analysis of the dynamic behavior of an electric vehicle using...

Analysis of the dynamic behavior of an electricvehicle using an equivalent roll stiffness model

J.L. Torres1, A. Gimenez1, J. Lopez-Martinez1, G. Carbone2 and M. Ceccarelli2

1University of Almeria, Spain, e-mail: [email protected], [email protected],[email protected] of Cassino, Italy, e-mail: [email protected], [email protected]

Abstract. This paper presents an analysis and simulation of the dynamic behavior of an electricvehicle. Governing dynamic equations are formulated and a three-dimensional prototype is built,which allows the collection of data on mass and inertia of its components. All these variables areimplemented in a Multibody System (MBS) model. This model is analyzed by using SimMechan-ics, a tool for MBS analysis. Some of the results of this analysis are used as an input to simulatethe suspension system in detail. The main contribution of this paper is the proposal, once validatedthe model, of a modification in the distribution of mass of the vehicle which improves its dynamicperformance. Moreover, due to the integration of this model in MATLAB/Simulink environment,it is possible to add control systems properly, such as electronic stability control and autonomouscontrol.

Key words: Electric vehicle, simulation, double-wishbone, multibody

1 Introduction

Electric vehicles are becoming a benchmark. In recent years, governments areadopting policies which encourage their developments. Climatic change and theuse of energy sources from carbon and petroleum motivate the use of renewable en-ergy from non polluting sources. In this way, the introduction of electric propulsionsystems in automobiles helps to reduce fossil fuel consumption and optimize theefficiency of new vehicles. Computer modeling and simulation may reduce costsand duration of the process of design in electric vehicle projects. This allows test-ing several settings and energy management strategies even before starting to buildthe prototype. Thus, it is interesting to study the parameters that characterize thedynamic behavior of such vehicles. An important issue to take into account is thattheir propulsion is carried out by an electric system, instead of the typical engineand transmission shaft. This involves the modification of several design parameter,such as the transfer case or the batteries location (it may represent up to the 40% ofthe vehicle’s weight). All these parameters affect considerably the dynamic behav-ior of a vehicle.

1

2 J. Torres, A. Gimenez, J. Lopez-Martinez, G. Carbone and M. Ceccarelli

Fig. 1 A prototype of an electric vehicle at University of Almeria

This problem has been treated with an object-oriented modeling approach. Thus,for example, Silva et al. [6]. proposed a model built in Dymola, an object-orientedmodeling tool based on Modelica language. They used the Multibody standard li-brary combined with Bond-Graph technique. In order to overcome the drawbacks ofthis technique applied to 3D-environments, it is used the Multi-Bond-Graph library,presented in [7]. Da Fonte et al. present a model, which is useful to examine the be-havior of the power electric system on starting, accelerating and braking maneuvers[2].In this work, an electric vehicle whose prototype is available at the University ofAlmeria (Fig. 1), is modeled to analyze its dynamic behavior. The use of this modelcould be used also for other electric vehicles of similar characteristics. The softwarefor the formulation and analysis of the full vehicle is Simmechanics, a modularMatlab toolbox based on block diagrams. Due to its integration on Matlab envi-ronment, one can properly enlarge and combine this model with others. Once themodel is validated, a modification is proposed for the vehicle design, which consistson changing the location of the main masses. Consequently, the weight of the bat-teries is distributed over the four wheels, instead of being concentrated at the centerpart of a vehicle. The mechanical aspects of the system are analyzed and discussedfrom simulation results.In section 2, the process carried out to develop the Multibody model is explained.Because the complexity derived of dealing with closed kinematic loops, an alter-native model of the suspension systems is proposed. Both models are simulated insection 3. Finally, results and conclusions are discussed in section 4.

2 Modeling

The model is based in the equivalent roll stiffness model, where the front and rearsuspensions are treated as rigid axles connected to the body by revolute joints [1].This model has 11 degrees-of-freedom and presents a tree-like structure. As men-tioned before, an alternative model of the suspension system is proposed. The pro-cess carried out to develop the model is described next. Firstly, a 3D-model of theis built in SolidWorks, based on the real vehicle available at the lab in Almeria, by

Title Suppressed Due to Excessive Length 3

considering only the relevant components from a dynamical viewpoint. This sim-plified model is then exported to Matlab through Simmechanics-link, a tool whichconnects Matlab and Solidworks. The dynamic equations are then formulated. Fi-nally a detailed model of the suspension system is implemented in Adams.View inorder to perform numerical simulations.

