Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 659873 15 pageshttpdxdoiorg1011552013659873

Research ArticleStudy on Driving Stability of Tank Trucks Based on EquivalentTrammel Pendulum for Liquid Sloshing

Xian-sheng Li Xue-lian Zheng Yuan-yuan Ren Yu-ning Wang and Zhu-qing Cheng

College of Traffic Jilin University No 5988 Renmin Street Changchun 130022 China

Correspondence should be addressed to Yuan-yuan Ren maggie170101gmailcom

Received 17 July 2013 Revised 27 September 2013 Accepted 27 September 2013

Academic Editor Wuhong Wang

Copyright copy 2013 Xian-sheng Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To investigate the driving stability of tank trucks an equivalent trammel pendulum was utilized to approximately demonstrate thedynamic characteristics of liquid sloshing in a partially filled tankThe oscillation movement of the trammel pendulum in the tankwas described under the tank-fixed coordinate system and its motion equation under the noninertia coordinate systemwas derivedusing a Lagrangian function The motion of the pendulum that expresses the fluid cargo dynamic behavior and that of the solidtruck was coupled with each other by the tank Therefore a tank truck dynamic model was established using Newtonrsquos first lawand the angular momentum A typical tank truck was selected and used to study its driving stability under steering angle step testThe study on tankers driving stability is of great importance for evaluating tankers driving safety investing the main impact factoraspecting tankers driving stability and developing activepassive roll control systems for them

1 Introduction

About 80 of global chemical and petroleum products aredelivered by road tank vehiclesThe transportation freight hasalready reached 4 billion tons per year In America tankersmake up more than 55 of all freight trucks Tankers arehugely convenient for fluidmaterial exchange and have a pos-itive effect on boosting the national economic developmentHowever they also create severe traffic safety problems whichwould result in huge people injury and property damageStatistical data collected by StatistiqueCanada has shown that83 of lorry rollover accidents on highways are caused bytank vehicles [1] Meanwhile in 2011 416 tanker accidentsoccurred in China resulting in more than 400 people beinginjured or killed as well as immense economic losses Besidesthis for the particularity of liquid cargoes the release of fluidcargo in tanker accidents could cause contamination to theroads the water and the air [2] Therefore great attentionmust be paid to tanker driving safety

Much works have been carried out on the characteristicsof tanker accidents in an attempt to investigate the primaryaccident type It was found by Treichel et al that rollover isthe most frequent accident type for tankers [3] comprising4516 of all tanker accidents in China in 2010 Furthermore

nearly 61 of tanker rollovers occurred on curved sections ofhighway Many researchers have studied the factors behindthis phenomenon and have concluded that liquid sloshingin a partially filled tank is the main cause [4ndash7] Due to thedifference in liquid cargo densities and limitations on vehicleaxle loads tanks are in a partially filled condition most ofthe time Due to the existence of liquid-free sloshing spacetransient liquid sloshing is produced when vehicle drivingstate changes [8 9] The lateral sloshing force that acts onthe tank wall increases vehicle rollover torque and degradesvehicle roll stability

To explore the influence of transient liquid sloshing ontanker roll stability many research works have been exploredand some important conclusions have been drawn Consid-ering the complexity of tank vehicle dynamic analysismdashastank vehicles are fluid-solid coupling multibody systemmdashsome important simplifications have to be made before theanalysis Strandberg et al assumed that liquid free surfaceis a tilted straight line whose gradient is a function ofvehiclersquos lateral acceleration and tankrsquos roll angle [10ndash13]Based on that the center of gravity (CG) of liquid bulk canbe obtained and a quasi-static (QS) dynamic analysis ofvehicle driving stability can be carried out The QS methodwas popular in the initial research phase of the late 1990s

2 Discrete Dynamics in Nature and Society

and many improved algorithms have been produced toincrease the analysis precision to some degree [14 15]

To investigate the accuracy of the QS method someresearchers studied the relationship between the evaluatedliquid sloshing effect obtained by calculating the movementof the CG of the liquid bulk and the observed effect Theyhave found that the mean values of sloshing forces and thecoordinates of the CG of the liquid bulk are quite closebetween the estimated and observed cases [16] This resultdemonstrates the correctness of the QS method which meanthat the worst vehicle driving state can be predicted by the QSmethod [17]

However the QS method cannot reflect the dynamiccharacteristics of liquid sloshing Therefore an equivalentmechanical model which can precisely demonstrate thedynamic characteristic of liquid sloshing in a partially filledtank must be modeled first [18ndash21] In estimating the tankerrsquosdynamic analysis using equivalent mechanical model forliquid sloshing the biggest difficulty for researchers is thecoupling of the fluid cargo and the solid vehicle To solvethis problem some people analyzed tanker driving stabilityinmultibody analysis software and someused the contractiveanalysis method [20 22] Neither method had the advantagesof the equivalent mechanical model for liquid sloshing as avehiclersquos driving stability cannot be precisely described by theanalytical model for vehicle roll stability control system

Therefore the purpose of this paper is to establish a tankvehicle dynamic model based on the equivalent mechanicalmodel for transient liquid sloshing in a partially filled tankThe trammel pendulum was used to describe liquid sloshingits motion was analyzed and the motion equation wasderived under the tank-fixed coordinate Then the couplingcondition for the fluid cargo and the solid vehicle wasstudied and a tanker dynamic model was established basedon this Finally MATLAB was used to simulate the rollstability of a typical tank truck under steering angle step testand some important conclusions were drawn The researchachievement is of great significance for commanding andevaluating tank trucks roll stability performance judging theconditions under which a tanker will lose its driving stabilityand designing roll stability control systems for tankers toimprove their driving safety

2 Equivalent Trammel Pendulum for LiquidSloshing in a Partially Filled Tank

21 The Basic Motion Equation and Parameter Values of theTrammel Pendulum Theoretical analysis and experimentalstudies had revealed that the first-order sloshingmode whichcan be described by the oscillation of liquid-free surfaceis the most important mode of liquid sloshing in partiallyfilled tanks [20 21] MATLABrsquos numerical method was usedto calculate the trajectory of the CG of the liquid bulkthat oscillates in a tank with an elliptical or circular crosssection The results showed that the trajectory of the CG ofthe liquid bulk remains parallel to the tankrsquos periphery [21]which is presented in Figure 1 According to this fact thetrammel pendulum model whose swing trajectory is elliptic

Trajectory ofliquid CG

Tank periphery

120579ap

ap

zp

bp yp

mp

mf

bf

A

Bacgbcg

Figure 1 Schematic diagram for the motion of the trammel pendu-lum

120579

ap

zp

bp

yp

rarrr

Figure 2 Analytical diagram for the motion of the trammel pendu-lum under an inertial coordinate system

is established to approximately simulate liquid sloshing in apartially filled tank

In Figure 1 119886119901 is half of the major axis of the pendulumrsquososcillation trajectory and 119887119901 is half of its minor axis 119886cg ishalf of the major axis of the elliptical trajectory of the CGof the liquid bulk and 119887cg is half of its minor axis 119898119901 is thependulum mass which is equal to the sloshing liquid mass119898119891 is the fixed liquid mass which does not participate insloshing 120579 is the pendulumrsquos amplitude

The Lagrangian function was used to derive the kineticequation of the trammel pendulum system The motionequation of the trammel pendulum that oscillates under tankfixed inertia coordinate system whose origin locates at thecenter of the tank (as shown in Figure 2) is written as follows

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) + 1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

+ 119892119887119901 cos 120579 = 0

(1)

The pendulum parameters in (1) are presented as follows

119887119901

119887

= 1089 + 0726Δ minus 01379Λ minus 0953Δ2minus 1216ΛΔ

+ 005141Λ2minus 006107Δ

3+ 05739ΛΔ

2+ 01632Λ

2Δ

(2)

Discrete Dynamics in Nature and Society 3

The tank periphery120579

zp

bp

yp

ap minus bp

120601

Z

Y

b

rarrx

rarrdrarr

R

rarrr

Figure 3 Analytical diagram for the motion of the trammel pendu-lum under the noninertial coordinate system

119898119901

119898

= 07844 minus 1729Δ + 03351Λ + 1156Δ2+ 07256ΛΔ

minus 01254Λ2minus 03219Δ

3minus 09152ΛΔ

2+ 008043Λ

2Δ

(3)

119898119891 = 119898 minus 119898119901 (4)

(119887 minus 119887119891)

119887

=

[119898 (119887 minus 119887cg) minus 119898119891 (119887 minus 119887119901)]

119898119891119887

(5)

where Δ is the liquid fill percentage which is the ratio of theheight of liquidrsquos level surface to that of the tank and Λ is theratio of the tankrsquos width to its height

The detailed derivation process for (1)ndash(5) is given byZheng et al [21]

According to Figure 1 the following equation holds

119886

119887

=

119886cg

119887cg=

119886119901

119887119901

= Λ (6)

From (6) the arm length of the pendulum can beexpressed as follows

119886119901 = 119887119901 times Λ (7)

As themass of liquid bulk can easily be acquiredwhile thetankrsquos size and liquid density are given the pendulum massand the fixed liquid mass can be obtained using (3) and (4)

22 Motion Equations of the Trammel Pendulum under theNoninertial Coordinate System During the driving processthe vehiclersquos driving state changes regularly under the influ-ence of a variety of factors It leads to the movement of thetank and in turn the coordinate system is being fixed relativeto the tank Therefore motion equations for a trammelpendulumunder a noninertial coordinate system (translationand rotation are included) should be derived in order to studythe motion characteristics of liquid bulk when tankers arebeing driven on curved sections of road avoiding obstaclesor changing lanes

The schematic diagram for the motion of the tank whenvehicle turns right is presented in Figure 3 For convenientanalysis of the vehiclersquos roll stability the origin of the tank-fixed coordinate system is located at the bottom of the tank

This location will not impact the motion analysis result Theorigin of the earth-fixed coordinate system also is located atthe bottom of the tank when the vehicle is static is thedistance vector from the origin of the earth-fixed coordinatesystem to that of the tank-fixed coordinate system is theabsolute position vector of the pendulum mass ball isthe absolute position vector of the pendulum mass ball 120601 isthe tank roll angle which is defined as the angle formed byrotating the 119884-axis counterclockwise till the position of the119910119901-axis

With reference to Figure 3 the absolute position of thependulum mass ball can be obtained as follows

= [119909 minus 119887 sin120601 + (119886119901 cos 120579 minus 119887119901 sin 120579 tan120601) cos120601] 119894

+ [119887 cos 120579 + (119887119901 sin 120579 + 119886119901 cos 120579 tan120601) cos120601] 119895

(8)

The velocity of the pendulum mass ball can be obtainedby solving the first-order derivative of (8) which may bepresented as follows

= [ minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601) 120601

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601) 120579] 119894

+ [(minus119887 sin120601 minus 119887119901 sin 120579 sin120601 + 119886119901 cos 120579 cos120601) 120601

+ (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601) 120579] 119895

(9)

Similarly the acceleration of the pendulummass ball canbe acquired by solving the differential of (9) which may bewritten as follows

= [minus

120601 (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

minus

120579 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

+ 2

120579

120601 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)

+

1206012(119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

+

1205792(minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601) + ] 119894

+ [

120601 (minus119887 sin120601 + 119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

+

120579 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1205792(119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

+

1206012(minus119887 cos120601 minus 119886119901 cos 120579 sin120601 minus 119887119901 sin 120579 cos120601)

minus 2

120579

120601 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)] 119895

(10)

The zero of potential energy is defined as the 119884-axisand the potential energy of the pendulum mass ball can bepresented as follows

119881 = 119898119892 (119887 + 119887119901 sin 120579 + 119886119901 cos 120579 tan120601) cos120601 (11)

4 Discrete Dynamics in Nature and Society

According to (10) and (11) a Lagrangian function can beused to obtain the motion equation of the pendulum systemwhich is written as follows

119871 = 119879 minus 119881 (12)

where 119879 is the kinetic energy of the pendulum mass ballThe motion of the trammel pendulum system can be

expressed by

120597

120597119905

(

120597119871

120597 119886119895

) minus

120597119871

120597119886119895

= 0 (13)

where 119886119895 is the degree of freedom for system 119871According to the motion analysis of the trammel pendu-

lum 120579 the pendulum amplitude and 120601 the tank roll angleare the two degrees of freedom for the trammel pendulumsystem

The differential of (12) with respect to 120579 can be expressedas follows

120597119871

120597120579

= 119898 [

1206012(minus1198862

119901sin 120579 cos 120579 + 119887

2

119901sin 120579 cos 120579 + 119887119901119887 cos 120579)

+

1205792(1198862

119901sin 120579 cos 120579 minus 119887

2

119901sin 120579 cos 120579)

+

120579

120601 (119886119901119887 cos 120579)

minus

120601 (minus119886119901 sin 120579 sin120601 + 119887119901 cos 120579 cos120601)

minus

120579 (119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

minus119892 (119887119901 cos 120579 minus 119886119901 sin 120579 tan120601) cos120601] (14)

The differential of (12) with respect to 120579 can be expressed

by

120597119871

120597

120579

= 119898 [

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) +

120601 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)] (15)

Finally the differential of (15) with respect to 119905 can be ex-pressed as follows

120597

120597119905

(

120597119871

120597

120579

) = 119898 [

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579)

+

1205792(21198862

119901sin 120579 cos 120579 minus 2119887

2

119901sin 120579 cos 120579)

+

120601 (119886119901119887119901 + 119886119901119887 sin 120579) + 120601

120579119886119901119887 cos 120579

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

minus

120579 (119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

+

120601 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)] (16)

Substituting (14)ndash(16) into (13) we get

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) +

120601 (119886119901119887119901 + 119886119901119887 sin 120579)

+

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

+

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

+ 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601) = 0

(17)

The differential of (12) with respect to 120601 can be expressedas follows

120597119871

120597120601

= 119898 [

120601 (119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

minus

120579 (minus119886119901 sin 120579 sin120601 + 119887119901 cos 120579 cos120601)

minus 119892119886119901 cos 120579 sec120601

+ 119892 (119887 + 119887119901 sin 120579 + 119886119901 cos 120579 tan120601) sin120601]

(18)

The differential of (12) with respect to 120601 can be expressed

by

120597119871

120597

120601

= 119898 [

120601 (1198872+ 1198862

119901cos2120579 + 119887

2

119901sin2120579 + 2119887119901119887 sin 120579)

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)] (19)

According to (19) the differential of (19) with respect to 119905can be obtained as follows

120597

120597119905

(

120597119871

120597

120601

)

= 119898 [

120601 (1198872+ 1198862

119901cos2120579 + 119887

2

119901sin2120579 + 2119887119901119887 sin 120579)

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579) + 119886119901119887

1205792 cos 120579

+

120601

120579 (minus21198862

119901sin 120579 cos 120579 + 2119887

2

119901sin 120579 cos 120579 + 2119887119901119887 cos 120579)

+

120579 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)

+

120601 (119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)] (20)

Discrete Dynamics in Nature and Society 5

Substituting (18)ndash(20) into (13) we get

120601 [(119887 + 119887119901 sin 120579)

2

+ 1198862

119901cos2120579]

+ 2

120579

120601 [

1

2

sin 2120579 (1198872119901minus 1198862

119901) + 119887119901119887 cos 120579]

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

+

1205792119886119901119887 cos 120579

minus 119892 [(119887 + 119887119901 sin 120579) sin120601 minus 119886119901 cos 120579 cos120601] = 0

(21)

Equations (17) and (21) are the equations of motionof the trammel pendulum under the noninertia coordinatesystem and the pendulum motion characteristics under thetranslation and rotation of the coordinate system can beanalyzed using (17) and (21)

3 Tank Truck Dynamic Model

31 Assumptions The roll stability of a tank truck is greatlyinfluenced by transient liquid sloshing in partially filledtanks Due to the flow characteristic of liquid bulk transientliquid sloshing is produced when the vehiclersquos driving statechanges which leads to the oscillation of the pendulummassball To simplify the analysis of the tank truckrsquos roll stabilitysome assumptions are made and listed as follows

(1) The sprung mass is the vehicle mass that lies over thesuspension system in which the mass of the liquidcargo is not included In this paper vehicle and liquidbulk as the solid and fluid parts respectively are notmixed with each other

(2) In the transverse direction the CG of the sprungmassis located at the bottom of the tank Moreover thesprung mass is symmetrical about the longitudinalaxis of the vehicle

(3) Transient liquid sloshing is merely produced alongthe transverse direction longitudinal liquid sloshingis not taken into consideration The movement andmass distribution of the liquid bulk are completelythe same at different tank cross sections whichmeansthat the CG of liquid bulk is located in the middle ofthe tank along the longitudinal direction

(4) The liquid-free surface is not broken during thevehiclersquos driving process

32 Force Analysis of the Tank Truck Based on the assump-tions given in Section 31 a reference coordinate system fortank truck dynamic analysis was selected first

The coordinate system being fixed relative to the vehiclewas chosen as the reference coordinate system for thedynamic analysis of the tank truck and the establishment ofthe vehiclersquos dynamic model The origin of the coordinatesystem is the point where the vertical line that goes through

the CG of the vehicle when it is static intersects with thevehiclersquos roll axis

The coordinate system being fixed relative to the vehiclersquossprung mass was defined as the 119909-119910-119911 system where the 119909-axis is parallel to the longitudinal axis of the vehicle andpoints in the direction in which the vehicle is being driventhe 119910-axis is perpendicular to the 119909-axis and points to the leftfrom the driverrsquos perspective and the 119911-axis is perpendicularto the 119909119910 plane and points straight up

The vehiclersquos unsprung mass is assumed not to produceroll movement

The top view and the back view of the vehiclersquos dynamicanalysis are plotted in Figures 4 and 5 According to Figure 5the lateral inertial forces act on the sprung mass theunsprung mass the fixed liquid mass and the pendulummass making up the vehiclersquos lateral inertial force The tirecornering forces make up the external lateral force that actson the vehicle according to Figure 4 On the basis of Newtonrsquosfirst law when the vehicle is steady its inertial lateral forceand external lateral force are equal in value and positive indirection which can be expressed as follows

(119898119905 + 119898119891) 119886119904 + 119898119901119886119891 + 119898119906119886119906 = 2 (119865119891 + 119865119903) (22)

where 119886119904 is the lateral acceleration of sprung mass 119886119906 isthe lateral acceleration of unsprung mass 119886119891 is the lateralacceleration of the pendulum mass ball 119898119905 is sprung mass119898119906 is unsprung mass 119865119891 is the front tire cornering force and119865119903 is the rear tire cornering force

Generally the fixed liquid mass does not stay static butslides slowly with the aid of gravity while the tank tiltsHowever the shift of the fixed liquid mass is quite smallcompared to that of the pendulum mass ball and its shiftamount drops greatly with the increase in the liquid fillpercentage Therefore the fixed liquid mass is assumed to bestatic and its position is assumed to stay constant as the tankrolls over Based on this assumption the fixed liquidmass canbe seen as the solid part and its lateral acceleration is equal tothat of the vehiclersquos sprung mass

According to the dynamic analysis of the vehicle thelateral acceleration of the sprung mass and unsprung masscan be presented by

119886119904 = 119881 (

120573 + 119903) minus ℎ119904

120601 + 119888 119903

119886119906 = 119881 (

120573 + 119903) minus 119890 119903

(23)

where 119881 is the vehiclersquos driving speed 120573 the vehiclersquos slipangle 119903 the vehiclersquos yaw angle ℎ119904 is the vertical distance fromthe CG of sprung mass to the roll center 119888 is the longitudinaldistance between the CG of sprung mass and that of the tanktruck and 119890 is the longitudinal distance between the CG ofunsprung mass and that of the tank truck

