crossing the block boundary, but not transferred Positive signals indicate an input Negative signals indicate a loss	PwrMechLoss	Total power loss	$$\begin{array}{l}{\dot{W}}_{loss}= -(P_t+P_d+P_c)+P_s\\ \\ P_t=\eta \left(T_d{\omega }_d+T_1{\omega }_1+T_2{\omega }_2\right)\end{array}$$
		PwrDampLoss	Power loss due to damping	$$P_d=-\left(b_1\left|{\omega }_1\right|+b_2\left|{\omega }_2\right|+b_d\left|{\omega }_d\right|\right)$$
		PwrCplngLoss	Power loss due to clutch	$$P_c=T_c\left|\overline{\omega }\right|$$
	PwrStored — Stored energy rate of change Positive signals indicate an increase Negative signals indicate a decrease	PwrStoredShft	Rate change of stored internal energy	$P_s=-({\omega }_1{\dot{\omega }}_1{J}_1+{\omega }_2{\dot{\omega }}_2{J}_2+{\omega }_d{\dot{\omega }}_d{J}_d)$

Dynamics

The Limited Slip Differential block implements these differential equations to represent the mechanical dynamic response for the crown gear, left axle, and right axle.

Mechanical Dynamic Response	Differential Equation
Crown Gear	$${\dot{\omega }}_d{J}_d=\eta T_d-{\omega }_d{b}_d-T_i$$
Left Axle	${\dot{\omega }}_1{J}_1=\eta T_1-{\omega }_1{b}_1-T_{i1}$
Right Axle	${\dot{\omega }}_2{J}_2=\eta T_2-{\omega }_2{b}_2-T_{i2}$

The block assumes rigid coupling between the crown gear and axles. These constraint equations apply.

$$\begin{array}{l}\eta T_1=\frac{N}{2}{T}_i-\frac{1}{2}{T}_c\\ \eta T_2=\frac{N}{2}{T}_i+\frac{1}{2}{T}_c\end{array}$$

$${\omega }_d=\frac{N}{2}({\omega }_1+{\omega }_2)$$

The equations use these variables.

N	Carrier-to-driveshaft gear ratio
Jd	Rotational inertia of the crown gear assembly
bd	Crown gear linear viscous damping
ωd	Driveshaft angular speed
$$\varpi$$	Slip speed
J1	Axle 1 rotational inertia
b1	Axle 1 linear viscous damping
ω1	Axle 1 speed
J2	Axle 2 rotational inertia
b2	Axle 2 linear viscous damping
ω2	Axle 2 angular speed
η	Efficiency
Td	Driveshaft torque
T1	Axle 1 torque
T2	Axle 2 torque
Ti	Axle internal resistance torque
Ti1	Axle 1 internal resistance torque
Ti2	Axle 2 internal resistance torque
μ	Coefficient of friction
$R_{eff}$	Effective clutch radius
$R_o$	Annular disk outer radius
Ri	Annular disk inner radius
Fc	Clutch force
Tc	Clutch torque
μ	Coefficient of friction

Table blocks in the Limited Slip Differential have these parameter settings:

• Interpolation method — Linear

• Extrapolation method — Clip

Ideal Clutch Coupling

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

$$T_c=F_{c}N\mu \left(\left|\varpi \right|\right)R_{eff}\tanh \left(4\left|\varpi \right|\right)$$

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

$R_{eff}=\frac{2(R_o{}^3-R_i{}^3)}{3(R_o{}^2-R_i{}^2)}$

The angular velocities of the axles determine the slip speed.

$$\varpi ={\omega }_1-{\omega }_2$$

Slip Speed Coupling

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

$$\varpi ={\omega }_1-{\omega }_2$$

Input Torque Coupling

To calculate the clutch torque, the input torque coupling model uses torque data that is a function of input torque.

