International journal of automotive technology, tập 10, số 4, 2009

coils, as shown in Figure 5, so the maximum number ofvalues along the horizontal axis of the pitch curve is always one greater than the number of active coils.. Hepta methylnonane2,2,4,4

DEVELOPMENT OF QUANTITATIVE MEASURING TECHNIQUE

TO FIND CRITICAL FLOW CONDITIONS FOR PREVENTING SOOT DEPOSIT ACCUMULATED IN THE DIESEL EXHAUST SYSTEM USING

MAIN MUFFLER COMPOSED OF THREE CHAMBERS

B.-H SONG 1) and Y.-H CHOI 2)*

1)Graduate School, Ajou University, Gyeonggi 443-749, Korea

2)Department of Mechanical Engineering, Ajou University, Gyeonggi 443-749, Korea

(Received 23 July 2008; Revised 28 October 2008)

ABSTRACT− If a vehicle that meets emission regulations operates sufficiently for a long time under low speed and low load conditions, soot contained in the exhaust gas is accumulated on the inner surface of the exhaust system This soot deposition problem occurs frequently in all diesel cars However, when a vehicle is placed under the conditions of sudden start and sudden acceleration after city mode driving for a long time, the deposited soot is abruptly blown up with the soot produced during fuel combustion In the present study, the main cause of the abrupt outburst of deposited soot is investigated to overcome this adverse phenomenon First, we developed a method to quantify the amount of the exhausted soot particles (or the accumulated soot particles) by measuring the opacity that represents the contamination level of the exhaust gas due to soot particles Using this measuring scheme for deposited soot, we found the critical conditions for engine speeds and load conditions at which soot particles are emitted into the air without accumulation in the exhaust system using main muffler composed of three chambers In order to meet these critical conditions and thus to drastically reduce soot accumulation, the exhaust system using the main muffler applied in this study must be designed to ensure that the flow velocity of the exhaust gas is higher than 62 m/s when the back pressure at the exit of the turbocharger is under 0.08 bars.

KEYWORDS : Deposited soot, Opacity, Particulate matter, NOx, Char, Coke, Soot, Spherule, PAH

NOMENCLATURE

FSN : filter soot number

N : opacity [%]

PM : particulate matter

SOF : soluble organic fraction

FTP : federal test procedure

PAH : polycyclic aromatic hydrocarbons

1 INTRODUCTION

The diesel engine has the potential to reduce the

environ-mental pollution problem The diesel engine is not only

more economical due to higher thermal efficiency

com-pared to other internal combustion engines, but also shows

lower concentrations of carbon dioxide in the exhaust gas

However, as the interest in both environmental pollution

and health increases dramatically, the harmfulness of diesel

exhaust emissions has been highlighted due to the bad

influence of its toxic components on the environment and

on the human body Therefore, the exhaust emission

stand-ards of each country are becoming stricter

In the harmful exhaust gas from the diesel engine, theconcentration of nitrogen oxides (NOx), whose maincomponents are nitrogen monoxide (NO) and nitrogendioxide (NO2), is similar to that of a spark-ignited engine,but particulate matter, in contrast, is considerably higher.The worst issue is that 0.2% to 0.5% of fuel weight isexhausted into the form of small particles below 0.1 µm indiameter (Heywood, 1988) In particular, ultrafine particlesbelow 0.1µm in diameter are exhausted into the atmos-phere, and they can penetrate deeply in the human bodythrough the respiratory organs Most medical expertswarn that these ultrafine soot particles can cause chronicpulmonary disease, lung cancer, cardiovascular disease,influenza, asthma, etc Also, ozone is a known toxicmaterial causing respiratory problems by damaging thelungs, and ground-level ozone (O3) is formed by chemicalreactions between NOx and volatile organic compounds(VOCs) in the presence of sunlight Here, VOCs areunburned hydrocarbons produced in the process of fuelcombustion Moreover, a chain reaction of NOx thatproduces ozone photochemically in daylight threatens todestroy the ozone layer within the stratosphere at night.This results in an increase of the quantity of ultravioletemissions from the sun and has a harmful influence on the

*Corresponding author. e-mail: ychoi@ajou.ac.kr

earth's atmosphere.

Therefore, it is important to reduce harmful emission

components from diesel engines, such as NOx and PM,

during the design and the development of diesel vehicles

Many studies have been made attempting to decrease

harmful exhaust gas dramatically in the fields of

pretreat-ment, after-treatpretreat-ment, and new combustion system

develop-ment, and now most automobile manufacturers are

produc-ing passenger cars that meet present emission standards

However, when diesel vehicles that satisfy the legal criteria

for exhaust gas emissions are placed under conditions of

low speeds and low loads for a long period of time, such as

in city driving, most of the vehicles have difficulty

avoid-ing the soot deposit phenomenon wherein soot particles

contained in exhaust gas collect on the inner surface of the

exhaust system In particular, the deposited soot particles

are abruptly emitted into the atmosphere when a driver

makes a sudden start or sudden acceleration, and a big soot

cloud is formed behind the car because the quantity of

emitted soot particles is extraordinarily greater than that of

soot particles produced automatically through the general

combustion of a direct injection (DI) diesel engine

Some-times this fine black soot cloud can even hide the vision of

a driver Therefore, it is necessary to study this problem

In the present study, we analyzed the cause of the soot

deposit phenomenon to find a solution to the abrupt

emission of deposited soot particles under conditions of

sudden start and sudden acceleration In addition, we have

developed a method to measure the quantity of soot

parti-cles deposited in the exhaust system, and we present a

design guide to remove the soot deposition problem

2 FORMATION AND OXIDATION OF SOOT

2.1 Soot Characteristics

Particulate matters consisting of carbon compounds are

produced in the combustion process, and their color in

general is black According to formation processes,

parti-culate matter is classified mainly into char, coke, and soot

Char is the particulate matter made of solid components ofresidue after the volatile components of fuel are evapo-rated Particulate matter formed by the pyrolysis of hydro-carbon in the liquid phase is called coke In particular, soot

is formed in the combustion process of gas phase fuel in ahigh temperature condition, and it includes a soluble organicfraction (Lee et al., 2006)

Figure 1 shows the soot configuration observed by anelectron microscope We can see that the soot spherules of10~80 nm in diameter (most of them are 15~30 nm) or itscollectives are connected in a chain shape (Haynes andWagner, 1981)

2.2 Soot FormationWhen the chemical energy contained in hydrocarbon fuel isnot completely released into available energy throughcombustion, soot is produced Therefore soot formationdue to incomplete combustion results in a decrease in fuelefficiency The process of soot formation also includesvarious physical and chemical steps Many theories havebeen proposed to explain these steps, and the generaltheory is as follows (Bockhorn and Schäfer, 1994).2.2.1 Formation step of macro-molecule precursorThe formation of molecular precursor that represents theorigin of soot formation is the most important step in theprocess of soot formation, and is achieved through thechemical condensation of gas phase material and thecoagulation reaction Even though it has not yet beenclarified how the molecular precursor is changed into asoot particle, two types of transition processes, such asphysical condensation and chemical condensation, havebeen suggested

The physical condensation process occurs when a highmolecular precursor is physically condensed into liquidphase soot when a high molecular precursor produced by areaction of the gas phase is supersaturated The chemicalcondensation process occurs when soot particles start to beproduced by the consecutive chemical reaction of a highmolecular precursor (Amann and Siegla, 1982)

2.2.2 Surface growth step via the coagulation of thegrowth agents of the gas state

The surface growth of soot particles occurs by the ing heterogeneous processes:

follow-(1) Through the deposit of hydrocarbon in the gas state onthe hot surface of a particle;

(2) Through the chemical reaction on the particle surfacewhere carbon atoms are added; and

(3) Through the desorption of products from the particles.Also, the mass of soot particles increases significantlythrough the surface growth process compared to the mass

at the initial stage of soot formation

Even though the surface growth process of soot particleshas not yet been clarified, we can assume that the surface

of a soot particle is similar to the border of polycyclicFigure 1 Photomicrograph of a soot particulate formed

during diesel combustion

aromatic hydrocarbons (PAH) covered with the chemical

bonds of carbon and hydrogen As a hydrogen atom from

the particle’s surface is isolated, the reaction of the particle

surface is activated, resulting in the production of radicals

that are highly reactive unpaired electrons The general

theory of the surface growth mechanism of soot particles is

that the surface is grown by the reaction of the produced

radicals and the gas state hydrocarbons (Karlsson et al.,

1998)

2.2.3 Collective coagulation step to form a bigger particle

Since coagulation is the process of becoming a bigger

particle through the merger of colliding particles, this

pro-cess takes place immediately after soot formation, or when

the soot particle is relatively small or immature The number

of total particles decreases in this process Figure 2 shows

the detailed structure of a soot particle We can see that

there are several fine particles in a soot particle

2.2.4 Agglomeration step to form a chain type collective

In the latter stage of soot formation, the particles stick

together and begin to agglomerate to make a collective

dump of soot particles Since the reaction region of particle

surfaces is reduced in this step, the chain-type unresolved

particle collective is built up due to the agglomeration of

small spherical particles (about 30~1800 particles) instead

of surface growth due to particle integration

2.3 Soot Oxidation

Oxidation reaction on the surface of a particle is found to

occur during the entire process of soot formation In

contrast to the growth process where atoms are added to

soot particles, the carbon deposited on soot particles is

consumed by the oxidation reaction

3 TEST CONDITIONS AND MEASURING

EQUIPMENT

Our test was performed on a 2700 cc, 5-cylinder, DI diesel

engine The specifications of the engine are shown in Table 1

An opacity measurement instrument made by AVL was

used to measure the pollution level of the exhaust gas in

real time Figure 3 shows the measuring principles of thisequipment which measures the opacity of the air con-taminated by diesel exhaust emissions When the measur-ing chamber consisting of a nonreflecting surface with alength of 0.43 m is homogeneously filled with exhaust gas,the intensity of the light source diminishes This loss oflight intensity is due to light dispersion from collisionsbetween the light and the particulate matter in the exhaustgas Therefore, the loss occurring from the light source tothe light receiver is measured as opacity The opacity of theexhaust gas is calculated using Equation (1) which is based

on the Beer-Lambert law Equation (2) defines the opacity

N [%] that is used to quantify the amount of exhaustedsoot

(1) (2)where

I 0= Intensity of the light source

I= Intensity of the light after traveling the measuring length

a vehicle operates at low speed and low load conditions for

I=I 0 ⋅ e – kL

I

I 0

-=1− N100

-Figure 2 Microstructure of a diesel soot particle

Table 1 Test engine specifications

a long period of time This is the main cause of soot

deposits, and results in much more soot than that produced

by normal combustion Figure 4 depicts the exhaust system

of a 2700 cc, 5-cylinder engine that meets the emission

regulations of Euro III Total soot quantity (f) emitted from

the exhaust can be described as a function of mass flow rate

of the exhaust gas (g) and soot concentration of exhaust gas

(h) Equation (3) represents this physical relationship

(3)The contamination level of exhaust gas (or the soot con-

centration) according to operating conditions is measured

to formulate a mathematical model for the relationship

between the opacity representing the soot concentration

and the mass flow rate of exhaust gas Figures 5 and 6

show the variation of exhaust soot mass flow rate and

opacity according to engine speed, and the correlation of

opacity and soot concentration in terms of the filter soot

number (FSN), respectively Using the experimental results

shown in Figures 5 and 6, the correlation formula between

the operating speed of the engine and the soot mass flow

rate at each operating speed (Equation (4)) is derived and is

depicted in Figure 7

Exhaust Soot Mass Flowrate [g/s]=

4.2 Method to Quantify the Deposited Soot

To quantitatively estimate the amount of soot deposited in

the exhaust system according to the operating conditions,

we perform experiments to evaluate soot accumulation

Figure 8 shows the operating mode to completely blow out

the soot remaining in the exhaust system before starting the

soot accumulation for this study After performing the blow-out mode for enough time to completely remove theremaining soot, the engine is operated for a certain time

f Exh Gas Flowrate,

Soot Concentration

⎛ ⎞ =g rpm ( ) h × Opacity ( )

Figure 4 Exhaust system used for Diesel Euro III engine

Figure 5 Variation of exhaust soot mass flow rate andopacity according to engine speed