According to (10) the lateral acceleration of the pendu-lum mass ball can be written as follows

119886119891 = 119881

120601 minus 1198631

120601 minus 1198632

120579 + 21198633

120601

120579 + 1198634

1206012+ 1198635

1205792 (24)

6 Discrete Dynamics in Nature and Society

Front wheel

CG of tank truck

CG of unsprungmass

CG of liquidcargo

Rear wheel

CG of sprung mass

Ff

Fr

lr

lf

c

e

e2

muau

mtas

mfas + mpaf

Figure 4 Top view of vehicle dynamic analysis

b

Central lineof the tank

Sprung mass

Pendulummass

Unsprung mass

Liquid fixed mass

Roll center

ΔFΔF

a

hs

muau

mpafmtas

mfas

Fr Fr

kr120601 + cr(d120601)

Figure 5 Back view of vehicle dynamic analysis

where

1198631 = 119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601

1198632 = 119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601

1198633 = 119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601

1198634 = 119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

1198635 = minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

(25)

The force analysis in the roll plane is plotted in Figure 5Due to the lateral inertia force the sprungmass and the liquid

Central line

of the tank

Coordinatefixed on the

tank

Coordinate fixedon the vehiclesprung mass

120579

ap

z

bp

y

AB

y998400

x998400

z998400

120601hs

1205792

b

x

o

Figure 6The schematic diagram for the pendulum rotating aroundthe roll axis

bulk rotate about the roll axis and bring about rollover torqueThe roll angle about the roll axis for sprungmass and the fixedliquid mass is 120601 which can be seen from Figure 5 Howeverowing to the tank and the pendulum mass ball movementthe roll angle about the roll axis for the pendulum mass ballis composed of two parts which is shown in Figure 6

According to Figure 6 the roll angle about the roll axis forthe pendulum mass ball can be expressed as follows

ang = 120601 + 1205792 (26)

where

1205792 = arctan(119860119861119861119874

) = arctan(minus119886119901 cos 120579

ℎ119904 + 119887 + 119887119901 sin 120579) (27)

Therefore the absolute roll rate and roll angular accelera-tion can be obtained as follows based on (26)

ang =

120601 +

1205792 ang =

120601 +

1205792(28)

1205792 =

1205791198611

119861

1205792 =

(1198601

120579 minus 1198602

1205792)

119860

(29)

where

119861 = (ℎ119904 + 119887 + 119887119901 sin 120579)2

+ (119886119901 cos 120579)2

1198611 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901]

119860 = 1198612 1198601 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901] 119861

1198602 = 2119886119901 cos 120579 [(ℎ119904 + 119887) sin 120579 + 119887119901]

times [119887119901 (ℎ119904 + 119887 + 119887119901 sin 120579) minus 1198862

119901sin 120579]

minus 119886119901119861 (ℎ119904 + 119887) cos 120579

(30)

Discrete Dynamics in Nature and Society 7

On the basis of (28) the vehiclersquos roll balance equationand yaw balance equation can be acquired by differentiatingthe angular momentum with respect to time

In general the angular momentum of a rotating object isdefined as the product of a bodyrsquos rotational inertia tensor androtational velocities about a particular coordinate(1) The Angular Momentum of Sprung Mass and the FixedLiquid Mass As the vehicle sprung mass and the fixed liquidmass have the same rotational velocity about the 119909-119910-119911coordinate system the angular momentum for the two partsis solved together

According to the definition of angular momentum thatfor the sprungmass and fixed liquidmass can be expressed asfollows

119867119904 =[

[

119868119909119909119904 + 119868119909119909119891 minus119868119909119910119904 minus 119868119909119910119891 minus119868119909119911119904 minus 119868119909119911119891

minus119868119910119909119904 minus 119868119910119909119891 119868119910119910119904 + 119868119910119910119891 minus119868119910119911119904 minus 119868119910119911119891

minus119868119911119909119904 minus 119868119911119909119891 minus119868119911119910119904 minus 119868119911119910119891 119868119911119911119904 + 119868119911119911119891

]

]

[

[

120601

0

119903

]

]

(31)

where 119868119909119909119904 119868119910119910119904 and 119868119911119911119904 are the moments of inertia for thesprungmass about the 119909-axis119910-axis and 119911-axis respectively119868119909119909119891 119868119910119910119891 and 119868119911119911119891 are the moments of inertia for the fixedliquid mass about the 119909-axis 119910-axis and 119911-axis respectively119868119909119910119904 = 119868119910119909119904 119868119909119911119904 = 119868119911119909119904 and 119868119910119911119904 = 119868119911119910119904 are the products ofinertia for the sprungmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively 119868119909119910119891 = 119868119910119909119891 119868119909119911119891 = 119868119911119909119891and 119868119910119911119891 = 119868119911119910119891 are the products of inertia for the fixed liquidmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively

According to the assumptions given in Section 31 thedistribution of the vehiclersquos sprungmass is symmetrical aboutthe119909119911-planeTherefore the products of inertia for the sprungmass can be expressed as follows

119868119909119910119904 = 0 119868119911119910119904 = 0 (32)

As the fixed liquidmass does notmovewhile the tank tiltsthe distribution of the fixed liquid mass is symmetrical aboutthe 119909119905119911119905-plane and the 119909119911-plane Therefore the products ofinertia for the fixed liquid mass can be expressed as follows

119868119909119910119891 = 0 119868119911119910119891 = 0 (33)

Therefore based on (31)ndash(33) the angularmomentum forthe vehiclersquos sprung mass and the fixed liquid mass can bewritten as follows

119867119904 = [(119868119909119909119904 + 119868119909119909119891)

120601 minus (119868119909119911119904 + 119868119909119911119891) 119903]119894

+ [minus (119868119911119909119904 + 119868119911119909119891)

120601 + (119868119911119911119904 + 119868119911119911119891) 119903]119895

(34)

(2)TheAngularMomentum of the PendulumMassAccordingto (28) the angular momentum of the pendulummass can beexpressed as follows

119867119901 =[

[

119868119909119909119901 minus119868119909119910119901 minus119868119909119911119901

minus119868119910119909119901 119868119910119910119901 minus119868119910119911119901

minus119868119911119909119901 minus119868119911119910119901 119868119911119911119901

]

]

[

[

120601 +

1205792

0

119903

]

]

(35)

where 119868119909119909119901 119868119910119910119901 and 119868119911119911119901 are the moments of inertia forthe pendulum mass about the 119909-axis 119910-axis and 119911-axis

respectively 119868119909119910119901 = 119868119910119909119901 119868119909119911119901 = 119868119911119909119901 and 119868119910119911119901 = 119868119911119910119901 are theproducts of inertia for the pendulum mass about the 119909- and119910-axis the 119909- and 119911-axis and the 119910- and 119911-axis respectively

The expanded form of (35) is written as

119867119901 = [119868119909119909119901 (

120601 +

1205792) minus 119868119909119911119901119903]119894 + [minus119868119909119910119901 (

120601 +

1205792) minus 119868119910119911119901119903]

119895

+ [minus119868119909119911119901 (

120601 +

1205792) + 119868119911119911119901119903]119896

(36)

(3) The Angular Momentum of the Unsprung Mass The yawmovement is the only rotational degree of freedom for theunsprung mass Therefore the angular momentum of theunsprung mass can be obtained by multiplying together themoment of inertia about the 119911-axis and the yaw rate aboutthat which is written as follows

119867119906 = 119868119911119911119906119903119896 (37)

where 119868119911119911119906 is the moment of inertia about the 119911-axis for theunsprung mass

The change in the vehicle fixed coordinate after it has beendriving for time Δ119905 is plotted in Figure 7 from which we cansee that the differentials of the 119894 119895 and

119896 unit vectors can beexpressed as follows

119894 = 119903 119895

119895 = minus119903 119894 +

120601

119896

119896 = minus

120601 119895 (38)

The first-order derivatives of (34) (36) and (37) can beobtained using (38) Thus the vehiclersquos roll moment and yawmoment have been acquired

According to the moment balance between the vehicleinertia moment and the external moment the vehiclersquos rolland yaw moment balance equation can be expressed asfollows

119868119911 119903 minus 119868119909119911

120601 minus 1198981198911198861199041198902 minus 1198981199011198861198911198902 minus 119868119909119911119901 (

120601 +

120579) + 119868119911119911119901 119903

minus 119868119909119910119901(

120601 +

120579)

2

minus 119868119910119911119901119903 (

120601 +

120579) = 2 (119865119891119897119891 minus 119865119903119897119903)

119868119909

120601 minus 119868119909119911 119903 + 119898119905ℎ119904119881(

120573 + 119903) + 1198671119898119891119886119904 + (ℎ119904 + 119887)119898119901119886119891

+ 119868119909119909119901 (

120601 +

120579) minus 119868119909119911119901 119903 + 119868119909119910119901119903 (

120601 +

120579) + 1198681199101199111199011199032

= minus119896120601120601 minus 119888120601

120601

+ 120601 (119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892 (ℎ119904 + 119887)) minus 119898119901119892119886119901 cos 120579(39)

where 119897119891 is the longitudinal distance from the CG of thetank truck to the front wheel 119897119903 is the longitudinal distancefrom the CG of the tank truck to the rear wheel 119896120601 is thesuspension roll stiffness 119888120601 is the suspension roll damping 1198902is the longitudinal distance from the CG of the tank truck tothe CG of liquid cargo and 1198671 is the vertical distance fromthe CG of the fixed liquid mass to the roll center

8 Discrete Dynamics in Nature and Society

Based on the parallel axis theorem of the moment ofinertia the following equation can be obtained

119868119911 = 119868119911119911119904 + 119868119911119911119906 + 1198981199051198882+ 119898119906119890

2+ 119868119911119911119891

119868119909 = 119868119909119909119904 + 119868119909119909119891 + 119898119905ℎ2

119904

119868119911119909 = 119868119909119911119904 + 119868119909119911119891 + 119898119905ℎ119904119888

(40)

The dynamic model of the tank truck is composed of (17)and (39)ndash(40)

The tire cornering force in the dynamic model can beobtained using the magic formula for tires which is given by

119865 = 119863 sin (119862 arctan (119861120572 minus 119864 (119861120572 minus arctan119861120572))) (41)

where 120572 is the tirersquos sideslip angle and 119861 119862 119863 and 119864 are theconstant parameters of the magic formula for tires

The tire sideslip angle in (41) can be expressed as follows

120572119891 = arctan(119881120573 + 119903119897119891

119881

) minus 120575119891

120572119903 = arctan(119881120573 minus 119903119897119903

119881

) minus 120575119903

(42)

where 120572119891 is the front tire sideslip angle and 120572119903 is the rear tiresideslip angle 120575119891 is the front wheel steering angle and 120575119903 isthe rear wheel steering angle

For the convenience of vehicle dynamic simulation thetank truck dynamic model is translated into the followingform

119872119909 = 119873 (43)

and (43) can be rewritten as follows

[

[

[

[

11989811 11989812 11989813 11989814

11989821 11989822 11989823 11989824

11989831 11989832 11989833 11989834

11989841 11989842 11989843 11989844

]

]

]

]

[

[

[

[

120573

119903

120601

120579

]

]

]

]

=

[

[

[

[

1198991

1198992

1198993

1198994

]

]

]

]

(44)

where

11989811 = 119881 (119898119905 + 119898119891 + 119898119901 + 119898119906)

11989812 = 119888 (119898119905 + 119898119891) minus 119890119898119906

11989813 = minusℎ119904 (119898119905 + 119898119891) minus 1198981199011198631

11989814 = minus1198981199011198632

11989821 = 1198811198902 (minus119898119891 minus 119898119901)

11989822 = 119868119911 minus 1198981198911198881198902 + 119868119911119911119901

11989823 = minus119868119909119911 + 119898119891ℎ1199041198902 + 11989811990111986311198902 minus 119868119909119911119901

11989824 = 11989811990111986321198902 minus

1198681199091199111199011198601

119860

11989831 = 119881 (119898119905ℎ119904 + 1198981198911198671 + 119898119901119867)

11989832 = minus119868119909119911 + 1198981198911198881198671 minus 119868119909119911119901

11989833 = 119868119909 minus 119898119891ℎ1199041198671 minus 1198981199011198671198631 + 119868119909119909119901

11989834 =

1198681199091199091199011198601

119860

minus 1198981199011198671198632

11989841 = 119881 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

11989842 = 0

11989843 = 119886119901119887119901 + 119886119901119887 sin 120579

11989844 = 1198862

119901sin2120579 + 119887

2

119901cos2120579

1198991 = 2 (119865119891 + 119865119903) minus 119903119881 (119898119905 + 119898119891 + 119898119906)

minus 119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

1198992 = 2 (119865119891119897119891 minus 119865119903119897119903) + 1198981198911198902119881119903

+ 1198981199011198902 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus

119868119909119911119901

12057921198602

119860

+ 119868119909119910119901(

120601 +

1205791198611

119861

)

2

+ 119868119910119911119901119903 (

120601 +

1205791198611

119861

)

1198993 = minus 119888120601

120601 + 120601 (minus119896120601 + 119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892119867)

minus 119886119901119898119901119892 cos 120579 minus 119881119903 (119898119905ℎ119904 + 1198981198911198671)

minus 119867119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus 119868119909119910119901119903 (

120601 +

120579) minus 1199032119868119910119911119901 +

12057921198681199091199091199011198602

119860

1198994 = minus 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

(45)

According to the Cramer rule the variables in (43) canbe expressed as follows while the matrix 119872 has a nonzerodeterminant

1199091 =1198721

119873

1199092 =1198722

119873

1199093 =1198723

119873

1199094 =1198724

119873

(46)

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

and many improved algorithms have been produced toincrease the analysis precision to some degree [14 15]

To investigate the accuracy of the QS method someresearchers studied the relationship between the evaluatedliquid sloshing effect obtained by calculating the movementof the CG of the liquid bulk and the observed effect Theyhave found that the mean values of sloshing forces and thecoordinates of the CG of the liquid bulk are quite closebetween the estimated and observed cases [16] This resultdemonstrates the correctness of the QS method which meanthat the worst vehicle driving state can be predicted by the QSmethod [17]

However the QS method cannot reflect the dynamiccharacteristics of liquid sloshing Therefore an equivalentmechanical model which can precisely demonstrate thedynamic characteristic of liquid sloshing in a partially filledtank must be modeled first [18ndash21] In estimating the tankerrsquosdynamic analysis using equivalent mechanical model forliquid sloshing the biggest difficulty for researchers is thecoupling of the fluid cargo and the solid vehicle To solvethis problem some people analyzed tanker driving stabilityinmultibody analysis software and someused the contractiveanalysis method [20 22] Neither method had the advantagesof the equivalent mechanical model for liquid sloshing as avehiclersquos driving stability cannot be precisely described by theanalytical model for vehicle roll stability control system

Therefore the purpose of this paper is to establish a tankvehicle dynamic model based on the equivalent mechanicalmodel for transient liquid sloshing in a partially filled tankThe trammel pendulum was used to describe liquid sloshingits motion was analyzed and the motion equation wasderived under the tank-fixed coordinate Then the couplingcondition for the fluid cargo and the solid vehicle wasstudied and a tanker dynamic model was established basedon this Finally MATLAB was used to simulate the rollstability of a typical tank truck under steering angle step testand some important conclusions were drawn The researchachievement is of great significance for commanding andevaluating tank trucks roll stability performance judging theconditions under which a tanker will lose its driving stabilityand designing roll stability control systems for tankers toimprove their driving safety

2 Equivalent Trammel Pendulum for LiquidSloshing in a Partially Filled Tank

21 The Basic Motion Equation and Parameter Values of theTrammel Pendulum Theoretical analysis and experimentalstudies had revealed that the first-order sloshingmode whichcan be described by the oscillation of liquid-free surfaceis the most important mode of liquid sloshing in partiallyfilled tanks [20 21] MATLABrsquos numerical method was usedto calculate the trajectory of the CG of the liquid bulkthat oscillates in a tank with an elliptical or circular crosssection The results showed that the trajectory of the CG ofthe liquid bulk remains parallel to the tankrsquos periphery [21]which is presented in Figure 1 According to this fact thetrammel pendulum model whose swing trajectory is elliptic

Trajectory ofliquid CG

Tank periphery

120579ap

ap

zp

bp yp

mp

mf

bf

A

Bacgbcg

Figure 1 Schematic diagram for the motion of the trammel pendu-lum

120579

ap

zp

bp

yp

rarrr

Figure 2 Analytical diagram for the motion of the trammel pendu-lum under an inertial coordinate system

is established to approximately simulate liquid sloshing in apartially filled tank

In Figure 1 119886119901 is half of the major axis of the pendulumrsquososcillation trajectory and 119887119901 is half of its minor axis 119886cg ishalf of the major axis of the elliptical trajectory of the CGof the liquid bulk and 119887cg is half of its minor axis 119898119901 is thependulum mass which is equal to the sloshing liquid mass119898119891 is the fixed liquid mass which does not participate insloshing 120579 is the pendulumrsquos amplitude

The Lagrangian function was used to derive the kineticequation of the trammel pendulum system The motionequation of the trammel pendulum that oscillates under tankfixed inertia coordinate system whose origin locates at thecenter of the tank (as shown in Figure 2) is written as follows

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) + 1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

+ 119892119887119901 cos 120579 = 0

(1)

The pendulum parameters in (1) are presented as follows

119887119901

119887

= 1089 + 0726Δ minus 01379Λ minus 0953Δ2minus 1216ΛΔ

+ 005141Λ2minus 006107Δ

3+ 05739ΛΔ

2+ 01632Λ

2Δ

(2)

Discrete Dynamics in Nature and Society 3

The tank periphery120579

zp

bp

yp

ap minus bp

120601

Z

Y

b

rarrx

rarrdrarr

R

rarrr

Figure 3 Analytical diagram for the motion of the trammel pendu-lum under the noninertial coordinate system

119898119901

119898

= 07844 minus 1729Δ + 03351Λ + 1156Δ2+ 07256ΛΔ

minus 01254Λ2minus 03219Δ

3minus 09152ΛΔ

2+ 008043Λ

2Δ

(3)

119898119891 = 119898 minus 119898119901 (4)

(119887 minus 119887119891)

119887

=

[119898 (119887 minus 119887cg) minus 119898119891 (119887 minus 119887119901)]

119898119891119887

(5)

where Δ is the liquid fill percentage which is the ratio of theheight of liquidrsquos level surface to that of the tank and Λ is theratio of the tankrsquos width to its height

The detailed derivation process for (1)ndash(5) is given byZheng et al [21]

According to Figure 1 the following equation holds

119886

119887

=

119886cg

119887cg=

119886119901

119887119901

= Λ (6)

From (6) the arm length of the pendulum can beexpressed as follows

119886119901 = 119887119901 times Λ (7)

As themass of liquid bulk can easily be acquiredwhile thetankrsquos size and liquid density are given the pendulum massand the fixed liquid mass can be obtained using (3) and (4)

22 Motion Equations of the Trammel Pendulum under theNoninertial Coordinate System During the driving processthe vehiclersquos driving state changes regularly under the influ-ence of a variety of factors It leads to the movement of thetank and in turn the coordinate system is being fixed relativeto the tank Therefore motion equations for a trammelpendulumunder a noninertial coordinate system (translationand rotation are included) should be derived in order to studythe motion characteristics of liquid bulk when tankers arebeing driven on curved sections of road avoiding obstaclesor changing lanes

The schematic diagram for the motion of the tank whenvehicle turns right is presented in Figure 3 For convenientanalysis of the vehiclersquos roll stability the origin of the tank-fixed coordinate system is located at the bottom of the tank