The Open Differential block assumes rigid coupling between the crown gear and axles. These constraint equations apply.

$$\eta T_{i1}= \eta T_{i2}=\frac{N}{2}{T}_i$$

$${\omega }_d=\frac{N}{2}({\omega }_1+{\omega }_2)$$

Ports

Inputs

DriveshftTrq — Torque

scalar

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

Axl1Trq — Torque

scalar

Axle 1 torque, T₁, in N·m.

Axl2Trq — Torque

scalar

Axle 2 torque, T₂, in N·m.

Temp — Temperature

scalar

Temperature, in K.

Dependencies

To enable this port:

• Set Efficiency factors to Driveshaft torque, speed and temperature.

• Select Input temperature.

Output

Info — Bus signal

bus

Bus signal containing these block calculations.

Signal			Description	Units
Driveshft	DriveshftTrq		Driveshaft torque	N·m
	DriveshftSpd		Driveshaft speed	rad/s
Axl1	Axl1Trq		Axle 1 torque	N·m
	Axl1Spd		Axle 1 speed	rad/s
Axl2	Axl2Trq		Axle 2 torque	N·m
	Axl2Spd		Axle 2 speed	rad/s
Cplng	CplngTrq		Torque coupling	N·m
	CplngSlipSpd		Slip speed	rad/s
PwrInfo	PwrTrnsfrd	PwrDriveshft	Mechanical power from driveshaft	W
		PwrAxl1	Mechanical power from axle 1	W
		PwrAxl2	Mechanical power from axle 2	W
	PwrNotTrnsfrd	PwrMechLoss	Total power loss	W
		PwrDampLoss	Power loss due to damping	W
		PwrCplngLoss	Power loss due to clutch	W
	PwrStoredShft	PwrStoredShft	Rate change of stored internal energy	W

DriveshftSpd — Angular speed

scalar

Driveshaft angular speed, ω_d, in rad/s.

Axl1Spd — Angular speed

scalar

Axle 1 angular speed, ω₁, in rad/s.

Axl2Spd — Angular speed

scalar

Axle 2 angular speed, ω₂, in rad/s.

Parameters

Block Options

Efficiency factors — Specify configuration

Constant (default) | Driveshaft torque, speed and temperature

To account for the block efficiency, use the Efficiency factors parameter. This table summarizes the block implementation for each setting.

Setting	Implementation
Constant	Constant efficiency that you can set with the Constant efficiency factor, eta parameter.
Driveshaft torque, temperature and speed	Efficiency as a function of base gear input torque, air temperature, and driveshaft speed. Use these parameters to specify the lookup table and breakpoints: Efficiency lookup table, eta_tbl Efficiency torque breakpoints, Trq_bpts Efficiency speed breakpoints, omega_bpts Efficiency temperature breakpoints, Temp_bpts For the air temperature, you can either: Select Input ambient temperature to create an input port. Set a Ambient temperature, Tamb parameter value. To select the interpolation method, use the Interpolation method parameter. For more information, see Interpolation Methods.

Interpolation method — Method

Flat | Nearest | Linear point-slope | Linear Lagrange | Cubic spline

For more information, see Interpolation Methods.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Input ambient temperature — Create input port

off (default) | on

Select to create input port Temp for the temperature.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Open Differential

Crown wheel (ring gear) located — Specify crown wheel connection

To the left of center-line (default) | To the right of center-line

Specify the crown wheel connection to the driveshaft.

Carrier to drive shaft ratio, NC/ND — Ratio

4 (default) | scalar

Carrier-to-driveshaft gear ratio, N.

Carrier inertia, Jd — Inertia

.1 (default) | scalar

Rotational inertia of the crown gear assembly, J_d, in kg·m^2. You can include the driveshaft inertia.

Carrier damping, bd — Damping

1e-3 (default) | scalar

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

Axle 1 inertia, Jw1 — Inertia

.1 (default) | scalar

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

Axle 1 damping, bw1 — Damping

1e-3 (default) | scalar

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

Axle 2 inertia, Jw2 — Inertia

.1 (default) | scalar

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

Axle 2 damping, bw2 — Damping

1e-3 (default) | scalar

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

Axle 1 initial velocity, omegaw1o — Angular velocity

0 (default) | scalar

Axle 1 initial velocity, ω_o1, in rad/s.