Figure 6 Correlation of opacity and soot concentrationversus FSN

Figure 7 Correlation of exhaust soot mass flow rate andengine speed

period to preheat the total system and to stabilize the oil

and the coolant temperature

Table 2 summarizes the operating conditions to simulate

the soot deposit phenomena These are four types of

operating conditions that appear often in the city drive

mode of the federal test procedure (FTP) After the soot is

accumulated during steady-state operation of two hours for

each experimental condition summarized in Table 2, the

opacity of the exhaust gas is measured during the blow-out

mode depicted in Figure 9

In Figure 9, the shaded part of the opacity curve

measured during this experiment can be defined as the

abrupt emission quantity of the deposited soot The opacitythat converges to constant value in the latter half of themeasurement can be defined as the quantity of soot nor-mally produced in the combustion process at a correspond-ing operating condition Therefore, the deposited sootquantity can be calculated using the opacity of exhaust gasmeasured during the blow-out mode (Figure 9) and thecorrelation of the mass flow rate of exhaust soot withengine speed (Equation (4)) Equation (5) describes thecalculation method of deposited soot

The total soot mass accumulated in the exhaust system=the total soot mass (calculated using Equation (3) and theopacity curve shown in Figure 9)−the soot mass produced

by engine combustion itself (calculated using Equation (3)

4.3 Calculation of Soot Quantities Deposited According toOperating Conditions

The morphological structure of soot particles produced inthe combustion process is very similar to that of graphite,but the chemical composition of a soot particle can not bedefined clearly as can that of graphite In addition, there is

a considerable difference in chemical composition betweenimmature soot and mature soot In particular, since theinternal fine structure of a soot particle is sensitive totemperature change, it is known that there is also a consi-derable difference in structure between a soot particle in theexhaust gas and a soot particle in the initial productionstage (Senda et al., 2002)

All soot particles produced at the operating conditionsdescribed in Table 2 are not completely emitted into the air

by passing through the exhaust system from the combustionchamber In the present study, the quantity of soot particlesaccumulated on the inner surface of the exhaust system iscalculated using the method described in Section 4.2

Figure 8 Operating mode for completely blowing out

residual deposited soot

Table 2 Selected operating conditions for investigating soot

deposit phenomena

Speed [rpm] 969.4 1598.1 1923.7 2233.2

Figure 9 Operating mode for quantitatively measuring the

mass of deposited soot after accumulation

Figure 10 Time history of opacity during blowout testmode after depositing soot for 2 hours at each test condi-tion

Figure 10 shows the real time variation of opacity

measured at the exit of the muffler during the blow-out

mode after the soot is deposited for two hours of operation

under the conditions shown in Table 2 With this, we can

evaluate the soot quantity quantitatively Table 3 shows the

calculated results for each curve in Figure 10 using the

method described in Section 4.2 We can see that the

amount of accumulated soot corresponds to the operating

speed and the smoke level that represents the combustion

characteristics of the engine used in this study

4.4 Critical Condition for Soot Particle Accumulation

The critical condition of soot deposit occurs when the soot

particles are emitted into the air without accumulating in

the exhaust system Experiments are performed to

deter-mine the critical operating condition In the soot deposit

phenomena, the most influential parameters are the flow

velocity of the exhaust gas and the back pressure condition

in exhaust system; these parameters are dominated by

change of engine speed Among the test conditions listed in

Table 2, an engine speed of 1598 rpm and torque of 108

N-m are used to accuN-mulate the soot particles After running

for one hour at this deposit mode, the engine speed is raised

to 2000 rpm and 2500 rpm We then tried to find the critical

condition at which the deposited soot particles start to blow

out by measuring the opacity variation at the exit of the

muffler with a gradual change of load conditions

Figure 11 shows the experimental results in our attempt

to find the critical condition of soot deposit From the

experimental results used for the derivation of the

corre-lation formula (Figure 5), the maximum opacity values that

appear naturally at the operating conditions of 2000 rpm

and 2500 rpm are 15 and 12, respectively In Figure 11, we

can define the critical condition as the point that exceeds

these natural maximum opacities Therefore, we can see

that the engine speed and load conditions of 2000 rpm,

58% and 2500 rpm, 50% are the critical conditions for soot

accumulation

Figure 12 shows the back pressure measured at the exit

of the turbocharger when the engine operated at 2000 rpm

and 2500 rpm It can be seen that the back pressure

condition at which soot particles are no longer deposited is

about 0.08 bars at the exit of the turbocharger Figure 13

shows the exhaust gas velocity measured just before the

exit of the muffler according to variation of engine speed It

can be seen that the exhaust gas velocity at the critical

Table 3 Accumulated soot quantity at selected operating

conditions to investigate the soot deposit phenomenon

condition is approximately 62 m/s This means that some

of the soot particles produced during the combustion

process is accumulated in the exhaust system at operating

conditions such that the flow velocity of exhaust gas is less

than 62 m/s Therefore, the solution for removing or

drastically reducing the soot accumulation phenomena in

the exhaust system is to decrease the flow resistance of the

exhaust system, and to increase the flow velocity of

ex-haust gas to be greater than 62 m/s when the back pressure

at the exit of the turbocharger is under 0.08 bars

5 CONCLUSIONS

When a vehicle operates at low speeds and low load

conditions for a long period of time, some of the soot

particles produced in the combustion process is not emitted

to the air, and they are deposited on the inner surface of

exhaust system This causes much more soot than that

produced by normal combustion and they are abruptly

exhausted when the vehicle is suddenly started or suddenly

accelerated In this study, the soot deposit phenomena were

investigated using a 2700cc, 5-cylinder engine having the

specifications in Table 1 and the main muffler depicted in

Figure 4 The results can be summarized as follows:

(1) We developed a method to quantify the amount of the

exhausted soot particles (or the accumulated soot

particles) by measuring the opacity that represents the

contamination level of the exhaust gas due to soot

particles Using this quantification method, the

deposit-ed soot quantities at the operating conditions shown in

Table 2 were calculated and the results were

summari-zed in Table 3 Since the accumulated soot quantity at

each operating condition in Table 3 corresponds to the

smoke level measured at each operating condition, we

succeeded in quantifying the exhaust smoke level

measured in terms of opacity which has a characteristic

that as the amount of particulate matter in the exhaust

increases, the opacity of the exhaust also increases

(2) Experiments were performed to find the critical

condi-tion at which soot particles are emitted into the air

without accumulation in the exhaust system using the

main muffler composed of three chambers as depicted

in Figure 4 From the experimental results, we found

that the critical conditions are the engine speeds and

the load conditions of 2000 rpm, 58% and 2500 rpm,50% At each critical condition, the back pressure atthe exit of the turbocharger was approximately 0.08bars, and the emission velocity of the exhaust gas at theexit of the muffler was about 62 m/s Therefore, inorder to remove or drastically reduce soot accumu-lation, the exhaust system using main muffler applied

in this study must be designed to decrease the flowresistance and to ensure that the flow velocity of theexhaust gas is higher than 62 m/s when the backpressure at the exit of the turbocharger is under 0.08bars

ACKNOWLEDGEMENT− This research was financially ported in part by the Ministry of Knowledge Economy (MKE) and Korea Industrial Technology Foundation (KOTEF) through the Human Resource Training Project for Strategic Technology REFERENCES

sup-Amann, C A and Siegla, D C (1982) Diesel what they are and why Aerosol Science and Technology,

particulates-1, 73−101

Bockhorn, H and Schäfer T (1994) Growth of Soot Particles in Premixed Flames by Surface Reactions.Soot formation in Combustion Springer-Verlag Berlin.Haynes, B S and Wagner, H G (1981) Soot formation

Progress in Energy and Combustion Science, 7, 229−

Lee, D G., Miller, A., Park, K H and Zachariah, M R.(2006) Effects of trace metals on particulate matterformation in a diesel engine: metal contents from ferro-cene and lube oil Int J Automotive Technology 7, 6,

667−673

Senda, J., Choi, D., Iwamuro, M., Fujimoto, H and Asai,

G (2002) Experimental analysis on soot formation cess in DI diesel combustion chamber by use of opticaldiagnostics SAE Paper No 2002-01-0893

EFFECTS OF END COILS ON THE NATURAL FREQUENCY

OF AUTOMOTIVE ENGINE VALVE SPRINGS

H LIU 1) and D KIM 2)*

1)Graduate School of Mechanical and Automotive Engineering, University of Ulsan, Ulsan 680-749, Korea

2)Department of Mechanical and Automotive Engineering, University of Ulsan, Ulsan 680-749, Korea

(Received 8 December 2008; Revised 10 April 2009)

ABSTRACT− In this paper, we present a method for estimating the natural frequencies of various engine valve springs such

as constant pitch, two-step variable pitch, three-step variable pitch, and progressive springs Since a valve spring’s surging amplitude is magnified when the spring’s natural frequency coincides with the frequency of the cam profile harmonic components, estimating the natural frequency of the spring is the first step in predicting valve spring surging phenomena A new method for calculating the valve spring’s natural frequency is proposed in this paper that considers the end coil effect This method predicts not only the natural frequency of a helical spring at a fixed number of active turns, but also the change

in the natural frequency as the spring is compressed The experimental results demonstrate that nonlinear characteristics of engine valve springs can be predicted from the given initial pitch curves.

KEY WORDS : Valve spring, Variable pitch spring, Natural frequency, Spring surging, End coil effect

NOMENCLATURE

d : spring wire diameter [mm]

D : spring mean coil diameter [mm]

R : spring mean coil radius [mm]

k : spring stiffness [N/mm]

k eq : equivalent spring constant [N/mm]

f : fundamental natural frequency [Hz]

N a : number of active coils

E : Young’s modulus [Pa]

G : spring material shear modulus [Pa]

ρ : density of spring material [kg/m3]

υ : Poisson ratio

I : transverse second moment of wire cross section [m4]

J : polar second moment of wire cross section [m4]

x n : number of spring turns

p(x n) : pitch, function of spring turns

g(x n) : accumulated gap, function of spring turns

γ : shear coefficient of wire cross section

1 INTRODUCTION

The valve spring is an important valve train component that

provides restoring forces for the intake and exhaust valves

(Liu et al., 2009) The internal vibration of a valve spring is

a critical factor that determines the dynamic characteristics

of valve trains Traveling waves in a spring may produce a

high level of dynamic stress, resulting in a reduced loading

capacity for the spring, which can cause premature failureand malfunctions in valve motion The fundamental naturalfrequency of the longitudinal vibration mode is the firstindicator in valve spring surging estimation (Kim andNguyen, 2007) However, because the number of activecoils changes according to spring height, the natural frequ-ency of the valve spring cannot be assumed to be constantduring operation; even the natural frequency estimation for

a fixed spring height is not a straightforward task due to theend coil effect at the boundaries

The natural frequencies of helical springs have beenstudied for many years Some early studies derived simpleequations for estimating the fundamental natural frequency

of a compressed helical spring (Wahl, 1949; SAE: SpringDesign Manual, 1990) The most comprehensive equationfor the dynamics of helical springs was derived by Wittrick(1966) A set of six ordinary differential equations and sixpartial differential equations were obtained based on theTimoshenko beam theory Following Wittrick’s work,some researchers introduced a transfer matrix method and afinite element method to predict the natural frequencies ofhelical springs (Pearson, 1982; Yildirim, 1996; Mottershead,1980; Kim, 1999) A pseudospectral method was employed

to investigate the free vibrations of cylindrical and cylindrical helical springs with fixed-fixed, free-free, fixed-free, and hinged-hinged boundary conditions (Lee, 2007a,2007b) Lin and Pisano (1987) also derived the generaldynamic equations of helical compression springs with avariable pitch angle and a variable helical radius.Almost all of the literature presented above assumed that

non-*Corresponding author. e-mail: djkim@ulsan.ac.kr

the boundary conditions of helical springs are fixed-fixed,

free-free or hinged-hinged However, for a practical valve

spring, the end coils are fixed only along the spring axis

direction, which is the main cause of the mismatch between

calculated and experimental results This paper focuses on

two topics: estimating the natural frequency of the valve

spring at a fixed height, and estimating the change in the

natural frequency due to spring compression A new method

for calculating the natural frequency of cylindrical valve

springs is proposed by considering the end coil effect

2 NATURAL FREQUENCY ESTIMATION OF A

COMPRESSED HELICAL SPRING

Figure 1 shows a typical engine valve spring that is

com-prised of active coils, end coils, and dead coils The

con-nection points between the active and end coils are defined

as boundary points

A simple equation for estimating the fundamental natural

frequency of a compressed helical spring is widely used in

the initial stages of valve spring designs By calculating the

stress wave propagation time between the fixed

bound-aries, the fundamental natural frequency is estimated as

follows (SAE: Spring Design Manual, 1990):

