Open Differential

Differential as a planetary bevel gear

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.

Transmission diagram

Efficiency

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

Setting Implementation

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.

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	$η T_{d} ω_{d}$
		`PwrAxl1`	Mechanical power from axle 1	$η T_{1} ω_{1}$
		`PwrAxl2`	Mechanical power from axle 2	$η T_{2} ω_{2}$
	`PwrNotTrnsfrd` — Power crossing the block boundary, but not transferred Positive signals indicate an input Negative signals indicate a loss	`PwrMechLoss`	Total power loss	$\begin{array}{l} {\dot{W}}_{l o s s} = - (P_{t} + P_{d}) + P_{s} \\ P_{t} = η T_{d} ω_{d} + η T_{1} ω_{1} + η T_{2} ω_{2} \end{array}$
		`PwrDampLoss`	Power loss due to damping	$P_{d} = - (b_{1} \| ω_{1} \| + b_{2} \| ω_{2} \| + b_{d} \| ω_{d} \|)$
	`PwrStored` — Stored energy rate of change Positive signals indicate an increase Negative signals indicate a decrease	`PwrStoredShft`	Rate change of stored internal energy	$P_{s} = - (ω_{1} {\dot{ω}}_{1} J_{1} + ω_{2} {\dot{ω}}_{2} J_{2} + ω_{d} {\dot{ω}}_{d} J_{d})$

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	${\dot{ω}}_{d} J_{d} = η T_{d} - ω_{d} b_{d} - T_{i}$
Left Axle	${\dot{ω}}_{1} J_{1} = η T_{1} - ω_{1} b_{1} - T_{i 1}$
Right Axle	${\dot{ω}}_{2} J_{2} = η T_{2} - ω_{2} b_{2} - T_{i 2}$

Mechanical Dynamic Response

Differential Equation

Crown Gear

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

Left Axle

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

Right Axle

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

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

$η T_{i 1} = η T_{i 2} = \frac{N}{2} T_{i}$

$ω_{d} = \frac{N}{2} (ω_{1} + ω_{2})$

The equations use these variables.

N	Carrier-to-driveshaft gear ratio
J_d	Rotational inertia of the crown gear assembly
b_d	Crown gear linear viscous damping
ω_d	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
T_d	Driveshaft torque
T₁	Axle 1 torque
T₂	Axle 2 torque
T_i	Driveshaft internal resistance torque
T_i1	Axle 1 internal resistance torque
T_i2	Axle 2 internal resistance torque

Ports

Inputs

expand all

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

expand all

Info — Bus signal
bus

Bus signal containing these block calculations.

Signal			Description	Units
`Driveshft`	`DriveshftTrq`		Driveshaft torque	N·m
`Driveshft`	`DriveshftSpd`		Driveshaft speed	rad/s
`Axl1`	`Axl1Trq`		Axle 1 torque	N·m
`Axl1`	`Axl1Spd`		Axle 1 speed	rad/s
`Axl2`	`Axl2Trq`		Axle 2 torque	N·m
`Axl2`	`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
	`PwrTrnsfrd`	`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

expand all

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

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.

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

Open Differential

Description

Efficiency

Power Accounting

Dynamics

Ports

Inputs

DriveshftTrq — Torque scalar

Axl1Trq — Torque scalar

Axl2Trq — Torque scalar

Temp — Temperature scalar

Dependencies

Output

Info — Bus signal bus

DriveshftSpd — Angular speed scalar

Axl1Spd — Angular speed scalar

Axl2Spd — Angular speed scalar

Parameters

Efficiency factors — Specify configuration Constant (default) | Driveshaft torque, speed and temperature

Interpolation method — Method Flat | Nearest | Linear point-slope | Linear Lagrange | Cubic spline

Dependencies

Input ambient temperature — Create input port off (default) | on

Dependencies

Crown wheel (ring gear) located — Specify crown wheel connection To the left of center-line (default) | To the right of center-line

Carrier to drive shaft ratio, Ndiff — Ratio 4 (default) | scalar

Carrier inertia, Jd — Inertia .1 (default) | scalar

Carrier damping, bd — Damping 1e-3 (default) | scalar

Axle 1 inertia, Jw1 — Inertia .1 (default) | scalar

Axle 1 damping, bw1 — Damping 1e-3 (default) | scalar

Axle 2 inertia, Jw2 — Inertia .1 (default) | scalar

Axle 2 damping, bw2 — Damping 1e-3 (default) | scalar

Axle 1 initial velocity, omegaw1o — Angular velocity 0 (default) | scalar

Axle 2 initial velocity, omegaw2o — Angular velocity 0 (default) | scalar

Constant efficiency factor, eta — Efficiency 1 (default) | scalar

Dependencies

Efficiency lookup table, eta_tbl — Lookup table M-by-N-by-L array

Dependencies

Efficiency torque breakpoints, Trq_bpts — Torque breakpoints [25, 50, 75, 100, 150, 200, 250] (default) | 1-by-M vector

Dependencies

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

Dependencies

Efficiency temperature breakpoints, Temp_bpts — Temperature breakpoints [290 358] (default) | 1-by-L vector

Dependencies

Ambient temperature, Tamb — Ambient temperature 297.15 (default) | scalar

Dependencies

Extended Capabilities

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

Version History

See Also

DriveshftTrq — Torque
`scalar`

Axl1Trq — Torque
`scalar`

Axl2Trq — Torque
`scalar`

Temp — Temperature
`scalar`

Info — Bus signal
bus

DriveshftSpd — Angular speed
`scalar`

Axl1Spd — Angular speed
`scalar`

Axl2Spd — Angular speed
`scalar`

Efficiency factors — Specify configuration
`Constant` (default) | `Driveshaft torque, speed and temperature`

Interpolation method — Method
`Flat` | `Nearest` | `Linear point-slope` | `Linear Lagrange` | `Cubic spline`

Input ambient temperature — Create input port
`off` (default) | `on`

Crown wheel (ring gear) located — Specify crown wheel connection
`To the left of center-line` (default) | `To the right of center-line`

Carrier to drive shaft ratio, Ndiff — Ratio
`4` (default) | `scalar`

Carrier inertia, Jd — Inertia
`.1` (default) | `scalar`

Carrier damping, bd — Damping
`1e-3` (default) | `scalar`

Axle 1 inertia, Jw1 — Inertia
`.1` (default) | `scalar`

Axle 1 damping, bw1 — Damping
`1e-3` (default) | `scalar`

Axle 2 inertia, Jw2 — Inertia
`.1` (default) | `scalar`

Axle 2 damping, bw2 — Damping
`1e-3` (default) | `scalar`

Axle 1 initial velocity, omegaw1o — Angular velocity
`0` (default) | `scalar`

Axle 2 initial velocity, omegaw2o — Angular velocity
`0` (default) | `scalar`

Constant efficiency factor, eta — Efficiency
`1` (default) | `scalar`

Efficiency lookup table, eta_tbl — Lookup table
`M`-by-`N`-by-`L` array

Efficiency torque breakpoints, Trq_bpts — Torque breakpoints
`[25, 50, 75, 100, 150, 200, 250]` (default) | `1`-by-`M` vector

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

Efficiency temperature breakpoints, Temp_bpts — Temperature breakpoints
`[290 358]` (default) | `1`-by-`L` vector

Ambient temperature, Tamb — Ambient temperature
`297.15` (default) | `scalar`

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