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MODELING APPLICATIONS

# Modeling and Analysis of Active Differential Dynamics

[+] Author and Article Information
Joško Deur

Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, I. Lučića 5, HR-10002 Zagreb, Croatiajosko.deur@fsb.hr

Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, I. Lučića 5, HR-10002 Zagreb, Croatiavladimir.ivanovic@fsb.hr

Matthew Hancock

Gaydon Engineering Centre, Jaguar Cars Ltd., Banbury Road, Gaydon, Warwick CV35 0XJ, UKmhancoc1@jaguarlandrover.com

Department of Automotive Engineering, School of Engineering, Cranfield University, Bedfordshire MK43 0AL, UKf.assadian@cranfield.ac.uk

Note that the maximum wheel speed difference may be higher for traction control applications.

Note that the half-shaft damping was neglected $(b=0)$, and the more dominant tire damping equals zero for the constant tire speed assumption 19 (see Ref. 12).

The equivalent power train output inertia referred to the differential input shaft, $Ii$, increases with the transmission gear ratio. Therefore, as the gear is changed from the assumed fourth gear toward the first gear, the natural frequency of 18.5 Hz would decrease to around 5 Hz (see Ref. 12).

According to Eq. 31, this assumption leads to $kt1=kt2$ (equal left and right tire static curve stiffnesses), which is not quite accurate for hard cornering maneuvers because of different normal loads of the outer and inner tires.

J. Dyn. Sys., Meas., Control 132(6), 061501 (Oct 29, 2010) (14 pages) doi:10.1115/1.4002482 History: Received September 30, 2008; Revised October 13, 2009; Published October 29, 2010; Online October 29, 2010

## Abstract

Active differentials are used to improve the overall performance of traction control and vehicle dynamics control systems. This paper presents the development of a unified mathematical model of active differential dynamics using the bond graph modeling technique. The study includes active limited slip differential and various common types of torque vectoring differentials. Different levels of model complexity are considered, starting from a second-order model with lumped input and output inertia toward higher-order models including the gear inertia and half-shaft compliance. The model is used for a theoretical analysis of drivability and time response characteristics of the active differential dynamics. The analysis is illustrated by simulation results.

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## Figures

Figure 1

(a) Kinematic scheme and (b) bond graph of open differential

Figure 2

Basic bond graph model of open differential dynamics

Figure 3

Full bond graph model of open differential dynamics

Figure 4

Longitudinal tire static curve

Figure 5

Kinematic schemes of active differentials: (a) ALSD, (b) superposition clutch TVD, (c) stationary clutch TVD, and (d) 4WD TVD

Figure 6

Bond graph models of active differentials dynamics: (a) ALSD, (b) superposition clutch TVD, (c) stationary clutch TVD, and (d) 4WD TVD

Figure 7

Normalized static curve of clutch friction

Figure 8

Principal block diagram of overall active differential model based on Karnopp clutch friction model

Figure 9

Illustration of ALSD torque transfer response for locked-clutch operating mode and constant wheel speed assumption

Figure 10

Illustration of ALSD response for locked-clutch operating mode and linearized wheel/tire model assumption

Figure 11

ALSD model response with respect to change in input torque

Figure 12

ALSD model response with respect to change in torque transfer demand (Δωt0=−2 rad/s)

Figure 13

Comparative ALSD model responses for different initial speeds ω0 and half-shaft stiffness k(Δωt0=−0.5 rad/s)

Figure 14

Comparative responses of different ALSD model for different initial speeds ω0(Δωt0=−0.5 rad/s)

Figure 15

Stationary clutch TVD model response with respect to change in torque transfer demand

Figure 16

4WD TVD model response with respect to change in torque transfer demand

Figure 17

Causal bond graph elements and related equations

Figure 18

Bond graph illustration of full ALSD model order reduction: (a) full model with ALSD inertia neglected and (b) basic model extended with half-shaft compliance lumped to clutch shaft

Figure 19

Reset integrator dynamic friction model for variable normal force and Coulomb sliding friction (TS=TC): (a) equivalent mechanical system and (b) block diagram of model

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