(1)where d is the diameter of the wire in mm, N a is the number

of active coils, and D is the mean coil diameter in mm

Since the spring material is assumed to be steel, the shear

modulus and density are set to 7.93×1010Pa and 7850 kg/

m3, respectively The SAE (SAE: Spring Design Manual,

1990) accepts equation (1) as a standard estimation method

for determining the fundamental natural frequency of a

helical spring However, the natural frequency of an engine

valve spring tends to be somewhat lower than the estimation

provided by equation (1) The error between the measured

and calculated frequencies may be as much as 10%, which

is due to the end coil effect Since the error in this type of

estimation is unacceptable, valve spring manufacturing

companies generally use their own empirical equations

Usually, 3/4~1.0 turns of engine valve springs are ground

at both sides in order to satisfy the system assemblycondition Since the ground surfaces are in contact with thecylinder head and retainer, the torsional deflection is tightlyconstrained Therefore, the ground coil can be assumed to

be fixed and is hence called a dead coil In this paper, wedefine the dead-end point as the connection point betweenthe end coils and dead coils

Even if the motion of the boundary point in the direction

of the spring axis is restrained, the torsional deflection ofthe spring wire may penetrate the end coils This phen-omenon seems to be a main source of the error between themeasured and estimated natural frequencies An FEMmodel was used to simulate the effect of a very smallcompression; a 3-D beam element was adopted and coil-to-coil contact was also considered in the model

Figure 2 shows the results of the FEM simulation of thetorsional strain along the wire The results indicate that thetorsional strain penetrates to the dead-end point Therefore,the end coils can be modeled by torsional spring elements

as shown in Figure 3 Finally, the valve spring can bedescribed by the active coils and two torsional springs atthe two extremities

The stiffness of the end coils can be determined byapplying Castigliano’s theorem (Ugural et al.,2003) Thebending and torsional moments for any cross section areshown in Figure 4

(3)The strain energy in the end coils can be obtained as

follows:

(4)The equivalent torsional stiffness of the coil can be

estimated as:

(5)

In this study, we modified the natural frequency

esti-mation method using a helical rod element (Kim, 1999)

The fixed boundary points are replaced by the torsional

stiffness element Finally, the natural frequencies of the

valve spring are determined by solving the characteristic

equation

3 NATURAL FREQUENCY CHANGES BY

COMPRESSION

Even if the natural frequency of a compressed helical

spring can be predicted precisely, it is still difficult to

understand the surging phenomenon of an engine valve

spring Since the valve spring height is different for the

open versus closed states of the valve, its natural frequency

changes significantly during operation This periodic change

in the natural frequency makes it difficult to predict spring

surging phenomena To understand the spring surgingcharacteristics, it is important to predict the natural frequ-ency change over the operating range The natural frequ-ency changes arise mostly from the change in the number

of active coils due to compression In general, the number

of active coils decreases during compression because some

of the active coils collapsed into dead coils at both sides

To control the surging amplitude, a valve spring is posely designed to have variable pitches Figure 6 showstypical pitch curves of various valve springs; namely, aconstant pitch spring, a two-step variable pitch spring, athree-step variable pitch spring, and a progressive pitchspring The pitch is defined as the gap between adjacent

Figure 3 Valve spring model considering the end coil effect

Figure 4 Bending and torsional moments of the end coils

Figure 5 Definition of the valve spring pitch

Figure 6 Various types of valve spring pitch curves

coils, as shown in Figure 5, so the maximum number of

values along the horizontal axis of the pitch curve is always

one greater than the number of active coils

Due to the complexity of the problem, an FEM simulation

was used to measure the torsional deflection of the end

coils The model accounts for how many effective end coils

should be considered in the analysis For this study, a

two-step variable pitch spring was used as an example There

are two different pitches, narrow and wide, in a two-step

variable pitch spring Since the narrow pitch part undergoes

contact earlier than the wide pitch part, our investigation is

focused on the narrow pitch part at the bottom of the

spring In fact, the wide pitch part at the top should have a

similar behavior during the contact process

Figure 7 shows the simulation results for the torsional

strain by compression of the two-step variable pitch spring

The percentages denote the ratio of the compressed

deflection to the initial free height When compression

begins, the torsional strain does not stop at the boundary

points but penetrates into the end coils up to the dead-end

point This result verifies the conclusion drawn above The

end coils from the boundary point to the dead-end point

should be considered in the initial stage of the compression

As the compression continues, the torsional strain increases

and the boundary points move to the active coil due to the

collapse into dead coils Therefore, the number of effective

end coils increases The results also show that the torsional

deflection does not stop at the boundary point but penetrates

up to the original dead-end point, regardless of the changingboundary point This result reveals the relationship betweenthe boundary points and the effective end coils Thus, thechange in the number of active coils can be determined bythe step-by-step compression procedure

To determine the change in the number of active coilsdue to compression, the effect of the end coils at the bottomand top should be considered simultaneously The pitch iscounted from the bottom boundary point to the top of thespring The maximum number of turns in the pitch curve isalways the total number of active turns plus one Figure 8(a)shows the initial pitch curve of a typical two-step variablepitch spring Figure 8(b) is the lower half of the pitchcurve

For the lower half of the pitch curve, the pitch can bedefined as a function of the number of spring turns:

Here, x n is the number of turns from the bottom ary point and N a is the number of active coils

bound-Now, the accumulated gap from the bottom is given by:

(6)Therefore, the accumulated gap can be found usingequation (6), as shown in Figure 9

For a given spring stiffness k, the equivalent stiffnessfrom the bottom to a certain point (x n) of the spring can beestimated as:

(7)

In equation (7), x end is the number of effective end coils

at the bottom Figure 10 shows the equivalent stiffness of

Figure 7 Analysis results of a valve spring under

the portion of the spring from the bottom to a certain

reference point in the spring The equivalent stiffness at the

boundary point without the end coil effect gives an infinite

value due to a zero divisor Therefore, if the end coil effect

is not considered, the boundary point remains unchanged

by the compression However, if the equivalent stiffness

with the end coil effect gives a finite value, the change in

the number of active coils varies with the compression As

the reference point approaches the middle part of the

spring, the equivalent stiffness converges to twice the valve

spring stiffness, because we only considered half of the

spring in the analysis

Now the stiffness from the bottom to a certain point of a

spring is calculated If an infinitesimal load δF is applied,

the deflection of the spring at an arbitrary point can also be

calculated as a function of the number of spring turns

(8)Because of the additional deflection, the accumulated

gap must be updated:

(9)Here, is an updated gap and is the exist-

ing gap before applying the infinitesimal load From the

updated accumulated gap, we can modify the pitch curve:

(10)Here, is the pitch curve updated with the increas-

ed load Since the pitch cannot be negative, the active coilcollapses at the boundary, which eventually reduces thenumber of active coils via compression If the updatedpitch is negative, where , it is set to 0 Usingthe updated pitch curve, the number of active coils is alsoupdated

Finally, the number of active coils and the number ofeffective end coils can be determined for each step Thestiffness of the equivalent torsional spring can be calculatedusing the updated end coils The system stiffness matrix isobtained by combining the equivalent torsional stiffnesswith the original stiffness matrix without the end coil effect.Since the torsional spring element does not account forinertia effects, the system mass matrix is unchanged

4 ILLUSTRATIVE EXAMPLES

The natural frequencies of various valve springs were

test-ed and compartest-ed with the estimattest-ed results using the methoddescribed in this paper Figure 11 shows a schematicdiagram of the experimental setup A valve spring is com-pressed to a known height and excited by the vibrationactuator The natural frequencies are estimated from thefrequency response function (FRF) relating the (input)acceleration and (output) strain on the spring wire Figure

12 shows the typical frequency response function sured from the test The natural frequencies of the valvesprings are also calculated both with and without consi-dering the end coils effect First, the number of active coilsare measured by using a gap gauge and are employed in thecalculation Secondly, the number of active coils are alsocalculated from the initial pitch curves Now we will dis-cuss some examples

mea-Example 1: Natural frequency of a constant pitch valvespring

To investigate the end coil effect, a constant pitch valvespring is selected and the natural frequencies are compared

In this example, the number of active coils is fixed Theproperties of the spring are as follows:

Figure 9 Accumulated gap of the lower half of the spring

Note: w/o means without the end coil effect; w/ means with the

end coil effect.

Figure 10 Equivalent stiffness of the lower half part

Figure 11 Schematic diagram of the experimental setup

N a= 5.9, d= 2.8 mm, R= 10.1 mm, ρ= 7850 kg/m3,

υ= 0.3, γ= 0.8863, x end1=x end2= 0.4

Table 1 shows the calculated and measured natural

fre-quencies The natural frequency, calculated using equation

(1) (SAE: Spring Design Manual, 1990), shows an error of

10.3%, while the finite element method, without

consider-ing the end coil effect, gives an error of 5.58% The

pro-posed method, which considers the end coil effect,

dramati-cally reduces the error The error between the experimental

and estimated natural frequencies is only 0.43% in this

example This result clearly shows that the effect of the end

coils must be specifically taken into account for the natural

frequency analysis

Example 2: Natural frequency of a two-step variable pitch

valve spring

To investigate the effect of spring compression on the

effective length of the end coils, a two-step variable pitch

valve spring is selected The number of active coils of the

two-step variable pitch spring changes significantly with

compression In this example, the changes in the boundary

points are measured and the number of active turns iscounted between the two boundary points The length ofthe effective end coils, x end1and x end2, are also counted fromthe boundary points to the dead-end points The properties

of the two-step valve spring are as follows:

d= 3.3 mm, R= 10.5 mm, ρ= 7850 kg/m3,

υ= 0.3, γ= 0.8863Table 2 shows the tested and calculated results of ex-ample 2 When the end coil effect is not accounted for,large errors are found in the calculated natural frequencies.Furthermore, the error increases to approximately 25% due

to the spring compression This result shows that theeffective length of the end coils increases consistently withthe amount of compression When the end coil effect isconsidered, the errors in the calculated natural frequenciesare bounded to a certain level The natural frequenciesobtained using the proposed method and considering theend coil effect are very accurate estimates Therefore,estimation of the effective length of end coils from theboundary point to the dead end point is reliable even whenthe narrow turns are collapsed

Example 3: Change in the natural frequency of a pitch valve spring due to compression

constant-Even for the case of a constant pitch valve spring, thenumber of active coils changes with the amount of com-pression due to the end coil effect The changes in thenatural frequencies of the spring are estimated from theinitial pitch curve

In this example, the number of active coils is not sured, but it is calculated by the method proposed in thispaper The same spring that was analyzed in example 1 isused for this example as well Figure 13 shows the initialpitch curve; the number of active coils in the free height is

Note: w/o means without end coil effect; w/ means with end coil effect.

Table 2 Results of example 2

compression If the end coil effect is not considered the

number of active coils is theoretically constant and affects

the natural frequency The experimental results clearly

show a change in the natural frequency under spring

compression for the constant pitch spring The estimated

natural frequencies given by the proposed method match

relatively well with the experimental results When the

spring is compressed close to the solid height, the

magnitude of the error increases Since the gap between the

coils is very small in this region, small errors in the initial

pitch curve will cause large errors in the number of active

coils The initial pitch used in the analysis is taken from the

drawing of the spring As a result of variations in mass

produced parts, such as the spring examined here, there

may be some error between the pitch curve in the drawing

and the pitch curve of the actual spring

Example 4: Change in the natural frequency of a two-step

variable pitch valve spring due to compression

Two-step variable pitch springs are most widely used in

automotive engine valve trains At an installation height,

close turns are in slight contact with each other, which

provides additional damping to suppress excessive spring

surging When the close turns are in contact, the number of

active coils changes instantaneously, resulting in a jump in

the natural frequencies during compression

This example analyzes the same spring that was analyzed

in example 2 Both the number of active coils and thenatural frequencies are calculated using the method proposed

in this paper Figure 15 shows the initial pitch curve Figure

16 shows the calculated and experimental results of thenatural frequencies while the spring is under compression

As shown in the figure, the proposed method can be apowerful tool for predicting the natural frequencies of avalve spring; both the calculated and tested results show ajump in the natural frequency at the spring installationheight of 35 mm

5 CONCLUSION

A new natural frequency estimation method for cylindricalvalve springs was developed in this study by consideringthe end coil effect The main results are:

(1) Even if the deflection of the wire is constrained alongthe direction of the spring axis, the torsional strainpenetrates into the end coils Regardless of the number

of close turns, the torsional strain penetrates to thedead-end point

(2) When the end coil effect is considered as an equivalenttorsional stiffness, the error between the estimated and

Figure 13 Initial pitch curve of a constant pitch valve

spring

Figure 14 Change in the natural frequency of a

constant-pitch valve spring due to compression

Figure 15 Initial pitch of a two-step variable pitch valvespring

Figure 16 Change in the natural frequency of a two-stepvariable pitch valve spring due to compression

tested natural frequencies is significantly reduced.

