International journal of automotive technology, tập 11, số 2, 2010

Four injector nozzles with approximately the same fuel injection rate were tested using the solenoid injection system 10 and 12 orifice configuration and piezoelectric system 6 and 12 or

COMPARISON OF TWO INJECTION SYSTEMS IN AN HSDI DIESEL ENGINE USING SPLIT INJECTION AND DIFFERENT INJECTOR NOZZLES

J BENAJES 1) , S MOLINA 1)* , R NOVELLA 1) , R AMORIM 1) ,

H BEN HADJ HAMOUDA 2) and J P HARDY 2)

1)Universidad Politécnica de Valencia, CMT-Motores Térmicos, Valencia 46022, Spain

2)Renault S.A.S., Rueil Malmaison 92508, France

(Received 28 January 2008; Revised 8 July 2009)

ABSTRACT− The demand for reduced pollutant emissions has motivated various technological advances in passenger car diesel engines This paper presents a study comparing two fuel injection systems and analyzing their combustion noise and pollutant emissions The abilities of different injection strategies to meet strict regulations were evaluated The difficult task

of maintaining a constant specific fuel consumption while trying to reduce pollutant emissions was the aim of this study The engine being tested was a 0.287-liter single-cylinder engine equipped with a common-rail injection system A solenoid and a piezoelectric injector were tested in the engine The engine was operated under low load conditions using two injection events, high EGR rates, no swirl, three injection pressures and eight different dwell times Four injector nozzles with approximately the same fuel injection rate were tested using the solenoid injection system (10 and 12 orifice configuration) and piezoelectric system (6 and 12 orifice design) The injection system had a significant influence on pollutant emissions and combustion noise The piezoelectric injector presented the best characteristics for future studies since it allows for shorter injection durations and greater precision, which means smaller fuel mass deliveries with faster responses

KEY WORDS : Diesel engines, Injection system, Nozzle, Pollutant emissions

NOMENCLATURE

ATDC : after top dead center

BGT : burned gas temperature

BSFC : brake specific fuel consumption

BTDC : before top dead center

CAD : crankshaft angle degrees

CO : carbon monoxide

CO2 : carbon dioxide

EGR : exhaust gas recirculation

EOI : end of injection

FSN : filter smoke number

HC : hydrocarbons

HRL : heat release law

IMEP : indicated mean effective pressure

MD : mass distribution

NOX : nitrogen oxides

RoHR : rate of heat release

SFC : specific fuel consumption

SOI : start of injection

1 INTRODUCTION

Many health and environmental problems have been

attri-buted to pollutant emissions, mainly NOX (Nitrogen Oxides)

and particulate matter, from petrol and diesel engines Airquality, especially in large urban areas, has been impacted

by engine exhaust gases and particulate matter To reducethis impact, a stricter emission regulation, Euro 5, began in

2009 Since 1996, when Euro 2 started to tighten emissionlimits, new technologies have been developed to meet thechallenge of complying with low emission limits In thefuture, regulations are predicted to become stricter for enginemanufacturers In the case of Euro 5, there is already anoticeable reduction in NOX and particulate matter com-pared to Euro 4 It is expected that diesel particulate filterswill be mandatory for all diesel cars by 2011 Euro 6,which will probably start being enforced in 2014, willsignificantly lower NOX emission limits from the current0.180 g/km (Euro 5) to 0.080 g/km This will force carmanufacturers to invest additional resources into research,and thus increase the final price of vehicles (EurActive,2004; The European Commission, 2006a, 2006b) Part of the Diesel engine’s great advances in performanceand control of pollutant emissions over the last decade can

be attributed to improvements in the injection systems Theintroduction of the high pressure common-rail injectionsystem has allowed for better control of the combustionprocess through flexible, more accurate control of theinjection parameters This system allows the number ofinjection pulses, the time interval between them, the injec-tion duration and the injection pressure (IP) to be precisely

140 J BENAJES et al.

controlled In newer injection systems, it is also possible to

control the injection rate shape (Robert Bosch GmbH,

2004) Despite the fact that this system increases control of

the fuel injection process, it makes finding the optimum

operating conditions more difficult (Desantes et al., 2007)

Due to the need to reduce emissions, extensive research

has focused on in-cylinder control of pollutant formation It

is well-known that reducing NOX, smoke and HC

(Hydro-carbons) emissions at the same time is a very difficult task

Some strategies focus on split injections as a way to control

emissions Nehmer and Reitz studied the effects of

rate-shape and split injection on diesel engine performance and

emissions They observed that the amount of fuel in the

first injection affected the engine-out emissions and

in-cylinder pressure rise rate, which are directly related to

combustion noise Higher NOX emissions and lower smoke

production were seen when more fuel was injected in the

pilot injection (Nehmer and Reitz, 1994)

Tow et al. investigated the effects of multiple injections

on combustion in heavy-duty Diesel engine operation at

medium and low load conditions Multiple injection

strate-gies reduced NOX emissions, and the dwell time between

injection events was shown to heavily influence

combus-tion process control (Tow et al., 1994)

Pierpont, Montgomery and Reitz tested multiple

injec-tion strategies involving EGR (Exhaust Gas Recirculainjec-tion)

in order to reduce NOX emissions without significant

pen-alties on smoke and BSFC (Brake Specific Fuel

Consump-tion) They observed that multiple injections could

effec-tively reduce particulate matter, NOX and combustion noise

They pointed out that the undesirable EGR collateral effects

of increased particulate emissions might be compensated

for by the use of multiple injections (Pierpont et al., 1995)

Montgomery et al. compared the behaviors of different

nozzles in relation to the flow exit area and number of

orifices and highlighted their influence on combustion

According to their study, nozzles with shorter spray

penet-rations produced more particulate matter and lower NOX

(Montgomery et al., 1996) Benajes et al also investigated

the influence of nozzle orifice number and the use of swirl

in a retarded split injection on gaseous emissions and

combustion noise They remarked that the low temperature

combustion obtained with a late injection is able to provide

ultra-low NOX emissions and reasonable combustion noise

at medium load conditions Their results also showed that a

high orifice number is prone to causing very high smoke

emissions due to an undesirable interaction among the fuel

spray jets, which could be intensified by swirl (Benajes et

al., 2006)

The use of multiple injections in a small diesel engine

was also discussed by Hotta et al They investigated how

an early pilot, close pilot and post-injection could affect the

combustion process and pollutant emissions In their work,

they observed that large early-pilot injections could increase

HC emissions due to a cylinder wall impingement It was

also found that the use of post-injection helped reduce

smoke emissions However, this phenomenon had ously been noticed (Hotta et al., 2005) Desantes et al.studied the usage of post-injections They concluded thatpost-injections were capable of reducing smoke emissionsconsiderably with no penalty on NOX emissions Their studywas focused on post-injection and the phenomenon of sootoxidation The results revealed that the post-injection redu-ced soot However, it was observed that post-injections didnot interact with the main injection Consequently, sootwould not be reduced by enhanced soot oxidation caused

previ-by the post-injection Furthermore, engine-out soot would

be the sum of the soot resulting from the combustion of themain and pilot injections separately Thus, the final level ofsoot decreased because the main pulse produced less sootand the post-injection did not produce significant additionalsoot (Desantes et al., 2007) Finally, Benajes et al. carriedout an investigation using a small single-cylinder enginebased on a statistical procedure called “Consecutive Screen-ings”, which showed significant improvements in pollutantemissions by substantially increasing the EGR rate, retard-ing the injection event and using variable dwell time (Benajes

et al., 2007)

The objective of this work was to compare the pollutantemissions using a piezoelectric injection system and asolenoid injection system in a light duty engine with a lowcompression ratio of 14:1 at constant SFC (Specific FuelConsumption), using split injections and running in a lowload engine mode The influence of dwell times and splitinjection mass distribution were also studied, in order toevaluate the possibilities of each injector in each case.Usually, in other studies, the influences of pilot- and post-injections on pollutant emissions are investigated, takinginto consideration that the use of split injections can affectcombustion efficiency and IMEP (Indicated Mean Effec-tive Pressure) In this work, the analysis the analysis wasperformed with the amount of injected fuel and IMEP heldconstant

2 EXPERIMENTAL FACILITY AND EQUIPMENT

The engine used in this work was a single-cylinder researchengine with a displacement volume of 0.287 liters, four valvesand low compression ratio, equipped with a common railinjection system This engine corresponds to a 1.2-liter, 4-cylinder engine

The engine was installed in a fully instrumented test cellwith all of the required facilities for the operations andcontrol of the engine The required boost pressure (BP) wasprovided by a screw compressor, and the intake air washeated to 40oC Exhaust gas recirculation (EGR) was keptconstant at 120oC NOX, CO (Carbon monoxide), HC, CO2

(Carbon Dioxide) and O2 (Oxygen) measurements wereperformed with a HORIBA 7100D gas analyzer Smokeemissions were measured with an AVL 415 variable sampl-ing smoke meter, which provided results directly in FSN

(filter smoke number) In-cylinder pressure was measured

with a piezoelectric transducer, and additional information,

such as IMEP and combustion noise, could be evaluated

during the tests using this data The experimental set-up is

presented in Figure 1

Combustion diagnosis software was used to calculate the

heat release (HRL), rate of heat release (RoHR), burned

gas temperature (BGT) and other valuable information Data

recorded from 50 consecutive engine cycles with a

resolu-tion of 0.2 crank angle degrees (CAD) was used for this

calculation The model is based on the solution of the energy

conservation equation in the cylinder, with the assumption

of uniform pressure and temperature over the instantaneous

volume This single-zone model enables the calculation of

the instantaneous average temperature in the burned gas, as

well as the heat released during the combustion (Lapuerta

et al., 1999) and (Desantes et al., 2004)

3 METHODOLOGY

The tests were carried out in two stages for each fuel

injec-tion system used in this work The first stage was to define

the engine’s operational conditions The second stage

con-sisted of the tests comparing dwell time and mass

distri-bution

3.1 Engine Operational Condition for the Preliminary Tests

In the preliminary tests, a 12-orifice nozzle was used for

the solenoid system and a 6-orifice nozzle was used for the

piezoelectric system Both nozzles had a conical orifice

shape and similar hydraulic mass flow The engine was

operated at 1500 rpm and 4.0 bar IMEP, and the dwell time

was set to be 1.0 ms In this study, the dwell time is

considered to be the time interval between the end of the

first injection event and the beginning of second injection

event

Ranges for injection pressure and EGR rate were defined

with the aim of performing a parametric study, and final

values are shown in Table 1 The mass distribution (MD) of

the split injection is presented in this format: 50/50 MD.This nomenclature indicates that 50% of the fuel mass isinjected in the first injection event and the remaining 50%

is injected in the second event

The engine mode and pollutant emission targets arebased on the EURO 4 cycle for a passenger diesel vehiclewith an aftertreatment particulate filter (Table 1) The pre-sence of the particulate filter in the exhaust allows for ahigh level of engine-out smoke emissions, as can be seen inthe Table 1

3.2 Engine TestThe engine test stage was characterized by sweeping thedwell time for various mass distributions The engine testsfor both fuel injection systems were performed with twodifferent nozzles, varying only the number of orifices.Although the nozzles differ in the number of orifices (10and 12 orifices), the theoretical hydraulic flow is verysimilar The ISFC was fixed at 250 g/kWh and an IMEP of4.0 bar was targeted It is important to point out that theSOI (Start of Injection) was varied in order to keep theIMEP at 4.0 bar

Engine tests using the solenoid and the piezoelectric tion systems were carried out with an injection pressure of

injec-900 bar and a 45% EGR rate, based on the results obtained

in the definition phase The boost pressure was set at 1.2bar

Using the solenoid system, the mass distribution ratioranged from 30/70 to 50/50, and the dwell time was sweptfrom 0.6 to 1.6 ms using both 12-orifice and 10-orificenozzles Dwell times shorter than 0.4 ms would be veryunstable, and thus were not tested Using the piezoelectric

Figure 1 Engine experimental laboratory set-up

Table 1 Engine operating conditions of the preliminarytests and pollutant emission targets

Engine operating conditionsSolenoid PiezoelectricEngine speed 1500 rpm 1500 rpmIMEP 4.0 bar 4.0 barSOI 1 f(SFC, IMEP) f(SFC, IMEP)Dwell time 1.0 0.6Injection pressure 600~1200 bar 900~1300 barEGR rate 40%~50% 40%

Nozzle 12 holes 6 holesMass fuel 8.0 mg/cc 8.0 mg/cc

MD 50%-50% 20%-80% to 80%-20%Smoke, noise and pollutant emission targets

Smoke < 2.00 FSNNoise < 80.0 dB

NOX < 0.25 g/kWh

CO < 6.50 g/kWh

HC < 1.50 g/kWh

142 J BENAJES et al.

injection system, the mass distribution ratio of the split

injection was swept from 20/80 to 50/50, and the dwell

times ranged from 0.2 to 1.4 ms The piezoelectric

injec-tion system responded faster and more accurately, allowing

for shorter injection durations and dwell times The same

tests were repeated with a constant SOI, instead of a

constant IMEP, in order to separately evaluate the effects of

different engine parameters on engine behavior

4 RESULTS AND ANALYSIS

4.1 Solenoid Injection System Preliminary Test Results

In the preliminary test phase, some engine tests were

carri-ed out to select the most suitable EGR rate and injection

pressure for the next part of this study The results and the

main observations of this part of the study are presented in

Figure 2(a) and (b) Graph (a) shows the pollutant

emi-ssions for each tested case Graph (b) shows HRL, RoHR

and BGT curves for 900 bar IP EGR swept and 45% EGR

rate injection pressure swept

As seen in Figure 2(a), for 600 bar IP, the levels of CO

and HC emissions were much higher than the targeted

values Unexpectedly, the NOX emissions did not seem to

be related to the injection pressure It is confirmed by the

BGT graph in Figure 2(b) that the burned gas temperature

peak did not change significantly with injection pressure,while it decreased considerably with increasing EGR rate.Smoke emissions for the lowest IP (blue circle) were underthe target value but very close to the limit Smoke was alsohigher than observed for the other injection pressures due

to worse mixing conditions This was not a good resultsince it did not leave margin to work on a possible trade-off For an injection pressure of 1200 bar, the obtained COvalues were slightly higher Combustion noise (in the greencircle) was also considered unsuitable, so the small reduc-tion in smoke emissions does not justify its use in the nextphase Using an injection pressure of 900 bar resulted insmoke emissions below the target value Moreover, thisconfiguration presented lower CO and HC emissions (dottedred lines) than the other IP values Thus, 900 bar seemed to

be the most reasonable injection pressure for further ment

develop-Of the tested EGR rates (40%~50%), 40% had thehighest NOX emissions because the higher O2 concentrationled to higher in-cylinder temperatures Thus, NOX formation(black arrow) was not inhibited enough to stay under thetarget value On the other hand, increasing the EGR rate to50% (grey arrows) significantly reduced the O2 concent-ration, enough to efficiently reduce the in-cylinder temper-atures and combustion noise However, this EGR rate hadunreasonably low combustion, which resulted in high COemissions Furthermore, it was concluded that the nextengine tests using the solenoid injection should be carriedout with the IP and EGR rate set to 900 bar and 45%,respectively

