SAE TECHNICALPAPER SERIES

2013-36-0335E

Vehicle Dynamics: a new way of understanding and optimizingvehicle performance.

ANA CRISTINA SOARES

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2013-36-0335

Vehicle Dynamics: A new way of understanding and optimizing

vehicle performance

Soares, Ana Cristina MAN Latin America

Quintiliano, Bibiana MAN Latin America

Dias, Rogério MAN Latin America

Copyright © 2013 SAE International

ABSTRACT

Evaluation of vehicle performance is one of the most

important phases of the new vehicle development. Start Ability

and Top Speed are factors that are noticed by users, therefore

are very important to the final product. Vehicle performance

evaluation has been largely benefited from the use of

simulation tools. In fact, MAN Latin America (ML) employs

simulation programs to evaluate the performance of its

vehicles (trucks and buses) achieving good results. However,

those programs are normally “closed codes” which makes

difficult the physical comprehension of results. Altogether,

this article presents Vehicle Dynamics, a macro developed by

ML engineering team. The aim of this macro is the automatic

calculation of Start Ability, Grade Ability, Top Speed, among

other performance parameters. In order to make Vehicle

Dynamics faster and more elegant the macro’s interface was

created in commercial graphical language since it allows the

use of tools like “Add”, “Replace” and “Select”, among

others. Those options allow it to be faster and help adding data

as well as registering several gear boxes, rear axle ratios and

tires that are used by ML. Hence, Vehicle Dynamics enables

immediate and reliable evaluation of new products and as a

consequence, the choice of the best powertrain configuration.

Furthermore, Vehicle Dynamics was also designed in order to

be easy and didactic for users who want to have a deeper

understanding of vehicle behavior.

INTRODUCTION

Vehicle performance evaluation consists in studying in an

analytical, numerical or experimental way the performance of

a given vehicle. In other words, to establish what the

maximum velocity that the vehicle will achieve is or the

maximum slope the vehicle will be able to negotiate, among

other performance parameters. Therefore, vehicle performance

is basically the evaluation of a vehicle longitudinal dynamics.

The aim of this study is quantifying the main forces acting in a

moving vehicle (or even in a vehicle about to move),

understanding their nature and, finally, establishing an

equation to relate all those forces.

LONGITUDINAL VEHICLE

DYNAMICS

According to Gillespie [1], determining the axle loading on a

vehicle’s under arbitrary conditions is a first simple

application of Newton’s Second Law:

∑ (1)

Where: Fx – Forces in the x-direction, Mveic – Vehicle mass, – Acceleration at x direction

In order to visualize all the forces acting in a vehicle, a free

body diagram can be very useful. In figure 1, the mentioned

forces are displayed.

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Figure 1 – Arbitrary forces acting in a vehicle [1].

Where:

Mv – Vehicle mass Vehicle weight, acting upon front wheels, Vehicle weight, acting upon rear wheels Faer – Force due to aerodynamic resistance Ftrat – Tractive force Rxf, Rxr –rolling resistances acting at frontal and rear

wheels.

Among the forces presented in figure 1, it can be seen the

tractive force which comes from engine torque and that is

responsible for vehicle movement. In contrast, the other

forces, like aerodynamics forces and rolling resistance forces,

act against movement. Since the comprehension of the nature

of those forces is vital to understanding vehicle dynamics, a

brief revision of them will be presented.

Tractive force

Engine torque is multiplied by gear box and rear axle ratio to

generate the propulsive force. According to Newton’s Third

Law, a reaction to this force, that is called Tractive Force, will

appear:

(2)

Where: Tractive force, N.

Torque generated by engine, Nm.

Gear box ratio, non-dimensional.

Rear axle ratio, non-dimensional.

– Tire dynamic radius, m.

It is important to point out that tractive force is responsible for

moving the vehicle; therefore it must be enough to overcome

all resistance forces.

Resistances

As mentioned before, the others forces showed on figure 1 act

against movement and, for that reason, they are called

Resistances. Those forces will be presented from now on.