This location will not impact the motion analysis result Theorigin of the earth-fixed coordinate system also is located atthe bottom of the tank when the vehicle is static is thedistance vector from the origin of the earth-fixed coordinatesystem to that of the tank-fixed coordinate system is theabsolute position vector of the pendulum mass ball isthe absolute position vector of the pendulum mass ball 120601 isthe tank roll angle which is defined as the angle formed byrotating the 119884-axis counterclockwise till the position of the119910119901-axis

With reference to Figure 3 the absolute position of thependulum mass ball can be obtained as follows

= [119909 minus 119887 sin120601 + (119886119901 cos 120579 minus 119887119901 sin 120579 tan120601) cos120601] 119894

+ [119887 cos 120579 + (119887119901 sin 120579 + 119886119901 cos 120579 tan120601) cos120601] 119895

(8)

The velocity of the pendulum mass ball can be obtainedby solving the first-order derivative of (8) which may bepresented as follows

= [ minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601) 120601

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601) 120579] 119894

+ [(minus119887 sin120601 minus 119887119901 sin 120579 sin120601 + 119886119901 cos 120579 cos120601) 120601

+ (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601) 120579] 119895

(9)

Similarly the acceleration of the pendulummass ball canbe acquired by solving the differential of (9) which may bewritten as follows

= [minus

120601 (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

minus

120579 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

+ 2

120579

120601 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)

+

1206012(119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

+

1205792(minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601) + ] 119894

+ [

120601 (minus119887 sin120601 + 119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

+

120579 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1205792(119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

+

1206012(minus119887 cos120601 minus 119886119901 cos 120579 sin120601 minus 119887119901 sin 120579 cos120601)

minus 2

120579

120601 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)] 119895

(10)

The zero of potential energy is defined as the 119884-axisand the potential energy of the pendulum mass ball can bepresented as follows

119881 = 119898119892 (119887 + 119887119901 sin 120579 + 119886119901 cos 120579 tan120601) cos120601 (11)

4 Discrete Dynamics in Nature and Society

According to (10) and (11) a Lagrangian function can beused to obtain the motion equation of the pendulum systemwhich is written as follows

119871 = 119879 minus 119881 (12)

where 119879 is the kinetic energy of the pendulum mass ballThe motion of the trammel pendulum system can be

expressed by

120597

120597119905

(

120597119871

120597 119886119895

) minus

120597119871

120597119886119895

= 0 (13)

where 119886119895 is the degree of freedom for system 119871According to the motion analysis of the trammel pendu-

lum 120579 the pendulum amplitude and 120601 the tank roll angleare the two degrees of freedom for the trammel pendulumsystem

The differential of (12) with respect to 120579 can be expressedas follows

120597119871

120597120579

= 119898 [

1206012(minus1198862

119901sin 120579 cos 120579 + 119887

2

119901sin 120579 cos 120579 + 119887119901119887 cos 120579)

+

1205792(1198862

119901sin 120579 cos 120579 minus 119887

2

119901sin 120579 cos 120579)

+

120579

120601 (119886119901119887 cos 120579)

minus

120601 (minus119886119901 sin 120579 sin120601 + 119887119901 cos 120579 cos120601)

minus

120579 (119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

minus119892 (119887119901 cos 120579 minus 119886119901 sin 120579 tan120601) cos120601] (14)

The differential of (12) with respect to 120579 can be expressed

by

120597119871

120597

120579

= 119898 [

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) +

120601 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)] (15)

Finally the differential of (15) with respect to 119905 can be ex-pressed as follows

120597

120597119905

(

120597119871

120597

120579

) = 119898 [

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579)

+

1205792(21198862

119901sin 120579 cos 120579 minus 2119887

2

119901sin 120579 cos 120579)

+

120601 (119886119901119887119901 + 119886119901119887 sin 120579) + 120601

120579119886119901119887 cos 120579

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

minus

120579 (119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

+

120601 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)] (16)

Substituting (14)ndash(16) into (13) we get

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) +

120601 (119886119901119887119901 + 119886119901119887 sin 120579)

+

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

+

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

+ 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601) = 0

(17)

The differential of (12) with respect to 120601 can be expressedas follows

120597119871

120597120601

= 119898 [

120601 (119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

minus

120579 (minus119886119901 sin 120579 sin120601 + 119887119901 cos 120579 cos120601)

minus 119892119886119901 cos 120579 sec120601

+ 119892 (119887 + 119887119901 sin 120579 + 119886119901 cos 120579 tan120601) sin120601]

(18)

The differential of (12) with respect to 120601 can be expressed

by

120597119871

120597

120601

= 119898 [

120601 (1198872+ 1198862

119901cos2120579 + 119887

2

119901sin2120579 + 2119887119901119887 sin 120579)

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)] (19)

According to (19) the differential of (19) with respect to 119905can be obtained as follows

120597

120597119905

(

120597119871

120597

120601

)

= 119898 [

120601 (1198872+ 1198862

119901cos2120579 + 119887

2

119901sin2120579 + 2119887119901119887 sin 120579)

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579) + 119886119901119887

1205792 cos 120579

+

120601

120579 (minus21198862

119901sin 120579 cos 120579 + 2119887

2

119901sin 120579 cos 120579 + 2119887119901119887 cos 120579)

+

120579 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)

+

120601 (119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)] (20)

Discrete Dynamics in Nature and Society 5

Substituting (18)ndash(20) into (13) we get

120601 [(119887 + 119887119901 sin 120579)

2

+ 1198862

119901cos2120579]

+ 2

120579

120601 [

1

2

sin 2120579 (1198872119901minus 1198862

119901) + 119887119901119887 cos 120579]

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

+

1205792119886119901119887 cos 120579

minus 119892 [(119887 + 119887119901 sin 120579) sin120601 minus 119886119901 cos 120579 cos120601] = 0

(21)

Equations (17) and (21) are the equations of motionof the trammel pendulum under the noninertia coordinatesystem and the pendulum motion characteristics under thetranslation and rotation of the coordinate system can beanalyzed using (17) and (21)

3 Tank Truck Dynamic Model

31 Assumptions The roll stability of a tank truck is greatlyinfluenced by transient liquid sloshing in partially filledtanks Due to the flow characteristic of liquid bulk transientliquid sloshing is produced when the vehiclersquos driving statechanges which leads to the oscillation of the pendulummassball To simplify the analysis of the tank truckrsquos roll stabilitysome assumptions are made and listed as follows

(1) The sprung mass is the vehicle mass that lies over thesuspension system in which the mass of the liquidcargo is not included In this paper vehicle and liquidbulk as the solid and fluid parts respectively are notmixed with each other

(2) In the transverse direction the CG of the sprungmassis located at the bottom of the tank Moreover thesprung mass is symmetrical about the longitudinalaxis of the vehicle

(3) Transient liquid sloshing is merely produced alongthe transverse direction longitudinal liquid sloshingis not taken into consideration The movement andmass distribution of the liquid bulk are completelythe same at different tank cross sections whichmeansthat the CG of liquid bulk is located in the middle ofthe tank along the longitudinal direction

(4) The liquid-free surface is not broken during thevehiclersquos driving process

32 Force Analysis of the Tank Truck Based on the assump-tions given in Section 31 a reference coordinate system fortank truck dynamic analysis was selected first

The coordinate system being fixed relative to the vehiclewas chosen as the reference coordinate system for thedynamic analysis of the tank truck and the establishment ofthe vehiclersquos dynamic model The origin of the coordinatesystem is the point where the vertical line that goes through

the CG of the vehicle when it is static intersects with thevehiclersquos roll axis

The coordinate system being fixed relative to the vehiclersquossprung mass was defined as the 119909-119910-119911 system where the 119909-axis is parallel to the longitudinal axis of the vehicle andpoints in the direction in which the vehicle is being driventhe 119910-axis is perpendicular to the 119909-axis and points to the leftfrom the driverrsquos perspective and the 119911-axis is perpendicularto the 119909119910 plane and points straight up

The vehiclersquos unsprung mass is assumed not to produceroll movement

The top view and the back view of the vehiclersquos dynamicanalysis are plotted in Figures 4 and 5 According to Figure 5the lateral inertial forces act on the sprung mass theunsprung mass the fixed liquid mass and the pendulummass making up the vehiclersquos lateral inertial force The tirecornering forces make up the external lateral force that actson the vehicle according to Figure 4 On the basis of Newtonrsquosfirst law when the vehicle is steady its inertial lateral forceand external lateral force are equal in value and positive indirection which can be expressed as follows

(119898119905 + 119898119891) 119886119904 + 119898119901119886119891 + 119898119906119886119906 = 2 (119865119891 + 119865119903) (22)

where 119886119904 is the lateral acceleration of sprung mass 119886119906 isthe lateral acceleration of unsprung mass 119886119891 is the lateralacceleration of the pendulum mass ball 119898119905 is sprung mass119898119906 is unsprung mass 119865119891 is the front tire cornering force and119865119903 is the rear tire cornering force

Generally the fixed liquid mass does not stay static butslides slowly with the aid of gravity while the tank tiltsHowever the shift of the fixed liquid mass is quite smallcompared to that of the pendulum mass ball and its shiftamount drops greatly with the increase in the liquid fillpercentage Therefore the fixed liquid mass is assumed to bestatic and its position is assumed to stay constant as the tankrolls over Based on this assumption the fixed liquidmass canbe seen as the solid part and its lateral acceleration is equal tothat of the vehiclersquos sprung mass

According to the dynamic analysis of the vehicle thelateral acceleration of the sprung mass and unsprung masscan be presented by

119886119904 = 119881 (

120573 + 119903) minus ℎ119904

120601 + 119888 119903

119886119906 = 119881 (

120573 + 119903) minus 119890 119903

(23)

where 119881 is the vehiclersquos driving speed 120573 the vehiclersquos slipangle 119903 the vehiclersquos yaw angle ℎ119904 is the vertical distance fromthe CG of sprung mass to the roll center 119888 is the longitudinaldistance between the CG of sprung mass and that of the tanktruck and 119890 is the longitudinal distance between the CG ofunsprung mass and that of the tank truck

According to (10) the lateral acceleration of the pendu-lum mass ball can be written as follows

119886119891 = 119881

120601 minus 1198631

120601 minus 1198632

120579 + 21198633

120601

120579 + 1198634

1206012+ 1198635

1205792 (24)

6 Discrete Dynamics in Nature and Society

Front wheel

CG of tank truck

CG of unsprungmass

CG of liquidcargo

Rear wheel

CG of sprung mass

Ff

Fr

lr

lf

c

e

e2

muau

mtas

mfas + mpaf

Figure 4 Top view of vehicle dynamic analysis

b

Central lineof the tank

Sprung mass

Pendulummass

Unsprung mass

Liquid fixed mass

Roll center

ΔFΔF

a

hs

muau

mpafmtas

mfas

Fr Fr

kr120601 + cr(d120601)

Figure 5 Back view of vehicle dynamic analysis

where

1198631 = 119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601

1198632 = 119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601

1198633 = 119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601

1198634 = 119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

1198635 = minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

(25)

The force analysis in the roll plane is plotted in Figure 5Due to the lateral inertia force the sprungmass and the liquid

Central line

of the tank

Coordinatefixed on the

tank

Coordinate fixedon the vehiclesprung mass

120579

ap

z

bp

y

AB

y998400

x998400

z998400

120601hs

1205792

b

x

o

Figure 6The schematic diagram for the pendulum rotating aroundthe roll axis

bulk rotate about the roll axis and bring about rollover torqueThe roll angle about the roll axis for sprungmass and the fixedliquid mass is 120601 which can be seen from Figure 5 Howeverowing to the tank and the pendulum mass ball movementthe roll angle about the roll axis for the pendulum mass ballis composed of two parts which is shown in Figure 6

According to Figure 6 the roll angle about the roll axis forthe pendulum mass ball can be expressed as follows

ang = 120601 + 1205792 (26)

where

1205792 = arctan(119860119861119861119874

) = arctan(minus119886119901 cos 120579

ℎ119904 + 119887 + 119887119901 sin 120579) (27)

Therefore the absolute roll rate and roll angular accelera-tion can be obtained as follows based on (26)

ang =

120601 +

1205792 ang =

120601 +

1205792(28)

1205792 =

1205791198611

119861

1205792 =

(1198601

120579 minus 1198602

1205792)

119860

(29)

where

119861 = (ℎ119904 + 119887 + 119887119901 sin 120579)2

+ (119886119901 cos 120579)2

1198611 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901]

119860 = 1198612 1198601 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901] 119861

1198602 = 2119886119901 cos 120579 [(ℎ119904 + 119887) sin 120579 + 119887119901]

times [119887119901 (ℎ119904 + 119887 + 119887119901 sin 120579) minus 1198862

119901sin 120579]

minus 119886119901119861 (ℎ119904 + 119887) cos 120579

(30)

Discrete Dynamics in Nature and Society 7

On the basis of (28) the vehiclersquos roll balance equationand yaw balance equation can be acquired by differentiatingthe angular momentum with respect to time

In general the angular momentum of a rotating object isdefined as the product of a bodyrsquos rotational inertia tensor androtational velocities about a particular coordinate(1) The Angular Momentum of Sprung Mass and the FixedLiquid Mass As the vehicle sprung mass and the fixed liquidmass have the same rotational velocity about the 119909-119910-119911coordinate system the angular momentum for the two partsis solved together

According to the definition of angular momentum thatfor the sprungmass and fixed liquidmass can be expressed asfollows

119867119904 =[

[

119868119909119909119904 + 119868119909119909119891 minus119868119909119910119904 minus 119868119909119910119891 minus119868119909119911119904 minus 119868119909119911119891

minus119868119910119909119904 minus 119868119910119909119891 119868119910119910119904 + 119868119910119910119891 minus119868119910119911119904 minus 119868119910119911119891

minus119868119911119909119904 minus 119868119911119909119891 minus119868119911119910119904 minus 119868119911119910119891 119868119911119911119904 + 119868119911119911119891

]

]

[

[

120601

0

119903

]

]

(31)

where 119868119909119909119904 119868119910119910119904 and 119868119911119911119904 are the moments of inertia for thesprungmass about the 119909-axis119910-axis and 119911-axis respectively119868119909119909119891 119868119910119910119891 and 119868119911119911119891 are the moments of inertia for the fixedliquid mass about the 119909-axis 119910-axis and 119911-axis respectively119868119909119910119904 = 119868119910119909119904 119868119909119911119904 = 119868119911119909119904 and 119868119910119911119904 = 119868119911119910119904 are the products ofinertia for the sprungmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively 119868119909119910119891 = 119868119910119909119891 119868119909119911119891 = 119868119911119909119891and 119868119910119911119891 = 119868119911119910119891 are the products of inertia for the fixed liquidmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively

According to the assumptions given in Section 31 thedistribution of the vehiclersquos sprungmass is symmetrical aboutthe119909119911-planeTherefore the products of inertia for the sprungmass can be expressed as follows

119868119909119910119904 = 0 119868119911119910119904 = 0 (32)

As the fixed liquidmass does notmovewhile the tank tiltsthe distribution of the fixed liquid mass is symmetrical aboutthe 119909119905119911119905-plane and the 119909119911-plane Therefore the products ofinertia for the fixed liquid mass can be expressed as follows

119868119909119910119891 = 0 119868119911119910119891 = 0 (33)

Therefore based on (31)ndash(33) the angularmomentum forthe vehiclersquos sprung mass and the fixed liquid mass can bewritten as follows

119867119904 = [(119868119909119909119904 + 119868119909119909119891)

120601 minus (119868119909119911119904 + 119868119909119911119891) 119903]119894

+ [minus (119868119911119909119904 + 119868119911119909119891)

120601 + (119868119911119911119904 + 119868119911119911119891) 119903]119895

(34)

(2)TheAngularMomentum of the PendulumMassAccordingto (28) the angular momentum of the pendulummass can beexpressed as follows

119867119901 =[

[

119868119909119909119901 minus119868119909119910119901 minus119868119909119911119901

minus119868119910119909119901 119868119910119910119901 minus119868119910119911119901

minus119868119911119909119901 minus119868119911119910119901 119868119911119911119901

]

]

[

[

120601 +

1205792

0

119903

]

]

(35)

where 119868119909119909119901 119868119910119910119901 and 119868119911119911119901 are the moments of inertia forthe pendulum mass about the 119909-axis 119910-axis and 119911-axis

respectively 119868119909119910119901 = 119868119910119909119901 119868119909119911119901 = 119868119911119909119901 and 119868119910119911119901 = 119868119911119910119901 are theproducts of inertia for the pendulum mass about the 119909- and119910-axis the 119909- and 119911-axis and the 119910- and 119911-axis respectively

The expanded form of (35) is written as

119867119901 = [119868119909119909119901 (

120601 +

1205792) minus 119868119909119911119901119903]119894 + [minus119868119909119910119901 (

120601 +

1205792) minus 119868119910119911119901119903]

119895

+ [minus119868119909119911119901 (

120601 +

1205792) + 119868119911119911119901119903]119896

(36)

(3) The Angular Momentum of the Unsprung Mass The yawmovement is the only rotational degree of freedom for theunsprung mass Therefore the angular momentum of theunsprung mass can be obtained by multiplying together themoment of inertia about the 119911-axis and the yaw rate aboutthat which is written as follows

119867119906 = 119868119911119911119906119903119896 (37)

where 119868119911119911119906 is the moment of inertia about the 119911-axis for theunsprung mass

The change in the vehicle fixed coordinate after it has beendriving for time Δ119905 is plotted in Figure 7 from which we cansee that the differentials of the 119894 119895 and

119896 unit vectors can beexpressed as follows

119894 = 119903 119895

119895 = minus119903 119894 +

120601

119896

119896 = minus

120601 119895 (38)

The first-order derivatives of (34) (36) and (37) can beobtained using (38) Thus the vehiclersquos roll moment and yawmoment have been acquired

According to the moment balance between the vehicleinertia moment and the external moment the vehiclersquos rolland yaw moment balance equation can be expressed asfollows

119868119911 119903 minus 119868119909119911

120601 minus 1198981198911198861199041198902 minus 1198981199011198861198911198902 minus 119868119909119911119901 (

120601 +

120579) + 119868119911119911119901 119903

minus 119868119909119910119901(

120601 +

120579)

2

minus 119868119910119911119901119903 (

120601 +

120579) = 2 (119865119891119897119891 minus 119865119903119897119903)

119868119909

120601 minus 119868119909119911 119903 + 119898119905ℎ119904119881(

120573 + 119903) + 1198671119898119891119886119904 + (ℎ119904 + 119887)119898119901119886119891

+ 119868119909119909119901 (

120601 +

120579) minus 119868119909119911119901 119903 + 119868119909119910119901119903 (

120601 +

120579) + 1198681199101199111199011199032

= minus119896120601120601 minus 119888120601

120601

+ 120601 (119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892 (ℎ119904 + 119887)) minus 119898119901119892119886119901 cos 120579(39)

where 119897119891 is the longitudinal distance from the CG of thetank truck to the front wheel 119897119903 is the longitudinal distancefrom the CG of the tank truck to the rear wheel 119896120601 is thesuspension roll stiffness 119888120601 is the suspension roll damping 1198902is the longitudinal distance from the CG of the tank truck tothe CG of liquid cargo and 1198671 is the vertical distance fromthe CG of the fixed liquid mass to the roll center

8 Discrete Dynamics in Nature and Society

Based on the parallel axis theorem of the moment ofinertia the following equation can be obtained

119868119911 = 119868119911119911119904 + 119868119911119911119906 + 1198981199051198882+ 119898119906119890

2+ 119868119911119911119891

119868119909 = 119868119909119909119904 + 119868119909119909119891 + 119898119905ℎ2