In Figure 2(b), the BGT graph shows that the maximumburned gas temperature did not vary as a function of injec-tion pressure, but decreased when the EGR rate increased

In order to maintain an IMEP of 4.0 bar, the SOI had to beadvanced when either injection pressure or EGR rateincreased The effect was that the RoHR of the first com-bustion was considerably reduced (see red arrows) At an

IP of 1200 bar, it seemed that the combustion of the pilotand main injections started almost simultaneously, increas-ing the RoHR slope and justifying the high values of com-bustion noise observed At a 50% EGR, the RoHR did notrise because increasing the EGR rate also caused a signifi-cant reduction in the mixing rate and, consequently, incombustion velocity (see brown arrows)

4.2 Engine Preliminary Test Results Using PiezoelectricInjection System

The preliminary tests using the piezoelectric injection systemintended to define an appropriate injection pressure andmass distribution range for the next engine tests This injec-tion system allows for injection durations as short as 140

µs, which permits injecting very small amounts of fuel,such as 20% of the total injected mass (1.6 mg/cc) Thus,the split injection was swept from 20/80 to 80/20 of themass distribution with a fixed dwell time of 0.6 ms (5.4CAD) There are two different ranges to be considered and

Figure 2 (a) Noise, soot and pollutant emissions from the

preliminary tests using the solenoid injection system; (b)

HRL, RoHR and BGT vs Crankshaft angle for solenoid

preliminary tests

observed separately: pilot injection (from 20/80 to 50/50)

and post injection (from 60/40 to 80/20)

Analyzing the results at 1300 bar of IP in Figure 3, the

pilot injection range presented higher levels of HC and CO

emissions than other injection pressures However, the

emissions tended to decrease when the fuel mass of the first

injection event (green arrow) was increased However, the

combustion noise increased in the same range because

more fuel mass was burned in the premixed combustion In

the post-injection range, it is possible to observe very high

combustion noise, up to unacceptable levels At 1100 bar

of IP, the smoke emissions stayed at very low levels, even

though the HC and CO emissions were high when using a

pilot injection The smoke emissions were slightly

decreas-ed, whereas CO and combustion noise did not increase

much The use of a 900 bar IP post-injection significantly

reduced smoke formation, as previously known, although

HC and CO emissions increased (blue circles) (Desantes et

al., 2007; Han et al., 1996) Moreover, the use of a pilot

injection at 900 bar of IP kept the CO and HC emissions at

a lower level than the other injection pressures Although

smoke emissions were higher than the other pressures, they

were still very far below the proposed target Based on this

analysis, the engine tests using the piezoelectric injector

were carried out with a 900 bar IP and a pilot injection The

EGR rate was increased to 45% in order to reduce NOX

formation since some of the test points did not meet their

targets

4.3 Engine Tests Analysis

The engine tests with the solenoid injector were performed

with a 10-orifice and 12-orifice nozzle The 12-orifice nozzle

tests had a range of mass distribution from 30/70 to 50/50

MD The same tests were repeated using the 10-orifice

nozzle, except for 30/70 MD because a high level of

combustion instability was found when using the 12-orifice

nozzle under those conditions The cause was the short

injection duration required to inject only 30% of the

inject-ed mass, which causinject-ed the injector neinject-edle to pulse as fast

as possible For both nozzles, the dwell time ranged from0.6 to 1.6 ms

Engine tests with the piezoelectric injector were carriedout with a 6-orifice and 12-orifice nozzle The mass distri-bution range was swept from 20/80 to 50/50, and the dwelltime ranged from 0.2 ms to 1.4 ms

Figure 4(a) and (b) present a comparison among all thenozzles used in this study, independently of the injectionsystem, with mass distributions of (a) 40/60 and (b) 50/50

A 45% EGR rate was used and the IMEP was isolated at4.0 bar The dashed lines are the proposed targets for eachpollutant It is important to make clear that 0.2 and 0.4 ms

of dwell time were not tested with the solenoid system and1.6 ms was not tested with the piezoelectric system The20/80 and 30/70 mass distribution graphs are not shownhere due to space limitations

Increasing the dwell time contributed to smoke mation (blue circles) when using the 6-orifice or 10-orificenozzles, as seen in Figure 4(a) and (b) The different levels

for-of smoke emissions depend on the number for-of orifices for-ofeach nozzle; the 12-orifice nozzles showed lower smokeemissions than the nozzles with fewer orifices When using12-orifice nozzles, more advanced SOI’s were necessary toreach a 4.0 bar IMEP, and they presented higher combus-tion noise

Figure 3 Noise, soot and pollutant emissions from the

preliminary tests using the piezoelectric injection system

Figure 4 Comparison of (a) 40/60 and (b) 50/50 massdistributions

144 J BENAJES et al.

NOX and smoke levels did not represent a problem since

they stayed below the target in the majority of the tested

points However, combustion noise, HC and CO emissions

did not fulfill the required limits with both nozzles and all

mass distributions In general, the 10-orifice nozzles

produ-ced more smoke than the 12-orifice nozzles However, the

12-orifice nozzle presented more combustion noise because

it required a slightly advanced injection timing to maintain

a 4.0 bar IMEP

As seen in Figure 4(b), the 6-orifice nozzle with reduced

dwell times, such as 0.2 or 0.4 ms, presented unsuitable

levels of combustion noise independently of the mass

di-stribution (black arrow) NOX emissions stayed below the

limit The combustion noise target was achieved for both

mass distributions between 0.6 and 1.0 ms of dwell time

The 20/80 and 30/70 mass distributions (not shown in the

picture) stayed close to the target However, HC and CO

emissions are still very high in all the cases Finally, for the

50/50 mass distribution, the smoke emissions were very

close to the limit of 2.0 FSN

Figure 5(a) and (b) present two cases in which the

configurations using the piezoelectric injector and solenoid

injector were equal In Figure 5(a), both injection systems

were tested using 12-orifice nozzles, 45% EGR, 4.0 bar

IMEP, 1.0 ms dwell time and −11.0 CAD ATDC SOI () (b)had the same configuration, except the dwell time and SOIwere changed to 1.2 ms and −11.5 CAD ATDC SOI,respectively The injector opening timings of the secondinjection event are represented in the graphs by the verticallines

Although the engine test configurations were exactly thesame in each graph, some differences are noticeable whencomparing the HRL, RoHR and BGT curves for both injec-tion systems The shorter time that the piezoelectric injectorrequired for opening and closing caused an advance of thesecond injection relative to the solenoid injector Conse-quently, the combustion process was advanced

Examining Figure 5, it can be seen that cool flame tions started before the second injection when using thesolenoid injector However, those cool flame reactions werenot seen when the piezoelectric injector was used Thesecond injections using the piezoelectric injector avoidedthe cool flame reactions In this case, the cool flame reac-tions were responsible for the temperature increase beforethe combustion process Furthermore, it can be seen thatthe maximum BGTs are very similar for both injectors, butthe average temperatures during the combustion were slight-

reac-ly higher using the solenoid injector, which could haveincreased NOX formation in the beginning and prolongedsmoke oxidation at the end of the combustion process Thiseffect is stronger for 1.0 ms of dwell time than for 1.2 ms.Figure 6 shows the graphs for HRL, RoHR and BGT fordifferent dwell times corresponding to the tests with 4.0 bar

of IMEP, using the 12-orifice nozzle with a mass tion of 50/50

distribu-It can be observed that the combustion occurs moresmoothly when the dwell time is increased due to retarda-tion of the center of the combustion With a dwell time of0.2 ms, the combustion is very similar to that of a singleinjection When the dwell time is increased to 1.4 ms (12.6CAD at 1500 rpm) the combustion of both injections seems

to be slower, and the premixed combustion less abrupt.Dwell times longer than 1.0 ms reduced the slope of the

Figure 5 Comparison of HRL, RoHR and BGT vs

crank-shaft angle curves for 40/60 MD using solenoid and

piezo-electric injectors with 1.0 ms and 1.2 dwell times The

engine configurations for each dwell time case were

exactly the same, including injection timing Figure 6 HRL, RoHR and BGT vs crankshaft angle for50/50 mass distributions

RoHR, causing less combustion noise The maximum BGT

was reduced by increasing the injection dwell time But the

time period that the BGT remained at NOX formation

temperatures increased with increasing injection dwell time

(Akihama et al., 2001) Thus, there was not a significant

change in NOX emissions

In order to study the isolated effects of mass distribution

and dwell time using the 12-orifice nozzle with the

piezo-electric injector, some Iso-SOI tests were carried out The

chosen SOI was -15.5 CAD ATDC, which was the one at

which the 20/80 mass distribution had an IMEP of 4.0 bar

Figure 7 represents the complete pollutant emission results

obtained from the Iso-SOI tests This graph presents all

tested dwell times and mass distributions

With increasing dwell time, combustion noise was

signi-ficantly reduced because the combustion started more

smoothly due to retardation of the center of the

combus-tion NOX formation was greatly reduced by retarding the

SOI However, there was a significant increase in HC,

smoke and CO emissions, mainly in the mass distributions

with smaller pilot injections

When the pilot fuel mass was increased (see Figure 8),

the center of the combustion (center of combustion is the

crank angle of 50% of heat release) was advanced towards

the TDC However, this did not result in a higher IMEP

The in-cylinder pressure rose earlier for longer pilot

injec-tions but also decreased earlier and remained lower during

expansion The IMEP was kept constant for all mass

distri-butions In Figure 8, the region where the pressure lines

cross is shown by the red circle This effect was attributed

to a completely premixed combustion, and changing the

mass distributions did not deteriorate the combustion

pro-cess For longer pilot injections, the RoHR is steeper,

lead-ing to an increase in combustion noise Larger pilot

injec-tions advanced the entire combustion However, it is

important to point out that the distance between the peaks

of the RoHR for the mass distributions of 20/80 and 50/50was less than 4 CAD The peak in-cylinder pressure alsoincreased with larger pilot injections This led to higher

NOX emissions However, the 50/50 mass distribution hibited a reduced slope in the HRL curve

ex-5 CONCLUSIONS

In this research, two different injection systems (based on asolenoid and a piezoelectric injector) were investigated.Preliminary tests were performed to select the best condi-tions for the main study In both cases, the chosen injectionpressure was 900 bar and the EGR rate was 45% Thesevalues were chosen to work with a pilot injection smallerthan 50% of the injected mass per cycle

Different injection strategies were tested for each tion system There were two injection events per cycle TheISFC was kept constant at 250 g/kW.h during the tests Theinjection strategies were characterized by sweeping themass distributions and dwell times A small sequence oftests was executed at constant SOI, in order to study theisolated effects of the mass distribution for different dwelltimes or combustion events

injec-Based on the results from the preliminary test and enginetest, it is possible to conclude:

(1) Independently of the injection system, 900 bar was themost suitable injection pressure for this engine Thelowest injection pressure of 600 bar resulted in higher

HC and smoke emissions This could have been due to

a longer fuel atomization process than the other tion pressures Higher injection pressures presentedlower smoke but higher combustion noise This could

injec-be attributed to the fact that more fuel mass is injectedbefore the combustion process is started, and the atomi-zation of the fuel is better at higher injection pressures

A 40% EGR rate presented excessive combustion noiseand high NOX emissions due to the fast premix com-bustion and high in-cylinder temperature On the otherhand, a 50% EGR rate caused excessive reduction of

O2 concentration, reduced NOX and combustion noise,

Figure 7 Pollutant emissions from the Iso-SOI tests

Figure 8 HRL, RoHR and BGT vs crank angle for Iso-SOItests with 1.0 ms of dwell time

146 J BENAJES et al.

but the HC and CO emissions increased to

unaccep-table levels

(2) With the injection pressure at 900 bar, mass

distribu-tions with the first injection larger than 50% of the

injected mass present high CO and HC emissions

How-ever, it has been observed that the use of a

post-injection smaller than 40% of the total injected mass

significantly reduces smoke formation

(3) An increase in dwell time with constant SOI produced a

smoother start of combustion and a cooler overall

com-bustion process, reducing noise and NOX emissions

However, the emissions of HC and CO increased under

these conditions

(4) Increasing the pilot injection quantity caused an increase

in combustion noise and NOX emissions due to faster,

hotter premixed combustion The opposite effect was

observed when the dwell time between injection events

was increased

(5) The solenoid injection system presented unsatisfactory

results due to high HC and CO emissions independent

of the number of nozzle orifices However, the

10-orifice nozzle resulted in levels of combustion noise

close to the target, while NOX remained under the limit

Smoke emission increased and stayed close to the limit

(6) The piezoelectric injection system with 6-orifice and

12-orifice nozzles, presented unacceptable results in

terms of HC and CO emissions The combustion noise

targets were achieved using both nozzles with dwell

times around 1.0 ms Moreover, the 6-orifice nozzle

showed higher smoke levels than the 12-orifice nozzle

(7) For the same engine test configuration, the solenoid

injector presented a slightly retarded combustion

pro-cess compared to the piezoelectric injector due to the

longer time needed for it to open completely This

behavior slightly changed the in-cylinder temperatures,

favoring NOX formation before the maximum BGT

was reached and soot oxidation at the end of the

com-bustion process

(8) The piezoelectric injection system presented better

results in terms of pollutant emissions It also permitted

more accurate control of the injection parameters,

includ-ing the possibility of injectinclud-ing very small quantities of

fuel in each injection event

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Han, Z., Uludogan, A and Hampson, G R (1996) anism of soot and nox emission reduction using multiple-injection in a diesel engine SAE Paper No 960633.Hotta, Y., Inayoshi, M and Nakakita, K (2005) Achievinglower exhaust emissions and better performance in anHSDI diesel engine with multiple injection SAE Paper

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Lapuerta, M., Armas, O and Hernandez, J (1999) nosis of DI diesel combustion from in-cylinder pressuresignal by estimation of mean thermodynamic properties

Diag-of gas Applied Thermal Engineering, 19, 513−529.Montgomery, D., Chan, M., Chang, C., Farrell, P andReitz, R (1996) Effect of injector nozzle hole size andnumber on spray characteristics and the performance ofheavy duty D.I diesel engine SAE Paper No 962002.Nehmer, D and Reitz, R (1994) Measurement of theeffect of injection rate and split injections on dieselengine soot and NOx emissions SAE Paper No 940668.Pierpont, D., Montgomery, D and Reitz, R (1995) Reduc-ing particulate and NOx using multiple injections andEGR in a D.I diesel SAE Paper No 962002

Robert Bosch GmbH (2004) Diesel-Engine Management.