Axle 2 initial velocity, omegaw2o — Angular velocity

0 (default) | scalar

Axle 2 initial velocity, ω_o2, in rad/s.

Constant efficiency factor, eta — Efficiency

1 (default) | scalar

Constant efficiency, η.

Dependencies

To enable this parameter, set Efficiency factors to Constant.

Efficiency lookup table, eta_tbl — Lookup table

M-by-N-by-L array

Dimensionless array of values for efficiency as a function of:

• M input torques

• N input speed

• L air temperatures

Each value specifies the efficiency for a specific combination of torque, speed, and temperature. The array size must match the dimensions defined by the torque, speed, and temperature breakpoint vectors.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Efficiency torque breakpoints, Trq_bpts — Torque breakpoints

[25, 50, 75, 100, 150, 200, 250] (default) | 1-by-M vector

Vector of input torque, breakpoints for efficiency, in N·m.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Efficiency speed breakpoints, omega_bpts — Speed breakpoints

[52.4 78.5 105 131 157 183 209 262 314 419 524] (default) | 1-by-N vector

Vector of speed, breakpoints for efficiency, in rad/s.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Efficiency temperature breakpoints, Temp_bpts — Temperature breakpoints

[290 358] (default) | 1-by-L vector

Vector of ambient temperature breakpoints for efficiency, in K.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Ambient temperature, Tamb — Ambient temperature

297.15 (default) | scalar

Ambient air temperature, T_air, in K.

Dependencies

To enable this parameter:

• Set Efficiency factors to Driveshaft torque, speed and temperature.

• Clear Input ambient temperature.

Slip Coupling

Coupling type — Torque coupling

Pre-loaded ideal clutch (default) | Slip speed dependent torque data | Input torque dependent torque data

Specify the type of torque coupling.

Number of disks, Ndisks — Torque coupling

4 (default) | scalar

Number of disks.

Dependencies

To enable the ideal clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Effective radius, Reff — Radius

.20 (default) | scalar

The effective radius, $R_{eff}$, used with the applied clutch friction force to determine the friction force. The effective radius is defined as:

$R_{eff}=\frac{2(R_o{}^3-R_i{}^3)}{3(R_o{}^2-R_i{}^2)}$

The equation uses these variables.

$R_o$	Annular disk outer radius
$R_i$	Annular disk inner radius

Dependencies

To enable the clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Nominal preload force, Fc — Force

500 (default) | scalar

Nominal preload force, in N.

Dependencies

To enable the clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Friction coefficient vector, muc — Friction

[.16 0.13 0.115 0.11 0.105 0.1025 0.10125] (default) | vector

Friction coefficient vector.

Dependencies

To enable the clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Slip speed vector, dw — Angular velocity

[0 10 20 40 60 80 100] (default) | vector

Slip speed vector, in rad/s.

Dependencies

To enable the clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Torque - slip speed vector, Tdw — Torque

[-100, -90, -50, -5, 0, 5, 50, 90, 100] (default) | vector

Torque vector, in N·m.

Dependencies

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

Slip speed vector, dwT — Angular velocity

[-200, -175, -100, - 50, 0, 50, 100, 175, 200] (default) | vector

Slip speed vector, in rad/s.

Dependencies

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

Torque - input torque vector, TTin — Torque

[-200 -175 -100 - 50 0 50 100 175 200] (default) | vector

Torque vector, in N·m.

Dependencies

To enable the input torque parameters, select Input torque dependent torque data for the Coupling type parameter.

Input torque vector, Tin — Torque

[-200 -175 -100 - 50 0 50 100 175 200] (default) | vector

Torque vector, in N·m.

Dependencies

To enable the input torque parameters, select Input torque dependent torque data for the Coupling type parameter.

Coupling time constant, tauC — Constant

.01 (default) | scalar

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.

Extended Capabilities

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Version History

Introduced in R2017a