119904

119868119911119909 = 119868119909119911119904 + 119868119909119911119891 + 119898119905ℎ119904119888

(40)

The dynamic model of the tank truck is composed of (17)and (39)ndash(40)

The tire cornering force in the dynamic model can beobtained using the magic formula for tires which is given by

119865 = 119863 sin (119862 arctan (119861120572 minus 119864 (119861120572 minus arctan119861120572))) (41)

where 120572 is the tirersquos sideslip angle and 119861 119862 119863 and 119864 are theconstant parameters of the magic formula for tires

The tire sideslip angle in (41) can be expressed as follows

120572119891 = arctan(119881120573 + 119903119897119891

119881

) minus 120575119891

120572119903 = arctan(119881120573 minus 119903119897119903

119881

) minus 120575119903

(42)

where 120572119891 is the front tire sideslip angle and 120572119903 is the rear tiresideslip angle 120575119891 is the front wheel steering angle and 120575119903 isthe rear wheel steering angle

For the convenience of vehicle dynamic simulation thetank truck dynamic model is translated into the followingform

119872119909 = 119873 (43)

and (43) can be rewritten as follows

[

[

[

[

11989811 11989812 11989813 11989814

11989821 11989822 11989823 11989824

11989831 11989832 11989833 11989834

11989841 11989842 11989843 11989844

]

]

]

]

[

[

[

[

120573

119903

120601

120579

]

]

]

]

=

[

[

[

[

1198991

1198992

1198993

1198994

]

]

]

]

(44)

where

11989811 = 119881 (119898119905 + 119898119891 + 119898119901 + 119898119906)

11989812 = 119888 (119898119905 + 119898119891) minus 119890119898119906

11989813 = minusℎ119904 (119898119905 + 119898119891) minus 1198981199011198631

11989814 = minus1198981199011198632

11989821 = 1198811198902 (minus119898119891 minus 119898119901)

11989822 = 119868119911 minus 1198981198911198881198902 + 119868119911119911119901

11989823 = minus119868119909119911 + 119898119891ℎ1199041198902 + 11989811990111986311198902 minus 119868119909119911119901

11989824 = 11989811990111986321198902 minus

1198681199091199111199011198601

119860

11989831 = 119881 (119898119905ℎ119904 + 1198981198911198671 + 119898119901119867)

11989832 = minus119868119909119911 + 1198981198911198881198671 minus 119868119909119911119901

11989833 = 119868119909 minus 119898119891ℎ1199041198671 minus 1198981199011198671198631 + 119868119909119909119901

11989834 =

1198681199091199091199011198601

119860

minus 1198981199011198671198632

11989841 = 119881 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

11989842 = 0

11989843 = 119886119901119887119901 + 119886119901119887 sin 120579

11989844 = 1198862

119901sin2120579 + 119887

2

119901cos2120579

1198991 = 2 (119865119891 + 119865119903) minus 119903119881 (119898119905 + 119898119891 + 119898119906)

minus 119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

1198992 = 2 (119865119891119897119891 minus 119865119903119897119903) + 1198981198911198902119881119903

+ 1198981199011198902 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus

119868119909119911119901

12057921198602

119860

+ 119868119909119910119901(

120601 +

1205791198611

119861

)

2

+ 119868119910119911119901119903 (

120601 +

1205791198611

119861

)

1198993 = minus 119888120601

120601 + 120601 (minus119896120601 + 119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892119867)

minus 119886119901119898119901119892 cos 120579 minus 119881119903 (119898119905ℎ119904 + 1198981198911198671)

minus 119867119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus 119868119909119910119901119903 (

120601 +

120579) minus 1199032119868119910119911119901 +

12057921198681199091199091199011198602

119860

1198994 = minus 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

(45)

According to the Cramer rule the variables in (43) canbe expressed as follows while the matrix 119872 has a nonzerodeterminant

1199091 =1198721

119873

1199092 =1198722

119873

1199093 =1198723

119873

1199094 =1198724

119873

(46)

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

The tank periphery120579

zp

bp

yp

ap minus bp

120601

Z

Y

b

rarrx

rarrdrarr

R

rarrr

Figure 3 Analytical diagram for the motion of the trammel pendu-lum under the noninertial coordinate system

119898119901

119898

= 07844 minus 1729Δ + 03351Λ + 1156Δ2+ 07256ΛΔ

minus 01254Λ2minus 03219Δ

3minus 09152ΛΔ

2+ 008043Λ

2Δ

(3)

119898119891 = 119898 minus 119898119901 (4)

(119887 minus 119887119891)

119887

=

[119898 (119887 minus 119887cg) minus 119898119891 (119887 minus 119887119901)]

119898119891119887

(5)

where Δ is the liquid fill percentage which is the ratio of theheight of liquidrsquos level surface to that of the tank and Λ is theratio of the tankrsquos width to its height

The detailed derivation process for (1)ndash(5) is given byZheng et al [21]

According to Figure 1 the following equation holds

119886

119887

=

119886cg

119887cg=

119886119901

119887119901

= Λ (6)

From (6) the arm length of the pendulum can beexpressed as follows

119886119901 = 119887119901 times Λ (7)

As themass of liquid bulk can easily be acquiredwhile thetankrsquos size and liquid density are given the pendulum massand the fixed liquid mass can be obtained using (3) and (4)

22 Motion Equations of the Trammel Pendulum under theNoninertial Coordinate System During the driving processthe vehiclersquos driving state changes regularly under the influ-ence of a variety of factors It leads to the movement of thetank and in turn the coordinate system is being fixed relativeto the tank Therefore motion equations for a trammelpendulumunder a noninertial coordinate system (translationand rotation are included) should be derived in order to studythe motion characteristics of liquid bulk when tankers arebeing driven on curved sections of road avoiding obstaclesor changing lanes

The schematic diagram for the motion of the tank whenvehicle turns right is presented in Figure 3 For convenientanalysis of the vehiclersquos roll stability the origin of the tank-fixed coordinate system is located at the bottom of the tank

This location will not impact the motion analysis result Theorigin of the earth-fixed coordinate system also is located atthe bottom of the tank when the vehicle is static is thedistance vector from the origin of the earth-fixed coordinatesystem to that of the tank-fixed coordinate system is theabsolute position vector of the pendulum mass ball isthe absolute position vector of the pendulum mass ball 120601 isthe tank roll angle which is defined as the angle formed byrotating the 119884-axis counterclockwise till the position of the119910119901-axis

With reference to Figure 3 the absolute position of thependulum mass ball can be obtained as follows

= [119909 minus 119887 sin120601 + (119886119901 cos 120579 minus 119887119901 sin 120579 tan120601) cos120601] 119894

+ [119887 cos 120579 + (119887119901 sin 120579 + 119886119901 cos 120579 tan120601) cos120601] 119895

(8)

The velocity of the pendulum mass ball can be obtainedby solving the first-order derivative of (8) which may bepresented as follows

= [ minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601) 120601

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601) 120579] 119894

+ [(minus119887 sin120601 minus 119887119901 sin 120579 sin120601 + 119886119901 cos 120579 cos120601) 120601

+ (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601) 120579] 119895

(9)

Similarly the acceleration of the pendulummass ball canbe acquired by solving the differential of (9) which may bewritten as follows

= [minus

120601 (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

minus

120579 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

+ 2

120579

120601 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)

+

1206012(119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

+

1205792(minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601) + ] 119894

+ [

120601 (minus119887 sin120601 + 119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

+

120579 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1205792(119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

+

1206012(minus119887 cos120601 minus 119886119901 cos 120579 sin120601 minus 119887119901 sin 120579 cos120601)

minus 2

120579

120601 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)] 119895

(10)

The zero of potential energy is defined as the 119884-axisand the potential energy of the pendulum mass ball can bepresented as follows

119881 = 119898119892 (119887 + 119887119901 sin 120579 + 119886119901 cos 120579 tan120601) cos120601 (11)

4 Discrete Dynamics in Nature and Society

According to (10) and (11) a Lagrangian function can beused to obtain the motion equation of the pendulum systemwhich is written as follows

119871 = 119879 minus 119881 (12)

where 119879 is the kinetic energy of the pendulum mass ballThe motion of the trammel pendulum system can be

expressed by

120597

120597119905

(

120597119871

120597 119886119895

) minus

120597119871

120597119886119895

= 0 (13)

where 119886119895 is the degree of freedom for system 119871According to the motion analysis of the trammel pendu-

lum 120579 the pendulum amplitude and 120601 the tank roll angleare the two degrees of freedom for the trammel pendulumsystem

The differential of (12) with respect to 120579 can be expressedas follows

120597119871

120597120579

= 119898 [

1206012(minus1198862

119901sin 120579 cos 120579 + 119887

2

119901sin 120579 cos 120579 + 119887119901119887 cos 120579)

+

1205792(1198862

119901sin 120579 cos 120579 minus 119887

2

119901sin 120579 cos 120579)

+

120579

120601 (119886119901119887 cos 120579)

minus

120601 (minus119886119901 sin 120579 sin120601 + 119887119901 cos 120579 cos120601)

minus

120579 (119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

minus119892 (119887119901 cos 120579 minus 119886119901 sin 120579 tan120601) cos120601] (14)

The differential of (12) with respect to 120579 can be expressed

by

120597119871

120597

120579

= 119898 [

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) +

120601 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)] (15)

Finally the differential of (15) with respect to 119905 can be ex-pressed as follows

120597

120597119905

(

120597119871

120597

120579

) = 119898 [

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579)

+

1205792(21198862

119901sin 120579 cos 120579 minus 2119887

2

119901sin 120579 cos 120579)

+

120601 (119886119901119887119901 + 119886119901119887 sin 120579) + 120601

120579119886119901119887 cos 120579

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

minus

120579 (119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

+

120601 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)] (16)

Substituting (14)ndash(16) into (13) we get

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) +

120601 (119886119901119887119901 + 119886119901119887 sin 120579)

+

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

+

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

+ 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601) = 0

(17)

The differential of (12) with respect to 120601 can be expressedas follows

120597119871

120597120601

= 119898 [

120601 (119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

minus

120579 (minus119886119901 sin 120579 sin120601 + 119887119901 cos 120579 cos120601)

minus 119892119886119901 cos 120579 sec120601

+ 119892 (119887 + 119887119901 sin 120579 + 119886119901 cos 120579 tan120601) sin120601]

(18)

The differential of (12) with respect to 120601 can be expressed

by

120597119871

120597

120601

= 119898 [

120601 (1198872+ 1198862

119901cos2120579 + 119887

2

119901sin2120579 + 2119887119901119887 sin 120579)

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)] (19)

According to (19) the differential of (19) with respect to 119905can be obtained as follows

120597

120597119905

(

120597119871

120597

120601

)

= 119898 [

120601 (1198872+ 1198862

119901cos2120579 + 119887

2

119901sin2120579 + 2119887119901119887 sin 120579)

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579) + 119886119901119887

1205792 cos 120579

+

120601

120579 (minus21198862

119901sin 120579 cos 120579 + 2119887

2

119901sin 120579 cos 120579 + 2119887119901119887 cos 120579)

+

120579 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)

+

120601 (119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)] (20)

Discrete Dynamics in Nature and Society 5

Substituting (18)ndash(20) into (13) we get

120601 [(119887 + 119887119901 sin 120579)

2

+ 1198862

119901cos2120579]

+ 2

120579

120601 [

1

2

sin 2120579 (1198872119901minus 1198862

119901) + 119887119901119887 cos 120579]

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

+

1205792119886119901119887 cos 120579

minus 119892 [(119887 + 119887119901 sin 120579) sin120601 minus 119886119901 cos 120579 cos120601] = 0

(21)

Equations (17) and (21) are the equations of motionof the trammel pendulum under the noninertia coordinatesystem and the pendulum motion characteristics under thetranslation and rotation of the coordinate system can beanalyzed using (17) and (21)

3 Tank Truck Dynamic Model

31 Assumptions The roll stability of a tank truck is greatlyinfluenced by transient liquid sloshing in partially filledtanks Due to the flow characteristic of liquid bulk transientliquid sloshing is produced when the vehiclersquos driving statechanges which leads to the oscillation of the pendulummassball To simplify the analysis of the tank truckrsquos roll stabilitysome assumptions are made and listed as follows

(1) The sprung mass is the vehicle mass that lies over thesuspension system in which the mass of the liquidcargo is not included In this paper vehicle and liquidbulk as the solid and fluid parts respectively are notmixed with each other

(2) In the transverse direction the CG of the sprungmassis located at the bottom of the tank Moreover thesprung mass is symmetrical about the longitudinalaxis of the vehicle

(3) Transient liquid sloshing is merely produced alongthe transverse direction longitudinal liquid sloshingis not taken into consideration The movement andmass distribution of the liquid bulk are completelythe same at different tank cross sections whichmeansthat the CG of liquid bulk is located in the middle ofthe tank along the longitudinal direction

(4) The liquid-free surface is not broken during thevehiclersquos driving process

32 Force Analysis of the Tank Truck Based on the assump-tions given in Section 31 a reference coordinate system fortank truck dynamic analysis was selected first

The coordinate system being fixed relative to the vehiclewas chosen as the reference coordinate system for thedynamic analysis of the tank truck and the establishment ofthe vehiclersquos dynamic model The origin of the coordinatesystem is the point where the vertical line that goes through

the CG of the vehicle when it is static intersects with thevehiclersquos roll axis

The coordinate system being fixed relative to the vehiclersquossprung mass was defined as the 119909-119910-119911 system where the 119909-axis is parallel to the longitudinal axis of the vehicle andpoints in the direction in which the vehicle is being driventhe 119910-axis is perpendicular to the 119909-axis and points to the leftfrom the driverrsquos perspective and the 119911-axis is perpendicularto the 119909119910 plane and points straight up

The vehiclersquos unsprung mass is assumed not to produceroll movement

The top view and the back view of the vehiclersquos dynamicanalysis are plotted in Figures 4 and 5 According to Figure 5the lateral inertial forces act on the sprung mass theunsprung mass the fixed liquid mass and the pendulummass making up the vehiclersquos lateral inertial force The tirecornering forces make up the external lateral force that actson the vehicle according to Figure 4 On the basis of Newtonrsquosfirst law when the vehicle is steady its inertial lateral forceand external lateral force are equal in value and positive indirection which can be expressed as follows

(119898119905 + 119898119891) 119886119904 + 119898119901119886119891 + 119898119906119886119906 = 2 (119865119891 + 119865119903) (22)

where 119886119904 is the lateral acceleration of sprung mass 119886119906 isthe lateral acceleration of unsprung mass 119886119891 is the lateralacceleration of the pendulum mass ball 119898119905 is sprung mass119898119906 is unsprung mass 119865119891 is the front tire cornering force and119865119903 is the rear tire cornering force

Generally the fixed liquid mass does not stay static butslides slowly with the aid of gravity while the tank tiltsHowever the shift of the fixed liquid mass is quite smallcompared to that of the pendulum mass ball and its shiftamount drops greatly with the increase in the liquid fillpercentage Therefore the fixed liquid mass is assumed to bestatic and its position is assumed to stay constant as the tankrolls over Based on this assumption the fixed liquidmass canbe seen as the solid part and its lateral acceleration is equal tothat of the vehiclersquos sprung mass

According to the dynamic analysis of the vehicle thelateral acceleration of the sprung mass and unsprung masscan be presented by

119886119904 = 119881 (

120573 + 119903) minus ℎ119904

120601 + 119888 119903

119886119906 = 119881 (

120573 + 119903) minus 119890 119903

(23)

where 119881 is the vehiclersquos driving speed 120573 the vehiclersquos slipangle 119903 the vehiclersquos yaw angle ℎ119904 is the vertical distance fromthe CG of sprung mass to the roll center 119888 is the longitudinaldistance between the CG of sprung mass and that of the tanktruck and 119890 is the longitudinal distance between the CG ofunsprung mass and that of the tank truck

According to (10) the lateral acceleration of the pendu-lum mass ball can be written as follows

119886119891 = 119881

120601 minus 1198631

120601 minus 1198632

120579 + 21198633

120601

120579 + 1198634

1206012+ 1198635

1205792 (24)

6 Discrete Dynamics in Nature and Society

Front wheel

CG of tank truck

CG of unsprungmass

CG of liquidcargo

Rear wheel

CG of sprung mass

Ff

Fr

lr

lf

c

e

e2

muau

mtas

mfas + mpaf

Figure 4 Top view of vehicle dynamic analysis

b

Central lineof the tank

Sprung mass

Pendulummass

Unsprung mass

Liquid fixed mass

Roll center

ΔFΔF

a

hs

muau

mpafmtas

mfas

Fr Fr

kr120601 + cr(d120601)

Figure 5 Back view of vehicle dynamic analysis

where

1198631 = 119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601

1198632 = 119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601

1198633 = 119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601

1198634 = 119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

1198635 = minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

(25)

The force analysis in the roll plane is plotted in Figure 5Due to the lateral inertia force the sprungmass and the liquid

Central line

of the tank

Coordinatefixed on the

tank

Coordinate fixedon the vehiclesprung mass

120579

ap

z

bp

y

AB

y998400

x998400

z998400

120601hs

1205792

b

x

o

Figure 6The schematic diagram for the pendulum rotating aroundthe roll axis

bulk rotate about the roll axis and bring about rollover torqueThe roll angle about the roll axis for sprungmass and the fixedliquid mass is 120601 which can be seen from Figure 5 Howeverowing to the tank and the pendulum mass ball movementthe roll angle about the roll axis for the pendulum mass ballis composed of two parts which is shown in Figure 6

According to Figure 6 the roll angle about the roll axis forthe pendulum mass ball can be expressed as follows

ang = 120601 + 1205792 (26)

where

1205792 = arctan(119860119861119861119874

) = arctan(minus119886119901 cos 120579

ℎ119904 + 119887 + 119887119901 sin 120579) (27)

Therefore the absolute roll rate and roll angular accelera-tion can be obtained as follows based on (26)

ang =

120601 +

1205792 ang =

120601 +

1205792(28)

1205792 =

1205791198611

119861

1205792 =

(1198601

120579 minus 1198602

1205792)

119860

(29)

where

119861 = (ℎ119904 + 119887 + 119887119901 sin 120579)2

+ (119886119901 cos 120579)2

1198611 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901]

119860 = 1198612 1198601 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901] 119861

1198602 = 2119886119901 cos 120579 [(ℎ119904 + 119887) sin 120579 + 119887119901]

times [119887119901 (ℎ119904 + 119887 + 119887119901 sin 120579) minus 1198862

119901sin 120579]

minus 119886119901119861 (ℎ119904 + 119887) cos 120579

(30)

Discrete Dynamics in Nature and Society 7

On the basis of (28) the vehiclersquos roll balance equationand yaw balance equation can be acquired by differentiatingthe angular momentum with respect to time

In general the angular momentum of a rotating object isdefined as the product of a bodyrsquos rotational inertia tensor androtational velocities about a particular coordinate(1) The Angular Momentum of Sprung Mass and the FixedLiquid Mass As the vehicle sprung mass and the fixed liquidmass have the same rotational velocity about the 119909-119910-119911coordinate system the angular momentum for the two partsis solved together

According to the definition of angular momentum thatfor the sprungmass and fixed liquidmass can be expressed asfollows

119867119904 =[

[

119868119909119909119904 + 119868119909119909119891 minus119868119909119910119904 minus 119868119909119910119891 minus119868119909119911119904 minus 119868119909119911119891

minus119868119910119909119904 minus 119868119910119909119891 119868119910119910119904 + 119868119910119910119891 minus119868119910119911119904 minus 119868119910119911119891

minus119868119911119909119904 minus 119868119911119909119891 minus119868119911119910119904 minus 119868119911119910119891 119868119911119911119904 + 119868119911119911119891