3rd edn SAE Warrendale PA

The European Commission (2006a) Euro 5 and 6 will Reduce Emissions from Cars Retrieved 2007, fromEUROPA: http://europa.eu/rapid/pressReleasesAction.do?reference=MEMO/06/409&format=HTML&aged=0&language=EN&guiLanguage=en

The European Commission (2006b) Tighter Wmission Limits for Cars After EP Adoption of Euro 5 and 6 Retrieved

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a heavy duty D.I diesel engine SAE Paper No 940897

DESIGN OF ACTIVE SUSPENSION AND ELECTRONIC STABILITY

PROGRAM FOR ROLLOVER PREVENTION

S YIM 1)* , Y PARK 2) and K YI 3)

1)BK21 School for Creative Engineering Design of Next Generation Mechanical and Aerospace Systems,

Seoul National University, Seoul 151-742, Korea

2)Department of Mechanical Engineering, KAIST, Daejeon 305-701, Korea

3)School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742, Korea

(Received 3 November 2008; Revised 2 September 2009)

ABSTRACT− This paper presents a method for the design of a controller for rollover prevention using active suspension and

an electronic stability program (ESP) Active suspension is designed with linear quadratic static output feedback control methodology to attenuate the effect of lateral acceleration on the roll angle and suspension stroke via control of the suspension stroke and tire deflection of the vehicle However, this approach has a drawback in the loss of maneuverability because the active suspension for rollover prevention produces in vehicles an extreme over-steer characteristic To overcome this drawback of the active suspension based method, ESP is designed Through simulations, the proposed method is shown to be effective in preventing rollover.

KEY WORDS : Rollover prevention, Active suspension, ESP, Optimal static output feedback control, Maneuverability

NOMENCLATURE

a y : lateral acceleration

g : gravitational acceleration constant

m : vehicle total mass

m s : sprung mass

m u : unsprung mass

I x : roll moment of inertia about roll axis

I z : yaw moment of inertia about yaw axis

k s : stiffness of a suspension spring

k t : stiffness of a tire

b s : damping coefficient of a suspension damper

h : height of C.G from ground

h s : height of C.G from a roll center

C f : cornering stiffness of a front tire

C r : cornering stiffness of a rear tire

t f : front track width

l f : distance from C.G to a front axle

l r : distance from C.G to a rear axle

r : radius of a wheel

K B : pressure-force constant

v x : longitudinal velocity of a vehicle

K g : gain of sliding mode controller

1 INTRODUCTION

Over the last decade, a widespread supply of SUVs (Sports

Utility Vehicles) with high centers of gravity (C.G.)

increas-ed the number of rollover accidents For example, in theUSA, there have been 4,045 fatalities and 88,000 injuriescaused by non-collision rollover accidents in 2004 (NHTSA,2004a) Most rollover accidents are fatal For instance, in

2003, the portion of rollovers in all crashes was mately 3%, but 33% of all fatalities were caused byrollovers (NHTSA, 2003)

approxi-As shown in Figure 1, the factors influencing rolloversare the lateral acceleration a y, the distance from the rollcenter to the center of gravity h s, and the lateral tire force

F y The untripped rollover occurs due to a large lateralacceleration generated by excessive steering at high speed

On a low-friction road or at low speed, the rollover cannotoccur because of insufficient lateral acceleration or lateraltire force Based on this observation, to prevent rollovers, it

is necessary to reduce the effect of the lateral accelerationand the lateral force on vehicles

Following the aforementioned idea, several controlschemes were proposed to prevent rollovers The mostcommon scheme is to reduce the lateral acceleration throughdecreasing a reference yaw rate with differential braking oractive front steering to produce a vehicle with under-steercharacteristics (Odenthal et al., 1999; Chen and Peng,2001; Ungoren and Peng, 2004; Yoon et al., 2006; Scho-field and Hagglund, 2008) However, this approach has thedrawback of deteriorated maneuverability, and the yaw ratetracking performance due to this loss of maneuverabilitymay cause another accident such as a crash or tripped rollover.Another approach for rollover prevention is to controlthe lateral load transfer with an active suspension, which

*Corresponding author. e-mail: thewait@naver.com

148 S YIM, Y PARK and K YI

has a direct effect on the rollover (Yang and Liu, 2003;

Duda and Berkner, 2004) In this approach, the

pre-computed roll moment is dynamically distributed to front

and rear axles through the active suspension However, the

active suspension is used not for rollover prevention but for

active roll compensation, that is, for roll angle and roll

stiffness at normal driving conditions without rollover danger

This type of active suspension has some limitations in

pre-venting rollovers because the attenuation of the effect of

lateral acceleration on roll angle or roll rate is weak

To prevent rollovers, it is necessary to reduce the effect

of the lateral acceleration on the roll angle and rate

As-suming the lateral acceleration as a disturbance, a

con-troller can be designed to attenuate its effect on the roll

angle or roll rate or suspension stroke For this purpose, an

active suspension controller is designed with the linear

quadratic (LQ) optimal control methodology With LQ

optimal control, it is easy to design a controller to regulate

a particular state or output variable against a disturbance In

LQ control methodology, it is assumed that all states are

available However, in real applications, the full states of a

system are not always available For practical considerations,

it is desirable to use available sensor signals Hence, the

optimal static output feedback (SOF) methodology is used

to design an active suspension controller (Levine and Athans,

1970; Toivonen and Makila, 1987)

The active suspension controller designed for rollover

prevention has a tendency to produce an over-steer

charac-teristic in the controlled vehicle, which deteriorates the

maneuverability of the vehicle (Lee et al., 1998) To

over-come this drawback, it is necessary to design an ESP ESP

is designed with direct yaw moment control (DYC) In

DYC, the yaw moment control is computed by a sliding

mode control methodology and is distributed to each wheel’s

braking force (Rajamani, 2006)

This paper is organized as follows Section 2 presents

the design procedure of active suspension for rollover

pre-vention In this section, LQ static output feedback control

methodology is adopted to design a controller, and

simu-lation is performed on a nonlinear vehicle model based on

commercial multi-body dynamics software, Carsim

(Mech-anical Simulation Corporation, 2001) ESP is designed with

direct yaw moment control, and simulation is also

perform-ed in section 3 Section 4 concludes this paper

2 DESIGN OF ACTIVE SUSPENSION FOR

ROLLOVER PREVENTION

2.1 Vehicle Model

The vehicle model for active suspension is a 4-DOF plane model, as shown in Figure 2 This model describesthe vertical and roll motion of a sprung mass and the verti-cal motion of an unsprung mass The disturbances acting

roll-on a vehicle are the road inputs z r1 and z r2 and the lateralacceleration a y

Equations of motion for this model can be obtained asfollows:

(1)

In Equation (1), f 1 and f 2 are suspension forces, defined asfollows:

(2)Using the above definitions and the assumption ,the equations of motion (1) are summarized as follows:

(3)where

Equation (3) can be rewritten in matrix form as Equation(4)

(4)Figure 1 Factors influencing rollovers

Figure 2 4-DOF roll-plane model

Rewriting Equation (4), Equation (5) is obtained as follows:

(5)where

With the definition of a state, the state-space equation for

the vehicle model is as follows:

(6)where

2.2 LQ Static Output Feedback Control

To design a controller for active suspension, LQ SOF

con-trol methodology is used The LQ objective function is given

in Equation (7), containing the terms that emphasize

verti-cal and roll acceleration, roll angle and rate, suspension

stroke, tire deflection, and control input

(7)For ride comfort, it is necessary to reduce the vertical

and roll acceleration However, this does not guarantee

rollover prevention Generally, it is known that it is

desir-able to reduce roll angle or roll rate to prevent rollover

However, the reduction of the roll angle or roll rate cannot

guarantee rollover prevention because these parameters

only weakly attenuate the effect of lateral acceleration on

the roll angle or roll rate To prevent rollover by attenuating

the effect of lateral acceleration on roll angle or roll rate, it

is essential to reduce the suspension stroke and tire

deflec-tion To demonstrate this, the following sets of weights are

proposed, as shown in Table 1 CASE1, CASE2, and CASE3

give large weights to the vertical and roll acceleration, the

roll angle and roll rate, and the suspension stroke and tire

deflection, respectively The weights of CASE2 and CASE3

are selected such that LQ objective functions have the

nearly same values

LQ SOF control objective is to find a controller with theform u=− Ky such that the LQ objective function (7) isminimized (Levine and Athans, 1970) In this paper, theoutput y is the roll rate, suspension stroke, and stroke rate,

as shown in Equation (8)

(8)

There have been several methods to compute the optimal

K (Toivonen and Makila, 1987) However, this problem hasnot been proven to have a global optimum In this situation,

a heuristic search is a good alternative to the classicalgradient-based search (Toivonen and Makila, 1987) Forthis reason, the evolutionary strategy, CMA-ES, is used tofind the optimal K (Hansen et al., 2003)

x = Ax + B 1 w + B 2 u

Table 1 Three cases for each control purpose

CASE1 1e3 1e3 1e1 1e1 1e1 1e1 1e-4

CASE2 1e1 1e1 1e7 1e7 1e1 1e1 1e-4

CASE3 1e1 1e1 1e1 1e1 5•1e7 5•1e7 1e-4

Table 2 Suspension parameters of SmallSUV in CarSim

150 S YIM, Y PARK and K YI

2.3 Evaluation of Active Suspension System

The parameters of the vehicle model are obtained from

SmallSUV given in CarSim, as shown in Table 2

For the three cases of weights in Table 1, the LQ SOF

gains are computed by evolutionary strategy, CMA-ES

With these gains, the Bode plots of the closed-loop system

from the road input and the lateral acceleration to each

output are shown in Figure 3

As shown in Figure 3(a) and (b), for the road input, the

responses of the vertical acceleration and roll angle are

improved for CASE1 However, the responses of the roll

angle and suspension stroke are deteriorated for CASE2

and CASE3 For the lateral acceleration input, CASE3

shows the best performance for the responses of the roll

angle and suspension stroke at the expense of deteriorating

those of the vertical acceleration and roll angle, as

com-pared with CASE 1

Table 3 shows the H ∞ norms of each controller and each

input-output channel As shown in Table 3, CASE3 has the

best performance in attenuating the effect of the lateral

acceleration on the roll angle and suspension stroke

There are several measures to assess the rollover danger,

such as lateral acceleration and lateral transfer ratio

Among these, the rollover index (RI) based method is very

simple and powerful (Yoon et al., 2006) In this paper, the

RI, calculated as shown in Equation (9), is used to assess

the rollover danger If the RI is equal to unity, the left or

right wheels lift off

(9)

To demonstrate the effect of the active suspension in

preventing rollovers, simulation is performed with the three

cases of controllers on the vehicle model SmallSUV given

in CarSim Steering input is the fishhook maneuver with a

maximum angle of 270 degree, as described in NHTSA

(NHTSA, 2004b) Initial vehicle speed is set to 80 km/h,

and there are no controls to maintain a constant speed The

tire-road friction coefficient is set to 1.1

Figure 4 shows the simulation results of each controller

As shown in Figure 4(a) and (b), CASE2 and CASE3 can

prevent the vehicle from rolling over However, CASE2shows severe chattering in control input (Figure 4(c)).These results show that the active suspension designed toreduce vertical/roll acceleration (CASE1) cannot mitigaterollovers and that the active suspension designed withCASE2 has severe chattering in control forces From theseresults, it can be concluded that the active suspensiondesigned with CASE 2 or CASE3 can prevent rollovers

To check the effect of CASE3, simulations are

perform-Table 3 H ∞ norms for each case and each input-output

Figure 4 Simulation results for each case

Figure 5 Trajectories for each speed

ed at various speeds Figure 5 shows the trajectories of

vehicles with the active suspension designed by CASE3

As shown in Figure 5, the controlled vehicle with CASE3

demonstrates severe over-steering because the controlled

vertical force on a tire results in increased lateral force (Lee

et al., 1998) This means that the maneuverability is

deteriorated The loss of maneuverability can cause other

accidents such as crashes or tripped rollovers Hence, it is

necessary to design an ESP to maintain maneuverability

3 DESIGN OF ESP FOR MANEUVERABILITY

3.1 ESP Design

An ESP is a device developed to maintain maneuverability,

that is, yaw rate tracking performance To design the ESP,

the linear 2DOF bicycle model is used, as shown in Figure

6 Assuming a linear lateral tire force, the equations of

motion for a linear bicycle model are given in Equation

For a fixed longitudinal speed v x, the reference yaw rate γ d

is given in Equation (12) (Rajamani, 2006)