(3) By considering the end coils effect, wire collapse at the

boundary point can be simulated The change in the

natural frequency due to spring compression can also

be predicted from the initial pitch curve Specific

numerical examples demonstrated that the method

proposed in this study agrees well with the test data

ACKNOWLEDGEMENTS−This work was supported by the

2007 Research Fund of the University of Ulsan.

REFERENCES

Kim, D (1999) Development of a finite element program

for dynamic analysis of helical springs IEEE Mechanics,

KORUS'99 309−314

Kim, D and Nguyen, V T (2007) Reduction of high

frequency excitations in a cam profile by using modified

smoothing spline curves. Int J Automotive Technology

8, 1, 59−66

Lee, J (2007a) Free vibration analysis of cylindrical

helical springs by the pseudospectral method J Sound

and Vibration, 302, 185−196

Lee, J (2007b) Free vibration analysis of non-cylindrical

helical springs by the pseudospectral method J Sound

and Vibration, 305, 543−551

Lin, Y Y and Pisano, A P (1987) General dynamic

equations of helical springs with static solution andexperimental verification. ASME J Applied Mechanics,

54, 910−917

Liu, J R., Jin, B., Xie, Y J., Chen, Y and Weng, Z T.(2009) Research on the electro-hydraulic variable valveactuation system based on a three-way proportionalreducting valve Int J Automotive Technology 10, 1,

SAE (1990) Spring Design Manual Prepared under theAuspices of the SAE Spring Committee

Ugural, A C and Fenster, S K (2003) Advanced Strength And Applied Elasticity Pearson Education, Inc London.Wahl, A M (1949) Mechanical Springs Penton Publish-ing Co., Cleveland, OH

Wittrick, W H (1966) On elastic wave propagation inhelical springs. Int J Mechanical Sciences,8, 25−47.Yildirim, V (1996) Investigation of parameters affectingfree vibration frequency of helical springs Int J Numeri- cal Methods in Engineering, 39, 99−114

EMPIRICAL APPROACH FOR PREDICTING THE CETANE NUMBER

OF BIODIESEL

P K BOSE *

Mechanical Engineering, Jadavpur University, Kolkata 700032, India

(Received 5 July 2007; Revised 26 July 2008)

ABSTRACT− The cetane number is an indicator of ignition quality and thus of fuel quality in the realm of diesel engines It

is conceptually similar to the octane number used for gasoline Generally, a compound that has a high octane number tends

to have a low cetane number and vice versa The cetane number of a diesel fuel is related to the ignition delay time In our work the first approach is a statistical one the accuracy of which depends upon the data obtained from various papers and literature sources, as all equations used were based on this data During prediction using more than one equation is a good approach, as it provides the accuracy as well as the relative error The second approach is also a statistical one, but its value depends upon the saponification number and iodine value Therefore the accuracy of this equation may be higher, since we can collect the data for saponification numbers and iodine values from literature, without needing to calculate them Using the saponification number and iodine value we can select an optimal biodiesel as generally a good biodiesel is selected using these three values Thus the second approach allows us the freedom to select a biodiesel

KEY WORDS : Cetane number, Biodiesel, Saponification value, Iodine value

1 INTRODUCTION

A long straight-chain hydrocarbon, hexadecane (C16H34;

trivial name of cetane, giving the cetane scale its name) is

the high quality standard on the cetane scale with an

assign-ed cetane number of 100 A highly branchassign-ed compound

2,2,4,4,6,8,8-heptamethylnonane (HMN, also C16H34), a

compound with poor ignition quality, is the low-quality

standard and with an assigned cetane number of 15 The

two reference compounds on the cetane scale show that the

cetane number decreases with decreasing chain length and

increasing branching Aromatic compounds, which occur

in significant amounts in conventional diesel fuel (DF),

especially DF2, have low cetane numbers but their cetane

number increases with increasing size of n-alkyl side

chains (Knothe and Dunn, 2001) The cetane scale is

arbitrary and compounds with CNN100 or CNb15 have

been identified The standard ASTM D975 for

conven-tional DF requires a minimum cetane number of 40 while

the standards for biodiesel prescribe a minimum of 47

(ASTM D6751) or 51 (EN14214)

Due to the high cetane numbers of many fatty

com-pounds, which can exceed the cetane scale, the term “lipid

combustion quality number” is suggested for these

com-pounds (Katwal and Soni, 2003) For conventional diesel

fuel, higher cetane numbers have been correlated with

reduced nitrogen oxides (NOx) in exhaust emissions This

correlation has led to efforts to improve the cetane number

of biodiesel fuels by means of additives known as cetaneimprovers Despite the inherently relatively high cetanenumbers of fatty compounds, NOx exhaust emissionsusually increase slightly when operating a diesel engine onbiodiesel The connection between the structure of fattyesters and exhaust emissions is investigated by studying theexhaust emissions caused by using enriched fatty acid alkylesters as fuel NOx exhaust emissions reportedly increasewith increasing unsaturation and decreasing chain length,which can also lead to a connection with the carbonnumbers of these compounds Particulate emissions, on theother hand, are hardly influenced by the aforementionedstructural factors The relationship between the carbonnumber and engine emissions is complicated by manyfactors including the technology level of the engine

In this paper we have concentrated in two methods

PREDICTING THE CETANE NUMBER OF ALCOHOLS AND METHYL ESTERS FROM THEIR PHYSICAL PROPERTIES

N-Hexadecane

Hexadecane (cetane), cetane Number=100

CETANE NUMBER CALCULATION FOR BOTH UNSATURATED AND SATURATED METHYL ESTERS USING THEIR IODINE VALUE AND SAPONIFICATION NUMBER

*Corresponding author. e-mail: pkb32@yahoo.com

Hepta methylnonane

2,2,4,4,6,8,8-Heptamethylnonane (Isooctane),

cetane number=15

PREDICTING THE CETANE NUMBER OF

ALCOHOLS AND METHYL ESTERS FROM

THEIR PHYSICAL PROPERTIES

Cetane numbers for the homologous series of straight

chain, saturated n-alcohols, C5-C12 and C14 have been

deter-mined according to ASTM D 613 (ASTM Standard

D1983-90, 1985) Cetane numbers ranged from 18.2 to 80.8

increases linearly with the carbon number Regression

ana-lysis was used to develop equations that relate various

physical properties or molecular characteristics of the alcohols

to the cetane number The following properties correlated

to the cetane number, in decreasing order: boiling Point >

melting point > carbon number > heat of combustion >

density Cetane numbers were also determined for straight

chain homologous of saturated methyl esters with carbon

numbers of 6, 10, 12, 14, 16 and 18 Equations were also

developed to relate physical properties of these esters to the

cetane number The level of correlation of these properties

with the cetane number was as follows, in descending

order: boiling point > viscosity > heat of vaporization > heat

of combustion > carbon number Knowing the values of

relevant physical properties and using these equations,

accurate predictions of the cetane number can be made for

saturated n-alcohols and methyl esters Cetane Numbers of

vegetable oil esters have been reported In the present

work, cetane numbers for straight-chain, saturated methyl

ester with carbon numbers of 6, 10, 12, 14, 16 and 18 were

determined

1.1 Procedures

Heat of combustion values for C5, C7 and C8 alcohols were

taken from Kharasch (1929) Heat of combustion values

for C10, C12 and C14 alcohols were previously reported

Heats of combustion for the C6, C9 and C11 alcohols were

determined by regression analysis For the alcohols, boiling

points, melting points and densities were taken from the

Aldrich catalog (1988, 1989), and Bailey (Swern, 1979)

For the esters, boiling points and melting points were taken

from Aldrich and Lange Data for viscosity, heat of

vaporization, and density were obtained from Bailey

1.2 Statistical Procedure

Matlab 7.1 was used to determine data regression and other

statistical data The output for regression analysis included

calculated Y, error (the difference between measured and

calculated Y), % error, R2 (correlation coefficient squared),the X coefficient(s), the Y intercept, the standard error of Yestimate and degrees of freedom From these statisticaldata, equations were developed that linked cetane number(Y) to the physical constants (X)

These equations were found to be linear, quadratic orcubic As the polynomial degree increased, so did the R2

To decide which polynomial degree was justified, a stical significance test was employed Briefly, a linear re-gression analysis was first performed and the R2 was deter-mined Next, a quadratic regression analysis was perform-

stati-ed which resultstati-ed in a higher R2 A t-value was calculatedwith a formula that used the increase in R2, the degrees offreedom This formula is:

t value = [(increase in R2) × (degree of freedom)/(1−R2)]1/2

(1)This calculated t-value was then compared to a tabular t-value at the 95% confidence level in Student’s t-distribu-tion table If the calculated t-value significantly exceededthe tabular value the process was repeated with the nextpolynomial degree until the calculated value no longerexceeded the tabular value The equation corresponding tothe last polynomial degree having significance was the oneused The procedure objectively allows the selection of theproper polynomial degree rather than guessing at a poly-nomial degree that may be incorrect

1.3 Results and DiscussionTable 1 lists the measured cetane number for the C5-C12 and

C14 alcohols and their physical properties Obviously, a veryclose relationship exists between the carbon number, theelectron number and the molecular weight for a givenalcohol

The electron number was determined as described byKharasch (1929) For example, the electron number of analcohol was defined as the number of valence electronsrelated to the residual carbon and hydrogen atoms aftersubtracting H2O from the empirical formula Thus CH3OHhas an electron number of 6.We have shown that thesethree characteristics for a given saturated compound may

be used interchangeably to produce the same lated y value, % error and R2, etc (Ma and Hanna, 1999;Martini and Shell, 1998) Therefore only the carbonnumber is discussed further in this paper as representative

regre-R2 (0.9931), with melting point a close second A ranking

of the physical properties of the n-alcohols in descending

order according to their ability to predict cetane number is:

boiling point > melting point > carbon number > density

Figure 1 and Figure 2 represent curves of cetane number

Vs boiling point and carbon number respectively

Regression equations for alcohols

The regression equations used to calculate the cetane number

are shown We found curvilinear relationships between

cetane number and boiling point, the melting point and

density, and a linear relationship between cetane number

and carbon number and heat of combustion As an example

the curvilinear relationship between cetane number andboiling point is shown in Figure 1 The close juxtaposition

of data points and the line of calculated values attest to theexcellent correlation (R2 of, 0.9931) between measured andcalculated cetane numbers The linear relationship betweencetane number and carbon number is illustrated in Figure 2.The goodness of fit is nearly equal to that of Figure 1 (R2 of0.9877)

The standard error of Y estimate indicates the amount oferror in the calculated cetane number R2 and standard errordata give assurance that the equations in Table 2 (except theone for density) can be used to predict the cetane numbersatisfactorily from the corresponding physical properties of

Table 1 Cetane number and physical properties of C5-C12 and C14 n-alcohols (Freedman, 1990)

Methyl esters stearateMethyl palmitateMethyl myristateMethyl Methyl laurate decanoateMethyl caproateMethyl

saturated n-alcohols The cetane numbers and physical

pro-perties of the saturated, straight chain methyl esters used

in our study are shown in Table 3 Examination of the ester

data for measured cetane numbers and carbon numbers

suggests a curvilinear relationship between these two

variables, in contrast to the linear relationship between the

same two variables shown in Figure 2 for the alcohols

Indeed, regression analysis and Students t-table blished that a quadratic equation was justified in describingthe relationship between cetane number and carbon numberfor these esters The small increase in cetane number, notedwhen progressing from methyl ester 14 to methyl ester 18

esta-is not significant The cetane numbers of straight chain,saturated methyl esters with carbon numbers of 8, 10, 12,

14, 16 and 18 have also been reported by Klopfenstein(1985) Our values are in good agreement with valuesreported for esters with carbon numbers of 10, 12 and 16.The cetane numbers of methyl ester 14 and methyl ester 18,however, differed from those in the table by 7 and 11,respectively Klopfenstein (1985) noted that, “It appearsthat for the methyl esters there is a nonlinear increase incetane number with increasing chain length of the fattyacid” For his methyl esters he gave the equation:

Y = 24.48 + 8.431X + (−0.1299 X2) (2)

To confirm the nonlinear relationship between the lated cetane number, Y, and the chain length of the fattyacid, X, his data was subjected to regression analysis andStudent’s t-test as discussed above, with the followingresults: i) the coefficient of 24.48 in the equation aboveshould be negative; and ii) only a linear equation is justi-fied, that equation being:

calcu-Figure 3 Cetane number vs boiling point (Esters)

Table 3 Important properties of some heavily used biodiesels (Biodiesel Standard, 1994, 1999, 2002, 2003)

Iodine

number Viscosity at 400CS oC CFPP ( o C) point Pour (o C)

Cloud point ( o C)

LHV (MJ/

Kg)

HHV (MJ/

Kg) Sp.Gr. (EPo C) T90 ( o C) T50 ( o C) T10 ( o C) IBP ( o C) Flash point (o C) numberCetane Fuel8.6 2.6 − 10 to

− 20 −25 to5 −25 to− 15 43.4 44.9 0.85 345 315 260 210 185 60∼72 40~52 D-2133.2 4.08 − 4.4 − 3.8 − 0.5 37 40.4 0.885 346 340 336 328 299 131 50.9 Soy ME

% Err

(Boiling point)

Cetane numbeer (Melting point)

% Err

(Melting point)

Cetane number (Carbon number)

% Err

(Carbon number)

Cetane number (Heat of combustion)

% Err (Heat of combus- tion).