Rolling resistance – this resistance is originated from the

interaction between tires and road surface and it can be

calculated as

( ) (3)

Where: Rolling resistance force, N. Vehicle mass, kg. Surface coeficient, non-dimensional.

Tire static coeficient, non-dimensional Tire dynamic coeficient, non-dimensional.

Gravity acceleration, m/s2 Slope, º

Aerodynamic Resistance – defined as the resistance

caused by air to the motion of a solid body moving through it.

It is calculated as:

(4)

Aerodynamic resitance, N. Aerodynamic resistance, N. – Air density, kg/m3. Front area, m2. Vehicle velocity, km/h

Grade Resistance – On a grade, vehicle weight may have

two components, a cosine component which is perpendicular

to the road surface and a sine component parallel to the road.

This sine component acts against movement; therefore it is a

resistance [1]. This force can be calculated by the expression:

(5)

Where:

Vehicle mass, kg. Gravity acceleration, m/s2 Slope, º

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Movement equation

Once the main forces acting longitudinally in a vehicle have

been defined, it is possible to apply Newton Second Law in

order to relate them. Then:

∑ (1)

(6)

From equations (2), (3), (4) and (5):

( ) -

(7)

VEHICLE PERFORMANCE

CALCULATION

After equation (7) has been defined, several parameters used

to describe vehicle performance will be calculated, applying in

each case, the proper boundary condition.

Start Ability

Star Ability is defined as the maximum slope that a vehicle is

able to negotiate, once it starts from rest. By definition, Start

Ability is calculated at 1000 rpm. Besides, the acceleration is

considered to be zero. In doing so, equation (7) becomes:

( )

(8)

Where:

T1000 – Available engine torque at 1000 rpm.

Grade Ability

Grade Ability is defined as the maximum slope that a vehicle

is able to negotiate, once it starts with a velocity different from

zero. Unlike Start Ability, which is calculated at 1000 rpm,

Grade Ability can be calculated over the entire engine range.

However, in general it is calculated at maximum torque rpm as

well as maximum power rpm. Applying the suitable boundary

condition, equation (7) becomes:

( )

( )

(9)

Power Speed

Defined as the vehicle speed that is calculated at maximum

power rpm, taking in account only kinematic factors like gear

box and rear axle ratios, or in other words, not taking in

consideration all resistances presented previously. Power

Speed is calculated like:

(

)

(10)

Where: –

Top Speed

It is the maximum velocity that a given vehicle can achieve at

a plane road, after having overcome all vehicle resistances.

Again, it is considered that vehicle acceleration is zero and, by

definition, that slope angle is zero. Altogether, equation (7)

becomes:

(

) (10)

Since torque and velocity are function of rpm, it is not

possible to know previously in which rpm equation (10) will

be verified, as a consequence, the solution is iterative, as it

schematically showed in figure 2:

Figure 2 – Iterative process for calculation Top Speed.

To begin with, a given rpm, rpm1, is chosen and the

correspondent torque, T1, and velocity, v1, are calculated.

Then, equation (10) is calculated in order to verify if the sum

result is zero. If not, another rpm, rpm2, is chosen and all the

process is repeated. There is only one rpm that verifies

equation (10) and to this specific rpm there is a specific

velocity. This velocity is Top Speed.

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VEHICLE DYNAMCIS MACRO

The calculation of the presented parameters is fundamental to

de analytical/numerical evaluation of a vehicle performance. It

is necessary to repeat those calculations every time that a

given vehicle powertrain configuration is modified. For

example, when gear box or tires are replaced. Nowadays,

naturally, those calculations are simplified by the use of

electronic macros. Even so, it is necessary to type long

equations, always taking care to modify a given parameters

every time this appears in an equation. For example, if vehicle

mass is modified it is necessary to update equations (2) and (4)

therefore, Start Ability, Grade Ability, and Top Speed will be

influenced. In face of it, it is clear that there is a need of an

automatic way of calculating those equations, in order to save

time (of typing) as well as make the calculation itself less

prone to errors. This was the main purpose of creating the

macro Vehicle Dynamics (in short, Dynamics), i.e., to make

the principal performance calculations automatic. To the

purpose of making the understanding of Vehicle Dynamics

easier, some sheets used to insert data, calculations and results

presentations will be presented.