(12)

To force a vehicle to track the reference yaw rate, the

direct yaw moment control is applied with sliding mode

control theory To force the error between the reference

yaw rate and actual one to zero, the sliding surface is

de-fined as given in Equation (13) For this sliding surface to

have stable dynamics, condition (14) must be satisfied

Combining Equations (10), (13), and (14), the control yaw

moment M B is obtained as Equation (15) (Uematsu and

Gerdes, 2002)

(13)(14)(15)After the control yaw moment is obtained by DYC, it isnecessary to distribute brake pressure to four wheels togenerate the given control yaw moment The given controlyaw moment is transformed into braking force of the frontwheel as follows:

(16)The relationship between the braking force F x,front andbrake pressure P B,front on the front wheel is assumed asfollows:

(17)From Equations (16) and (17), the brake pressures ofeach wheel can be obtained for a given control yaw moment

M B The relationship between braking pressures of frontand rear wheels can be obtained as follows:

(18)The brake pressure is applied to only one set of either theleft or the right wheels For example, if the sign of the con-trol yaw moment is positive, then braking pressure is ap-plied only to the left wheels, and vice versa

3.2 Evaluation of Active Suspension and ESPThe simulation conditions are identical to those of the pre-vious section except that the active suspension is designedwith CASE3 and that ESP is applied The parameters of

SmallSUV used in ESP are given in Table 4 Figure 7 showsthe simulation results of the vehicle with an active suspen-sion and ESP In Figure 7, the legends AS Only, ESP Only,and AS+ESP indicate a vehicle with an active suspension,with ESP, and with both, respectively

As shown in Figure 7(b), the rollover index is over unity

if only the ESP is applied This means that the vehicle is indanger of a rollover In comparison, a vehicle with activesuspension and ESP has a small roll angle and rolloverindex, as shown in Figure 7(a) and (b) With active suspen-sion and ESP, a rollover cannot occur at any speed As thetrajectories of the controlled vehicle show in Figure 7(c),the vehicle with ESP is not drifted while preventing a

Figure 6 2-DOF bicycle model including the control yaw

152 S YIM, Y PARK and K YI

rollover This means that maneuverability is not

deterio-rated due to the ESP Figure 8 shows the trajectories of the

controlled vehicle for various speeds Contrary to theresults in the previous section, the ESP can maintain themaneuverability of the controlled vehicle without rollover

at high speeds From these results, it can be concluded thatthe proposed method is effective in preventing rollovers

4 CONCLUSION

In this paper, a rollover prevention controller was proposedfor vehicles with a high C.G., such as SUVs and vans.Active suspension with lateral acceleration as a disturbancewas designed The controller gains were obtained throughthe LQ SOF method for several weightings Through Bodeplot analysis and simulation, the controller, with high em-phasis on the suspension stroke and tire deflection, caneffectively prevent the vehicle from rolling over Despitethe remarkable performance in mitigating the rollover, thiscontroller resulted in over-steer characteristics for thevehicle, deteriorating maneuverability ESP was designed

to maintain the maneuverability of the controlled vehicle.Through simulations, it is concluded that the proposedmethod can effectively prevent rollover at any speed

As shown in Figure 3, the active suspension designed forrollover prevention deteriorated the ride comfort To over-come this drawback, the proposed active suspension should

be activated under rollover situations To accomplish this, aswitching scheme between normal and rollover situationwill be developed in future research

ACKNOWLEDGEMENT− This work was supported by the second stage BK21 Project and the Korea Science and Eng- ineering Foundation (KOSEF) through the National Research Laboratory Program (R0A-2005-000-10112-0).

REFERENCES

Chen, B and Peng, H (2001) Differential-braking-basedrollover prevention for sports utility vehicles with human-in-the-loop evaluations Vehicle System Dynamics 36, 4-

Lee, J S., Kwon, H J and Oh, C Y (1998) A study ofeffects on the active suspension upon vehicle handling

Trans KSME, Part A, 22, 3, 603−610

Levine, W S and Athans, M (1970) On the determination

of optimal constant output feedback gains for linearmultivariable systems IEEE Trans Automatic Control,

15, 44−48

Mechanical Simulation Corporation (2001) CarSim User Manual Version 5

National Highway Traffic Safety Administration (2003)

Figure 7 Simulation results for each case

Figure 8 Trajectories of the controlled vehicle at various

speeds

Motor Vehicle Traffic Crash Injury and Fatality Estimates,

2002 Early Assessment, NCSA (National Center for

Stati-stics and Analysis) Advanced Research and Analysis

National Highway Traffic Safety Administration (2004a)

Traffic Safety Facts 2004, US Department of

Transpor-tation.

National Highway Traffic Safety Administration (2004b)

Testing the Dynamic Rollover Resistance of Two

15-passenger Vans with Multiple Load Configurations, US

Department of Transportation

Odenthal, D., Bunte, T and Ackermann, J (1999)

Non-linear steering and braking control for vehicle rollover

avoidance European Control Conf., Karlsruhe, Germany

Rajamani, R (2006) Vehicle Dynamics and Control New

York Springer

Schofield, B and Hagglund, T (2008) Optimal control

allocation in vehicle dynamics control for rollover

miti-gation American Control Conf., Westin Seattle Hotel,Seattle, Washington, USA, June 11-13, 3231−3236.Toivonen, H T and Makila, P M (1987) Newton's methodfor solving parametric linear quadratic control problems

Yang, H and Liu, Y U (2003) A robust active suspensioncontroller with rollover prevention SAE Paper No 2003-01-0959

Yoon, J., Yi, K and Kim, D (2006) Rollover index-basedrollover mitigation system Int J Automotive Technology

7, 7, 821−826

International Journal of Automotive Technology , Vol 11, No 2, pp 155 − 166 (2010)

155

APPROXIMATIONS TO THE MAGIC FORMULA

A LÓPEZ * , P VÉLEZ and C MORIANO

Industrial Engineering Department, Universidad Antonio de Nebrija, C/Pirineos 55, Madrid 28040, Spain

(Recevied 5 December 2008; Revised 8 June 2009)

ABSTRACT− Pacejka’s tire model is widely used and well-known by the automotive engineering community The magic formula describes the brake force, side force and self-aligning torque in terms of the longitudinal slip and slip angle, plus several corrections This paper uses approximation theory to obtain different types of approximations to the magic formula: rational functions (RA) resulting from the Remez algorithm, expansions in a series of Chebyshev polynomials (ACh), a series

of Chebyshev rational polynomials (ARChPs), a series of rational orthogonal functions (ORF) and a series of ARChPs that result from grade-1 ORFs The last expansion shows the fastest convergence and most effective computation Jacobi rational polynomials can also be obtained to complement this expansion and facilitate fine-tuning in specific areas of the error curve This work is complemented by obtaining the original rational approximations to the inverse tangent function, which take advantage of the curve symmetry to reduce the computation load and provide models that include the influence of the vertical load The convergence properties of the development in series and the error values resulting from numeric examples for the three types of stress are shown The proposed final ARChP expressions show very low error (1%) compared to the original magic formula They can be computed 20 times faster; they can be evaluated, derived and integrated analytically easily; and their coefficients can be obtained from tests using common least-squares algorithms

KEY WORDS : Magic formula, Tire model, Approximation theory

1 INTRODUCTION

This paper searches for a Chebyshev series expansion of

Pacejka’s tire model in order to obtain a more efficient

mathematical expression with enhanced analytical

proper-ties that can be integrated in the series expansion of the

equations that describe vehicular dynamics The final aim

is to advance toward analytic solutions of those equations

using symbolic computing

López et al. (2006) has provided an example of

expan-sion in the power series of a simple longitudinal dynamics

of a vehicle The resulting polynomial expressions

facili-tate very fast computation of the dynamic equations in real

time Moreover, pre-computation of answers dependent on

the model entries can be achieved simply with the use of

symbolic computation tools (MAPLE) The need to obtain

simple and accurate formulations for the tire model to be

integrated into the previous dynamic model led to the

publication (López et al., 2007) of a first paper, which in

turn led to the development of RA, ACh and ARChP

approximations for longitudinal stress and bivariate

ap-proximations to the magic formula Now, that report has

been completed and further expanded with the addition of

rational orthogonal functions theory and ORF and ARChP

expansions stemming from ORFs Expansions in Jacobi

polynomials are also added for exact shift adjustments at

the origin and to obtain the error in different sections of thecurve This article covers three types of stress in addition tosimplifications based on curve symmetry

A new, efficient bivariate expansion is presented cients are easily obtained from the tests using standard leastsquares algorithms

Coeffi-2 REVIEW OF THEORETICAL BASIS

2.1 Approximation of a Function in a Chebyshev Series(ACh)

Chebyshev polynomials (Fox and Parker, 1968) of the firstkind are defined by

and are orthogonal to the function w(x)=(1− x 2)−1/2 on theinterval [−1, 1]

To work in different [a, b] intervals, shifted polynomialsmust be used:

Their general expression (Abramowitz and Stegun, 1972)

where is the largest whole number less than or equal

to n/2

They fulfill the following recursive property:

n=1, 2, (1)Chebyshev polynomials can be computed and manipulated

using the MAPLE Orthopoly library

The expansion of a function in a Chebyshev series

(ACh) has the following form:

,The single comma in the summation indicates that the first

term must be divided by 2

This expansion usually converges faster than the power

series, and the coefficients are described by the following:

If we truncate the series at N degrees, we get an

approxi-mation to the function: the accuracy of the approxiapproxi-mation

improves as N increases Because of the properties of

Chebyshev polynomials, truncating the function at N-1

degrees is the best N-1-degree polynomial approximation

to the function with N degrees

The coefficients a n can be assessed with direct

integ-ration in some functions, but, in general, this calculation is

not possible, and the previous integral must be

approximat-ed by some other quadrature formula MAPLE uses

quad-rature algorithms that first analyze the singularities and

then use Clenshaw-Curtis quadrature (Clenshaw and Curtis,

1960; Waldvogel, 2006); if the result is not satisfactory,

Newton-Cotes adaptive formulae are used All of these

algorithms are carried out in the Chebpade function from

the MAPLE Numapprox library of approximation of

func-tions

2.2 Approximation Using Rational Functions (RA)

RA approximations are more efficient when the function

varies rapidly in some areas but not in others, which occurs

in tire behavior, especially when longitudinal stress is

con-sidered

The Padé approximation provides rational expressions

with their numerators and denominators developed in power

series They are processed efficiently as a continuous

frac-tion Chebyshev-Padé developments generate more

com-pact and accurate rational expressions with Chebyshev

polynomials in their numerator and denominator The

MAPLE Numapprox library also implements the rational

approximations Its Chebpade function turns the initial

Chebyshev function into a power series, carries out a Padé

approximation and turns the resulting numerator and

de-nominator into Chebyshev series again

Chebyshev-Padé functions obtain good approximations,

but not those of minimum-maximum error (known as

minimax) To find the latter, the second Remez algorithm

(Remez, 1934) is used, which is a modified Padé approximation; it fine-tunes the result with numericiterations and converges to an improved minimax approxi-mation

Chebyshev-The second Remez algorithm produces optimal resultsthat approximate both rational and polynomial functions.This function allows the minimum error of any given func-tion f(t) weighted with any weight term w(t) to be calcu-lated If w(t)=1/|f(t)| is used, the minimum relative error isobtained The minimax approximation with n-degree poly-nomials in the numerator and m-degree polynomials in thedenominator requires (n+m) additions and (n+m) multipli-cations for its evaluation, which are indicated as a minimaxapproximation [n,m]

These methods are described in many books on mation theory (Powel, 1981)

approxi-In MAPLE, the Remez algorithm is implemented by theminimax function that is included in the Numapprox library

;where

The development of a function in a series of Chebyshevrational polynomials is:

for Chebyshev polynomials are orthogonal on the interval [−1,1], but our independent variables (slip K and lateral slip)vary between 0 and 100 and between −15o and 15o, respec-tively (because the formula is the same, we will genericallycall both of them x, and their initial and end points x in and

x fin, respectively) Thus, the Chebyshev expansion in series

on the original variable x cannot be performed because itsdomain exceeds the orthogonality of Chebyshev polynomials

Therefore, shifted polynomials at v (Fox and Parker, 1968,

p 49), , with the following variable changemust be used:

T nd( )=T v n ( ) u

APPROXIMATIONS TO THE MAGIC FORMULA 157

where u shifts between −1 and 1 Therefore, the final

development is:

We can see that a double transformation, from the x to y

domains and from the v to u domains, was required The

function in the u domain is approximated by a Chebyshev

development in series, and in the resulting approximate

function, the two previous transformations are undone to

obtain the approximate function in the original domain x

(slip or lateral slip)

2.4 Rational Orthogonal Functions (ORF) Theory

2.4.1 Introduction

According to Bultheel et al. (1999), if A={α 1, α 2 }, is a

sequence of real numbers other than zero, the linear vector

space of n-degree rational functions with poles at {α 1 , …,

α n } is defined by the space of functions L={b0, b1…, bn},

where the base functions are defined by:

If we orthonormalize and assume an interval on the real

line that excludes every pole, then these functions meet the

recurrence relation

Van Deun et al. (2004) has obtained the coefficients of the

recurrence relation (E n, F n) for the case of Chebyshev ORF

functions with Chebyshev weight functions A Chebyshev

weight function is a Jacobi weight function of the type:

where δ y γ= ±1/2

When the Joukowski transformation is introduced,

Van Deun gets the coefficients:

If n=1, we get:

;where

Examples and application to the magic formula are shown

in section 3.6

2.4.2 Expansion in ORFs seriesThe best least-square approximation obtained after truncat-ing the expansion in a series of orthogonal functions F(x)(of any type) of a function f(x) is (Burden and Douglas,

1998);

(2)where the coefficients are:

The weight function w(x) defines the importance of theapproximation of different sections of the interval [x 1, x 2].For example, the Chebyshev weight function:

has very little influence in the center of the interval andmore influence at its ends

In this particular case, F j(x) are ORFs The value of r j

represents the norm of the ORF function, and for ORFfunctions with Chebyshev weight functions, it takes theconstant value r j=π We recall that, in the case of Cheby-shev polynomials F j(x)=T(j,x), this value was r j=π/2.MAPLE does not support any function related to ORFs,neither the generation nor expansion of functions Expan-sion of the magic formula in ORF is presented in section3.6

2.4.3 Expansion in a series of Jacobi polynomialsWithin the families of classic orthogonal polynomials gene-rated from the Sturm-Liouville differential equation, fromwhich Chebyshev polynomials are also derived, we consi-der Jacobi polynomials (Totik, 2005) The weight function

v=1/2 v [ ( fin – v in )u+ v ( in + v fin ) ]; ⇒

v fin – v in - –

F 1a=− 2β 1 c; F 1b=−β 1 c F 1c= 1 β ( – 1 )c c= 1 β – 1

1 β + 12 -

in this type of polynomial

is controlled by two parameters, δ and γ, which allow the

area of a best approximation in the orthogonality interval to

be chosen In practice, this is very interesting because it

allows us to improve the error adjustment in any area of the

longitudinal stress, lateral stress, or self-aligning torque

curves, depending on the application in which the

approxi-mation is used: for instance, looking either for a more

reduced error in slip values close to zero or for values close

to the maximum stress or the maximum slip point (100%)

The norm r j in Jacobi polynomials is not constant, and it

is a function of δ, γ and the degree of the n-polynomial

The recurrence relation seen for the Chebyshev

polyno-mials (1) in section 2.1 is made more general in the case of

Jacobi polynomials:

where the recurrence coefficients are now:

Jacobi polynomials can also be computed and manipulated

using the MAPLE Orthopoly library

The expansion of a function in a series of Jacobi

poly-nomials uses the same expression (2) as in section 2.4.2,

but with a Jacobi weight function The integral must be

programmed in MAPLE A library for expansions of

func-tions in Jacobi series is not available

2.5 Magic Formula

The well-known model proposed by Bakker et al. (1987,

1989) and Pacejka (2002), is a semi-empirical tire model

based on the “magic” formula:

Y=D.sin[C.arctan(BX–E.[BX-arctan(BX)])]

The shape of the curve is controlled by four parameters: B,

C, D and E The equation can calculate the following:

• Lateral forces in a tire, Fy, as a function of the slip angle

of the tire, α, (in degrees)

• Braking force, Fx, as a function of longitudinal slip K (%)

• Self-aligning torque, Mz, as a function of the slip angle α

B, C, D and E are constants that describe the inclination of

the curve at the origin (BCD), the peak value (D), the

curvature (E) and the basic form (C) for each case (lateral,

braking or self-aligning torque) In addition, the curve can

have vertical (Sv) or horizontal (Sh) shifts at the origin.The full expression is:

Y=D.sin[C.arctan(B(X+Sh)–E.[B(X+Sh)

−arctan(B(X+Sh))])]+SvCoefficients B, D and E are functions of the vertical load inthe tire, Fz:

BCD 2=a 3.sin(a 4(arctan(a 5.F)));

BCD1 is valid for the longitudinal force and the aligning torque with C=1.65 and C=2.4, respectively.BCD2 is valid for the lateral force with C=1.3.The Camber angle γ in the wheel modifies the shifts Shand Sv and the stiffness BCD:

arctan(x)≈

which is valid , anti-symmetrical, has a very low solute maximum error ε |ε| < 0.0025, is far more accuratethan other pseudoarctan(x) formulations presented in theliterature , and has “nice” coefficients

ab-In continuous fraction form, it can be expressed as:arctan(x)≈sign(x)

The function requires four additions and two divisionsplus the sign Even lower errors can be obtained by ap-proximations [2, 3] or higher

x 4.66 8 x ( + )

5 6 x 5.1.x + + 2 - x

∀

x 0.4741 – + -x 1.6505741.762934+

–

APPROXIMATIONS TO THE MAGIC FORMULA 159

3.1.2 Sin(x)

The function sin(x) also appears with −4.1 < x < 4.1 rad If

we proceed in the same way as with arctan(x), we get:

|ε| < 0.018, 9 Op (5 Mul, 1Div and 3 Add)+2.abs(x)

There are more efficient approximations [2,1] at the

longitudinal and lateral stress ranks:

(−2.3 < x < 2.3)

|ε| < 0.014, 6 Op (3 Mul, 1Div 2 Add)

3.2 Direct Approximations ACh to the Magic Formula

In this article, we consider relative error to be the absolute

error divided by the maximum absolute value of the

func-tion This approach is convenient because approximations

with a minimax classic relative error (divided by the

modulus in each x value) give good results in low force

sections of the curve close to 0 (the least interesting

section), but very poor results in the rest of the curve (the

most interesting part) Our definition allows us to compare

errors for different vertical loads easily

The expansion in the Chebyshev series of the magic

formula does not allow the use of low degree polynomials;

the following table shows the polynomial degree and the

relative error (as defined in the previous paragraph), with a

vertical weight Fz=8 kN

The high values of the normal weight are those that need

a higher polynomial degree

Regarding lateral force, the direct ACh of the magic

formula requires n≥5 polynomials at the rank 0 < x < 15o

to cover all the weight values in our sample tire (from 0 to

10 kN) For the rank −15 < x < 15o, we need at least n≥20

degree polynomials: low normal weight values require a

higher polynomial degree

For the self-aligning torque we need n≥13 for the half

interval and n≥45 for the complete interval 15 < x < 15o

Therefore, using direct expansions in Chebyshev series

in the magic formula is not a good idea, because

conver-gence is not fast enough

3.3 Rational Approximations (RA) to the Magic Formula

3.3.1 Longitudinal force

The minimax [2,2] adjustments carried out with the MAPLE

Numapprox library related to our sample tire give relative

error values that increase with the vertical weight Fz from

Rel.Error=0.9% for Fz=1 kN, to 1.36% for Fz=8 kN

Using minimax approximations [2,3], we get Relative

Error values that fluctuate between 0.36% for Fz=1 kN and

0.79% for Fz=8 kN:

Figures 1 and 2 show both the adjustment of the mation [2,3] and the error For the case of [2,2], the curvehas a similar shape but a slightly higher error

approxi-If we accept an acceptable maximum relative errorcriterion of less than 1%, we must work with minimaxapproximations [2,3]

If we delete the independent term from the numerator inthe previous approximation [2,3], we get a slightly highererror; however, this error is zero at the origin

3.3.2 Lateral force and self-aligning torque

If we proceed in the same way, we can see Error < 1% inminimax adjustments [5,3] for self-aligning torque, where

−15 < x < 15o The figures show the lateral stress curve andthe error curve with Fz=8 kN

We will show how to take advantage of the symmetryusing the results of the approximation between 0 and 15o insection 3.9

Regarding the self-aligning torque, if we want to

proximate the whole rank between −15 and 15 with a Rel

Error < 1%, we must use an approximation [5,5]

We check the resulting curves for Fz=3 kN

If we focus on the rank 0 15, the approximations [2,3]

produce results with a Relative Error < 1%

3.4 Approximations ARChP to the Magic Formula with

Constant Fz

Applying the expansion described in section 2.3, if we

include the suggested double transformation, we can see

that the ARChPs of the form:

give results with a Relative Error < 1% with polynomials:

n≥8 for Fx with 0≤x≤100, Fy with 0≤ α ≤15 and

0≤Fz≤8 kN

n≥12 for Mz with 0≤ α ≤15, and 0≤Fz≤8 kN

It is evident that convergence is faster than in the case ofACh direct expansions; however, it is still not satisfactorybecause the polynomials have at least eight degrees 3.5 ARChP Approximations from ORFs

The convergence speed can be improved if we expand in aChebyshev series of rational functions of the type:The optimal factor b in each case varies with Fz

Figure 3 Rational approximation [5,3] of the lateral force

as a function of the lateral slip

Figure 4 Absolute error (N) at RA [5,3] of the lateral force

versus lateral slip

Figure 5 Rational approximation [5,5] of the self-aligningtorque

Figure 6 Absolute error (N.m) at the rational mation [5,5] of the self-aligning torque

approxi-APPROXIMATIONS TO THE MAGIC FORMULA 161

We obtain Relative Error < 1% in the following

expan-sions:

n≥4 with b=4 for Fx with 0≤x≤100 and 0≤Fz≤8 kN

n≥4 with b=4 for Fy with 0≤ α ≤15 and 0≤Fz≤8 kN

n≥9 with b=3.5 for Mz with 0≤ α ≤15 and 0≤Fz≤8 kN

The specified values of b guarantee a RelError < 1% for

0≤Fz≤8 kN However, we can improve the error for

every value of Fz by modifying b slightly

In Fx, for 4≤Fz≤7 kN, the result is n≥3 with Error < 1%

This expansion calculation in MAPLE is performed with

the Minimax-Remez function, which produces more

accu-rate results than the Chebpade function, as has already been

stated

These results are excellent: for example,

Fz=6 kN (constant)

C=1.65; D=6097.2; B=0.2064; E=0.606; BCD=2076.600

a1=-21.3; a2=1144; a3=49.6; a4:=226; a5=0.69e-1;

a6=-0.6e-2; a7=0.56e-1; a8=.486

xin=0; xfin=100; vin=0; vfin=100/104; b=4

Original equation (Magic formula)

If we execute expansions in x for different values of Fz,

look for the optimal b in each case and calculate the gression of b=f(Fz), we increase the convergence speed; inthe case of our sample tire, we can get:

re-n≥3 with b=5.4629-0.2829.Fz for Fx with 0≤x≤100 and 0≤Fz≤8 kN

3.6 Monopole ORF Approximations to the Magic FormulaWhen applying the results given in section 2.4, we can getthe base of the monopole ORF functions from the values ofb

As an example, the three ORFs and the approximationfor the braking force are shown for the same tire data andmaximum force (6 kN)

In this case, the error curve is very similar to that shown inFigure 8, although it fluctuates between −60 N and +60 N.This approximation is less accurate and requires morecomputation than the previous one because the previousapproximation was a minimax (this can be seen whenchecking the maximum local error leveled in Figure 8), andthis approximation is a minimum squared expansion,which is less precise

3.7 Bipole ORF Approximations to the Magic FormulaUsing the same tire data as in the previous example, start-ing from pole 4, the third pole increases and the initialbipole decreases until the minimum error is observed:

52 25 -

v= xx 4 -+

Figure 7 Braking Force (N) versus Longitudinal slip in

Chebyshev series of the rational function x/(x+b)

Figure 8 Absolute Error (N) versus Longitudinal slip in aChebyshev series of the rational function x/(x+b)

In this case, the error is better than the previous ones, and

because it is similar to Figure 8, it fluctuates between −40

N and +40 N

This bipolar ORF approximation has the same error as

RA minimax [2,3], although the minimax has complex poles

Similar ORF expansions can be performed for lateral force

and self-aligning torque

3.8 Approximations in a Series of Jacobi Rational

Poly-nomials

If we use the expressions in section 2.4.4., we can start

from the values δ=−1/2 and γ=1/2, which correspond to

the Chebyshev weight function (which, in turn, is a

parti-cular case of Jacobi polynomial) Increasing both values

reduces the error in the central area of the curve and

increases it at the ends The error can also be adjusted to

zero at the ends by keeping one of the two parameters fixed

and changing the other, while keeping the maximum error

values constant for the whole curve Thus, for instance, the

value of shifts from the origin to the values Sv and Sh

indicated in section 2.5 can be adjusted

For example, for the same tire with Fz=6 kN, an

adjust-ment with a null error is shown at the origin (Sv=Sh=0)

with the values δ=−1/2 and γ=−0.4685/2 The resulting

approximation (with b=3.85) is the following:

Fxap=(7369.26+(15916.3-19867.52.v).v).v

where

The error curve looks very similar to that shown in Figure

8, which is also between ±50 N, but the current error curve

has a null error at the origin

We can adjust the null error at the end of the curve or at

its maximum using this method

We can also adjust the slope at the origin (the value BCD

in the original Pacejka formula) to obtain an exact value or,

with a moderate error, to achieve global maximum error

values around 1% For example, the following

In this type of approximation:

self-of the performed computation with a lower degree nomial

poly-To calculate the approximate expression we do thefollowing:

(1) Calculate the approximation to the original functionwithout shifts with a null error at the origin in theinterval 0 vfin using Jacobi polynomials

(2) Calculate the valid expression for the interval -xfin xfin,which passes through the origin

(3) Apply Sv and Sh shifts to the approximate expression.The following is an example for the lateral force Fy:a1=−22.1; a2=1011; a3=1078; a4=1.82; a5=0.208; a6=0;a7=−0.354; a8=0.707;

D=5270.4; BCD=1076.149; B=0.1571: E=−1.417Sh:=−0.126:Sv:=−181:Fz=6 kN

(1) Approximation using a series of Jacobi rational nomials of the function with Sv=Sh=0 and a null error

poly-at the origin (calculpoly-ated with α=−1/2 and β=0.71)

APPROXIMATIONS TO THE MAGIC FORMULA 163

The maximum absolute error is 68 N:

The resulting curves are shown in Figures 9 and 10

4 INFLUENCE OF THE VERTICAL LOAD

For the longitudinal force, we approximate the influence of

the vertical load from the curve Fap1 obtained for 1 kN by

adding the peak value factor D(Fz) that coincides with that

of the original formula (now it gives the approximate peak

value) and a second shape factor Ff Both factors are

func-tions of the vertical load Fz We associate the linear

coeffi-cient A’1 with stiffness at the origin We calculate the

regre-ssion with optimal values of Fs and A’1 for each value of

Fz

Fxap=D.(

We show an example of the braking force with three degrees

and a maximum relative error of 1.1% for 1 kN≤ Fz≤8 kN:

From the Jacobi approximation with Fz=1;