Cetane number (Density) (Density)% Err

Y = (−3.96) + 5.045X (3)

Using data reported by Ryan and Stapper (1987) for

hexadecane, we obtained a curvilinear relationship by

regression analysis of cetane number vs carbon number

For these hydrocarbons the equation was:

Y = (−30.90) + 15.446X + (−0.4562X2) (4)

Where Y is cetane number and X is carbon number

Thus, the change in cetane number with carbon number

can be described by quadratic equations for both esters and

hydrocarbons within the reported range of carbon numbers

As noted earlier, the relationship between cetane number

and carbon number for alcohols was linear over the C5 to

C14 range studied For esters over the C6 to C14 range, the

relationship between cetane number and carbon number

was also linear (R2 of 0.9981)

Only methyl ester 16 and methyl ester 18 caused a

change in the slope of the curve This suggests that had it

been possible to determine the cetane number for C16 and

C18 alcohols, the alcohol curve shape might have been

similar in shape to that of the ester curve By the same

reasoning, had the cetane numbers for the C16 and C18

alcohols been available, the cetane numbers vs boiling

point plot for the alcohols may have leveled off at higher

cetane number, as it did for the esters

1.4 Prediction of Cetane Number from the Physical

> heating value > carbon number > surface tension > ing point > density As with the alcohols the boiling pointpredicted the cetane number most precisely The R2 wasessentially perfect (0.9999), and the average % error, 0.1,was at least an order of magnitude lower than the othervalues shown in the table Regression analysis of the cetanenumber (boiling point data and the Student’s t-test) showedthat a cubic equation was justified for relating these twovariables These properties may also be used to predictcetane number

melt-1.5 Influence of Compound Structure on Cetane NumberFor mono-alkyl esters of the fatty acids comprising bio-diesel (Da Silva Ramos, 1984), long ignition delay timeswith low cetane numbers and subsequent poorer com-bustion have been associated with more highly unsaturatedcomponents such as the esters of linoleic (9Z,12Z-octa-decadienoic; C18:2) and linolenic (9Z,12Z,15Z-octadecatri-enoic; C18:3) acids High cetane numbers were observedfor esters of saturated fatty acids such as palmitic (hexa-

Table 5 Cetane number and physical properties (Methyl esters) (Dunn and Bagby, 1995)

Methyl esters Melting point numberCarbon bustion.kg-calHeat of com- Heat of vaporiza-tion in cal/g Viscosity cst @40oC @760mmB.P Measured cetane

Figure 4 Variations of cetane number with respect to

different physical properties for certain carbon numbers

Figure 5 Percentage of error with respect to calculated fordifferent carbon numbers

decanoic; C16:0) and stearic (octadecanoic; C18:0) acids.

Generally, cetane number increases with chain length

(Harrington, 1986)

As the data in Table 6 shows, the cetane number of fatty

compounds is not negatively affected by branching in the

alcohol moiety (Kalayasiri, 1996) Apparently, the effect of

the long, unbranched fatty acid chains more than

compen-sates any cetane-lowering effect of the branched alcohol

moiety The only cetane number reported in the literaturefor the branched ester of a vegetable oil, iso-propyl soyate,was 52.6, comparing well with methyl soyate with a report-

ed cetane number around 46 or higher, although anotherstudy showed reduced ‘cetane response’ for iso-propyl pal-mitate and iso-propyl stearate That result also coincideswith the present ones on pure esters, showing that branch-ing in the alcohol moiety does not negatively affect the

Table 6 Effect of chemical structure on melting and boiling points of fatty acids and their methyl esters

Acid chain No of carbons Structure boiling point, Methyl ester o

C

Methyl ester melting point, o

C

Acid ing point, o

boil-C

Acid ing point, o

Table 7 Regression equations for cetane numbers of saturated methyl esters w.r.t their physical properties

Y estimationBoiling point Y=(−41.30)+0.2785×X+0.001209×X2+(−3×10−6×X3) 0.9999 0.1

Table 8 Calculated and measured values and %error derived from their regression equations

Fames Measured calculatedBoiling err Calculated err Calculated err Calculated err Calculated errCalculated err Calculated err

point Viscosity Heating value combustionHeat of Carbon number Melting point Density

cetane number Thus, a general statement that can be

derived from the present results is that one saturated, long

straight chain in a fatty ester suffices to impart a high

cetane number, regardless of branching in the other chain

This statement, derived here from esters of straight-chain

fatty acids with branched alcohols, would need to be

confirmed for esters of branched acids with straight-chain

alcohols

1.6 Regression Equations for the Esters

Regression equations and associated statistical data that

relate the cetane numbers of the esters to select physical

properties are shown in Table 7 Cetane number is related

to boiling point and viscosity by cubic equations to heating

Value, heat of combustion, carbon number, and surface

tension by quadratic equations; and to melting point, and

density by linear equations The cubic equations have the

highest R2 and the lowest standard errors Thus the boiling

point equation for viscosity with its high R2 (0.9985) and

relatively low standard error (1.4) is a good second choice

Because the ASTM permits a reproducibility range of

2.5~3.3, equations for heating value, heat of combustion,

carbon number and surface tension could also be employed

2 CETANE NUMBER CALCULATION FOR

BOTH UNSATURATED AND SATURATED

METHYL ESTERS THROUGH THEIR IODINE

VALUES AND SAPONIFICATION NUMBERS

Fatty acid profiles of seed oils of plant species having 30%

or more fixed oil in their see has been examined

Saponifi-cation number, iodine value and cetane number of fatty

acid methyl esters of oil are empirically determined, which

has been previously proven by Krisnangkura (1986) Iodine

value and cetane number are used to predict the quality of

fatty acid methyl esters of oil for use as biodiesel The Fatty

acid methyl esters of some plants meet the specification of

biodiesel standard of USA only These selected species

have great potential for biodiesel

The raw materials being exploited commercially by the

biodiesel countries constitute the edible fatty oils derivedfrom rapeseed, soybean, palm sunflower, coconut, linseedetc (Tyagi and Kakkar, 1991) The use of such edible oil toproduce biodiesel in India is not feasible due to a big gapbetween demand and supply of such oils in the country.There is a long list of trees, shrubs and herbs which areplentiful in India and can be exploited for use as diesel fuel(Mohibbe et al., 2005) The saponification number, iodinevalue and cetane number of fatty acids of methyl esters ofthese oils were calculated empirically and were used toestablish their suitability for use as biodiesel

2.1 Materials and MethodsSeed oil contents and fatty acid compositions of oils werecollected from the literature Saponification number (SN)and iodine value (IV) of oils were obtained from literature

or calculated from reported fatty acid of methyl ester positions of oils with the help of Equation (5) and Equation(6) respectively

where, Ai is the percentage, D is the number of doublebonds and MWi is the molecular mass of each component.Cetane numbers of fatty acids of methyl esters werecalculated from Equation (7)

2.2 Iodine ValueIodine number is a measure of the degree of unsaturation ofthe fuel (DIN 53241, IP 84/81) Unsaturation can lead todeposit formation and storage stability problems with bio-diesel Soy and rape methyl esters have iodine numbers ofapproximately 133 and 97 respectively Data for tallowesters were not found, but a lower iodine number is expect-

ed based on the greater degree of saturation Research at

∑ 254X D XA i

Mw i

-5458 SN

-Figure 6 Variation of cetane number with different

physi-cal properties of esters at different carbon numbers

Figure 7 Percentage of error with respect to our formula atdifferent carbon numbers

Mercedes-Benz (Shafer, 1994) suggests that biodiesel with

iodine number greater than 115 is not acceptable because

of excessive carbon deposits

2.3 Results and Discussion

The saponification number and iodine value were

calcu-lated using the above formula Saponification number

depends upon the molecular weight and the percentage

concentration of fatty acid components present in fatty

acids of methyl esters of oils However iodine value,

according to equation 2 depends upon three

variables-percentage concentrations of unsaturated fatty acid

compo-nents, their molecular weight and the number of double

bonds present in them The calculated saponification

number and iodine value are in good agreement with the

experimentally determined respective values

Equation (3) predicts the cetane numbers of fatty acids

of methyl esters of seed oils with reasonable accuracy For

example, equation 3 predicts the cetane numbers of fatty

acids of methyl esters of seed oils with reasonable accuracy

and the actual cetane number of fatty acids of methyl esters

of bambasu, palm peanut, soybean and sunflower oils are

bin good agreement :the values are 64.40, 63; 63.3, 62;

56.4, 54; 42.5, 45; and 50.6 and 49, respectively From the

example, the expected correlation of predicted cetane

numbers with their actual carbon numbers will be

some-what less than ±2.5 Equation (3) may also be used to

predict the cetane numbers of fatty acids of methyl esters of

unusual oils if the iodine value of such oils calculated with

the help of equation (2) matches well with the

experi-mentally determined iodine value of the oils However, this

is yet to be confirmed by actual experimentation The

calculated saponification number and iodine value ranged

from 169.2 to 312.5 and 4.8 to 212, respectively Cetane

number values among the species varied from 20.56 to

67.47 (Knothe, 2002)

The above formulas not only determined the cetane

numbers of fatty acids of methyl esters of oil, but they can

also be used to predict the cetane numbers of individual

fatty acids of methyl esters irrespective of their saturation

During combustion different intermediate species are

form-ed, which have a great influence on the cetane number of

the fuel Thus we can say that a fuel cetane number is

determined by its constituents or the type of species formed

during combustion Therefore, if we know the individual

cetane number of different fatty acids of methyl esters of a

fuel, we can predict the cetane number of that fuel by the

formula given below

It has been proven in constant volume combustion

apparatus (CVCA) that the cetane number of intermediate

species determines the cetane number of the original fuel

For a two component system:

Thus by knowing the individual cetane number we can

calculate the cetane number of the main fuel This dual cetane number can be achieved through either previ-ously recorded data or the derived equations from thephysical properties as shown in the table Since the formulahas the limitation to only saturated methyl esters, we canuse other methods to determine the cetane numbers ofunsaturated fatty acids of methyl esters

indivi-2.4 Determining the Cetane Numbers of UnsaturatedMethyl Esters

Plant oils consist of a number of fatty acids of methyl esters.Based on this work the same group of equations can beused to predict the cetane number of such oils or biodiesel

We can predict their cetane numbers by establishing theiriodine values and their saponification numbers and usingthese values in Equation (3) The only necessary know-ledge is the percentage of each fatty acids of methyl esters

in the oil This now becomes a very easy procedure

To evaluate the cetane number of an unsaturated FAME,

a slightly different approach is needed To consider a diesel consisting of only unsaturated fatty acids of methylesters, as before first its saponification number and iodinevalue are obtained Subsequently in case of Ai the value of

bio-100 should be used, as it is the only FAME in the oil Thus

to find out the cetane number of unsaturated fatty acids ofmethyl esters the equations are

as well as relative error The second approach is also astatistical one, but its result depends upon saponificationnumbers and iodine values Hence the accuracy of theseequations is expected to be higher, because saponificationnumber and iodine value data may be gathered from liter-ature Using the saponification number and iodine value agood biodiesel choice can be made, as generally a good

a

( ) High CN Component− b ( ) low CN Component

100 -

∑ 254X D XA i

Mw i

-5458 SN

-biodiesel is selected via these three values Therefore the

second approach provides the freedom to select a biodiesel

ACKNOWLEDGEMENT− This paper is based upon work

supported by the JADAVPUR UNIVERSITY, Kolkata, India We

extends our great thanks and acknowledgement to all, who has

been a active part of this project

ASTM Standard D1983-90 (1995) Standard Test Method

for Fatty Acid Composition by Gas-Liquid

Chromato-graphy of Methyl Esters ASTM West Conshohocken

PA

Biodiesel Standard (1994) DIN V51606, Germany

Biodiesel Standard (1999) ASTMPS121, USA

Biodiesel Standard (2002) ASTMD 6751, USA

Biodiesel Standard (2003) EN 14214, European Standard

Organization

Da Silva Ramos, L C., Tango, J S., Savi, A and Leal, N

R (1984) Variability for oil and fatty acid composition

in castorbean varieties JAOCS 61, 12, 1841–1843

Dunn, R O and Bagby, M O (1995) Low-Temperature

properties of triglyceride-based diesel fuels:

Transesteri-fied methyl esters and petroleum middle distillate/ester

blends J Am Oil Chem Soc.,72, 895–904

Freedman, B., Bagby, M O., Callahan, T J and Ryan III,

T W (1990) Cetane numbers of fatty esters, fatty alcohols

and triglycerides determined in a constant volume

com-bustion bomb SAE Paper No 900343

Harrington, K J (1986) Chemical and physical properties

of vegetable oil esters and their effect in diesel fuel

per-formance Biomass, 9, 1–17

Kalayasiri, P., Jayashke, N and Krisnangkura, K (1996)

Survey of seed oils for use as diesel fuels J American

Oil Chemical Society, 73, 471–474

Katwal, R P S and Soni, P L (2003) Biofuels: An

opportunity for socioeconomic development and cleanerenvironment Indian Forester ISSN 0019-4816 Source/ Source 129, 8, 939–949

Kharasch, M S (1929) J Res Natl Bur Stand (U.S.)2:359

Klopfenstein, W E (1985) Effect of molecular weights offatty acid esters on cetane numbers as diesel fuels J American Oil Chem Soc 65, 6, 1029–1031

Knothe, G and Dunn, R O (2001) Biofuels derived fromvegetable oils and fats Gunstone, F D., Hamilton, R J.editors Oleochemical Manufacture and Applications,

UK, Sheffield Academic Press, 106–163

Knothe, G., Dunn, R O., Shockley, M W and Bagby, M

O (2000) Synthesis and characterization of some chain diesters with branched or bulky moieties J American Oil Chem Soc 77, 8, 865–871

long-Krisnangkura, K (1986) A simple method for estimation

of cetane index of vegetable oil ethyl esters J American Oil Chem Soc., 63, 552–553

Ma, F and Hanna, M A (1999) Biodiesel production: Areview Bioresour Technology, 70, 1–15

Martini, N and Shell, J S editors (1998) Plant Oils as Fuels-Present State of Science and Future Development.Springer Berlin 276

Mohibbe, M., Azam, A W and Nahar, N M (2005).Prospects and potential of fatty acid methyl esters ofsome non-traditional seed oils for use as biodiesel inIndia Biomass and Bioenergy 29, 293−302

Ryan III, T W and Stapper, B (1987) Diesel fuel ignitionquality as determined in a constant volume combustionbomb SAE Paper No. 870586

Shafer, A (1994) Biodiesel Research Mercedes Engine Warrenty Policy, Presented at Commercialization

Benz-of Biodiesel: Established Benz-of Engine Warranties sity of Idaho National Center for Advanced Transporta-tion Technology 125

Univer-Swern, D (1979) Bailey's Industrial Oil and Fat Products,

1,4th edn., John Wiley & Sons New York 159−177.Tyagi, P D and Kakkar, K K (1991) Non-Conventional Vegetable Oils. Batra Book Service New Delhi India

VEHICLE STEERING RETURNABILITY WITH MAXIMUM STEERING

WHEEL ANGLE AT LOW SPEEDS

Y G CHO *

Hyundai Motor Company, 772-1 Jangduk-dong, Hwaseong-si, Gyeonggi 445-706, Korea

(Received 30 June 2008; Revised 21 October 2008)

ABSTRACT − In this paper, an analytical model with suitable vehicle parameters, together with a multi-body model is proposed to predict steering returnability in low-speed cornering with what is expected to be adequate precision as the steering wheel moves from lock to lock This model shows how the steering response can be interpreted in terms of vertical force, lateral force with aligning moment, and longitudinal force The simulation results show that vertical steering rack forces increase in the restoring direction according to steering rack displacement for both the inner and outer wheels As lateral forces due to side-slip angle are directed toward the medial plane of the vehicle in both wheels, the outer wheel pushes the steering wheel in the returning direction while the inner wheel does not In order to improve steering returnability, it is possible to increase the total steering rack force in both road wheels through adjustments to the kingpin axis and steering angle This approach is useful for setting up a proper suspension geometry during conceptual chassis design.

KEY WORDS : Steering returnability, Kingpin axis, Steering effort, Wheel alignment, Suspension geometry

NOMENCLATURE

τ : caster angle

σ : kingpin inclination angle

λ : spatial kingpin angle with relation to vertical direction

Well-designed vehicle chassis systems require good

steer-ing performance with regard to understeer characteristics,

response, feedback, on-center feel, steering torque

build-up, and steering returnability In particular, steering

return-ability involves two maneuvers: maximum steering wheel

returnability (which happens during low-speed cornering),

and on-center area returnability (which occurs during

high-speed driving with a small steering wheel angle) In speed cornering, a driver controls the steering wheel withprecision (to take the car out of the parking lot, forexample), and then releases the steering wheel to make thevehicle go straight Upon release, the steering wheel is ex-pected to return automatically to a straight-ahead position.Therefore, it is important to design a steering returnabilitycharacteristic adequate to move the road wheels againstvarious forms of resistance at low vehicle speeds and toprovide adequate steering wheel torque during handlingmaneuvers in order to give the driver appropriate feedback.Steering returnability at low speeds is heavily influenced

low-by the restoring moment that originates from the sion and steering geometry, in contrast to the restoringmoment from lateral acceleration at high speeds Gough(1953) shows that steering geometry and frictional forceinfluence the maximum steering wheel torque when avehicle is stationary Pitts and Wildig (1978) discuss theinfluence of steering geometry between parallel steeringand full Ackerman steering with an experimental valida-tion Sharp and Granger (2003) devised a static tire test-rig

suspen-to measure tire properties, and developed a stationarysuspension model They show that kingpin offset at theground has a very small influence on steering wheel torquebecause of the small vehicle lifting effect However, they

do not discuss the turning effect or suspension geometrysuch as the kingpin and caster angles Kim et al (2007)propose a more accurate multi-body dynamic approach toestimate steering wheel torque with experimental validation,and the work includes some analysis of suspension geo-metry effects Pfeffer and Harrer (2008) present a method

*Corresponding author. e-mail: multivehicle@yahoo.co.kr

that allows the steady-state steering wheel torque

charac-teristics to be laid out analytically with respect to the

steer-ing wheel angle near the on-center area without

conside-ration of caster trail migconside-ration for inner or outer road

wheels Schmitt (2003) introduces a steering torque

simu-lation model that includes suspension geometry changes

according to road wheel steering angle with a focus on

vertical and braking forces during parking maneuvers Cho

and Lee (2004) presents a kingpin axis moment analysis

and reveals the directional contribution of vertical, lateral,

and longitudinal tire forces The current increase in the use

of electrical assistance like MDPS makes this subject

topical, as evidenced by the increased number of studies in

recent literature (Park et al., 2007) Kurishige et al (2000)

propose an electric power steering control strategy to

improve steering wheel returnability Most papers focus on

steering effort rather than returnability, and there have been

few studies on suspension geometry for automatic

return-ability

This paper suggests an analytical method to acquire

returning steering wheel torque through an alteration of the

kingpin axis in the suspension geometry Section 2

de-scribes the characteristics of tire force and moment at low

speeds Section 3 introduces the kingpin axis moment

mechanism including the kingpin axis migration on

steer-ing Section 4 describes the analytical model simulating the

restoring moment in the vertical, lateral, and longitudinal

directions The model is validated by a full vehicle

simu-lation with geometric sensitivity results presented in Section 5

2 TIRE FORCES AND MOMENTS AT LOW

SPEEDS

Since tire force and moment have a considerable effect on a

vehicle’s cornering performance, they are important factors

for understanding steering returnability The tire lateral

force (Fy), aligning moment (Mz), and overturning moment

(Mx) are generated by tire slip angle, camber angle, axle

weight (Fz), and their coupling effects Driving force (Fx),which depends on the slip ratio and axle weight, is alsogenerated to maintain constant speed on a front-wheel-drive vehicle

To understand the tire characteristics at low speeds, wemeasured tire force and moment using an MTS® flat trackmachine at the 1.0 deg slip angle region and at variousspeeds We found that cornering stiffness at 5 km/h is 6%lower than that at high speeds, and aligning moment stiff-ness is 8% lower than that at high speeds Since tire proper-ties change considerably at low speeds, tire properties are

of consequence in steering returnability Table 1 lists themeasured tire parameters; cornering stiffness in the frontand rear tires was 1,354 N/deg and 1,232 N/deg respec-tively, and the pneumatic trail was 22.1 mm

3 KINGPIN AXIS MOMENT WITH REGARD

TO STEERING RETURNABILITY

3.1 Trajectory of the Kingpin Axis

A tire connected to the suspension system through a hubbearing is steered freely around a rotating axis called thekingpin axis In a McPherson strut system (Figure 2), thekingpin axis is defined as a vector between the strut topmounting point and the lower ball joint point at a lowercontrol arm On a local tire coordinate system, kingpin axistrajectories migrate on a contact patch plane as well as awheel center plane according to the tire turning angle, even

Figure 1 Tire cornering stiffness and aligning moment

stiffness according to speed

Table 1 Tire parameters measured at flat track machine.Cornering stiffness (C f, C r) 1354, 1232 N/degCamber thrust coefficient (K f) 141 N/degOverturning coefficient (K o) 40 Nm/deg

Figure 2 McPherson strut system (front suspension)

though the kingpin axis is fixed in a global coordinate

system The trajectories on a contact patch plane (where

lateral and vertical forces are applied) are drawn as caster

trail and kingpin offset at the ground The trajectories on a

wheel center plane (where driving force is applied) are

represented as wheel center trail and kingpin offset at the

wheel center These trajectories are dependent on the

suspension hard point and contribute to the kingpin axis

moment by the moment lever arm corresponding to each

directional force

The simulated kingpin axis trajectories (Figure 3) show

that as the kingpin axis moves forward, the caster trail of

the inner road wheel increases up to 40 mm, while the

caster trail of the outer road wheel decreases up to 40 mm

with the rear movement of the kingpin axis The caster trail

difference between the inner and outer road wheel increases

according to the tire turning angle On the other hand, the

kingpin axis on the wheel center plane migrates laterally

slightly while fore-aft migration is little

3.2 Moment Around Steering Axis

Tire forces and moments at the inner and outer road wheels

generate a kingpin axis moment, where the moment lever

arm is the vector from the kingpin axis to a point of force

action The kingpin moment transforms into the steering

rack force (Frack), and finally transfers to the driver in the

form of steering wheel torque

If there is no input for the steering wheel angle, the

steer-ing rack forces of the inner and outer road wheel

counter-balance each other due to symmetry between the left and

right sides As the steering wheel rotates, however,

suspen-sion asymmetry creates a difference in the kingpin moment

between the inner and outer road wheels, and the kingpin

moment causes the steering rack force to rotate the steering

wheel When this steering rack force opposes a driver’s

steering input and is capable of overcoming steering

fric-tion (steering gear box fricfric-tion, strut bearing fricfric-tion, and

tire friction), the steering wheel automatically restores to

the straight-ahead position

4 ANALYTIC RETURNABILITY MODEL FOR STEERING RACK FORCE

In this section, we propose an analytic model that considersreturnability with an analysis of steering rack force for eachdirectional tire force (vertical, lateral, and longitudinal).4.1 Vertical Force

A local coordinate system that follows the vehicle nate system is defined on the tire contact patch plane withorigin A (contact patch point) in Figure 4, where D is thepoint where the kingpin axis crosses the contact patchplane, r c is the kingpin offset at the ground, n c is the castertrail, and δ is the tire turning angle Moment (M V) byvertical force is defined as the cross-product of the dis-placement vector (r A) and the vertical axle force (F V)

(1), where

As the weight of a vehicle body transfers due to lateralacceleration, the vertical axle load changes This meansthat the inner axle load decreases and the outer axle loadincreases, even though lateral acceleration is less than 1.0m/s2 for low-speed cornering Lateral acceleration of avehicle can be derived from a vehicle model with twodegrees of freedom (Hwang, 2008), where the lateralacceleration in an equilibrium state is the product of yawrate and lateral velocity, as in Equation (3) We selected atarget speed of 5 km/h in order to mimic the creep speed ofour target vehicle This speed depends on engine and clutch

Figure 3 Locus of kingpin axis (right road wheel side)

Figure 4 Vehicle coordinate system with regard to thekingpin axis

characteristics of the target vehicle, and the lateral

accele-ration at maximum steering wheel angle was calculated to

be 0.46 m/s2

Road wheel lifting effects can be ignored with little

kingpin offset, since the contact patch and wheel center

height do not change on steering Therefore, since the

stroke of the spring and shock absorber are very small in

this system, the variation of spring and shock absorber

forces do not need to be considered Equation (2) describes

the vertical forces at each road wheel, where δ f represents

the average value of the inner and outer steer angles, and m f

g is the weight supported by the suspension spring at each

front road wheel

for the inner road wheel (2) for the outer road wheel

, where

(3)

The moment (M V) is transferred to the kingpin moment

(M KV) by the projection to the kingpin axis inclination

composed of the caster angle (τ) and the kingpin angle (σ)