2.1 Longitudinal dynamics

All acting forces on the vehicle’s behavior are identified. Then, a force equilibriumcondition is formulated. Four main forces has been considered. Assuming M as thevehicle mass, the opposing forces to its motion are the aerodynamic drag force, thestatic friction, the force of viscous friction and the weight assigned to each wheel.All these forces act along the X axis and in opposed way to the vehicle motion. The

FXweight

FZweight

FZtotal

MX

vy

vxα

v

ba

FY

wb

a

lXZ Y

X

Z

θ

a) b)

Fig. 2 a) Longitudinal dynamics; b) lateral dynamics

fundamentals of those forces are analyzed next.

Static friction force: This force is only dependent on the friction coefficient, andweight. This coefficient depends on the contact surface. In this work, it is considereda typical case with a tire over dry asphalt in the form:

Fre = M ·g ·µe (1)

Aerodynamic drag force: This braking force is derived from the displaced massof fluid (air) that the vehicle has to replace when moving. This force depends on thesquare of the vehicle speed, and can be expressed as

Fra =ρ · cd ·A f · (v)2

2(2)

where ρ represents the air density, cd is the drag coefficient specific to the vehicle(a typical value for small vehicles is used), and A f is the frontal area given in square


meters.Weight action This is the force with a larger contribution on the sum of brakingforces. It is calculated according to Figure 2 in the form

FZweight = M · sin(θ) (3)

Thus, the total braking force can be expressed as

FRT = Fre +Fra +FXweight (4)

2.2 Lateral dynamics

When a vehicle turns, several forces are produced due to centrifugal acceleration.Roll moment and reactions on the wheels are analyzed, together with lateral forces[4]. The adopted convention for the calculation of these forces on each wheel isshown in Fig. 2, where:

• vx,vy, Components X and Y of vehicle’s speed• MX , Torque roll (along X axis)• α , Steer angle• w , Vehicle width• a , Longitudinal distance from CG to front axle• b , longitudinal distance from CG to rear axle• FZ , Wheel vertical reaction force• FY , Wheel lateral force

The value of FY can be computed as, [3]

FY =(−a1 ·FZ +a2 ·F2

Z)·α (5)

where a1 and a2 represent the proportional and quadratic no-lineal coefficient ofthe tire model, respectively. Thus, it is possible to obtain the lateral force acting oneach wheel, as function of the steer angle. Used parameters for this model are listedin Table 1

2.3 Double-wishbone suspension

The initial model of the vehicle does not include a detailed suspension system sinceit is based on the equivalent-roll model. However, the data obtained from the simula-tions are implemented in a separate suspension system. In this way, it is possible toanalyze the effects of the full-vehicle dynamics without including closed kinematicloops, and eventually focus on the suspension system separately. An scheme of thissubsystem is shown in Fig. 3


Table 1 Values for model parameters

Parameter Symbol Value Units

vehicle weight m 1670 kglongitudinal distance from CG to front axle a 1.50 mlongitudinal distance from CG to rear axle b 0.90 mheight of sprung mass above ground hs 0.40 msprung front roll stiffness k f 90 N/mmsprung rear roll stiffness kr 55 N/mmnonlinear tire model proportional coefficient a1 1.8 ·10−5 -nonlinear tire model quadratic coefficient a2 9.3 ·10−4 -

Fig. 3 Double-wishbone suspension

3 Simulation

3.1 Longitudinal dynamics

The vehicle is driven along a straight line from steady state to a velocity of 10 m/swith constant acceleration. The effect of the resistance forces can be analyzed toevaluate how these forces increase as the vehicle is reaching the desired velocity.The necessary propulsion force increases correspondingly. Eventually, a variationof the inclination is produced from t=6s till t=8s. The results can be seen in thenext figures. Fig. 4 shows the evolution of the velocity, traction force and generatedresistance as obtained by the numerical simulation for the full vehicle.

As expected, the mayor contribution to the braking forces is due to the change ofpendant. The static friction remains constant throughout the simulation whereas theinfluence of the velocity-dependent braking force (aerodynamic drag) is negligible.

3.2 Lane change maneuver

The vehicle is driven at constant speed of 10m/s. Then the driver steers to the leftin order to carry out a lane change. Two different settings are simulated in orderto compare the results. The first one belongs to the original model. The second


0 1 2 3 4 5 6 7 8 9 100

5

10

Time (s)V

eloc

ity(m

/s)

0 1 2 3 4 5 6 7 8 9 100

500

1000

Time (s)

Forc

e(N

)

0 1 2 3 4 5 6 7 8 9 10

0

Time(s)

Forc

e(N

)

Inclination of 2.5 %

c)

b)

a)

-500

-1000

Fig. 4 Results of numerical simulation: a) velocity; b) propulsion; c) resistance forces

one belongs to a proposed modification, consisting of an alternative distribution ofmasses along the vehicle. The batteries are distributed over the four wheels, insteadof being placed at the center of the vehicle, which implies a better use of space andan increase of rotational inertia of the vehicle about its longitudinal axis.