Data Insertion

The first sheet of Dynamics, which will be called data-sheet,

is used to insert all necessary data to carry out vehicle

performance evaluation. The data-sheet consists of three

tables, which will be presented in the following.

Figure 3 – Data-sheet, table 1: insertion of GVW and engine

curve.

As it is presented at figure 3, at the first table from data-sheet,

vehicle type, GVW (Gross Vehicle Weight) are informed as

well as engine-torque curve. On figure 4, it is showed table 2,

where transmission type can be chosen. By clicking “Choose

New Gear Box”, a new dialog box opens, where several kinds

of gear boxes can be chosen. Additionally, there is a sheet

where it is possible to insert new gear boxes1.

Figure 4 – Data-sheet, table 2: inclusion of gear boxes

ratios.

Figure 5 – Data-sheet, table 2: choosing of desired gear box.

1 Not shown in this paper.

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Finally, table 3 is showed on figure 6. As it can be seen, there

are four buttons to select automatically rolling resistance

coefficients, RCest and RCdyn, tire dimensions, rear axle ratio

(RAR), and road type. In the several other fields showed it can

be inserted front area, aerodynamic resistance coefficient, air

density, in short, all vehicle data. Furthermore, on figure 7, it

is presented a dialog box for choosing RAR: at first user

chooses between single or double axle and then, the rear axle

ratio is chosen.

Figure 6 – Data-sheet, table 3: insertion of road type,

efficiency, front area, etc.

Figure 7 – Data-sheet: dialog box used for choosing rear

axle type as well as its ratios.

AUTOMCATIC CALCULATIONS

Once all data has been inserted, performance calculation is

almost automatic. In Dynamics there are 5 sheets, each one

related to one of performance parameters: Start Ability, Grade

Ability, Top Speed, Velocity and rpm. In each of those sheets it

is possible to visualize which data was used in the calculation.

For example, in Start Ability sheet, figure 8, it can be seen

GVW, torque at 1000 rpm, gear box and rear axle ratios,

among others. This possibility of consulting the calculations

allows the user to understand how a given parameter is

calculated besides knowing which factors are taken in account.

As a consequence, the user is able to evaluate how to improve

that parameter. For example, in Start Ability case, if the user

needs to enhance this factor, it is clear from figure 8, that the

easiest possibilities are choosing a gear box or rear axle with

different ratios. (Of course, it is possible to reduce GVW, but

this is not a viable alternative).

Figure 8 – Start Ability sheet

There is only one sheet where calculation is not fully

automatic, that is, Top Speed, since it was explained before,

Top Speed calculation is iterative. Because of this, Top Speed

sheet was created in a way that the calculation presented on

equation 10 is performed over all engine rpm range. To each

pair torque/rpm, the macro calculates automatically the

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available tractive force (function of torque/rpm), vehicle

velocity (function of rpm) and resistance forces (function of

rpm). Then equation (10) is calculated repeatedly, until it is

verified:

(

) (10)

This iterative process is showed schematically on figure 9.

Figure 9 – Top Speed iterative calculation.

RESULTS PRESENTATION

The last sheet of Dynamics is the one where all results are

presented in a clear and self-understanding way. This sheet,

named Results, is showed on figure 10. The presented results

are Start Ability, Grade Ability, Top Speed, Power Speed,

besides the rpms to achieve the velocities of 60, 70, 80 e 90

km/h. Results sheet is very useful for the user that wants only

to have fast and reliable results, not needing to understand the

physics behind them.

Figure 10 – Results sheet.