]

]

[

[

120601

0

119903

]

]

(31)

where 119868119909119909119904 119868119910119910119904 and 119868119911119911119904 are the moments of inertia for thesprungmass about the 119909-axis119910-axis and 119911-axis respectively119868119909119909119891 119868119910119910119891 and 119868119911119911119891 are the moments of inertia for the fixedliquid mass about the 119909-axis 119910-axis and 119911-axis respectively119868119909119910119904 = 119868119910119909119904 119868119909119911119904 = 119868119911119909119904 and 119868119910119911119904 = 119868119911119910119904 are the products ofinertia for the sprungmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively 119868119909119910119891 = 119868119910119909119891 119868119909119911119891 = 119868119911119909119891and 119868119910119911119891 = 119868119911119910119891 are the products of inertia for the fixed liquidmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively

According to the assumptions given in Section 31 thedistribution of the vehiclersquos sprungmass is symmetrical aboutthe119909119911-planeTherefore the products of inertia for the sprungmass can be expressed as follows

119868119909119910119904 = 0 119868119911119910119904 = 0 (32)

As the fixed liquidmass does notmovewhile the tank tiltsthe distribution of the fixed liquid mass is symmetrical aboutthe 119909119905119911119905-plane and the 119909119911-plane Therefore the products ofinertia for the fixed liquid mass can be expressed as follows

119868119909119910119891 = 0 119868119911119910119891 = 0 (33)

Therefore based on (31)ndash(33) the angularmomentum forthe vehiclersquos sprung mass and the fixed liquid mass can bewritten as follows

119867119904 = [(119868119909119909119904 + 119868119909119909119891)

120601 minus (119868119909119911119904 + 119868119909119911119891) 119903]119894

+ [minus (119868119911119909119904 + 119868119911119909119891)

120601 + (119868119911119911119904 + 119868119911119911119891) 119903]119895

(34)

(2)TheAngularMomentum of the PendulumMassAccordingto (28) the angular momentum of the pendulummass can beexpressed as follows

119867119901 =[

[

119868119909119909119901 minus119868119909119910119901 minus119868119909119911119901

minus119868119910119909119901 119868119910119910119901 minus119868119910119911119901

minus119868119911119909119901 minus119868119911119910119901 119868119911119911119901

]

]

[

[

120601 +

1205792

0

119903

]

]

(35)

where 119868119909119909119901 119868119910119910119901 and 119868119911119911119901 are the moments of inertia forthe pendulum mass about the 119909-axis 119910-axis and 119911-axis

respectively 119868119909119910119901 = 119868119910119909119901 119868119909119911119901 = 119868119911119909119901 and 119868119910119911119901 = 119868119911119910119901 are theproducts of inertia for the pendulum mass about the 119909- and119910-axis the 119909- and 119911-axis and the 119910- and 119911-axis respectively

The expanded form of (35) is written as

119867119901 = [119868119909119909119901 (

120601 +

1205792) minus 119868119909119911119901119903]119894 + [minus119868119909119910119901 (

120601 +

1205792) minus 119868119910119911119901119903]

119895

+ [minus119868119909119911119901 (

120601 +

1205792) + 119868119911119911119901119903]119896

(36)

(3) The Angular Momentum of the Unsprung Mass The yawmovement is the only rotational degree of freedom for theunsprung mass Therefore the angular momentum of theunsprung mass can be obtained by multiplying together themoment of inertia about the 119911-axis and the yaw rate aboutthat which is written as follows

119867119906 = 119868119911119911119906119903119896 (37)

where 119868119911119911119906 is the moment of inertia about the 119911-axis for theunsprung mass

The change in the vehicle fixed coordinate after it has beendriving for time Δ119905 is plotted in Figure 7 from which we cansee that the differentials of the 119894 119895 and

119896 unit vectors can beexpressed as follows

119894 = 119903 119895

119895 = minus119903 119894 +

120601

119896

119896 = minus

120601 119895 (38)

The first-order derivatives of (34) (36) and (37) can beobtained using (38) Thus the vehiclersquos roll moment and yawmoment have been acquired

According to the moment balance between the vehicleinertia moment and the external moment the vehiclersquos rolland yaw moment balance equation can be expressed asfollows

119868119911 119903 minus 119868119909119911

120601 minus 1198981198911198861199041198902 minus 1198981199011198861198911198902 minus 119868119909119911119901 (

120601 +

120579) + 119868119911119911119901 119903

minus 119868119909119910119901(

120601 +

120579)

2

minus 119868119910119911119901119903 (

120601 +

120579) = 2 (119865119891119897119891 minus 119865119903119897119903)

119868119909

120601 minus 119868119909119911 119903 + 119898119905ℎ119904119881(

120573 + 119903) + 1198671119898119891119886119904 + (ℎ119904 + 119887)119898119901119886119891

+ 119868119909119909119901 (

120601 +

120579) minus 119868119909119911119901 119903 + 119868119909119910119901119903 (

120601 +

120579) + 1198681199101199111199011199032

= minus119896120601120601 minus 119888120601

120601

+ 120601 (119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892 (ℎ119904 + 119887)) minus 119898119901119892119886119901 cos 120579(39)

where 119897119891 is the longitudinal distance from the CG of thetank truck to the front wheel 119897119903 is the longitudinal distancefrom the CG of the tank truck to the rear wheel 119896120601 is thesuspension roll stiffness 119888120601 is the suspension roll damping 1198902is the longitudinal distance from the CG of the tank truck tothe CG of liquid cargo and 1198671 is the vertical distance fromthe CG of the fixed liquid mass to the roll center

8 Discrete Dynamics in Nature and Society

Based on the parallel axis theorem of the moment ofinertia the following equation can be obtained

119868119911 = 119868119911119911119904 + 119868119911119911119906 + 1198981199051198882+ 119898119906119890

2+ 119868119911119911119891

119868119909 = 119868119909119909119904 + 119868119909119909119891 + 119898119905ℎ2

119904

119868119911119909 = 119868119909119911119904 + 119868119909119911119891 + 119898119905ℎ119904119888

(40)

The dynamic model of the tank truck is composed of (17)and (39)ndash(40)

The tire cornering force in the dynamic model can beobtained using the magic formula for tires which is given by

119865 = 119863 sin (119862 arctan (119861120572 minus 119864 (119861120572 minus arctan119861120572))) (41)

where 120572 is the tirersquos sideslip angle and 119861 119862 119863 and 119864 are theconstant parameters of the magic formula for tires

The tire sideslip angle in (41) can be expressed as follows

120572119891 = arctan(119881120573 + 119903119897119891

119881

) minus 120575119891

120572119903 = arctan(119881120573 minus 119903119897119903

119881

) minus 120575119903

(42)

where 120572119891 is the front tire sideslip angle and 120572119903 is the rear tiresideslip angle 120575119891 is the front wheel steering angle and 120575119903 isthe rear wheel steering angle

For the convenience of vehicle dynamic simulation thetank truck dynamic model is translated into the followingform

119872119909 = 119873 (43)

and (43) can be rewritten as follows

[

[

[

[

11989811 11989812 11989813 11989814

11989821 11989822 11989823 11989824

11989831 11989832 11989833 11989834

11989841 11989842 11989843 11989844

]

]

]

]

[

[

[

[

120573

119903

120601

120579

]

]

]

]

=

[

[

[

[

1198991

1198992

1198993

1198994

]

]

]

]

(44)

where

11989811 = 119881 (119898119905 + 119898119891 + 119898119901 + 119898119906)

11989812 = 119888 (119898119905 + 119898119891) minus 119890119898119906

11989813 = minusℎ119904 (119898119905 + 119898119891) minus 1198981199011198631

11989814 = minus1198981199011198632

11989821 = 1198811198902 (minus119898119891 minus 119898119901)

11989822 = 119868119911 minus 1198981198911198881198902 + 119868119911119911119901

11989823 = minus119868119909119911 + 119898119891ℎ1199041198902 + 11989811990111986311198902 minus 119868119909119911119901

11989824 = 11989811990111986321198902 minus

1198681199091199111199011198601

119860

11989831 = 119881 (119898119905ℎ119904 + 1198981198911198671 + 119898119901119867)

11989832 = minus119868119909119911 + 1198981198911198881198671 minus 119868119909119911119901

11989833 = 119868119909 minus 119898119891ℎ1199041198671 minus 1198981199011198671198631 + 119868119909119909119901

11989834 =

1198681199091199091199011198601

119860

minus 1198981199011198671198632

11989841 = 119881 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

11989842 = 0

11989843 = 119886119901119887119901 + 119886119901119887 sin 120579

11989844 = 1198862

119901sin2120579 + 119887

2

119901cos2120579

1198991 = 2 (119865119891 + 119865119903) minus 119903119881 (119898119905 + 119898119891 + 119898119906)

minus 119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

1198992 = 2 (119865119891119897119891 minus 119865119903119897119903) + 1198981198911198902119881119903

+ 1198981199011198902 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus

119868119909119911119901

12057921198602

119860

+ 119868119909119910119901(

120601 +

1205791198611

119861

)

2

+ 119868119910119911119901119903 (

120601 +

1205791198611

119861

)

1198993 = minus 119888120601

120601 + 120601 (minus119896120601 + 119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892119867)

minus 119886119901119898119901119892 cos 120579 minus 119881119903 (119898119905ℎ119904 + 1198981198911198671)

minus 119867119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus 119868119909119910119901119903 (

120601 +

120579) minus 1199032119868119910119911119901 +

12057921198681199091199091199011198602

119860

1198994 = minus 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

(45)

According to the Cramer rule the variables in (43) canbe expressed as follows while the matrix 119872 has a nonzerodeterminant

1199091 =1198721

119873

1199092 =1198722

119873

1199093 =1198723

119873

1199094 =1198724

119873

(46)

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

According to (10) and (11) a Lagrangian function can beused to obtain the motion equation of the pendulum systemwhich is written as follows

119871 = 119879 minus 119881 (12)

where 119879 is the kinetic energy of the pendulum mass ballThe motion of the trammel pendulum system can be

expressed by

120597

120597119905

(

120597119871

120597 119886119895

) minus

120597119871

120597119886119895

= 0 (13)

where 119886119895 is the degree of freedom for system 119871According to the motion analysis of the trammel pendu-

lum 120579 the pendulum amplitude and 120601 the tank roll angleare the two degrees of freedom for the trammel pendulumsystem

The differential of (12) with respect to 120579 can be expressedas follows

120597119871

120597120579

= 119898 [

1206012(minus1198862

119901sin 120579 cos 120579 + 119887

2

119901sin 120579 cos 120579 + 119887119901119887 cos 120579)

+

1205792(1198862

119901sin 120579 cos 120579 minus 119887

2

119901sin 120579 cos 120579)

+

120579

120601 (119886119901119887 cos 120579)

minus

120601 (minus119886119901 sin 120579 sin120601 + 119887119901 cos 120579 cos120601)

minus

120579 (119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

minus119892 (119887119901 cos 120579 minus 119886119901 sin 120579 tan120601) cos120601] (14)

The differential of (12) with respect to 120579 can be expressed

by

120597119871

120597

120579

= 119898 [

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) +

120601 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)] (15)

Finally the differential of (15) with respect to 119905 can be ex-pressed as follows

120597

120597119905

(

120597119871

120597

120579

) = 119898 [

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579)

+

1205792(21198862

119901sin 120579 cos 120579 minus 2119887

2

119901sin 120579 cos 120579)

+

120601 (119886119901119887119901 + 119886119901119887 sin 120579) + 120601

120579119886119901119887 cos 120579

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

minus

120579 (119886119901 cos 120579 cos120601 minus 119887119901 sin 120579 sin120601)

+

120601 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)] (16)

Substituting (14)ndash(16) into (13) we get

120579 (1198862

119901sin2120579 + 119887

2

119901cos2120579) +

120601 (119886119901119887119901 + 119886119901119887 sin 120579)

+

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

+

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

+ 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601) = 0

(17)

The differential of (12) with respect to 120601 can be expressedas follows

120597119871

120597120601

= 119898 [

120601 (119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

minus

120579 (minus119886119901 sin 120579 sin120601 + 119887119901 cos 120579 cos120601)

minus 119892119886119901 cos 120579 sec120601

+ 119892 (119887 + 119887119901 sin 120579 + 119886119901 cos 120579 tan120601) sin120601]

(18)

The differential of (12) with respect to 120601 can be expressed

by

120597119871

120597

120601

= 119898 [

120601 (1198872+ 1198862

119901cos2120579 + 119887

2

119901sin2120579 + 2119887119901119887 sin 120579)

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)] (19)

According to (19) the differential of (19) with respect to 119905can be obtained as follows

120597

120597119905

(

120597119871

120597

120601

)

= 119898 [

120601 (1198872+ 1198862

119901cos2120579 + 119887

2

119901sin2120579 + 2119887119901119887 sin 120579)

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579) + 119886119901119887

1205792 cos 120579

+

120601

120579 (minus21198862

119901sin 120579 cos 120579 + 2119887

2

119901sin 120579 cos 120579 + 2119887119901119887 cos 120579)

+

120579 (119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601)

+

120601 (119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)] (20)

Discrete Dynamics in Nature and Society 5

Substituting (18)ndash(20) into (13) we get

120601 [(119887 + 119887119901 sin 120579)

2

+ 1198862

119901cos2120579]

+ 2

120579

120601 [

1

2

sin 2120579 (1198872119901minus 1198862

119901) + 119887119901119887 cos 120579]

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

+

1205792119886119901119887 cos 120579

minus 119892 [(119887 + 119887119901 sin 120579) sin120601 minus 119886119901 cos 120579 cos120601] = 0

(21)

Equations (17) and (21) are the equations of motionof the trammel pendulum under the noninertia coordinatesystem and the pendulum motion characteristics under thetranslation and rotation of the coordinate system can beanalyzed using (17) and (21)

3 Tank Truck Dynamic Model

31 Assumptions The roll stability of a tank truck is greatlyinfluenced by transient liquid sloshing in partially filledtanks Due to the flow characteristic of liquid bulk transientliquid sloshing is produced when the vehiclersquos driving statechanges which leads to the oscillation of the pendulummassball To simplify the analysis of the tank truckrsquos roll stabilitysome assumptions are made and listed as follows

(1) The sprung mass is the vehicle mass that lies over thesuspension system in which the mass of the liquidcargo is not included In this paper vehicle and liquidbulk as the solid and fluid parts respectively are notmixed with each other

(2) In the transverse direction the CG of the sprungmassis located at the bottom of the tank Moreover thesprung mass is symmetrical about the longitudinalaxis of the vehicle

(3) Transient liquid sloshing is merely produced alongthe transverse direction longitudinal liquid sloshingis not taken into consideration The movement andmass distribution of the liquid bulk are completelythe same at different tank cross sections whichmeansthat the CG of liquid bulk is located in the middle ofthe tank along the longitudinal direction

(4) The liquid-free surface is not broken during thevehiclersquos driving process

32 Force Analysis of the Tank Truck Based on the assump-tions given in Section 31 a reference coordinate system fortank truck dynamic analysis was selected first

The coordinate system being fixed relative to the vehiclewas chosen as the reference coordinate system for thedynamic analysis of the tank truck and the establishment ofthe vehiclersquos dynamic model The origin of the coordinatesystem is the point where the vertical line that goes through

the CG of the vehicle when it is static intersects with thevehiclersquos roll axis

The coordinate system being fixed relative to the vehiclersquossprung mass was defined as the 119909-119910-119911 system where the 119909-axis is parallel to the longitudinal axis of the vehicle andpoints in the direction in which the vehicle is being driventhe 119910-axis is perpendicular to the 119909-axis and points to the leftfrom the driverrsquos perspective and the 119911-axis is perpendicularto the 119909119910 plane and points straight up

The vehiclersquos unsprung mass is assumed not to produceroll movement

The top view and the back view of the vehiclersquos dynamicanalysis are plotted in Figures 4 and 5 According to Figure 5the lateral inertial forces act on the sprung mass theunsprung mass the fixed liquid mass and the pendulummass making up the vehiclersquos lateral inertial force The tirecornering forces make up the external lateral force that actson the vehicle according to Figure 4 On the basis of Newtonrsquosfirst law when the vehicle is steady its inertial lateral forceand external lateral force are equal in value and positive indirection which can be expressed as follows

(119898119905 + 119898119891) 119886119904 + 119898119901119886119891 + 119898119906119886119906 = 2 (119865119891 + 119865119903) (22)

where 119886119904 is the lateral acceleration of sprung mass 119886119906 isthe lateral acceleration of unsprung mass 119886119891 is the lateralacceleration of the pendulum mass ball 119898119905 is sprung mass119898119906 is unsprung mass 119865119891 is the front tire cornering force and119865119903 is the rear tire cornering force

Generally the fixed liquid mass does not stay static butslides slowly with the aid of gravity while the tank tiltsHowever the shift of the fixed liquid mass is quite smallcompared to that of the pendulum mass ball and its shiftamount drops greatly with the increase in the liquid fillpercentage Therefore the fixed liquid mass is assumed to bestatic and its position is assumed to stay constant as the tankrolls over Based on this assumption the fixed liquidmass canbe seen as the solid part and its lateral acceleration is equal tothat of the vehiclersquos sprung mass

According to the dynamic analysis of the vehicle thelateral acceleration of the sprung mass and unsprung masscan be presented by

119886119904 = 119881 (

120573 + 119903) minus ℎ119904

120601 + 119888 119903

119886119906 = 119881 (

120573 + 119903) minus 119890 119903

(23)

where 119881 is the vehiclersquos driving speed 120573 the vehiclersquos slipangle 119903 the vehiclersquos yaw angle ℎ119904 is the vertical distance fromthe CG of sprung mass to the roll center 119888 is the longitudinaldistance between the CG of sprung mass and that of the tanktruck and 119890 is the longitudinal distance between the CG ofunsprung mass and that of the tank truck

According to (10) the lateral acceleration of the pendu-lum mass ball can be written as follows

119886119891 = 119881

120601 minus 1198631

120601 minus 1198632

120579 + 21198633

120601

120579 + 1198634

1206012+ 1198635

1205792 (24)

6 Discrete Dynamics in Nature and Society

Front wheel

CG of tank truck

CG of unsprungmass

CG of liquidcargo

Rear wheel

CG of sprung mass

Ff

Fr

lr

lf

c

e

e2

muau

mtas

mfas + mpaf

Figure 4 Top view of vehicle dynamic analysis

b

Central lineof the tank

Sprung mass

Pendulummass

Unsprung mass

Liquid fixed mass

Roll center

ΔFΔF

a

hs

muau

mpafmtas

mfas

Fr Fr

kr120601 + cr(d120601)

Figure 5 Back view of vehicle dynamic analysis

where

1198631 = 119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601

1198632 = 119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601

1198633 = 119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601

1198634 = 119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

1198635 = minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

(25)

The force analysis in the roll plane is plotted in Figure 5Due to the lateral inertia force the sprungmass and the liquid

Central line

of the tank

Coordinatefixed on the

tank

Coordinate fixedon the vehiclesprung mass

120579

ap

z

bp

y

AB

y998400

x998400

z998400

120601hs

1205792

b

x

o

Figure 6The schematic diagram for the pendulum rotating aroundthe roll axis

bulk rotate about the roll axis and bring about rollover torqueThe roll angle about the roll axis for sprungmass and the fixedliquid mass is 120601 which can be seen from Figure 5 Howeverowing to the tank and the pendulum mass ball movementthe roll angle about the roll axis for the pendulum mass ballis composed of two parts which is shown in Figure 6