The path followed by vehicle according to both solutions is represented in Fig. 5:a), and the sprung mass roll angle is shown on Fig. 5: b). The simulation shows thatthe vehicle reacts better under the same steering signal and the roll angle decreaseswhen the vehicle is configured according to the second setting. As a consequence,the vehicle stability and comfort increase, with a positive effect on the driver andpassengers security too.

0 5 10 153

2

1

0

1

2

3

0 50 100 1500.5

0

0.5

1

1.5

2

2.5

3

3.51 Bat.4 Bat.

a) b)Time (s)X distance (m)

Ydi

stan

ce(m

)

Rol

lang

le(◦

)

1 Bat.4 Bat.

Fig. 5 a) Path followed by the car; b) roll angle


3.3 Double-wishbone suspension

When modeling a multibody system like the double-wishbone suspension using aset of redundant generalized coordinates, a system containing ODEs and non-linearalgebraic constraint equations governing its motion is generated. [5] The commer-cial package Adams.View has been used to perform numerical simulation of multi-body system dynamicsOriginally, the system presented in Fig. 3 has three degrees-of-freedom:the rotationof the wheel, the displacement of the rack, and the vertical of the wheel carrier.However, if the wheel axis is blocked, and the displacement of the rack is consid-ered as a known input, the number of D.o.F is reduced to one. Consequently, thesimulation is carried out taking into account the vertical wheel displacement.

200 250 300 350 4000.5

0

0.5

1

1.5

2

Length (mm)

Ang

le(d

eg)

200 250 300 350 4000.15

0.1

0.05

0

0.05

Length (mm)

Ang

le(d

eg)

a) b)

0 0.5 1 1.5 2 2.5 3 3.5

-40

0

40

80 Original conf.Alternative conf.

-80

Time (s)

No

Uni

ts

c)

Fig. 6 a) Path followed by the car; b) roll angle

Fig. 6: a) and b) show the evolution of the TOE and camber angle respectively asa function of the vertical wheel displacement. These kinematic analyses are commonfor both configurations. On the other hand, Fig. 6: c) shows a comparison of thedamper displacement under the action of the vertical forces calculated using thefull-vehicle model. These forces are introduced as an s-force input in Adams.View.As can be seen, due to the reduction on the roll angle when the vehicle is configuredaccording to the second setting, the vertical forces acting on the wheels are lower,which implies a shorter shock travel.


4 Conclusions

A model is presented to analyze the dynamic behavior of an electric vehicle. Theproposed low-cost solution may be manipulated by non-experts in multibody dy-namics. Thus, control engineers may benefit of its simplicity and the use of theblock-diagram language. Despite the simplifications, the roll equivalent model isdemonstrated to be accurate, with an acceptable loss of fidelity when analyzing thefull-vehicle dynamics. Moreover, the model may be enhanced as much as requiredsince its modular and multidomain configuration. Hence, it is possible to incorporatethe behavior of the electric motor and batteries. A process to combine the advantagesof a CAD and mathematical/modeling software has also been presented so that thevehicle model simulates different maneuvers. The results are compared with thoseobtained from an alternative vehicle solution, where the batteries are distributedover the four wheels. The advantages of this proposed solution are discussed withnumerical results from simulations for the roll angle, the path followed by the caror vertical reaction forces on the wheels. These last data are used as an input in aseparate model of the double-wishbone suspension system to discuss the effects ofdynamics on the suspensions. By using Adams/View, the variation of the camber,caster and toe parameters are analyzed with respect to the vertical displacement, aswell as the damper displacement vs. wheel vertical force.

Acknowledgements The first author, who has carried out a period of study at LARM in 2011, isvery grateful to Andalusia Regional Government, Spain, for financing this work through the Pro-gramme FPDU 2008 and 2009. This is a programme co-financed by the European Union throughthe European Regional Development Fund (ERDF).

References

1. Blundell, M. and Harty, D.: The Multibody Systems Approach to Vehicle Dynamics. Elsevier(2004)

2. da Fonte Terras, J. M. et al..: Modelling and simulation of a commercial electric vehicle.In: 13th International IEEE Conference on Intelligent Transportation Systems (ITSC), 1588-1593 (2010)

3. Gillespie, T.D.: Fundamentals of vehicle dynamics. Society of Automotive Engineers, 495(1992)

4. Milliken, W. et al..: Race car vehicle dynamics workbook. SAE International (1998)5. Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press (1998)6. Silva, L.I. et al..: Vehicle dynamics using multi-bond graphs: Four wheel electric vehicle

modeling. In: 34th Annual Conference of IEEE Industrial Electronics, 2846-2851 (2008)7. Zimmer, D.: A Modelica library for Multibond Graphs and its Applications in 3D-Mechanics.

ETH Zurich (2006)

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