The stiffness factor with minimum error is:

A’1=−0.00102906.Fz2+0.0092337.Fz+1.104The slope at the origin is D.A’1/FS,where D the approxi-mate peak value

Figures 11 and 12 show the error in this approximation

If we use the original BCD value to calculate A’1, whereA’1=Fs.BCD/D, we can use existing data; however, theerror is three times greater

Clearly, we can integrate the factor D into the A’i, cients by finding the products D.A’2 and D.A’3, and thenreducing the product D.A’1 to two degrees using Minimax-Remez [2,0] We can also consider A’2 and A’3 to vary with

coeffi-Fz for a longer period of time to obtain more accurateexpressions: shorter expressions of D.A’1(Fz)have a largererror There are many possibilities One of the simplest isthe following:

Figure 10 Absolute error (N) versus slip angle in a degree

3 symmetrical Jacobi approximation, with shifts Sv, Sh

Figure 11 Braking force versus longitudinal slip for able vertical load (1 kN≤Fz≤8 kN)

Vari-more with Fz; thus, we must use variable coefficients A2

and A3 with Fz The expression is the same:

We also use the original peak value D for Fy:

D=a1 Fz2+a2Fz; a1=−22.1; a2=1011;

Relative Error curves are similar to those in Figure 12, with

a maximum error of 2.6% for a 3-degree expansion with 1

kN < Fz < 8 kN

Once again we can integrate the products D.A’i=f(Fz),

with different results in terms of complexity and error

In all these examples, for both Fx and Fy, the error is 0 at

the origin and we can add original shifts Sv and Sx, as seen

in section 3.9, using existing data

5 COMPUTATIONAL EFFICIENCY

In order to compare the computational efficiency, we

consi-der the expressions shown in section 3.5:

Original equation (Magic formula)

Our proposed approximated formula computes 20 times

faster than the original Magic formula

(C-Compiler: MinGW (c); Intel Core2 CPU T5600 at1.83 GHz; 987 MHz, 1,99 GB RAM)

If we compare the expressions shown in section 3.9,which are valid for both the positive and negative sides:

F1xap=sign(x).(5166.26+(24265.833−30220.43.v1).v1).v1

F2xap=F1xap(x+Sx)+Svwith the full magic formula, including shifts:

Y=D.sin[C.arctan(B(X+Sh)–E.[B(X+Sh)

−arctan(B(X+Sh))])]+SvThen the approximated expression runs eleven times faster

In addition, integrating the model into the Chebyshevseries expansion of the equations that describe the vehi-cular dynamics is also possible Thus, analytic solutions with

a very high computational efficiency can be determined

6 DETERMINING THE COEFFICIENTS FROM TESTS

The previous equation:

Fxap=A0+(A1+(A2+A3.v).v).v

is a polynomial; thus, it is easy to obtain the coefficients Aifrom tests using the Least-Squares standard algorithms.Previously, we had to transform data from the original vari-able (for example, slip) to the transformed variable, v=x/(x+b) For example:

The slip data vector (%) (22 values, simulated example) isthe following:

[0,1,2,3,4,5,6,7,9,13,17,21,25,31,37,43,49,58,68,78,88,100]The braking force data vector Fx (22 values, simulated ex-ample) is the following:

Figure 12 Relative error for variable vertical load

Figure 13 Transformation of slip values; v=x/(x+b)

APPROXIMATIONS TO THE MAGIC FORMULA 165

The optimal value b used in the transformation is

un-known, but at the end of section 3.5, we were able to obtain

the optimal values of b for a given tire, which can be

expressed with a linear expression in terms of Fz:

b=5.46−0.28.Fz

The optimal value of the coefficient b for this tire, can take

values from 3.2 to 5.2 when the normal load Fz changes

from 1 to 8 kN

If b takes values higher or lower than the optimal, the

addition of quadratic deviations from the test values (curve

4 in Figure 14) is always bigger The minimum addition of

quadratic deviations is found when b is optimal We only

need to program a loop to calculate the following for every

step:

−The transformed points (Curve 2 at Figure 13) for the

given b value

−Curve 3 of Figure 14 using a common least-squares

algorithm with three degrees

−Curve 4, which undoes the transformation from v to x

−The addition of quadratic deviations from curve 4 to test

values

In the loop, we vary the value of b in a wide range (from 1

to 20 for example), to find the optimal value of b for agiven normal load Fz Corresponding Ai values for theoptimal b are also optimal

Computing this loop takes 3 or 4 seconds The full cess is automatic

pro-Determining coefficients for variable Fz is accomplished

as described in Section 4 from the set of constant Fz curves

7 CONCLUSION

From the analysis of the different types of approximations

to the magic formula with constant Fz, we propose anefficient and accurate calculation using the following type

of expression:

being which results from expansion in a series of shifted Jacobirational polynomials as they converge at relative errorsaround 1% with low degrees (3 or 4); they can be adjusted

in specific curve areas, especially at the origin, and theyallow the use of the same expression for both sides of thesymmetric curve for Fy and Mz, as seen in section 3.9.Additionally, we can obtain analytic derivatives and inte-grals of this expression easily The latter is not possible inthe original magic formula The slope at the origin is alsocalculated easily (A1/b), and if we use 3-degree polyno-mials as proposed, we can calculate the abscissa of themaximum value analytically

If maximum accuracy is the main goal for a constantvalue of Fz, the use of Minimax-Remez rational approxi-mations, such as those seen in section 3.3, is recommended.The described techniques use state of the art theories offunction approximation together with the symbolic compu-tation programs (MAPLE) and show the advantages ofhandling the equations analytically, especially for Fx and

Fy, for which we can use 3-degree polynomials with verylow error Self aligning torque requires higher degrees

We can obtain different expressions of Fx and Fy pending on the normal load Fz, as in Pacejka’s originalformulation, and use the same peak value factor D, to takeadvantage of the already existing data

de-The proposed expressions can be computed much faster(20 times faster) than the original magic formula In addi-tion, integration in the Chebyshev series expansion of theequations that describe vehicular dynamics is also possiblewith high computational efficiency

Obtaining the coefficients from test samples is also easybecause the proposed expressions are polynomials and auto-matic Least-Squares algorithms can be used

ACKNOWLEDGEMENTS− This report has been financed by the Dirección General de Universidades (Comunidad de Madrid) and the Instituto Madrileño de Desarrollo IMADE.

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International Journal of Automotive Technology , Vol 11, No 2, pp 167 − 172 (2010)

167

METHOD FOR ANALYZING EPICYCLIC GEARBOXES

T CIOBOTARU 1)* , D FRUNZETI 2) and L JÄNTSCHI 2)

1)Military Technical Academy, 81-83 G Cosbuc Bvd., Bucharest 050141, Romania

2)Technical University of Cluj-Napoca, 103-105 Muncii Bvd., Cluj-Napoca 400641, Romania

(Received 19 March 2009; Revised 25 June 2009)

ABSTRACT− This paper presents a method for analyzing epicyclic gearboxes by evaluating the speeds, torques and power

of the external elements in epicyclic gear mechanisms, as well as the total ratios of the gear box The method is based on the equations that describe each epicyclic gear mechanism and rules that assign appropriate codes to the external elements The method emphasizes how power flows are transmitted through the epicyclic gears, as well as power losses Analysis of an epicyclic gear box is performed to illustrate the proposed method.

KEY WORDS : Epicyclic gear, Epicyclic gearbox

1 INTRODUCTION

Principles of Mechanisms (Willis, 1841) is widely

consi-dered to be the first publication dedicated solely to the field

now called kinematics, and discusses analytical modeling

of an epicyclic gear train for the first time in published

literature Later, Analysis and Design of Mechanisms (Lent,

1970) presents the methodology proposed by Willis for

finding the rotational speeds of each branch of the epicyclic

gear train in detail, along with specific methods for

designing three- and four-gear trains Further developments

of the method were made by Tsai (2001), by using graph

theory, and extending “the traditional concept of a lever

representation of a planetary gear set to one that includes

negative lever ratios” (Raghavan, 2007) These methods

have been found to be suitable in a series of applications

(Ashmore, 2006) and (Karaivanov and Popov, 2008)

Epicyclic gearboxes have been used as an alternative to

other methods (Kim et al., 2008b, 2008a) more frequently

because of their advantages: compact design, automatic

shifting, and increased control possibilities, for example

The goal of analyzing an epicyclic gearbox is to evaluate

the gear ratios, the power flow and the efficiency for every

stage (Lee, 2007) This paper presents a methodology to

analyze epicyclic gearboxes using elements of power flow

theory This methodology was established for the use a

computer for efficient calculations, and to evaluate the

symbolic expressions for gear ratios, torques and power

flows for all the elements This paper presents the

metho-dology applied to a stationary working regime

characteriz-ed by invariant specharacteriz-eds (dω/dt=0); consequently, the inertia

of gears and shafts is neglected

2 APPLICATION OF POWER FLOW THEORY

The epicyclic gear box consists of one or more epicyclicgear mechanisms (EGM), clutches and freewheels Everyelement may be described using linear equations to calcu-late speeds, torques and powers (Fogarasy and Smith,1995; Du and Zhang, 2007)

A power flow graph is a graphical representation of asystem of linear equations The power flow graph establi-shes the link between the input variables (speed, torque andpower) and the output variables and emphasizes how thepower flow is transmitted (Ryu et al., 2009) Consequently,the power flow graph represents a visual tool to easilygenerate an epicyclic gearbox diagram, as well as ananalysis tool

The power transmitted by an element may be calculatedusing the following formula:

where P is power (W); M istorque (Nm); and ω is angularvelocity (rad/sec)

The above formula indicates that the power is computed

as the product of two factors: the load factor (torque) andkinematic factor (velocity) Both torque and velocity may

be positive or negative; consequently, the power is positive(input or convergent power) or negative (output or emer-gent power)

The power flow graph uses the following basic elements:

• nodes that transform at least one factor (load or kinematic);

• arcs that transmit the power flow without changing thefactors

The graph elements obey the following two rules:(1) For every node, the sum of convergent power balancesthe sum of emergent power and dissipated power; thusthe total sum of power is zero:

*Corresponding author. e-mail: cticusor2004@yahoo.com

, (2)where P τi represents the dissipated power in node i.

This rule is derived from the general law of energy

conservation applied to the epicyclic gearbox structure

(2) For every node, the sum of the torques is zero (the law

of torque balance):

There are several types of nodes used to describe the

kinematic structure of epicyclic gearboxes The most

fre-quently used nodes are summarized in Table 1 Using the

graphical symbols listed in the table below, any kinematic

gearbox can be transformed into power flow graphs

3 EPICYCLIC GEAR MECHANISM

FUNDAMENTALS

The epicyclic gearbox consists of one or more epicyclic

gear mechanism clutches and brakes Its kinematic diagram

is presented in Figure 1

Consequently, the epicyclic gear consists of 3 elements

with fixed axles (sun gear, ring gear and carrier arm) and

one element with a rotating axle (planet gear) The

epi-cyclic gear mechanism presented in Figure 1 Epiepi-cyclic

gear has 2 degrees of freedom (DOF), but there are

mechanisms with more DOF, for instance, the Ravigneux

mechanism

According to the Willis principle, the equation

describ-ing the kinematics of EGM is as follows:

where is the ratio when the power is transmitted from

element x to element y and the axle of element z is fixed

The indices that indicate the elements of EGM have the

following rules (where n represents the total number of

EGMs in the gearbox):

• for sun gears: 1, 4, …, 3j – 2, …, 3n – 2;

• for ring gears: 2, 5, …, 3j – 1, …, 3n – 1;

• for the carrier arm: 0, 3, …, 3j – 3, …, 3n – 3.Applying the index rules listed above, equation (4), ingeneral, becomes the following:

For a single EGM (in this case j=1), equation takes thefollowing form:

If the carrier arm is fixed (ω 0=0), EGM transforms into

an ordinary gear mechanism with fixed axles;

consequent-ly, equation defines the ratio :

-Table 1 Main types of nodes

Node Equations

METHOD FOR ANALYZING EPICYCLIC GEARBOXES 169

(9)or:

In conclusion, the kinematics of an EGM with 2 DOF is

completely described by a single equation with 3 variables

The distribution of torques among the EGM elements is

given by the following system of equations:

From the first equation, the following relation results:

For an ideal EGM (no power losses), equation (12) is

similar to equation (7), but it has a different meaning

Equation (7) defines the kinematic ratio, and equation (12)

defines the torque transformation ratio For an ideal EGM,

the two ratios (kinematic and torque transformations,

respectively) are equal numerically

For each external element of the EGM, the power is

calculated using the following equation:

An EGM with 2 DOF has the representation shown in

Figure 1 It is assumed that there is an ideal meshing of

gears, so there are no power losses

In a real situation, the meshing of gears generates power

losses, mainly due to the friction The lost power is

transformed into heat and is dissipated

The existence of power loss modifies the torques but has

no influence on the speeds of the external elements of the

EGM Consequently, equation (7) remains valid, but the

real torque transformation ratio is given by the following

equation:

where the sign “~” (tilde) applies to the real torques and

powers to indicate that they differ from their respective

ideal values

Considering only nodes with two external elements, the

efficiency of the transmission is given by the following

equation:

(15)where the value of the exponent u takes into considerationthe sense of the power flow For a transmission includingepicyclic gears, the value of the exponent u could becalculated using the following relation:

where indices a and b represent the input and output shafts,respectively

4 ANALYSIS OF THE EPICYCLIC GEARBOX

Within an epicyclic gearbox, the power flows from input a

to output b using one or more paths, according to thestructure of the transmission

Analyzing the epicyclic gearbox requires calculating thetotal ratios of the gearbox for each stage, the distribution ofthe power flows among the EGMs, and the overall effici-ency for each stage The proposed method calculates allthese data in an efficient manner using commonly availablesoftware for solving algebra problems