According to Figure 6 the roll angle about the roll axis forthe pendulum mass ball can be expressed as follows

ang = 120601 + 1205792 (26)

where

1205792 = arctan(119860119861119861119874

) = arctan(minus119886119901 cos 120579

ℎ119904 + 119887 + 119887119901 sin 120579) (27)

Therefore the absolute roll rate and roll angular accelera-tion can be obtained as follows based on (26)

ang =

120601 +

1205792 ang =

120601 +

1205792(28)

1205792 =

1205791198611

119861

1205792 =

(1198601

120579 minus 1198602

1205792)

119860

(29)

where

119861 = (ℎ119904 + 119887 + 119887119901 sin 120579)2

+ (119886119901 cos 120579)2

1198611 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901]

119860 = 1198612 1198601 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901] 119861

1198602 = 2119886119901 cos 120579 [(ℎ119904 + 119887) sin 120579 + 119887119901]

times [119887119901 (ℎ119904 + 119887 + 119887119901 sin 120579) minus 1198862

119901sin 120579]

minus 119886119901119861 (ℎ119904 + 119887) cos 120579

(30)

Discrete Dynamics in Nature and Society 7

On the basis of (28) the vehiclersquos roll balance equationand yaw balance equation can be acquired by differentiatingthe angular momentum with respect to time

In general the angular momentum of a rotating object isdefined as the product of a bodyrsquos rotational inertia tensor androtational velocities about a particular coordinate(1) The Angular Momentum of Sprung Mass and the FixedLiquid Mass As the vehicle sprung mass and the fixed liquidmass have the same rotational velocity about the 119909-119910-119911coordinate system the angular momentum for the two partsis solved together

According to the definition of angular momentum thatfor the sprungmass and fixed liquidmass can be expressed asfollows

119867119904 =[

[

119868119909119909119904 + 119868119909119909119891 minus119868119909119910119904 minus 119868119909119910119891 minus119868119909119911119904 minus 119868119909119911119891

minus119868119910119909119904 minus 119868119910119909119891 119868119910119910119904 + 119868119910119910119891 minus119868119910119911119904 minus 119868119910119911119891

minus119868119911119909119904 minus 119868119911119909119891 minus119868119911119910119904 minus 119868119911119910119891 119868119911119911119904 + 119868119911119911119891

]

]

[

[

120601

0

119903

]

]

(31)

where 119868119909119909119904 119868119910119910119904 and 119868119911119911119904 are the moments of inertia for thesprungmass about the 119909-axis119910-axis and 119911-axis respectively119868119909119909119891 119868119910119910119891 and 119868119911119911119891 are the moments of inertia for the fixedliquid mass about the 119909-axis 119910-axis and 119911-axis respectively119868119909119910119904 = 119868119910119909119904 119868119909119911119904 = 119868119911119909119904 and 119868119910119911119904 = 119868119911119910119904 are the products ofinertia for the sprungmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively 119868119909119910119891 = 119868119910119909119891 119868119909119911119891 = 119868119911119909119891and 119868119910119911119891 = 119868119911119910119891 are the products of inertia for the fixed liquidmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively

According to the assumptions given in Section 31 thedistribution of the vehiclersquos sprungmass is symmetrical aboutthe119909119911-planeTherefore the products of inertia for the sprungmass can be expressed as follows

119868119909119910119904 = 0 119868119911119910119904 = 0 (32)

As the fixed liquidmass does notmovewhile the tank tiltsthe distribution of the fixed liquid mass is symmetrical aboutthe 119909119905119911119905-plane and the 119909119911-plane Therefore the products ofinertia for the fixed liquid mass can be expressed as follows

119868119909119910119891 = 0 119868119911119910119891 = 0 (33)

Therefore based on (31)ndash(33) the angularmomentum forthe vehiclersquos sprung mass and the fixed liquid mass can bewritten as follows

119867119904 = [(119868119909119909119904 + 119868119909119909119891)

120601 minus (119868119909119911119904 + 119868119909119911119891) 119903]119894

+ [minus (119868119911119909119904 + 119868119911119909119891)

120601 + (119868119911119911119904 + 119868119911119911119891) 119903]119895

(34)

(2)TheAngularMomentum of the PendulumMassAccordingto (28) the angular momentum of the pendulummass can beexpressed as follows

119867119901 =[

[

119868119909119909119901 minus119868119909119910119901 minus119868119909119911119901

minus119868119910119909119901 119868119910119910119901 minus119868119910119911119901

minus119868119911119909119901 minus119868119911119910119901 119868119911119911119901

]

]

[

[

120601 +

1205792

0

119903

]

]

(35)

where 119868119909119909119901 119868119910119910119901 and 119868119911119911119901 are the moments of inertia forthe pendulum mass about the 119909-axis 119910-axis and 119911-axis

respectively 119868119909119910119901 = 119868119910119909119901 119868119909119911119901 = 119868119911119909119901 and 119868119910119911119901 = 119868119911119910119901 are theproducts of inertia for the pendulum mass about the 119909- and119910-axis the 119909- and 119911-axis and the 119910- and 119911-axis respectively

The expanded form of (35) is written as

119867119901 = [119868119909119909119901 (

120601 +

1205792) minus 119868119909119911119901119903]119894 + [minus119868119909119910119901 (

120601 +

1205792) minus 119868119910119911119901119903]

119895

+ [minus119868119909119911119901 (

120601 +

1205792) + 119868119911119911119901119903]119896

(36)

(3) The Angular Momentum of the Unsprung Mass The yawmovement is the only rotational degree of freedom for theunsprung mass Therefore the angular momentum of theunsprung mass can be obtained by multiplying together themoment of inertia about the 119911-axis and the yaw rate aboutthat which is written as follows

119867119906 = 119868119911119911119906119903119896 (37)

where 119868119911119911119906 is the moment of inertia about the 119911-axis for theunsprung mass

The change in the vehicle fixed coordinate after it has beendriving for time Δ119905 is plotted in Figure 7 from which we cansee that the differentials of the 119894 119895 and

119896 unit vectors can beexpressed as follows

119894 = 119903 119895

119895 = minus119903 119894 +

120601

119896

119896 = minus

120601 119895 (38)

The first-order derivatives of (34) (36) and (37) can beobtained using (38) Thus the vehiclersquos roll moment and yawmoment have been acquired

According to the moment balance between the vehicleinertia moment and the external moment the vehiclersquos rolland yaw moment balance equation can be expressed asfollows

119868119911 119903 minus 119868119909119911

120601 minus 1198981198911198861199041198902 minus 1198981199011198861198911198902 minus 119868119909119911119901 (

120601 +

120579) + 119868119911119911119901 119903

minus 119868119909119910119901(

120601 +

120579)

2

minus 119868119910119911119901119903 (

120601 +

120579) = 2 (119865119891119897119891 minus 119865119903119897119903)

119868119909

120601 minus 119868119909119911 119903 + 119898119905ℎ119904119881(

120573 + 119903) + 1198671119898119891119886119904 + (ℎ119904 + 119887)119898119901119886119891

+ 119868119909119909119901 (

120601 +

120579) minus 119868119909119911119901 119903 + 119868119909119910119901119903 (

120601 +

120579) + 1198681199101199111199011199032

= minus119896120601120601 minus 119888120601

120601

+ 120601 (119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892 (ℎ119904 + 119887)) minus 119898119901119892119886119901 cos 120579(39)

where 119897119891 is the longitudinal distance from the CG of thetank truck to the front wheel 119897119903 is the longitudinal distancefrom the CG of the tank truck to the rear wheel 119896120601 is thesuspension roll stiffness 119888120601 is the suspension roll damping 1198902is the longitudinal distance from the CG of the tank truck tothe CG of liquid cargo and 1198671 is the vertical distance fromthe CG of the fixed liquid mass to the roll center

8 Discrete Dynamics in Nature and Society

Based on the parallel axis theorem of the moment ofinertia the following equation can be obtained

119868119911 = 119868119911119911119904 + 119868119911119911119906 + 1198981199051198882+ 119898119906119890

2+ 119868119911119911119891

119868119909 = 119868119909119909119904 + 119868119909119909119891 + 119898119905ℎ2

119904

119868119911119909 = 119868119909119911119904 + 119868119909119911119891 + 119898119905ℎ119904119888

(40)

The dynamic model of the tank truck is composed of (17)and (39)ndash(40)

The tire cornering force in the dynamic model can beobtained using the magic formula for tires which is given by

119865 = 119863 sin (119862 arctan (119861120572 minus 119864 (119861120572 minus arctan119861120572))) (41)

where 120572 is the tirersquos sideslip angle and 119861 119862 119863 and 119864 are theconstant parameters of the magic formula for tires

The tire sideslip angle in (41) can be expressed as follows

120572119891 = arctan(119881120573 + 119903119897119891

119881

) minus 120575119891

120572119903 = arctan(119881120573 minus 119903119897119903

119881

) minus 120575119903

(42)

where 120572119891 is the front tire sideslip angle and 120572119903 is the rear tiresideslip angle 120575119891 is the front wheel steering angle and 120575119903 isthe rear wheel steering angle

For the convenience of vehicle dynamic simulation thetank truck dynamic model is translated into the followingform

119872119909 = 119873 (43)

and (43) can be rewritten as follows

[

[

[

[

11989811 11989812 11989813 11989814

11989821 11989822 11989823 11989824

11989831 11989832 11989833 11989834

11989841 11989842 11989843 11989844

]

]

]

]

[

[

[

[

120573

119903

120601

120579

]

]

]

]

=

[

[

[

[

1198991

1198992

1198993

1198994

]

]

]

]

(44)

where

11989811 = 119881 (119898119905 + 119898119891 + 119898119901 + 119898119906)

11989812 = 119888 (119898119905 + 119898119891) minus 119890119898119906

11989813 = minusℎ119904 (119898119905 + 119898119891) minus 1198981199011198631

11989814 = minus1198981199011198632

11989821 = 1198811198902 (minus119898119891 minus 119898119901)

11989822 = 119868119911 minus 1198981198911198881198902 + 119868119911119911119901

11989823 = minus119868119909119911 + 119898119891ℎ1199041198902 + 11989811990111986311198902 minus 119868119909119911119901

11989824 = 11989811990111986321198902 minus

1198681199091199111199011198601

119860

11989831 = 119881 (119898119905ℎ119904 + 1198981198911198671 + 119898119901119867)

11989832 = minus119868119909119911 + 1198981198911198881198671 minus 119868119909119911119901

11989833 = 119868119909 minus 119898119891ℎ1199041198671 minus 1198981199011198671198631 + 119868119909119909119901

11989834 =

1198681199091199091199011198601

119860

minus 1198981199011198671198632

11989841 = 119881 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

11989842 = 0

11989843 = 119886119901119887119901 + 119886119901119887 sin 120579

11989844 = 1198862

119901sin2120579 + 119887

2

119901cos2120579

1198991 = 2 (119865119891 + 119865119903) minus 119903119881 (119898119905 + 119898119891 + 119898119906)

minus 119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

1198992 = 2 (119865119891119897119891 minus 119865119903119897119903) + 1198981198911198902119881119903

+ 1198981199011198902 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus

119868119909119911119901

12057921198602

119860

+ 119868119909119910119901(

120601 +

1205791198611

119861

)

2

+ 119868119910119911119901119903 (

120601 +

1205791198611

119861

)

1198993 = minus 119888120601

120601 + 120601 (minus119896120601 + 119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892119867)

minus 119886119901119898119901119892 cos 120579 minus 119881119903 (119898119905ℎ119904 + 1198981198911198671)

minus 119867119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus 119868119909119910119901119903 (

120601 +

120579) minus 1199032119868119910119911119901 +

12057921198681199091199091199011198602

119860

1198994 = minus 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

(45)

According to the Cramer rule the variables in (43) canbe expressed as follows while the matrix 119872 has a nonzerodeterminant

1199091 =1198721

119873

1199092 =1198722

119873

1199093 =1198723

119873

1199094 =1198724

119873

(46)

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

Substituting (18)ndash(20) into (13) we get

120601 [(119887 + 119887119901 sin 120579)

2

+ 1198862

119901cos2120579]

+ 2

120579

120601 [

1

2

sin 2120579 (1198872119901minus 1198862

119901) + 119887119901119887 cos 120579]

+

120579 (119886119901119887119901 + 119886119901119887 sin 120579)

minus (119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601)

+

1205792119886119901119887 cos 120579

minus 119892 [(119887 + 119887119901 sin 120579) sin120601 minus 119886119901 cos 120579 cos120601] = 0

(21)

Equations (17) and (21) are the equations of motionof the trammel pendulum under the noninertia coordinatesystem and the pendulum motion characteristics under thetranslation and rotation of the coordinate system can beanalyzed using (17) and (21)

3 Tank Truck Dynamic Model

31 Assumptions The roll stability of a tank truck is greatlyinfluenced by transient liquid sloshing in partially filledtanks Due to the flow characteristic of liquid bulk transientliquid sloshing is produced when the vehiclersquos driving statechanges which leads to the oscillation of the pendulummassball To simplify the analysis of the tank truckrsquos roll stabilitysome assumptions are made and listed as follows

(1) The sprung mass is the vehicle mass that lies over thesuspension system in which the mass of the liquidcargo is not included In this paper vehicle and liquidbulk as the solid and fluid parts respectively are notmixed with each other

(2) In the transverse direction the CG of the sprungmassis located at the bottom of the tank Moreover thesprung mass is symmetrical about the longitudinalaxis of the vehicle

(3) Transient liquid sloshing is merely produced alongthe transverse direction longitudinal liquid sloshingis not taken into consideration The movement andmass distribution of the liquid bulk are completelythe same at different tank cross sections whichmeansthat the CG of liquid bulk is located in the middle ofthe tank along the longitudinal direction

(4) The liquid-free surface is not broken during thevehiclersquos driving process

32 Force Analysis of the Tank Truck Based on the assump-tions given in Section 31 a reference coordinate system fortank truck dynamic analysis was selected first

The coordinate system being fixed relative to the vehiclewas chosen as the reference coordinate system for thedynamic analysis of the tank truck and the establishment ofthe vehiclersquos dynamic model The origin of the coordinatesystem is the point where the vertical line that goes through

the CG of the vehicle when it is static intersects with thevehiclersquos roll axis

The coordinate system being fixed relative to the vehiclersquossprung mass was defined as the 119909-119910-119911 system where the 119909-axis is parallel to the longitudinal axis of the vehicle andpoints in the direction in which the vehicle is being driventhe 119910-axis is perpendicular to the 119909-axis and points to the leftfrom the driverrsquos perspective and the 119911-axis is perpendicularto the 119909119910 plane and points straight up

The vehiclersquos unsprung mass is assumed not to produceroll movement

The top view and the back view of the vehiclersquos dynamicanalysis are plotted in Figures 4 and 5 According to Figure 5the lateral inertial forces act on the sprung mass theunsprung mass the fixed liquid mass and the pendulummass making up the vehiclersquos lateral inertial force The tirecornering forces make up the external lateral force that actson the vehicle according to Figure 4 On the basis of Newtonrsquosfirst law when the vehicle is steady its inertial lateral forceand external lateral force are equal in value and positive indirection which can be expressed as follows

(119898119905 + 119898119891) 119886119904 + 119898119901119886119891 + 119898119906119886119906 = 2 (119865119891 + 119865119903) (22)

where 119886119904 is the lateral acceleration of sprung mass 119886119906 isthe lateral acceleration of unsprung mass 119886119891 is the lateralacceleration of the pendulum mass ball 119898119905 is sprung mass119898119906 is unsprung mass 119865119891 is the front tire cornering force and119865119903 is the rear tire cornering force

Generally the fixed liquid mass does not stay static butslides slowly with the aid of gravity while the tank tiltsHowever the shift of the fixed liquid mass is quite smallcompared to that of the pendulum mass ball and its shiftamount drops greatly with the increase in the liquid fillpercentage Therefore the fixed liquid mass is assumed to bestatic and its position is assumed to stay constant as the tankrolls over Based on this assumption the fixed liquidmass canbe seen as the solid part and its lateral acceleration is equal tothat of the vehiclersquos sprung mass

According to the dynamic analysis of the vehicle thelateral acceleration of the sprung mass and unsprung masscan be presented by

119886119904 = 119881 (

120573 + 119903) minus ℎ119904

120601 + 119888 119903

119886119906 = 119881 (

120573 + 119903) minus 119890 119903

(23)

where 119881 is the vehiclersquos driving speed 120573 the vehiclersquos slipangle 119903 the vehiclersquos yaw angle ℎ119904 is the vertical distance fromthe CG of sprung mass to the roll center 119888 is the longitudinaldistance between the CG of sprung mass and that of the tanktruck and 119890 is the longitudinal distance between the CG ofunsprung mass and that of the tank truck

According to (10) the lateral acceleration of the pendu-lum mass ball can be written as follows

119886119891 = 119881

120601 minus 1198631

120601 minus 1198632

120579 + 21198633

120601

120579 + 1198634

1206012+ 1198635

1205792 (24)

6 Discrete Dynamics in Nature and Society

Front wheel

CG of tank truck

CG of unsprungmass

CG of liquidcargo

Rear wheel

CG of sprung mass

Ff

Fr

lr

lf

c

e

e2

muau

mtas

mfas + mpaf

Figure 4 Top view of vehicle dynamic analysis

b

Central lineof the tank

Sprung mass

Pendulummass

Unsprung mass

Liquid fixed mass

Roll center

ΔFΔF

a

hs

muau

mpafmtas

mfas

Fr Fr

kr120601 + cr(d120601)

Figure 5 Back view of vehicle dynamic analysis

where

1198631 = 119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601

1198632 = 119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601

1198633 = 119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601

1198634 = 119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

1198635 = minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

(25)

The force analysis in the roll plane is plotted in Figure 5Due to the lateral inertia force the sprungmass and the liquid

Central line

of the tank

Coordinatefixed on the

tank

Coordinate fixedon the vehiclesprung mass

120579

ap

z

bp

y

AB

y998400

x998400

z998400

120601hs

1205792

b

x

o

Figure 6The schematic diagram for the pendulum rotating aroundthe roll axis

bulk rotate about the roll axis and bring about rollover torqueThe roll angle about the roll axis for sprungmass and the fixedliquid mass is 120601 which can be seen from Figure 5 Howeverowing to the tank and the pendulum mass ball movementthe roll angle about the roll axis for the pendulum mass ballis composed of two parts which is shown in Figure 6

According to Figure 6 the roll angle about the roll axis forthe pendulum mass ball can be expressed as follows

ang = 120601 + 1205792 (26)

where

1205792 = arctan(119860119861119861119874

) = arctan(minus119886119901 cos 120579

ℎ119904 + 119887 + 119887119901 sin 120579) (27)

Therefore the absolute roll rate and roll angular accelera-tion can be obtained as follows based on (26)

ang =

120601 +

1205792 ang =

120601 +

1205792(28)

1205792 =

1205791198611

119861

1205792 =

(1198601

120579 minus 1198602

1205792)

119860

(29)

where

119861 = (ℎ119904 + 119887 + 119887119901 sin 120579)2

+ (119886119901 cos 120579)2

1198611 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901]

119860 = 1198612 1198601 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901] 119861

1198602 = 2119886119901 cos 120579 [(ℎ119904 + 119887) sin 120579 + 119887119901]

times [119887119901 (ℎ119904 + 119887 + 119887119901 sin 120579) minus 1198862

119901sin 120579]

minus 119886119901119861 (ℎ119904 + 119887) cos 120579

(30)

Discrete Dynamics in Nature and Society 7

On the basis of (28) the vehiclersquos roll balance equationand yaw balance equation can be acquired by differentiatingthe angular momentum with respect to time