Equation (5) gives the real angle of the kingpin axis (λ),

which is the angle between the z-axis and the kingpin axis

Therefore, the restoring moment (M KV) due to vertical force

is described as the product of the moment (M V) and the

kingpin axis unit vector (e K), as in Equation (4)

(4)

, where

(5)

(6)

Finally, Equation (7) calculates the steering rack force

(Cho and Lee, 2004), since M KV is the cross product of the

steering rack force (F s) and the effective arm vector (r eff), as

shown in Figure 2

(7)The simulation results (Figure 5) show that at zero

steering rack displacement, the total steering rack forces of

the inner and outer road wheels cancel one another throughleft and right wheel symmetry However, as the steeringrack moves laterally, load lever arms change according tocaster angle and kingpin inclination Steering rack forces atboth the inner and outer wheel increase in the restoringdirection As a result, we found that the vertical force atsteering is beneficial to the restoring moment Additionally,

we considered the overturning moment obtained from theoverturning coefficient (K o), vertical force, and slip angle,but we found that it makes only a minor contribution due tolow slip during low-speed turning

4.2 Lateral ForceThe lateral tire force at the maximum steering wheel angle

is a result of the geometric slip angle, the centripetal forcefrom lateral acceleration, and the camber thrust fromcamber alteration on turning First, the geometric slip angle

is defined as follows: the normal vectors on the two frontroad wheels must intersect the extension of the rear axlecenter line (neglecting the small slip angle at the rear roadwheels) The vehicle turning center (O) then becomes thegeometric center of the front and rear axles for low-speedcornering (Matschinsky, 2000) In chassis system design,the inner tire turning angle (δ i) is usually set to be largerthan the outer tire turning angle (δ o) to avoid tire scrub oncornering,which is called the Ackermann characteristic

= F V ( – n c sin δ + r c cos δ )tan τ cos λ

+ F V ( n c cos δ r + c sin δ )tan σ cos λ

tan λ = lh - = w -2h+t2 = tan 2 σ + tan 2 τ

r eff × F rack = M KV

Table 2 Vehicle parameters

m f (half mass of front wheels) 530 kg

H (height of gravity center) 0.5 m

Second, vehicle lateral acceleration produces a centripetal

force that acts on both road wheels For instance, a left

directional force obtained as the mass multiplied by the

lateral acceleration occurs during left cornering Third,

camber thrust force (Equation 9) is included in this model

due to high alteration in the lock-to-lock steering condition

in a McPherson strut system Camber angle with respect to

the road varies over the range 0o~−1o at the outer road

wheel and +5o~ +7o at the inner wheel When there is

negative camber at the outer front wheel and positive

camber at the inner front wheel, the lateral forces of both

wheels are in the direction of the turning center

(outer wheel)

(9)Equations (10) and (11) describe the moments around

the tire contact point for geometric slip and camber angle,

respectively The tire side slip force (F S) acts on point B in

Figure 4 with the lever arm (n c+n r), while the camber

force (F C) acts on point A with the caster trail lever arm

(n c) The kingpin axis moments in Equations (12) and (13)

are obtained from the product of the moments (M S and M c)

and the kingpin axis vector (e k) The moment caused by tire

side slip force is of consequence in steering returnability,

because the caster trail (n c) varies considerably according

to the tire turning angle It means that the outer wheel acts

on a smaller caster trail during turning, while the inner

wheel acts on a larger caster trail (Figure 3) In this model,

the pneumatic trail (n r) is assumed to be the same for both

wheels in the low slip angle region

(10) (11) (12) (13)The simulation results (Figure 8) show that forces due to

geometric slip angle are directed towards the medial plane

of the vehicle for both wheels, while tire forces due tolateral acceleration are both in the same direction Theforce F s, which is the sum of forces due to geometric slipangle and lateral acceleration, shows that the outer roadwheel pushes the steering wheel in the returning direction,while the inner wheel pushes in the non-returning direc-tion The inflection of F s is found at the inner wheeloccuring at around 70 mm of steering rack stroke, becausethe term C f(δ i − θ i) is linearly decreasing At the same time,

m f a y is nonlinearly increasing due to the nonlinear steeringangle, as in Figure 7 The camber thrust force caused by alarge positive camber at the inner wheel rotates the steeringwheel in the restoring direction, but the camber thrust force

at the outer wheel is small due to the small camber ation during turning

alter-4.3 Longitudinal Force

In order to maintain constant speed, driving force should beapplied at the wheel center position to overcome rollingresistance as well as the braking portion of the lateral force,since the longitudinal component of the lateral force ap-plies like a braking force Equations (14) and (15) describethe kingpin moment caused by the longitudinal force,where r C is the wheel center vector and F X is the drivingforce at the wheel center

(14) (15)

On a wheel suspension with a fixed kingpin axis, thewheel-center offset r C remains nearly constant irrespective

of tire turning angle, so that the kingpin moments of thetraction force and the rolling-resistance force depend not

on moment lever arm but on the longitudinal force of eachroad wheel

4.4 Combined ForceThe total kingpin axis moment obtained by the summation

Figure 6 Turning center of vehicle at low speeds

Figure 7 Tire turning angle according to steering rackdisplacement

of each directional kingpin moment transforms to a

steering rack force with an effective steering arm vector as

shown in Equation (17) The steering rack force enables the

prediction of steering returnability according to its

direc-tion If the direction of the steering rack force opposes that

of a driver’s steering input, then the steering wheel

auto-matically restores to the straight-ahead position If not, then

there is no restoration

The contribution of the component force (Table 3) atmaximum steering wheel angle was analyzed, and wefound that steering returnability benefits from the innerwheel for vertical force, the outer wheel for side slip angle,the inner wheel for camber angle, and the inner wheel fordriving force This system, with a total steering rack force

of +1,318 N in the lock-to-lock condition, can return matically to the center position

auto-(16) (17)

5 VALIDATION

5.1 Multi-body Vehicle Model

To correlate with the analytic model, we developed a plete, detailed, multi-body vehicle model in the multi-bodysoftware environment, MSC/ADAMS The front McPhersonstrut suspension is composed of a single lower control arm,

com-a knuckle, com-a tie rod, com-and com-a strut com-assembly joined by com-a coilspring and shock absorber They are connected to eachother by a suspension joint or bushing The steering rackand pinion systems were modeled with an appropriate gear

M TK = ΣM K = M KV +M KS +M KC +M KX +M KO

r eff × F rack = M TK

Figure 8 Steering rack force by lateral force (analytic

model)

Table 3 Contribution to steering rack force at maximum

steering wheel angle

Direction InnerSteering rack force (N)Outer Total

Lateral force (camber) 1,368 −28 1,340

ratio, a link from pinion to rack, and two universal joints at

the steering column The steering gear box located behind

the wheel center was mounted on a subframe, and

hydr-aulic power assisted the steering effort The rear

suspen-sion, which is mounted on a cross member, was a

multi-link type with three rigid multi-links and one flexible multi-link We

developed suspension and steering assemblies with exact

hard points representing the kingpin axis, which defined

the orientation of the steering axis From the side view, the

steering axis passes through the contact patch, thus

defin-ing the caster trail through the road wheel radius From the

rear view, the tire-to-road contact center is offset from the

kingpin steering axis The tire and wheel are connected by

a suspension system through flexible hub bearings The tire

model was devised using a ‘magic formula’ format with

measurement data acquired from the flat track machine at

low speeds We also modeled the engine map, brake

pre-ssure, and transmission gear assembly for a virtual driver to

keep the vehicle speed constant

The multi-body model was validated on a SPMD

(Sus-pension Parameter Measurement Device) testing machine

(Lee et al., 2004), since the returning response at low

speeds depends considerably on suspension and steering

kinematics Testing was conducted with the steering wheel

rotated by a steering robot at lock-to-lock, and the tire

con-tact patch freely steered on a roller plate Wheel alignment

variation was measured by an optical sensor The measured

tire turning angle and camber angle were shown to be well

correlated with the full vehicle model, as in Figure 9

5.2 Response of the Steering System

The input into the system was the steering wheel angle

using a step function The steering wheel angle changed

slowly from straight ahead to maximum steering left lock

(approximately 500 degrees) and was then released after 25

seconds for returnability evaluation Until the release of the

steering wheel, the steering rack force generated from tire

force was supported by hydraulic assisting power as well as

the driver’s steering torque Upon the release of the

steer-ing wheel, the driver’s steersteer-ing effort and the hydraulic

power force disappeared simultaneously, since the power

assisting force is proportional to steering wheel torque,

even though the power-assisted force augments the driver’s

steering effort during cornering Consequently, only the tire

force pushes the steering rack upon release of the steering

wheel, and the movement of the steering rack determines

steering returnability performance

Figure 10 shows that the steering rack force increases in

proportion to the steering wheel angle in the restoring

direction, so that the steering wheel angle can return to the

center position automatically over the whole steering range

If the driver speeds up, the steering wheel will return faster

due to lateral acceleration The total steering rack forces

(Figure 11) of the analytical and full vehicle models are

well-correlated according to steering rack displacement It

is noted that the inflection occurs around 60~70 mm of the

steering rack stroke owing to the inflection of lateral forceshown in Figure 8

5.3 Parameter Sensitivity StudyThe total steering rack forces (∆F rack) were studied using a

Figure 10 Steering response by full vehicle model

Figure 11 Steering rack force comparison between analyticand full vehicle model

parameter analysis of the variation in the kingpin axis

vector (e k) The simulation results shown in Figure 12

(derived from the caster and kingpin angle variation)

indi-cate that steering rack force is dependent on vertical force

In the case that caster angle increases with the fixed caster

trail, the outer road wheel favors the restoring moment, but

the inner road wheel does not Additionally, the

contribu-tion of the inner wheel is greater than that of the outer

wheel; therefore, the increase in caster angle causes the

total steering rack force to decrease When the kingpinangle increases with the fixed kingpin offset, both roadwheels favor the restoring moment Therefore, increasingthe kingpin angle causes the total steering rack force toincrease These sensitivities are dependent on the suspen-sion geometry as well as the suspension link type

(18)

In order to validate the model sensitivity, suspension

∆M K = Σ i ( r i × F i ) ∆e ⋅ k = r eff × ∆F rack

Figure 12 Kingpin axis sensitivity according to wheel alignment

Figure 13 Steering returnability by suspension geometry alteration

geometry in the full vehicle model was altered and

simu-lated, as shown in Figure 13 The abbreviation MOD1

indicates a suspension having high caster and low kingpin

angle, obtained by a −3 mm movement of the lower ball

joint both longitudinally and laterally The MOD2 indicates

a suspension with low caster angle and high kingpin angle,

obtained by a lower ball joint movement of +3 mm both

longitudinally and laterally The results show that MOD2

has a more favorable steering rack force in both the

analytical and full vehicle models Therefore, the analytic

model correlates well with the full vehicle model with

regard to steering returnability at the chosen sensitivity

level

6 CONCLUSIONS

We developed an analytic suspension model that considers

the kingpin axis moment, where the kingpin axis trajectory

varies considerably in a lock-to-lock condition, up to ±40

mm The kingpin moment of the model transforms into the

steering rack force (Frack), and finally transfers to the driver

as steering wheel torque We also developed a tire model

for low speed steering returnability, since the measured tire

cornering and aligning stiffness at low speeds are lower

than those at high speeds by 6% and 8%

The steering rack force for evaluating returnability was

analytically determined for each directional tire force

Steering rack forces due to vertical force increased in the

restoring direction according to rack displacement for both

the inner and outer wheels Lateral forces due to side slip

angle were directed toward the medial plane of the vehicle

for both wheels, with the outer wheel pushing the steering

wheel in the returning direction and the inner wheel

push-ing in the non-returnpush-ing direction The large camber at the

inner wheel drives the steering wheel in the returning

direction, while the small camber at the outer wheel makes

little contribution From the simulation, the maximum

steering rack force in this system was 1,152 N for the inner

wheel, 165 N for the outer wheel, and 1,318 N for both

wheels Therefore, the steering wheel can be returned

automatically, thus validating the multi-body model

In order to improve steering returnability, it is important

to increase the total steering rack force at both wheels To

achieve enough steering rack force, development of an

adequate suspension geometry (low caster angle, high

kingpin angle, and a steering angle between the inner out

outer wheels) is required, especially during the vehicledevelopment stage

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87−96

METHOD FOR CONTROL OF STEERING ANGLES FOR ARTICULATED

VEHICLES USING VIRTUAL RIGID AXLES

K.-H MOON 1) , S.-H LEE 2) , S CHANG 1) , J.-K MOK 1) and T.-W PARK 3)*

1)Korea Railroad Research Institute, 360-1 Woram-dong, Uiwang-si, Gyeonggi 437-757, Korea

2)Graduate School of Mechanical Engineering Ajou University, Gyeonggi 443-749, Korea

3)Department of Mechanical Engineering Ajou University, Gyeonggi 443-749, Korea

(Received 24 June 2008; Revised 27 November 2008)

ABSTRACT− Many methods we have been developed to control the rear wheels of a vehicle, but most of them are designed for automobiles with four wheels The AWS (all wheel steering) control method for articulated vehicles is currently applied only to Phileas vehicles developed by APTS, but the control algorithm for this system has yet to be reported In the present paper, a new algorithm is proposed after the AWS ECU (electronic control unit) of the Phileas vehicle was tested and analyzed

in order to understand the existing steering algorithm The new algorithm considers the vehicle geometry, stability of handling, and safety, and can be easily applied to multi-axle vehicles In order to verify the AWS algorithm, the trajectory and steering angles of each algorithm were compared using the commercial software ADAMS Turning radius, swing-out, and swept path width were also investigated to determine the turning performance of the proposed algorithm.