The method is best illustrated using a gearbox as anexample; a kinematic diagram is presented in Figure 3.The considered gearbox consists of 3 epicyclic gearswith the constants K 1…K 3, 3 brakes denoted by B1…B3,and a clutch Moreover, it can achieve 4 forward speeds.The input shaft is denoted by a, and the output shaft isdenoted by b This gearbox is part of the ZF 4 HP 250transmission; the reverse is realized using an EGM as aninverter

The sequence of the brakes and clutch usage is presented

-=−M˜y ⋅ ω y M˜ x ⋅ ω x

-=i˜x y ,

i x yz, -; u=±1

Figure 2 Power flow graph of the EGM with 2 DOF

Figure 3 Kinematic diagram of the gearbox

Table 2 Sequence of brakes and clutch

Stage B1 B2 B3 C1st operated

2nd operated3rd operated4th operated

The equivalent power flow graph of the epicyclic gearbox

is presented in Figure 4

The epicyclic gearbox consists of 3 epicyclic gear

mech-anisms with constants K 1, K 2, and K 3, respectively; 3 brakes,

F1, F2 and F3; and the clutch A The power flow diagram

includes the branched nodes n1…n3 The input shaft is

labeled a, and the output shaft is labeled b

The power flow diagram allows links between the

ex-ternal elements of the EGMs to be identified quickly, and

facilitates calculation of the gearbox ratios

The first step in analyzing the epicyclic gear box is

calculating the gearbox ratios

For each EGM, a single kinematic equation may be

written By rearranging the order of terms, the following

equations are obtained:

The links between the external EGM’s elements are

described by the following equations:

The input element a is presumed to have a known speed:

for convenience, it is assumed that:

Consequently, the gear box ratio will be:

Equations (17)~(24) are valid for each stage of the

gear-box A specific stage is obtained as a single brake or the

clutch is activated according to the shifting diagram

pre-sented in Table 2 Consequently, the following equations

may be written for each of the four stages:

• for the 1st stage:

Adding one equation from relations (26)~(20) to the set

of equations (17)~(24), a system of 9 simultaneous tions is obtained Consequently, the nine variables ω 0…ω 8

equa-can be calculated

The calculation could be performed symbolically; thus,the ratio of the gearbox is expressed as a rational functionwhose variables are functions of the constants of the EGMs.Solving the systems of simultaneous equations for eachstage, the following speeds were obtained

In order to calculate the torques, the following equationsare derived for each node:

Figure 4 Power flow diagram of the epicyclic gearbox

Table 3 Speeds for EGM’s external elements

Gear 1 Gear 2 Gear 3

METHOD FOR ANALYZING EPICYCLIC GEARBOXES 171

• for the branched node 1:

Equations (30)~(35) together with one of the restrictions

(36)~(39) compose a system of simultaneous equations; for

instance, for the first stage, the system is the following:

(40)

Taking into consideration equations (9), (15), and (16)

and a real situation with power losses, the system of

simultaneous equations (40) becomes:

(41)

The exponents u 1, u 2, and u 3 can be calculated using the

relations determined for gearbox ratios presented in Table

3

To calculate the efficiency of the gearbox, equation (15)

is applied to the overall structure of EGMs:

(42)The gearbox ratio is expressed as a rational function

whose variables are the EGM constants:

In the presence of power losses, the torque formation ratio is given by the same rational function butwith different variables:

i a b , 1=F K ( 1 , K 2 , K 3 ), i ∈ { 1, 2, 3 }

Table 4 Speed, torque and power flow calculation results

1st stage 2nd stage 3rd stageNo

losses Real losses RealNo losses RealNo

1 1 0.560 0.562 0.359 0.3633.64 3.514 2.040 1.975 1.308 1.274

(44)

To perform the numerical calculations, an average value

was adopted for the efficiency of the EGM: η=0.9653 The

final numerical results are presented in Table 4

For the first stage, the reduced power flow diagram is

presented in Figure 5 It can be observed that only the

epicyclic gear mechanism denoted K 1 contributes to the

transmission of the power flow The same conclusion could

be formulated by analyzing the structure of the function

derived for the first stage:

Because the constant K 1 is the only variable, this EGM

only transmits the power flow

For the second stage, the ratio is calculated by:

,and the transmission of the power flow is realized by two

EGMs; the same conclusion results from the diagram

presented in Figure 6

For this stage, the power flow is transmitted through two

paths It may be observed that the power flow is

trans-mitted by each link in the same direction; in other words,

there is no internal power flow within the closed loop

For the third stage, brake F3 is engaged The power flow

diagram for this stage is presented in Figure 7 It may be

observed that, in the third stage, all epicyclic gear

mech-anisms contribute to power flow transmission The input

power splits among the three EGMs, and the EGMs noted K2 and K1 act as summing mechanisms

Additionally, the proposed method may generate quate software applications that benefit both undergraduateautomotive students and gearbox specialists

Fogarasy, A A and Smith, M R (1995) New simplifiedapproach to the kinematic analysis and design of epicyclicgearboxes. Proc Institution of Mechanical Engineers, Part C: J Mechanical Engineering Science 209, 1, 49−53.Karaivanov, A and Popov, R (2008) Computer aidedkinematic analysis of planetary gear trains of the 3ktype Proc 3rd Int Conf Manufacturing Engineering (ICMEN), Chalkidiki, Greece, 571−578

Kim, B S., Ha, S B., Lim, W S and Cha, S W (2008a).Performance estimation model of a torque converter Part1: Correlation between the internal flow field and energyloss coefficient Int J Automotive Technology 9, 2, 141−

148

Kim, C W., Jung, S N and Choi, J H (2008b) Automotivestructure vibration with component mode synthesis on amulti-level Int J Automotive Technology 9, 1, 119−122.Lee, C.-H (2007) Measurement and characterization offriction in automotive driveshaft joints Int J Automotive Technology 8, 6, 723−730

Lent, D (1970) Analysis and Design of Mechanisms.Prentice-Hall. New Jersey

Raghavan, M (2007) Efficient computational techniquesfor planetary gear train analysis 12th IFToMM World Cong., Besançon, France,1−5

Ryu, W., Cho, N., Yoo, I., Song, H and Kim, H (2009).Performance analysis of a CVT clutch system for ahybrid electric vehicle Int J Automotive Technology

10, 1, 115−121

Tsai, L W (2001) Enumeration of Kinematic Structures According to Function CRC Press LLC 155−182.Willis, R (1841) Principles of Mechanism CambridgeUniversity Press

Figure 5 Power flow diagram for the 1st stage

Figure 6 Power flow diagram for the 2nd stage

Figure 7 Power flow diagram for the 3rd stage

International Journal of Automotive Technology , Vol 11, No 2, pp 173 − 185 (2010)

173

OPTIMIZATION OF BUS ROLLOVER STRENGTH BY CONSIDERATION

OF THE ENERGY ABSORPTION ABILITY

C.-C LIANG * and G.-N LE

Department of Mechanical and Automation Engineering, Da-Yeh University, Changhua 51591, Taiwan

(Received 16 September 2008; Revised 5 March 2009)

ABSTRACT− Buses are an integral part of the national transportation system of each country A rollover event is one of the most important hazards that concerns the safety of the passengers and the crew in a bus In the past, it was observed after the accident that the deforming superstructure seriously threatens the lives of the passengers Thus, the stiffness of the bus frame

is the first thing that needs to be considered The unfortunate side of strengthening the bus superstructure is that it usually causes the bus weight to increase This paper presents an efficient and robust analysis method with which to design the bus superstructure for a reduction in occupant injuries from rollover accidents while the weight of the strengthened bus is maintained at the same level First, the absorbed energy of the bus frame and its components during a rollover were investigated by using a LS-DYNA numerical study The highest energy absorption region, which is the side section of the bus frame, was found and focused on for the investigation of a means to re-distribute the energy-absorption ability of the side frame component Then the thickness parameters that were obtained from the re-distribution of the energy-absorption ability were used in the analysis to optimize the design Finally, a prototype of the bus with a reasonable thickness for the window pillars and the side wall bars, which was based on the optimized parameters, was verified to ensure it satisfied ECE R66 In this paper, an effective usage of materials and an efficient and robust analysis method were presented to design the bus superstructure Although the optimization process for increasing the stiffness is simple, this study improves the upper displacement by 39.9% and the lower displacement by 49.3% (versus the bus survivor space) while maintaining the bus weight at the existing level.

KEY WORDS : Rollover, Bus superstructure, Survivor space, LS-DYNA, Absorbed energy, ECE R66, Optimization

1 INTRODUCTION

There are many heart-breaking bus accidents Although

bus rollovers are less likely than other kinds of accidents,

they are very severe (Table 1) According to worldwide

rollover accident statistics from 1973 to the present, there

have been more than 570 bus rollover accidents (UNECE,

2007) For this reason, since 1987, the Economic

Com-mission for Europe has enforced Regulation No.66 (ECE

R66), which regulates the strength of the bus’s structure, to provide protection to the bus and the coach’soccupants during rollover accidents through the maintenance

super-of a survival space (JASIC, 1998, 2006)

ECE R66 has almost become an international regulationfor all motorcoaches It allows bus manufacturers todetermine the crashworthiness in rollover events followingreal tests or computer simulations (JASIC, 2006) Thus, thegoal of the design is to strictly satisfy ECE R66 whilecarrying the required load with a minimum componentweight without failing The process of producing the bestbus superstructure has been termed as structural optimization.With advances in both computation and structural analysisvia the finite element method, this paper, which is based onECE R66, presents an optimization study of the bus’srollover strength by considering the energy absorptionability

In recent years, automotive industries are concentratingmore on vehicle rollovers The literature offers many paperswith which to study the structural strength and rolloverstrength of buses Indeed, only a few papers have studiedthe optimal structural design of bus rollovers The rollover

of a bus is simulated by using a full FEA program and theresearchers (Kumagai et al., 1994; Niii and Nakagawa,

*Corresponding author. e-mail: ccliang@mail.dyu.edu.tw

Table 1 Motorcoach crashes and fatalities by most harmful

event of 48 crashes and 146 fatalities, (FARS 1996-2005)

Event Motorcoach crashes Motorcoach fatalitiesQuantity Rate [%] Quantity Rate [%]

Rollover 14 29 49 34

Roadside 15 31 53 36

Multi-vehicle 17 36 20 14

1996; Castejon et al., 2001; Elitok et al., 2005a; 2005b)

have shown good agreement between the test and the

analysis technique The optimization studies are based on

an analysis of the FE bus model for the maximum stiffness,

which includes the torsional and bending stiffnesses of the

full structure, while maintaining the weight at the same

level of the specific bus (Kim, 1992, 1993) Lan et al.

(2004) have comparatively analyzed the bus side structures

and lightweight optimization The ANSYS solver was used

to perform sensitivity studies and structural optimization to

reduce the body weight without losing the overall strength

and rigidity To study the relationship between the lowest

shear mode and the weight of the bus, Lin and Nian (2006)

built the CAE model and used HYPERSTUDY for the

sensitivity and optimization analysis methodology to determine

the optimized parameters for building a new model that

meets the ECE R66 Boada et al. (2007) combined ANSYS

and MATLAB to structurally optimize a simplified bus

structure’s weight and torsional stiffness with the genetic

algorithm

The optimization process may be performed by design

calculations, the trial and error design practice (with the help

of intuition), or the automated optimization analysis method

Although many studies have been done on bus structure

strength for lightweight buses, only a few studies have

investigated the optimum design of the bus superstructure

that is based on ECE R66 Only Lin et al (2006) carried out

research that follows ECE R66 That study, however, is still

limited to the body section Thus, this paper will present an

optimal structure throughout the whole bus that is based on

the ECE R66 process and the DOE technique that uses

LS-DYNA for the FE solver and Excel for the experimental

regression

In this paper, an ECE R66 calculation procedure that is

performed for a certain bus is described This

12.6-meter-long bus (see Figure 1) is constructed with steel material

and has special reinforcement bars at the roof - side and

side - floor joins where the large displacements can be

happened (Toni Roca et al., 1997) The FEA modeling is

done by the FEMB pre-processing software, calculations

are made by means of a non-linear, explicit, 3-D dynamic

FE computer code, LS-DYNA The calculation technique

has been checked by verifying the calculations on both a

breast knot of the side-body and a roof edge knot of the

vehicle; then, subsequent numerical simulations were

performed A high degree of theoretical and experimental

correlation is obtained, which confirms its validity (Chiu,

2007) With an assessed method in the ECE R66, a

complete vehicle rollover test simulation was carried out,

and the deformation results were observed with respect to

the residual space It is inferred that the structure of the bus

followed the required regulations This numerical study

also inferred the absorbed-energy distribution throughout

the whole bus Then the highest energy absorption region

was determined and focused on At this region, the

investigation of the absorbed-energy distribution and the

re-distribution of the energy absorption ability derived theset of bus-frame thickness parameters for the optimalanalysis of the design By using the Design of Experiment(DOE) with the regression technique of the Excel tool, areasonable thickness parameter set was found, and aprototype of the bus with an optimized thickness was built

to verify the satisfaction of ECE R66

In this paper, an optimization technique that is based onthe re-distribution of the energy absorption ability to thehighest energy absorption region of the bus frame isdescribed While maintaining the bus weight at the existinglevel, the results indicate an improvement in the upper andlower displacements by 39.9% and 49.3%, respectively,compared to the bus survivor space This study is acontribution to the automotive industry to reduce theproduction cost and number of occupant injuries throughpassive safety of the rollover event

2 ECE R66 REGULATION

The ECE R66 regulation was issued on Jan 30th, 1987, and

is enforced by the Economic Commission of Europe Theregulation was created because of the serious nature ofrollover accidents It applies to all single-decked vehiclesthat are constructed for carrying more than 22 passengers(whether seated or standing), in addition to the driver andcrew “Superstructure” refers to the parts of a vehicle’sstructure that contribute to the strength of the vehicle in theevent of a rollover accident

2.1 Residual SpaceThe purpose of the ECE R66 regulation is to ensure that thevehicle’s superstructure has sufficient strength so that theresidual space during and after the rollover test on thecomplete vehicle remains unharmed This means that nopart of the vehicle that is outside of the residual space at thestart of the rollover, like luggage, intrudes into the residualspace and that no part of the residual space projects outside

of the deformed structure The envelope of the vehicle’sresidual space is defined by creating a vertical transverseFigure 1 Full-scale bus model and bus frame components