In general the angular momentum of a rotating object isdefined as the product of a bodyrsquos rotational inertia tensor androtational velocities about a particular coordinate(1) The Angular Momentum of Sprung Mass and the FixedLiquid Mass As the vehicle sprung mass and the fixed liquidmass have the same rotational velocity about the 119909-119910-119911coordinate system the angular momentum for the two partsis solved together

According to the definition of angular momentum thatfor the sprungmass and fixed liquidmass can be expressed asfollows

119867119904 =[

[

119868119909119909119904 + 119868119909119909119891 minus119868119909119910119904 minus 119868119909119910119891 minus119868119909119911119904 minus 119868119909119911119891

minus119868119910119909119904 minus 119868119910119909119891 119868119910119910119904 + 119868119910119910119891 minus119868119910119911119904 minus 119868119910119911119891

minus119868119911119909119904 minus 119868119911119909119891 minus119868119911119910119904 minus 119868119911119910119891 119868119911119911119904 + 119868119911119911119891

]

]

[

[

120601

0

119903

]

]

(31)

where 119868119909119909119904 119868119910119910119904 and 119868119911119911119904 are the moments of inertia for thesprungmass about the 119909-axis119910-axis and 119911-axis respectively119868119909119909119891 119868119910119910119891 and 119868119911119911119891 are the moments of inertia for the fixedliquid mass about the 119909-axis 119910-axis and 119911-axis respectively119868119909119910119904 = 119868119910119909119904 119868119909119911119904 = 119868119911119909119904 and 119868119910119911119904 = 119868119911119910119904 are the products ofinertia for the sprungmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively 119868119909119910119891 = 119868119910119909119891 119868119909119911119891 = 119868119911119909119891and 119868119910119911119891 = 119868119911119910119891 are the products of inertia for the fixed liquidmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively

According to the assumptions given in Section 31 thedistribution of the vehiclersquos sprungmass is symmetrical aboutthe119909119911-planeTherefore the products of inertia for the sprungmass can be expressed as follows

119868119909119910119904 = 0 119868119911119910119904 = 0 (32)

As the fixed liquidmass does notmovewhile the tank tiltsthe distribution of the fixed liquid mass is symmetrical aboutthe 119909119905119911119905-plane and the 119909119911-plane Therefore the products ofinertia for the fixed liquid mass can be expressed as follows

119868119909119910119891 = 0 119868119911119910119891 = 0 (33)

Therefore based on (31)ndash(33) the angularmomentum forthe vehiclersquos sprung mass and the fixed liquid mass can bewritten as follows

119867119904 = [(119868119909119909119904 + 119868119909119909119891)

120601 minus (119868119909119911119904 + 119868119909119911119891) 119903]119894

+ [minus (119868119911119909119904 + 119868119911119909119891)

120601 + (119868119911119911119904 + 119868119911119911119891) 119903]119895

(34)

(2)TheAngularMomentum of the PendulumMassAccordingto (28) the angular momentum of the pendulummass can beexpressed as follows

119867119901 =[

[

119868119909119909119901 minus119868119909119910119901 minus119868119909119911119901

minus119868119910119909119901 119868119910119910119901 minus119868119910119911119901

minus119868119911119909119901 minus119868119911119910119901 119868119911119911119901

]

]

[

[

120601 +

1205792

0

119903

]

]

(35)

where 119868119909119909119901 119868119910119910119901 and 119868119911119911119901 are the moments of inertia forthe pendulum mass about the 119909-axis 119910-axis and 119911-axis

respectively 119868119909119910119901 = 119868119910119909119901 119868119909119911119901 = 119868119911119909119901 and 119868119910119911119901 = 119868119911119910119901 are theproducts of inertia for the pendulum mass about the 119909- and119910-axis the 119909- and 119911-axis and the 119910- and 119911-axis respectively

The expanded form of (35) is written as

119867119901 = [119868119909119909119901 (

120601 +

1205792) minus 119868119909119911119901119903]119894 + [minus119868119909119910119901 (

120601 +

1205792) minus 119868119910119911119901119903]

119895

+ [minus119868119909119911119901 (

120601 +

1205792) + 119868119911119911119901119903]119896

(36)

(3) The Angular Momentum of the Unsprung Mass The yawmovement is the only rotational degree of freedom for theunsprung mass Therefore the angular momentum of theunsprung mass can be obtained by multiplying together themoment of inertia about the 119911-axis and the yaw rate aboutthat which is written as follows

119867119906 = 119868119911119911119906119903119896 (37)

where 119868119911119911119906 is the moment of inertia about the 119911-axis for theunsprung mass

The change in the vehicle fixed coordinate after it has beendriving for time Δ119905 is plotted in Figure 7 from which we cansee that the differentials of the 119894 119895 and

119896 unit vectors can beexpressed as follows

119894 = 119903 119895

119895 = minus119903 119894 +

120601

119896

119896 = minus

120601 119895 (38)

The first-order derivatives of (34) (36) and (37) can beobtained using (38) Thus the vehiclersquos roll moment and yawmoment have been acquired

According to the moment balance between the vehicleinertia moment and the external moment the vehiclersquos rolland yaw moment balance equation can be expressed asfollows

119868119911 119903 minus 119868119909119911

120601 minus 1198981198911198861199041198902 minus 1198981199011198861198911198902 minus 119868119909119911119901 (

120601 +

120579) + 119868119911119911119901 119903

minus 119868119909119910119901(

120601 +

120579)

2

minus 119868119910119911119901119903 (

120601 +

120579) = 2 (119865119891119897119891 minus 119865119903119897119903)

119868119909

120601 minus 119868119909119911 119903 + 119898119905ℎ119904119881(

120573 + 119903) + 1198671119898119891119886119904 + (ℎ119904 + 119887)119898119901119886119891

+ 119868119909119909119901 (

120601 +

120579) minus 119868119909119911119901 119903 + 119868119909119910119901119903 (

120601 +

120579) + 1198681199101199111199011199032

= minus119896120601120601 minus 119888120601

120601

+ 120601 (119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892 (ℎ119904 + 119887)) minus 119898119901119892119886119901 cos 120579(39)

where 119897119891 is the longitudinal distance from the CG of thetank truck to the front wheel 119897119903 is the longitudinal distancefrom the CG of the tank truck to the rear wheel 119896120601 is thesuspension roll stiffness 119888120601 is the suspension roll damping 1198902is the longitudinal distance from the CG of the tank truck tothe CG of liquid cargo and 1198671 is the vertical distance fromthe CG of the fixed liquid mass to the roll center

8 Discrete Dynamics in Nature and Society

Based on the parallel axis theorem of the moment ofinertia the following equation can be obtained

119868119911 = 119868119911119911119904 + 119868119911119911119906 + 1198981199051198882+ 119898119906119890

2+ 119868119911119911119891

119868119909 = 119868119909119909119904 + 119868119909119909119891 + 119898119905ℎ2

119904

119868119911119909 = 119868119909119911119904 + 119868119909119911119891 + 119898119905ℎ119904119888

(40)

The dynamic model of the tank truck is composed of (17)and (39)ndash(40)

The tire cornering force in the dynamic model can beobtained using the magic formula for tires which is given by

119865 = 119863 sin (119862 arctan (119861120572 minus 119864 (119861120572 minus arctan119861120572))) (41)

where 120572 is the tirersquos sideslip angle and 119861 119862 119863 and 119864 are theconstant parameters of the magic formula for tires

The tire sideslip angle in (41) can be expressed as follows

120572119891 = arctan(119881120573 + 119903119897119891

119881

) minus 120575119891

120572119903 = arctan(119881120573 minus 119903119897119903

119881

) minus 120575119903

(42)

where 120572119891 is the front tire sideslip angle and 120572119903 is the rear tiresideslip angle 120575119891 is the front wheel steering angle and 120575119903 isthe rear wheel steering angle

For the convenience of vehicle dynamic simulation thetank truck dynamic model is translated into the followingform

119872119909 = 119873 (43)

and (43) can be rewritten as follows

[

[

[

[

11989811 11989812 11989813 11989814

11989821 11989822 11989823 11989824

11989831 11989832 11989833 11989834

11989841 11989842 11989843 11989844

]

]

]

]

[

[

[

[

120573

119903

120601

120579

]

]

]

]

=

[

[

[

[

1198991

1198992

1198993

1198994

]

]

]

]

(44)

where

11989811 = 119881 (119898119905 + 119898119891 + 119898119901 + 119898119906)

11989812 = 119888 (119898119905 + 119898119891) minus 119890119898119906

11989813 = minusℎ119904 (119898119905 + 119898119891) minus 1198981199011198631

11989814 = minus1198981199011198632

11989821 = 1198811198902 (minus119898119891 minus 119898119901)

11989822 = 119868119911 minus 1198981198911198881198902 + 119868119911119911119901

11989823 = minus119868119909119911 + 119898119891ℎ1199041198902 + 11989811990111986311198902 minus 119868119909119911119901

11989824 = 11989811990111986321198902 minus

1198681199091199111199011198601

119860

11989831 = 119881 (119898119905ℎ119904 + 1198981198911198671 + 119898119901119867)

11989832 = minus119868119909119911 + 1198981198911198881198671 minus 119868119909119911119901

11989833 = 119868119909 minus 119898119891ℎ1199041198671 minus 1198981199011198671198631 + 119868119909119909119901

11989834 =

1198681199091199091199011198601

119860

minus 1198981199011198671198632

11989841 = 119881 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

11989842 = 0

11989843 = 119886119901119887119901 + 119886119901119887 sin 120579

11989844 = 1198862

119901sin2120579 + 119887

2

119901cos2120579

1198991 = 2 (119865119891 + 119865119903) minus 119903119881 (119898119905 + 119898119891 + 119898119906)

minus 119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

1198992 = 2 (119865119891119897119891 minus 119865119903119897119903) + 1198981198911198902119881119903

+ 1198981199011198902 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus

119868119909119911119901

12057921198602

119860

+ 119868119909119910119901(

120601 +

1205791198611

119861

)

2

+ 119868119910119911119901119903 (

120601 +

1205791198611

119861

)

1198993 = minus 119888120601

120601 + 120601 (minus119896120601 + 119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892119867)

minus 119886119901119898119901119892 cos 120579 minus 119881119903 (119898119905ℎ119904 + 1198981198911198671)

minus 119867119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus 119868119909119910119901119903 (

120601 +

120579) minus 1199032119868119910119911119901 +

12057921198681199091199091199011198602

119860

1198994 = minus 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

(45)

According to the Cramer rule the variables in (43) canbe expressed as follows while the matrix 119872 has a nonzerodeterminant

1199091 =1198721

119873

1199092 =1198722

119873

1199093 =1198723

119873

1199094 =1198724

119873

(46)

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Front wheel

CG of tank truck

CG of unsprungmass

CG of liquidcargo

Rear wheel

CG of sprung mass

Ff

Fr

lr

lf

c

e

e2

muau

mtas

mfas + mpaf

Figure 4 Top view of vehicle dynamic analysis

b

Central lineof the tank

Sprung mass

Pendulummass

Unsprung mass

Liquid fixed mass

Roll center

ΔFΔF

a

hs

muau

mpafmtas

mfas

Fr Fr

kr120601 + cr(d120601)

Figure 5 Back view of vehicle dynamic analysis

where

1198631 = 119887 cos120601 + 119886119901 cos 120579 sin120601 + 119887119901 sin 120579 cos120601

1198632 = 119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601

1198633 = 119886119901 sin 120579 sin120601 minus 119887119901 cos 120579 cos120601

1198634 = 119887 sin120601 minus 119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

1198635 = minus119886119901 cos 120579 cos120601 + 119887119901 sin 120579 sin120601

(25)

The force analysis in the roll plane is plotted in Figure 5Due to the lateral inertia force the sprungmass and the liquid

Central line

of the tank

Coordinatefixed on the

tank

Coordinate fixedon the vehiclesprung mass

120579

ap

z

bp

y

AB

y998400

x998400

z998400

120601hs

1205792

b

x

o

Figure 6The schematic diagram for the pendulum rotating aroundthe roll axis

bulk rotate about the roll axis and bring about rollover torqueThe roll angle about the roll axis for sprungmass and the fixedliquid mass is 120601 which can be seen from Figure 5 Howeverowing to the tank and the pendulum mass ball movementthe roll angle about the roll axis for the pendulum mass ballis composed of two parts which is shown in Figure 6

According to Figure 6 the roll angle about the roll axis forthe pendulum mass ball can be expressed as follows

ang = 120601 + 1205792 (26)

where

1205792 = arctan(119860119861119861119874

) = arctan(minus119886119901 cos 120579

ℎ119904 + 119887 + 119887119901 sin 120579) (27)

Therefore the absolute roll rate and roll angular accelera-tion can be obtained as follows based on (26)

ang =

120601 +

1205792 ang =

120601 +

1205792(28)

1205792 =

1205791198611

119861

1205792 =

(1198601

120579 minus 1198602

1205792)

119860

(29)

where

119861 = (ℎ119904 + 119887 + 119887119901 sin 120579)2

+ (119886119901 cos 120579)2

1198611 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901]

119860 = 1198612 1198601 = 119886119901 [(ℎ119904 + 119887) sin 120579 + 119887119901] 119861

1198602 = 2119886119901 cos 120579 [(ℎ119904 + 119887) sin 120579 + 119887119901]

times [119887119901 (ℎ119904 + 119887 + 119887119901 sin 120579) minus 1198862

119901sin 120579]

minus 119886119901119861 (ℎ119904 + 119887) cos 120579

(30)

Discrete Dynamics in Nature and Society 7

On the basis of (28) the vehiclersquos roll balance equationand yaw balance equation can be acquired by differentiatingthe angular momentum with respect to time

In general the angular momentum of a rotating object isdefined as the product of a bodyrsquos rotational inertia tensor androtational velocities about a particular coordinate(1) The Angular Momentum of Sprung Mass and the FixedLiquid Mass As the vehicle sprung mass and the fixed liquidmass have the same rotational velocity about the 119909-119910-119911coordinate system the angular momentum for the two partsis solved together

According to the definition of angular momentum thatfor the sprungmass and fixed liquidmass can be expressed asfollows

119867119904 =[

[

119868119909119909119904 + 119868119909119909119891 minus119868119909119910119904 minus 119868119909119910119891 minus119868119909119911119904 minus 119868119909119911119891

minus119868119910119909119904 minus 119868119910119909119891 119868119910119910119904 + 119868119910119910119891 minus119868119910119911119904 minus 119868119910119911119891

minus119868119911119909119904 minus 119868119911119909119891 minus119868119911119910119904 minus 119868119911119910119891 119868119911119911119904 + 119868119911119911119891

]

]

[

[

120601

0

119903

]

]

(31)

where 119868119909119909119904 119868119910119910119904 and 119868119911119911119904 are the moments of inertia for thesprungmass about the 119909-axis119910-axis and 119911-axis respectively119868119909119909119891 119868119910119910119891 and 119868119911119911119891 are the moments of inertia for the fixedliquid mass about the 119909-axis 119910-axis and 119911-axis respectively119868119909119910119904 = 119868119910119909119904 119868119909119911119904 = 119868119911119909119904 and 119868119910119911119904 = 119868119911119910119904 are the products ofinertia for the sprungmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively 119868119909119910119891 = 119868119910119909119891 119868119909119911119891 = 119868119911119909119891and 119868119910119911119891 = 119868119911119910119891 are the products of inertia for the fixed liquidmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively

According to the assumptions given in Section 31 thedistribution of the vehiclersquos sprungmass is symmetrical aboutthe119909119911-planeTherefore the products of inertia for the sprungmass can be expressed as follows

119868119909119910119904 = 0 119868119911119910119904 = 0 (32)

As the fixed liquidmass does notmovewhile the tank tiltsthe distribution of the fixed liquid mass is symmetrical aboutthe 119909119905119911119905-plane and the 119909119911-plane Therefore the products ofinertia for the fixed liquid mass can be expressed as follows

119868119909119910119891 = 0 119868119911119910119891 = 0 (33)

Therefore based on (31)ndash(33) the angularmomentum forthe vehiclersquos sprung mass and the fixed liquid mass can bewritten as follows

119867119904 = [(119868119909119909119904 + 119868119909119909119891)

120601 minus (119868119909119911119904 + 119868119909119911119891) 119903]119894

+ [minus (119868119911119909119904 + 119868119911119909119891)

120601 + (119868119911119911119904 + 119868119911119911119891) 119903]119895

(34)

(2)TheAngularMomentum of the PendulumMassAccordingto (28) the angular momentum of the pendulummass can beexpressed as follows

119867119901 =[

[

119868119909119909119901 minus119868119909119910119901 minus119868119909119911119901

minus119868119910119909119901 119868119910119910119901 minus119868119910119911119901

minus119868119911119909119901 minus119868119911119910119901 119868119911119911119901

]

]

[

[

120601 +

1205792

0

119903

]

]

(35)

where 119868119909119909119901 119868119910119910119901 and 119868119911119911119901 are the moments of inertia forthe pendulum mass about the 119909-axis 119910-axis and 119911-axis

respectively 119868119909119910119901 = 119868119910119909119901 119868119909119911119901 = 119868119911119909119901 and 119868119910119911119901 = 119868119911119910119901 are theproducts of inertia for the pendulum mass about the 119909- and119910-axis the 119909- and 119911-axis and the 119910- and 119911-axis respectively

The expanded form of (35) is written as

119867119901 = [119868119909119909119901 (

120601 +

1205792) minus 119868119909119911119901119903]119894 + [minus119868119909119910119901 (

120601 +

1205792) minus 119868119910119911119901119903]

119895

+ [minus119868119909119911119901 (

120601 +

1205792) + 119868119911119911119901119903]119896

(36)

(3) The Angular Momentum of the Unsprung Mass The yawmovement is the only rotational degree of freedom for theunsprung mass Therefore the angular momentum of theunsprung mass can be obtained by multiplying together themoment of inertia about the 119911-axis and the yaw rate aboutthat which is written as follows

119867119906 = 119868119911119911119906119903119896 (37)

where 119868119911119911119906 is the moment of inertia about the 119911-axis for theunsprung mass

The change in the vehicle fixed coordinate after it has beendriving for time Δ119905 is plotted in Figure 7 from which we cansee that the differentials of the 119894 119895 and

119896 unit vectors can beexpressed as follows

119894 = 119903 119895

119895 = minus119903 119894 +

120601

119896

119896 = minus

120601 119895 (38)

The first-order derivatives of (34) (36) and (37) can beobtained using (38) Thus the vehiclersquos roll moment and yawmoment have been acquired

According to the moment balance between the vehicleinertia moment and the external moment the vehiclersquos rolland yaw moment balance equation can be expressed asfollows

119868119911 119903 minus 119868119909119911

120601 minus 1198981198911198861199041198902 minus 1198981199011198861198911198902 minus 119868119909119911119901 (

120601 +

120579) + 119868119911119911119901 119903

minus 119868119909119910119901(

120601 +

120579)

2

minus 119868119910119911119901119903 (

120601 +

120579) = 2 (119865119891119897119891 minus 119865119903119897119903)

119868119909

120601 minus 119868119909119911 119903 + 119898119905ℎ119904119881(

120573 + 119903) + 1198671119898119891119886119904 + (ℎ119904 + 119887)119898119901119886119891

+ 119868119909119909119901 (

120601 +

120579) minus 119868119909119911119901 119903 + 119868119909119910119901119903 (

120601 +

120579) + 1198681199101199111199011199032

= minus119896120601120601 minus 119888120601

120601

+ 120601 (119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892 (ℎ119904 + 119887)) minus 119898119901119892119886119901 cos 120579(39)

where 119897119891 is the longitudinal distance from the CG of thetank truck to the front wheel 119897119903 is the longitudinal distancefrom the CG of the tank truck to the rear wheel 119896120601 is thesuspension roll stiffness 119888120601 is the suspension roll damping 1198902is the longitudinal distance from the CG of the tank truck tothe CG of liquid cargo and 1198671 is the vertical distance fromthe CG of the fixed liquid mass to the roll center