EVALUATION OF VEHICLE

DYNAMICS RESULTS

Before starting to use any simulation program it is

fundamental that the user can trust the results generated by

this program. There are basically two ways of achieving a

significant grade of confidence in simulation programs. The

first is comparing simulation results with experimental ones

while the other is using another program that is reliable and

well established as a standard. In this paper, Dynamics results

were compared to a former macro used by ML, whose name is

Performance. This macro has been used for several years by

ML engineers, with excellent adherence with experimental

results. As a result, the comparison between Dynamics and

Performance was enough to validating the former. It is

important to point out that both macros, Dynamics and

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Performance, were created based on classical statics and

dynamics equations. Those equations do not need any kind of

validation. What needs validation, after all, is Dynamic macro

itself, beginning with the typing of all presented equations till

the automatic fulfillment of Start Ability, Grade Ability, Top

Speed, velocity e rpm sheets.

Table 1 – Comparison between normalized results from

Performance and Dynamics. Several products from MAN

Latina America have been simulated.

Bus

Start Ability Grade Ability Top Speed

Performance 1,0000 1,0000 1,0000

Dynamics 1,0005 1,0000 1,0005

Light Truck

Performance 1,0000 1,0000 1,0000

Dynamics 1,0006 0,9998 0,9988

Medium Truck

Performance 1,0000 1,0000 1,0000

Dynamics 1,0004 1,0020 1,0050

Heavy Truck

Performance 1,0000 1,0000 1,0000

Dynamics 1,0007 0,9998 1,0006

From table 1, it can be seen that the agreement between

Dynamics and Performance was excellent, for buses and

trucks. Based on those results, Vehicle Dynamics was

validated, being used nowadays by ML engineering team.

CONCLUSION

Numerical/analytical evaluation of vehicle performance is

based on several equations originated from statics and vehicle

dynamics. In some cases, those equations are virtually huge

besides taking in account several data like vehicle mass, front

area, engine torque, etc. On the other hand, the choice of the

best powertrain configuration requires the repetitive

evaluation of several gear boxes, rear axle and tires.

Altogether, it is evident that creating a macro able to calculate

all performance parameters in an automatic way is more than

desirable. Consequently, MAN Latin America engineering

team developed Vehicle Dynamics, a macro where parameters

like Start Ability, Grade Ability, Top Speed and Power Speed

are automatically calculated. Vehicle Dynamics was validated

based on analytical results, presenting excellent agreement

with them. For this reason, Vehicle Dynamics is being used

successfully by MAN Latin America engineering team.

REFERENCES

1. Gillespie, Thomas D., “Fundamental of Vehicle

Dynamics”, Society of Automotive Engineers, Inc.

2. Azevedo Jr, G., “Apostila Performance Veicular”,

Associação Eduacional Dom Bosco.

CONTACT INFORMATION

Ana Cristina Cosme Soares

Engenheiro Alan da Costa Batista, 100

Pedra Selada, CPI 4227

Cep 27511-970 Resende, RJ

Telefone: 5524-33811421

e-mail: ana.soares@volkswagen.com.br

Bibiana Quintiliano

e-mail: bibianaquintiliano@hotmail.com

ACKNOWLEDGMENTS

Authors would like to acknowledge engineer Geraldo

Azevedo Junior, for support with Performance macro and

engineer Frederico Braz e Silva for technical support. Finally,

authors acknowledge MAN Latin America for general

support.

DEFINITIONS, ACRONYMS,

ABBREVIATIONS

Ftrat Tractive Force

Tmotor Engine Torque

ig Gear box ratio Id Rear axle ratio

Rdyn Tire dynamics radius

Frol Rolling reistance force Mveic Vehicle mass RCest Tire rolling resistance, static RCdyn Tire rossling resistance, dynamic g Gravity acceleration Slope angle Faer Aerodynamic force Cw Aerodynamic drag coefficient Air density v Vehicle velocity Fgrad Grade force GVW Gross Vehicle Weight RAR Rear Axle Ratio

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