KEY WORDS : Articulated vehicle, All wheel steering, Virtual rigid axle

1 INTRODUCTION

Steering systems are classified as FWS (front wheel

ing), RWS (rear wheel steering), and AWS (all wheel

steer-ing) according to the position of the steering wheels In

general, AWS is applied to improve stability and turning

performance (Tokihiko and Katsuhiko, 2003; Jang et al.,

1995) Most automobiles are mainly controlled by the FWS

system, although some cars use AWS to improve stability

Articulated vehicles with a pivoting joint have difficulty in

making sharp turns because of their long body The AWS

system for articulated vehicles is effective in reducing the

turning radius as well as the platform length because it

steers all wheels simultaneously

There are many methods used to control the rear wheels

such as front wheel steering angle proportion, steering

force feedback, yaw rate feedback and zero side slip angle

(Duane and Eric, 1995; An et al., 2008) Various rear wheel

control methods for AWS have been developed, but they

are generally applied only to four wheel steering cars

Therefore, new control methods must be developed for

articulated vehicles The AWS control method for

articulat-ed vehicles is currently appliarticulat-ed only to Phileas vehicles in

the Netherlands A design was published for a controller

used to guide an articulated vehicle along a path, but the

control algorithm for manual driving has yet to be reported

(Bruin and Bosch, 1999; Bruin and Damen, 2000)

In the present study, the steering characteristics of thePhileas vehicles are analyzed, and a new algorithm is pro-posed The proposed algorithm contains an equation used

to control the rear wheels of an articulated vehicle and amethod used to set up virtual rigid axles geometrically Toverify the proposed AWS algorithm, the commercial soft-ware ADAMS was used to validate the dynamic model andalgorithm The proposed method for setting and controllingthe virtual rigid axles can also be applied to multi-axlevehicles and can aid in eliminating structural errors becausethe setting of the virtual rigid axles is suitable for thegeometry of vehicles Virtual rigid axles can also be easilychanged according to road conditions and safety standards

2 ANALYSIS OF THE EXISTING STEERING ALGORITHM

The AWS ECU of the Phileas vehicle was tested andanalyzed to understand the existing steering algorithm Themethod of virtual rigid axles used to steer rear axles is alsointroduced

2.1 Testing of Existing Algorithm

An articulated vehicle with a single articulated joint sists of three axles, as shown in Figure 1 In the case of anarticulated vehicle with AWS, two ECUs are installed tocontrol the 2nd and 3rd axles

con-Figure 2 shows a schematic of the AWS ECU test paratus The steering device and hydraulic system consists

ap-*Corresponding author. e-mail: park@ajou.ac.kr

of components used in real vehicles, shown in Figure 3.

Vehicle speed signals with square-wave shape were

recorded on the ECU through the emulator and the data

acquisition board

Figure 4 shows the test results for the 2nd axle angle as a

function of the 1st axle angle for various vehicle speeds

The 3rd axle angles according to the articulation angle are

shown in Figure 5

2.2 Method of Virtual Rigid axle and Analysis of Existing

Algorithm

The virtual rigid axle is defined as the virtual axle which is

not steered at the point where the turn center and vehicle

body meet perpendicularly A vehicle with an FWS system

has a fixed rear axle, whose extended line meets the turn

center In the case of an AWS vehicle, the virtual rigid axle

determines its role and controls the angles of the rear

wheels

In existing algorithms, the equation of the steering anglescan be deduced by using the value of virtual rigid axles.Equation (1) for the 2nd axle angle according to the 1staxle angle was derived from the bicycle model, which canrepresent the vehicle effectively, as shown in Figure 6

Figure 1 Articulated vehicle

Figure 2 Schematic of AWS ECU test apparatus

Figure 3 AWS ECU test apparatus

Figure 4 Test results of second axle angle

Figure 5 Test results of third axle angle

Figure 6 Bicycle model for existing algorithm

Equation (1) for the 3rd axle angle was expressed as

equation (2), and finally, was expanded into equation (3),

which can be applied to multi-articulated vehicles Figure 7

shows the comparison of the 3rd axle angles for calculated

and measured values without additional functions

It was found that equation (2) was used for AWS algorithm

of the Phileas vehicle, which was obtained simply by

applying equation (1) to the 3rd axle When these equations

are used, the turn center does not become geometrically

coincident, resulting in small errors in the steering angle

Parameters used in the algorithm are listed in Table 1,

where P 1 and P 2 were determined from experience

(1)(2)(3)According to the test results, the existing algorithm

added suppression of small steering and a limit on the

speed For a small pilot angle, the rear wheel angles were

suppressed by 5o on the 1st axle angle and 2o on thearticulated angle To obtain secure performance even athigh speeds, the steering angle of the rear axle was reducedaccording to the speed The steering angle of the rear wheelmaintained the maximum steering angle at speeds up to 30km/h and then proportionally decreased to zero steeringangle at 45 km/h The 3rd axle angle did not increase andretained the maximum value when the articulation angleexceeded 36o, as shown in Figure 5

3 NEW ALGOLITHM FOR AWS

An efficient method for setting the virtual axles was posed in addition to a new steering angle equation Consi-dering the steering limits which are used for suppression ofsmall steering angles and rear swing-out as well as steeringangle limit depending on the speed, the new algorithm wasfinally determined, as shown in Figure 8

pro-3.1 Equation for Rear Steering AngleNew equations for the rear steering angle were proposed byusing the bicycle model, as shown in Figure 9 The equa-tion of the 2nd axle angle is similar to the existing equation,but the equation of the 3rd axle angle is different from theexisting equation, as shown in equation (5) Although thesuggested equation has two more parameters than the

w 1 Wheel base between axle 1 and axle 2

w 2 Distance between axle 3 and articulation point

virtual rigid axle and axle 2

virtual rigid axle and axle 3 Figure 8 Procedure to determine rear steering angles.

existing algorithm, the angle is correct because of the

geometric coincidence with the turn center This equation

can also be applied to multi-articulated vehicles, as shown

in equation (6)

(4)(5)(6)

3.2 Setting Method for Virtual Rigid Axle

The virtual rigid axle is geometrically related to the angles

of each axle and the articulation (Moon et al., 2007; Jeon et

al., 2008) For a vehicle with two axles, such as an

automobile, the virtual rigid axle can be set up within the

limits of the 2nd axle angle by considering the 1st axle

angle and the trunk space In the case of an articulated

vehicle, however, it is not easy to set up the virtual rigid

axle because the articulation angle becomes an additional

variable An articulation device can easily be broken by

compulsory steering Thus, the virtual rigid axle must be

set up according to the angle of articulation first Then, the

turn center should be made coincident by considering the

maximum steering angle of each axle A flow chart for

setting the virtual rigid axles is shown in Figure 10

Assuming that the rear axles are fixed, the value of the

virtual rigid axles is zero and the turn center is located

at the O position However, when the rear axles are

steered, the turn center moves from the O position to the O'

position with the value of the virtual rigid axle, as shown in

Figure 11

The maximum steering angle of the 1st axle in equation(7) was obtained by using the maximum articulated anglewhen the rear axles are fixed The value of the virtual rigidaxles in equations (8) and (10) were taken from themaximum steering angle of the rear axles Equations (9)and (11) were obtained from the relation between virtualrigid axle 1 and virtual rigid axle 2

Figure 9 Bicycle model for proposed algorithm

Figure 10 Flow chart for setting virtual rigid axles

Figure 11 Movement of turn center according to the virtualrigid axle

(8)(9)

(10)(11)The maximum steering angle in Table 2 is the mean of

the maximum steering angles of the right wheel and the left

wheel Using the variables of Table 1, Table 2, and L1, the

maximum value of the virtual rigid axles can be found If

P1 and P2 obtained for each algorithm are substituted into

equation (6) and equation (3), the 3rd axle angles can be

found, as shown in Figure 12

3.3 Suppression of Small Steering Angles

For small pilot angles or steering angles of the 1st axle, the

wheels of the rear axle are not steered in order to improve

the stability of handling The steering angle of the rear axle

varies due to the virtual rigid axle, the value of which is

zero to keep the rear axle in the 0o-position The point of

discontinuity occurs when the value of P suddenly changesfrom zero to the set point Therefore, the transition functionwas derived based on the response of the first-order system

to a step-function input shown in equation (13) Figure 13shows the relation between the pilot angle and the steeredangle for the limit values of the steering angles, 5o and 2o

If (12)

If (13)

(14)where ε is the permissible value, θ is the steering angle, θ 0

is the limit value of the steering angle, and Pmax is themaximum value of the virtual rigid axle

3.4 Suppression of Rear Swing-out The rear of the vehicle generally overhangs the rear axle

As a result, the rear of a vehicle swings to the outside of therear axle, as shown in Figure 14 In the case of a vehiclewith AWS, the swing-out increases because of the rearsteering at a reverse phase angle To prevent rear swing-out, the angles of the rear wheel were suppressed for aconstant distance and then the values of the virtual rigidaxle were moved to maximum setting values, as shown inequation (15)

(15)(16)

where ω is the angular velocity of the steering axle and δmax

is the maximum steering angle

3.5 Steering Angle according to Vehicle SpeedWhen a vehicle travels around a curve, speed is restrictedfor safety For railway vehicles, UIC (international union ofrailways) regulations limit lateral acceleration to 3 m/s2

(UIC 518, 2005) An articulated vehicle with rear wheel

P 1 ′=f 2 ( δ 1 , δ 2max )= l tan δ × 2 max

tan δ 1 + tan δ 2 max

× tanα cosα ×

tanδ 3max

- cosα tanδ1

tanα tanδ + 1

( ) cosα × - –

+ -

max

ω

-

-Table 2 Maximum angle of articulated vehicle

δ1max Maximum Steering Angle of Axle1

δ2max Maximum Steering Angle of Axle2

δ3max Maximum Steering Angle of Axle3

Figure 12 Comparison of 3rd axle angle depending on the

algorithm

Figure 13 2nd steering angle according to the pilot angle

steering is more unstable than one with only front wheel

steering due to an increase in the yaw angle Since lateral

acceleration by centrifugal force is proportional to the

squared velocity and inversely proportional to the radius,

the minimum radius Rmin is expressed in equation (17) The

lateral acceleration in equation (17) was obtained from the

center point of the articulation between both carriages

(17)For the maximum lateral acceleration of 3 m/s2, the

steering angle limit according to vehicle speed is shown in

Figure 15 The graphs for the steering angles were

lineari-zed while maintaining the same integral value Then, the

value of the virtual rigid axle could be expressed with the

vehicle speed by the following equations

If (18)

If (19)

If (20)

4 APPLICATION OF DEVELOPED

ALGORITHM USING ADAMS

The steering angle of the 1st axle is controlled by the

driver, but the articulation angle is geometrically

determin-ed according to the steering state of each wheel Therefore,

a multi-body dynamic simulation was performed to gate the steering angles and the trajectory of the vehicle.4.1 Multi-body Dynamics Model

investi-To verify the AWS algorithm, the commercial softwareADAMS was used to validate the dynamic model andalgorithm After modeling each component, such as sus-pension, steering system, and tire, the dynamic model ofthe full vehicle was assembled, as shown in Figure 16 Thedata used for the model were based on the data from a realvehicle, such as that required for the dimensions of thevehicle, characteristics of the dampers, air-springs, andtires Dry asphalt with the friction coefficient of 0.8 wasapplied to the road surface and the lateral forces on the tirewere obtained from the Magic Formula model (Pacejka,2002)

4.2 Dynamics Model VerificationThe reliability of the multibody dynamics model whichwas applied to the existing AWS control algorithm has

Figure 14 Rear edge swing-out

Figure 15 Steering angle limit according to vehicle speed

Figure 16 Dynamics model of full vehicle

Figure 17 Input value of 1st axle angles