OPTIMIZATION OF BUS ROLLOVER STRENGTH BY CONSIDERATION OF THE ENERGY ABSORPTION ABILITY 175

plane within the vehicle, which has the periphery that is

described in Figure 2 The SR points are located on the

seatback of each forward or rearward facing seat, which is

500 mm above the floor under the seat, 150 mm from the

inside surface of the sidewall of the vehicle (JASIC, 1998,

2006)

2.2 Rollover Test

This regulation is not only continuously updated based on

the actual requirements but also used as an international

bus rollover regulation The current version was issued on

Feb 22nd, 2006 The rollover test is a lateral tilting test (see

Figure 3) The complete vehicle stands on a tilting platform

(the suspension is blocked) and is slowly tilted to an

unstable equilibrium position If the vehicle type does not

fit with the occupant restraints, it will be tested at an

unladen curb mass If the vehicle is fitted with occupant

restraints, it will be tested as the total effective vehicle

mass The rollover test starts in this unstable vehicle

position with zero angular velocity, and the axis of rotation

passes through the wheel-ground contact points The

vehicle tips over into a ditch that has a horizontal, dry,

smooth concrete ground surface with a nominal depth of

800 mm (JASIC, 1998, 2006)

The rollover test is carried out on the side of the vehicle

that is more dangerous with respect to the residual space

This decision was made by the technical service on the

basis of the manufacturer’s proposals by considering the

following

· The lateral eccentricity of the center of gravity and its

effect on the reference energy in the unstable starting

position of the vehicle

· The asymmetry of the residual space

· The different asymmetrical construction features of the

two sides of the vehicle, and the support that is given by

the partition or inner boxes (e.g wardrobe, toilet, and

kitchenette)

The side with less support shall be chosen as the direction

of the rollover test

2.3 Test MethodsThe latest version of ECE R66, version 2006, with theabove requirements describes a test to be chosen amongstthese five different methods:

(a) Complete Vehicle Rollover Test(b) Body Section Rollover Test(c) Body section test with Quasi-static load(d) Component testing based on Quasi-static calculation(e) Complete vehicle rollover test based on computer simu-lation

Method (a) was accepted as the standard method while theothers are equivalent methods Amongst them, methods (c)and (d) are the new methods in ECE R66, version 2006.The methods (a), (b), and (c) are experimental methods thatare based on the real test Method (e) is officially acceptedfor full-scale computer simulations (JASIC, 2006) In thispaper, method (e) is used to perform the numericalanalysis

2.4 Computer Simulation of a Rollover TestComputer simulation of a rollover test on a completevehicle is an equivalent approval method It allowsmanufacturers to virtually test designs and safety features

in crash scenarios until they obtain the safe and optimumdesign When developing costly prototypes, this techniquesaves time and money

The analysis processes are as follows

(1) Construct the test model from a full-scale bus modeland tilting-platform model

(2) Based on the testing conditions, set up the materialcard, boundary condition card, contact card, and apply

a load to the test model

(3) Determine the tilting angle to reduce the computingtime

(4) Use numerical analysis software to carry out the lation of the rollover accident for this testing model.(5) Evaluate the status of the testing model based on theobtained simulation results

simu-Figure 2 Residual space, all measurements are in

milli-meters (JASIC, 2006)

Figure 3 Rollover test process (JASIC, 2006)

3 LS-DYNA AND NON-LINEAR EXPLICIT

ALGORITHM

3.1 LS-DYNA General Description

LS-DYNA was developed by LSTC (Livermore Software

Technology Cooperation) It is a multifunctional applicable

explicit and implicit Finite-Element program that simulates

and analyzes highly-nonlinear physical phenomena that are

obtained in real world problems Usually, the phenomena

are subjected to large deformations within short time

durations, e.g crashworthiness simulations The significant

features of the LS-DYNA are the fully-automatic definitions

of the contact areas, the large library of constitutive models,

the large library of element types, and the special

im-plementation for the automobile industry (Hallquist, 2006;

LSTC, 2007)

This paper uses finite element software to carry out the

bus rollover simulation The behavior of the bus rollover

simulation is a transient, dynamic, nonlinear, large-deformation

problem The Finite Element Analysis (FEA) code,

LS-DYNA, is a favorite tool for this problem, which often

includes contact and impact The main solution is based on

explicit time integration

3.2 Non-linear Explicit Algorithm

The explicit method was originally developed and primarily

used to solve dynamic problems that involve deformable

bodies Accelerations and velocities at a particular point in

time are assumed to be constant during a time increment

and are used to solve for the next point in time For the

explicit method, a central difference time integration

method is used The word ‘implicit’ here refers to the

method by which the state of a finite element model is

updated from time t to t + ∆t A fully implicit procedure

means that the state at t + ∆t is determined based on the

information at time t + ∆t while the explicit method solves

for t + ∆t based on information at time t Accelerations that

are evaluated at time t are given by: ,

where F is the vector of externally applied forces, I is the

vector of internal element forces, and M is the lumped mass

matrix It is a trivial process to invert the lumped mass

because the matrix is diagonalized, unlike the global stiffness

matrix in the implicit solution method Therefore, each time

the increment is computationally inexpensive to solve The

velocities and displacements are then evaluated as:

,where u is the displacement and the superscripts refer to the

time increment The term ‘explicit’ refers to the fact that

the state of the analysis is advanced by assuming constant

values for the velocities, , and the accelerations, , across

the half-time intervals

The geometry is updated by adding the displacement

increments to the initial geometry:

, For nonlinear problems being solved through explicitsolutions, a lumped mass matrix is required for simpleinversions The equations become uncoupled and can bedirectly (explicit) solved No inversion of the stiffnessmatrix is required All nonlinearities (including contact) areincluded in the internal force vector The major com-putational expense is in calculating the internal forces Noconvergence checks are needed since the equations areuncoupled Very small time steps are required to maintainthe stability limit

3.2.1 Time step sizeLS-DYNA checks all of the elements when calculating therequired time step For stability reasons, a scale factor of0.9 (default) is used to decrease the time step:

.The characteristic length, l, and the wave propagationvelocity, c, are dependent on the element type

3.2.2 Stability limitThe explicit solution is only stable if the time step size issmaller than the critical time step size: ,where

ω max is the largest natural circular frequency

Due to this very small time step size, the explicit method isuseful only for very short transients

3.3 Process of numerical studyThis paper concerns the numerical analyses of the busrollover simulation The numerical study is based on theLS-DYNA finite element analysis procedure, as shown inFigure 4 The package LS-DYNA software contains a Pre-

OPTIMIZATION OF BUS ROLLOVER STRENGTH BY CONSIDERATION OF THE ENERGY ABSORPTION ABILITY 177

processing Finite Element Model Builder (FEMB), a

LS-DYNA solver, and a post-processing LS-PREPOST (Hallquist,

2006; LSTC, 2007) With LS-DYNA, the standard inputs

(such as geometry, mesh density, materials, element

properties, boundary conditions, and contact modes) can be

used The LS-DYNA solver will perform solutions The

output results (such as stress and strain of elements,

displacement, velocity and acceleration of nodes, energy

distribution, etc.) can be clearly shown through the user

interface (Hallquist, 2006; LSTC, 2007) The models of

elements, materials and contacts are specified in the following

sections

3.3.1 Element models

Spatial discrimination is achieved by the use of a four-node

shell element, an eight-node solid element, and rigid

bodies A variety of element formulations are available for

each element type Adaptive re-meshing is available for

shell elements and is widely used The mesh density is

nearly between 25 and 50 mm The shell elements with

Belytschko-Tsay formulation are used for the structures of

the whole bus A solid element is used for the tilting

platform of the rollover simulation

(a) Solid element model

A solid element is used for the 3-D modeling of solid

structures The element is defined by 8 nodes that have the

following degrees of freedom at each node: translations,

velocities, and accelerations in the nodal x, y, and z

directions The geometry of this solid element is shown in

Figure 5 with Nodes (I, J, K, L, M, N, O, P) and degrees of

Freedom (UX, UY, UZ, VX, VY, VZ, AX, AY, AZ)

(b) Shell element model

This shell element is a four-node element with both bending

and membrane capabilities Both in-plane and normal loads

are permitted

The element has 12 degrees of freedom at each node:

translations, accelerations, and velocities in the nodal x, y,and z directions and rotations about the nodal x, y, and z-axes The geometry of this shell element is shown in Figure

6with Nodes (I, J, K, L) and degrees of Freedom (UX, UY,

UZ, VX, VY, VZ, AX, AY, AZ, ROTX, ROTY, ROTZ).Both of the two element types above are only used inexplicit dynamic analyses (Hallquist, 2006; LSTC, 2007)where V(X, Y, Z) refers to the nodal velocity and A(X, Y,Z) refers to the nodal acceleration Although V(X, Y, Z)and A(X, Y, Z) appear as DOFs, they are not actuallyphysical DOFs However, these quantities are computed asDOF solutions and are stored for post processing.3.3.2 Materials and contacts models

To obtain the material data, tension tests were applied toseveral specimens at the Automotive Research & TestingCentre, Taiwan, R.O.C (ARTC) (Chiu, 2007) The true stress-strain curves were obtained and imposed in the LS-DYNA,accordingly The material model for the deformable structure

in LS-DYNA was the ‘MAT_TYPE_024, _LINEAR_ISOTROPIC_PLASTICITY_MODEL’ (Hallquist,2006; LSTC, 2007) This is an elastic plastic materialmodel that uses the Young’s modulus if stresses are belowthe yield stress and the measured stress-strain curve if thestresses are above the yield stress Rigid parts (engine, gearbox, tilting platform for ECE R66 test, etc.) were modeledwith ‘MAT_TYPE_001, RIGID MATERIAL’ ‘MAT_TYPE_

PIECEWISE-020, ELASTIC MATERIAL’ was simulated for the vehicletires

For explicit analysis, there are no contact elements Wesimply indicate the contact surfaces, the type of contactbetween them, and other parameters that are related to thecontact type Owing to the complicated large-deformationdynamics or quasi-statics conditions that typically occurduring an explicit analysis, determining the contact betweenthe components in a model can be extremely difficult Forthis reason, special features have been included in the LS-DYNA program to make defining the contact between thesurfaces as efficient as possible All contacts are defined inLS-DYNA through the use of the CONTACT pull down

Figure 5 Geometry of solid element Figure 6 Geometry of shell element

menu The contact called CONTACT_AUTOMATIC_SINGLE

_SURFACE is established when a surface of one body

contacts itself or the surface of another body This contact

type is easy to use because no contact or target surface

definitions are required It is efficient for self-contacting

problems or large deformation problems where the general

areas of contact are not known beforehand This contact

was used for simulating the contacting mode of the vehicle

and the tilting plate in the rollover tests The contact called

CONTACT_AUTOMATIC_SURFACE_TO-_SURFACE

is established when a surface of one body penetrates the

surface of another body This is commonly used for

arbitrary bodies that have large contact areas and it is very

efficient for bodies that experience large amounts of

relative sliding with friction, such as block sliding on a

plane This contact was used for simulating the contacting

mode of the bus frame with some reinforcements The

contact called CONTACT_RIGID_WALL_PLANAR was

used to set the contacting mode of the vehicle and ground

4 ECE R66 SIMULATION AND NUMERICAL

STUDY

4.1 Computational Model

The FM vehicle model that is used for simulation is based

on the full-scale bus model that has been developed at

Da-Yeh University, Taiwan, for rollover crashworthiness

investigation and evaluation of reinforcement structures

(Chai, 2005; Chang, 2006; Chiu, 2007) It includes 62,617

nodes and 59,652 elements There are four types of

elements that include 67,084 quadrilateral elements, 914

triangular elements, 35 hexagons, and 99 mass The

description is based on the element shells and their materials,

see Table 2 The formulation of the shell elements that are

used for this paper is based on the Belytschko-Tsay theory

(Belytschko et al., 1984; Hallquist, 2006) The unloaded

vehicle weight is 7,716.47 [kg] (7.71647 [ton]), and its

capacity is 49 passengers The vehicle sizes and the

position of the CG (Center of Gravity) are shown in Figure

1 The center of gravity (CG) of the vehicle was measured

by means of a test platform at ARTC The measured values

were in good agreement with the ones that come from the

FEA model To match the measurements and calculate CG

exactly, the CG of the engine, gearbox, and axles was

fine-tuned in the FEA model

The FE modeling was done by the pre-processing Finite

Element Builder (FEMB) of LS-DYNA and calculations

were made by means of a nonlinear, explicit, 3-D dynamic

FE computer code, LS-DYNA The calculation techniquewas checked by verifying the calculations on a breast knot

of the side-body (Figure 7(a)), on a roof-edge knot (Figure7(b)), and on a floor-pillar knot (Figure 7(c)), which wereextracted from the vehicle superstructure Those threeseparate specimens were subjected to certain boundaryconditions and quasi-static loads at ARTC The same testscenarios were simulated with the LS-DYNA Force-deflection curves for both the experiment and the simulationwere compared and a good correlation was seen betweenthe experiment and the simulation results (Figure 7)

In this paper, LS-DYNA 971MPP in the 4 NodesWindow Cluster was used to investigate the absorbed-energy distribution as well as the displacement of thewindow pillars and side wall bars on this full-scale busmodel for rollover safety, according to ECE R66.Table 2 Material for simulating bus superstructure, chassis, tires, axis and tilting plate

Bus supers-tructure Chassis of bus Axis and tilting plate TiresDensity (ton/mm3) 7.83E-09 2.783E-08 7.83E-09 2.85E-09Young’s modulus (N/mm2) 210,000 205,000 200,000 11,000

Yield stress (N/mm2) 282 270

Plastic fracture pressure (N/mm2) 3.76E08 1.0E08

Figure 7 Tests, simulations and results of the body knot formodel validation