Open Differential

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 Open 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

Use the Open Differential block to:

• Dynamically couple the post-transmission driveshaft to the wheel axles or universal joints

• Model simplified or older drivetrains when optimal traction control does not require passive or active torque vectoring

• Model mechanical power splitting in generic gearbox and drive line scenarios

The block is suitable for use in hardware-in-the-loop (HIL) and optimization workflows. All the parameters are tunable.

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	$$\begin{array}{l}{\dot{W}}_{loss}= -(P_t+P_d)+P_s\\ \\ P_t=\eta T_d{\omega }_d+\eta T_1{\omega }_1+\eta T_2{\omega }_2\end{array}$$
		PwrDampLoss	Power loss due to damping	$$P_d=-\left(b_1\left|{\omega }_1\right|+b_2\left|{\omega }_2\right|+b_d\left|{\omega }_d\right|\right)$$
	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 Open 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 Open Differential block assumes rigid coupling between the crown gear and axles. These constraint equations apply.

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

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

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
η	Differential efficiency
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
Td	Driveshaft torque
T1	Axle 1 torque
T2	Axle 2 torque
Ti	Driveshaft internal resistance torque
Ti1	Axle 1 internal resistance torque
Ti2	Axle 2 internal resistance torque

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
PwrInfo	PwrTrnsfrd	PwrDriveshft	Mechanical power from driveshaft	W
		PwrAxl1	Mechanical power from axle 1	W
		PwrAxl2	Mechanical power from axle 2	W
	PwrTrnsfrd	PwrMechLoss	Total power loss	W
		PwrDampLoss	Power loss due to damping	W
	PwrStored	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.

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, Ndiff — Ratio

4 (default) | scalar

Carrier-to-driveshaft gear ratio, N, dimensionless.

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.

Efficiency

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.

Extended Capabilities

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

Version History

Introduced in R2017a