8 Discrete Dynamics in Nature and Society

Based on the parallel axis theorem of the moment ofinertia the following equation can be obtained

119868119911 = 119868119911119911119904 + 119868119911119911119906 + 1198981199051198882+ 119898119906119890

2+ 119868119911119911119891

119868119909 = 119868119909119909119904 + 119868119909119909119891 + 119898119905ℎ2

119904

119868119911119909 = 119868119909119911119904 + 119868119909119911119891 + 119898119905ℎ119904119888

(40)

The dynamic model of the tank truck is composed of (17)and (39)ndash(40)

The tire cornering force in the dynamic model can beobtained using the magic formula for tires which is given by

119865 = 119863 sin (119862 arctan (119861120572 minus 119864 (119861120572 minus arctan119861120572))) (41)

where 120572 is the tirersquos sideslip angle and 119861 119862 119863 and 119864 are theconstant parameters of the magic formula for tires

The tire sideslip angle in (41) can be expressed as follows

120572119891 = arctan(119881120573 + 119903119897119891

119881

) minus 120575119891

120572119903 = arctan(119881120573 minus 119903119897119903

119881

) minus 120575119903

(42)

where 120572119891 is the front tire sideslip angle and 120572119903 is the rear tiresideslip angle 120575119891 is the front wheel steering angle and 120575119903 isthe rear wheel steering angle

For the convenience of vehicle dynamic simulation thetank truck dynamic model is translated into the followingform

119872119909 = 119873 (43)

and (43) can be rewritten as follows

[

[

[

[

11989811 11989812 11989813 11989814

11989821 11989822 11989823 11989824

11989831 11989832 11989833 11989834

11989841 11989842 11989843 11989844

]

]

]

]

[

[

[

[

120573

119903

120601

120579

]

]

]

]

=

[

[

[

[

1198991

1198992

1198993

1198994

]

]

]

]

(44)

where

11989811 = 119881 (119898119905 + 119898119891 + 119898119901 + 119898119906)

11989812 = 119888 (119898119905 + 119898119891) minus 119890119898119906

11989813 = minusℎ119904 (119898119905 + 119898119891) minus 1198981199011198631

11989814 = minus1198981199011198632

11989821 = 1198811198902 (minus119898119891 minus 119898119901)

11989822 = 119868119911 minus 1198981198911198881198902 + 119868119911119911119901

11989823 = minus119868119909119911 + 119898119891ℎ1199041198902 + 11989811990111986311198902 minus 119868119909119911119901

11989824 = 11989811990111986321198902 minus

1198681199091199111199011198601

119860

11989831 = 119881 (119898119905ℎ119904 + 1198981198911198671 + 119898119901119867)

11989832 = minus119868119909119911 + 1198981198911198881198671 minus 119868119909119911119901

11989833 = 119868119909 minus 119898119891ℎ1199041198671 minus 1198981199011198671198631 + 119868119909119909119901

11989834 =

1198681199091199091199011198601

119860

minus 1198981199011198671198632

11989841 = 119881 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

11989842 = 0

11989843 = 119886119901119887119901 + 119886119901119887 sin 120579

11989844 = 1198862

119901sin2120579 + 119887

2

119901cos2120579

1198991 = 2 (119865119891 + 119865119903) minus 119903119881 (119898119905 + 119898119891 + 119898119906)

minus 119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

1198992 = 2 (119865119891119897119891 minus 119865119903119897119903) + 1198981198911198902119881119903

+ 1198981199011198902 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus

119868119909119911119901

12057921198602

119860

+ 119868119909119910119901(

120601 +

1205791198611

119861

)

2

+ 119868119910119911119901119903 (

120601 +

1205791198611

119861

)

1198993 = minus 119888120601

120601 + 120601 (minus119896120601 + 119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892119867)

minus 119886119901119898119901119892 cos 120579 minus 119881119903 (119898119905ℎ119904 + 1198981198911198671)

minus 119867119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus 119868119909119910119901119903 (

120601 +

120579) minus 1199032119868119910119911119901 +

12057921198681199091199091199011198602

119860

1198994 = minus 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

(45)

According to the Cramer rule the variables in (43) canbe expressed as follows while the matrix 119872 has a nonzerodeterminant

1199091 =1198721

119873

1199092 =1198722

119873

1199093 =1198723

119873

1199094 =1198724

119873

(46)

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

On the basis of (28) the vehiclersquos roll balance equationand yaw balance equation can be acquired by differentiatingthe angular momentum with respect to time

In general the angular momentum of a rotating object isdefined as the product of a bodyrsquos rotational inertia tensor androtational velocities about a particular coordinate(1) The Angular Momentum of Sprung Mass and the FixedLiquid Mass As the vehicle sprung mass and the fixed liquidmass have the same rotational velocity about the 119909-119910-119911coordinate system the angular momentum for the two partsis solved together

According to the definition of angular momentum thatfor the sprungmass and fixed liquidmass can be expressed asfollows

119867119904 =[

[

119868119909119909119904 + 119868119909119909119891 minus119868119909119910119904 minus 119868119909119910119891 minus119868119909119911119904 minus 119868119909119911119891

minus119868119910119909119904 minus 119868119910119909119891 119868119910119910119904 + 119868119910119910119891 minus119868119910119911119904 minus 119868119910119911119891

minus119868119911119909119904 minus 119868119911119909119891 minus119868119911119910119904 minus 119868119911119910119891 119868119911119911119904 + 119868119911119911119891

]

]

[

[

120601

0

119903

]

]

(31)

where 119868119909119909119904 119868119910119910119904 and 119868119911119911119904 are the moments of inertia for thesprungmass about the 119909-axis119910-axis and 119911-axis respectively119868119909119909119891 119868119910119910119891 and 119868119911119911119891 are the moments of inertia for the fixedliquid mass about the 119909-axis 119910-axis and 119911-axis respectively119868119909119910119904 = 119868119910119909119904 119868119909119911119904 = 119868119911119909119904 and 119868119910119911119904 = 119868119911119910119904 are the products ofinertia for the sprungmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively 119868119909119910119891 = 119868119910119909119891 119868119909119911119891 = 119868119911119909119891and 119868119910119911119891 = 119868119911119910119891 are the products of inertia for the fixed liquidmass about the 119909- and 119910-axis 119909- and 119911-axis and 119910- and 119911-axis respectively

According to the assumptions given in Section 31 thedistribution of the vehiclersquos sprungmass is symmetrical aboutthe119909119911-planeTherefore the products of inertia for the sprungmass can be expressed as follows

119868119909119910119904 = 0 119868119911119910119904 = 0 (32)

As the fixed liquidmass does notmovewhile the tank tiltsthe distribution of the fixed liquid mass is symmetrical aboutthe 119909119905119911119905-plane and the 119909119911-plane Therefore the products ofinertia for the fixed liquid mass can be expressed as follows

119868119909119910119891 = 0 119868119911119910119891 = 0 (33)

Therefore based on (31)ndash(33) the angularmomentum forthe vehiclersquos sprung mass and the fixed liquid mass can bewritten as follows

119867119904 = [(119868119909119909119904 + 119868119909119909119891)

120601 minus (119868119909119911119904 + 119868119909119911119891) 119903]119894

+ [minus (119868119911119909119904 + 119868119911119909119891)

120601 + (119868119911119911119904 + 119868119911119911119891) 119903]119895

(34)

(2)TheAngularMomentum of the PendulumMassAccordingto (28) the angular momentum of the pendulummass can beexpressed as follows

119867119901 =[

[

119868119909119909119901 minus119868119909119910119901 minus119868119909119911119901

minus119868119910119909119901 119868119910119910119901 minus119868119910119911119901

minus119868119911119909119901 minus119868119911119910119901 119868119911119911119901

]

]

[

[

120601 +

1205792

0

119903

]

]

(35)

where 119868119909119909119901 119868119910119910119901 and 119868119911119911119901 are the moments of inertia forthe pendulum mass about the 119909-axis 119910-axis and 119911-axis

respectively 119868119909119910119901 = 119868119910119909119901 119868119909119911119901 = 119868119911119909119901 and 119868119910119911119901 = 119868119911119910119901 are theproducts of inertia for the pendulum mass about the 119909- and119910-axis the 119909- and 119911-axis and the 119910- and 119911-axis respectively

The expanded form of (35) is written as

119867119901 = [119868119909119909119901 (

120601 +

1205792) minus 119868119909119911119901119903]119894 + [minus119868119909119910119901 (

120601 +

1205792) minus 119868119910119911119901119903]

119895

+ [minus119868119909119911119901 (

120601 +

1205792) + 119868119911119911119901119903]119896

(36)

(3) The Angular Momentum of the Unsprung Mass The yawmovement is the only rotational degree of freedom for theunsprung mass Therefore the angular momentum of theunsprung mass can be obtained by multiplying together themoment of inertia about the 119911-axis and the yaw rate aboutthat which is written as follows

119867119906 = 119868119911119911119906119903119896 (37)

where 119868119911119911119906 is the moment of inertia about the 119911-axis for theunsprung mass

The change in the vehicle fixed coordinate after it has beendriving for time Δ119905 is plotted in Figure 7 from which we cansee that the differentials of the 119894 119895 and

119896 unit vectors can beexpressed as follows

119894 = 119903 119895

119895 = minus119903 119894 +

120601

119896

119896 = minus

120601 119895 (38)

The first-order derivatives of (34) (36) and (37) can beobtained using (38) Thus the vehiclersquos roll moment and yawmoment have been acquired

According to the moment balance between the vehicleinertia moment and the external moment the vehiclersquos rolland yaw moment balance equation can be expressed asfollows

119868119911 119903 minus 119868119909119911

120601 minus 1198981198911198861199041198902 minus 1198981199011198861198911198902 minus 119868119909119911119901 (

120601 +

120579) + 119868119911119911119901 119903

minus 119868119909119910119901(

120601 +

120579)

2

minus 119868119910119911119901119903 (

120601 +

120579) = 2 (119865119891119897119891 minus 119865119903119897119903)

119868119909

120601 minus 119868119909119911 119903 + 119898119905ℎ119904119881(

120573 + 119903) + 1198671119898119891119886119904 + (ℎ119904 + 119887)119898119901119886119891

+ 119868119909119909119901 (

120601 +

120579) minus 119868119909119911119901 119903 + 119868119909119910119901119903 (

120601 +

120579) + 1198681199101199111199011199032

= minus119896120601120601 minus 119888120601

120601

+ 120601 (119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892 (ℎ119904 + 119887)) minus 119898119901119892119886119901 cos 120579(39)

where 119897119891 is the longitudinal distance from the CG of thetank truck to the front wheel 119897119903 is the longitudinal distancefrom the CG of the tank truck to the rear wheel 119896120601 is thesuspension roll stiffness 119888120601 is the suspension roll damping 1198902is the longitudinal distance from the CG of the tank truck tothe CG of liquid cargo and 1198671 is the vertical distance fromthe CG of the fixed liquid mass to the roll center

8 Discrete Dynamics in Nature and Society

Based on the parallel axis theorem of the moment ofinertia the following equation can be obtained

119868119911 = 119868119911119911119904 + 119868119911119911119906 + 1198981199051198882+ 119898119906119890

2+ 119868119911119911119891

119868119909 = 119868119909119909119904 + 119868119909119909119891 + 119898119905ℎ2

119904

119868119911119909 = 119868119909119911119904 + 119868119909119911119891 + 119898119905ℎ119904119888

(40)

The dynamic model of the tank truck is composed of (17)and (39)ndash(40)

The tire cornering force in the dynamic model can beobtained using the magic formula for tires which is given by

119865 = 119863 sin (119862 arctan (119861120572 minus 119864 (119861120572 minus arctan119861120572))) (41)

where 120572 is the tirersquos sideslip angle and 119861 119862 119863 and 119864 are theconstant parameters of the magic formula for tires

The tire sideslip angle in (41) can be expressed as follows

120572119891 = arctan(119881120573 + 119903119897119891

119881

) minus 120575119891

120572119903 = arctan(119881120573 minus 119903119897119903

119881

) minus 120575119903

(42)

where 120572119891 is the front tire sideslip angle and 120572119903 is the rear tiresideslip angle 120575119891 is the front wheel steering angle and 120575119903 isthe rear wheel steering angle

For the convenience of vehicle dynamic simulation thetank truck dynamic model is translated into the followingform

119872119909 = 119873 (43)

and (43) can be rewritten as follows

[

[

[

[

11989811 11989812 11989813 11989814

11989821 11989822 11989823 11989824

11989831 11989832 11989833 11989834

11989841 11989842 11989843 11989844

]

]

]

]

[

[

[

[

120573

119903

120601

120579

]

]

]

]

=

[

[

[

[

1198991

1198992

1198993

1198994

]

]

]

]

(44)

where

11989811 = 119881 (119898119905 + 119898119891 + 119898119901 + 119898119906)

11989812 = 119888 (119898119905 + 119898119891) minus 119890119898119906

11989813 = minusℎ119904 (119898119905 + 119898119891) minus 1198981199011198631

11989814 = minus1198981199011198632

11989821 = 1198811198902 (minus119898119891 minus 119898119901)

11989822 = 119868119911 minus 1198981198911198881198902 + 119868119911119911119901

11989823 = minus119868119909119911 + 119898119891ℎ1199041198902 + 11989811990111986311198902 minus 119868119909119911119901

11989824 = 11989811990111986321198902 minus

1198681199091199111199011198601

119860

11989831 = 119881 (119898119905ℎ119904 + 1198981198911198671 + 119898119901119867)

11989832 = minus119868119909119911 + 1198981198911198881198671 minus 119868119909119911119901

11989833 = 119868119909 minus 119898119891ℎ1199041198671 minus 1198981199011198671198631 + 119868119909119909119901

11989834 =

1198681199091199091199011198601

119860

minus 1198981199011198671198632

11989841 = 119881 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

11989842 = 0

11989843 = 119886119901119887119901 + 119886119901119887 sin 120579

11989844 = 1198862

119901sin2120579 + 119887

2

119901cos2120579

1198991 = 2 (119865119891 + 119865119903) minus 119903119881 (119898119905 + 119898119891 + 119898119906)

minus 119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

1198992 = 2 (119865119891119897119891 minus 119865119903119897119903) + 1198981198911198902119881119903

+ 1198981199011198902 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus

119868119909119911119901

12057921198602

119860

+ 119868119909119910119901(

120601 +

1205791198611

119861

)

2

+ 119868119910119911119901119903 (

120601 +

1205791198611

119861

)

1198993 = minus 119888120601

120601 + 120601 (minus119896120601 + 119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892119867)

minus 119886119901119898119901119892 cos 120579 minus 119881119903 (119898119905ℎ119904 + 1198981198911198671)

minus 119867119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus 119868119909119910119901119903 (

120601 +

120579) minus 1199032119868119910119911119901 +

12057921198681199091199091199011198602

119860

1198994 = minus 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

(45)

According to the Cramer rule the variables in (43) canbe expressed as follows while the matrix 119872 has a nonzerodeterminant

1199091 =1198721

119873

1199092 =1198722

119873

1199093 =1198723

119873

1199094 =1198724

119873

(46)

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Based on the parallel axis theorem of the moment ofinertia the following equation can be obtained

119868119911 = 119868119911119911119904 + 119868119911119911119906 + 1198981199051198882+ 119898119906119890

2+ 119868119911119911119891

119868119909 = 119868119909119909119904 + 119868119909119909119891 + 119898119905ℎ2

119904

119868119911119909 = 119868119909119911119904 + 119868119909119911119891 + 119898119905ℎ119904119888

(40)

The dynamic model of the tank truck is composed of (17)and (39)ndash(40)

The tire cornering force in the dynamic model can beobtained using the magic formula for tires which is given by

119865 = 119863 sin (119862 arctan (119861120572 minus 119864 (119861120572 minus arctan119861120572))) (41)

where 120572 is the tirersquos sideslip angle and 119861 119862 119863 and 119864 are theconstant parameters of the magic formula for tires

The tire sideslip angle in (41) can be expressed as follows

120572119891 = arctan(119881120573 + 119903119897119891

119881

) minus 120575119891

120572119903 = arctan(119881120573 minus 119903119897119903

119881

) minus 120575119903

(42)

where 120572119891 is the front tire sideslip angle and 120572119903 is the rear tiresideslip angle 120575119891 is the front wheel steering angle and 120575119903 isthe rear wheel steering angle

For the convenience of vehicle dynamic simulation thetank truck dynamic model is translated into the followingform

119872119909 = 119873 (43)

and (43) can be rewritten as follows

[

[

[

[

11989811 11989812 11989813 11989814

11989821 11989822 11989823 11989824

11989831 11989832 11989833 11989834

11989841 11989842 11989843 11989844

]

]

]

]

[

[

[

[

120573

119903

120601

120579

]

]

]

]

=

[

[

[

[

1198991

1198992

1198993

1198994

]

]

]

]

(44)

where

11989811 = 119881 (119898119905 + 119898119891 + 119898119901 + 119898119906)

11989812 = 119888 (119898119905 + 119898119891) minus 119890119898119906

11989813 = minusℎ119904 (119898119905 + 119898119891) minus 1198981199011198631

11989814 = minus1198981199011198632

11989821 = 1198811198902 (minus119898119891 minus 119898119901)

11989822 = 119868119911 minus 1198981198911198881198902 + 119868119911119911119901

11989823 = minus119868119909119911 + 119898119891ℎ1199041198902 + 11989811990111986311198902 minus 119868119909119911119901

11989824 = 11989811990111986321198902 minus

1198681199091199111199011198601

119860

11989831 = 119881 (119898119905ℎ119904 + 1198981198911198671 + 119898119901119867)

11989832 = minus119868119909119911 + 1198981198911198881198671 minus 119868119909119911119901

11989833 = 119868119909 minus 119898119891ℎ1199041198671 minus 1198981199011198671198631 + 119868119909119909119901

11989834 =

1198681199091199091199011198601

119860

minus 1198981199011198671198632

11989841 = 119881 (119886119901 sin 120579 cos120601 + 119887119901 cos 120579 sin120601)

11989842 = 0

11989843 = 119886119901119887119901 + 119886119901119887 sin 120579

11989844 = 1198862

119901sin2120579 + 119887

2

119901cos2120579

1198991 = 2 (119865119891 + 119865119903) minus 119903119881 (119898119905 + 119898119891 + 119898119906)

minus 119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

1198992 = 2 (119865119891119897119891 minus 119865119903119897119903) + 1198981198911198902119881119903

+ 1198981199011198902 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus

119868119909119911119901

12057921198602

119860

+ 119868119909119910119901(

120601 +

1205791198611

119861

)

2

+ 119868119910119911119901119903 (

120601 +

1205791198611

119861

)

1198993 = minus 119888120601

120601 + 120601 (minus119896120601 + 119898119905119892ℎ119904 + 1198981198911198921198671 + 119898119901119892119867)

minus 119886119901119898119901119892 cos 120579 minus 119881119903 (119898119905ℎ119904 + 1198981198911198671)

minus 119867119898119901 (21198633

120579

120601 + 1198634

1206012+ 1198635

1205792)

minus 119868119909119910119901119903 (

120601 +

120579) minus 1199032119868119910119911119901 +

12057921198681199091199091199011198602

119860

1198994 = minus 119892 (119887119901 cos 120579 cos120601 minus 119886119901 sin 120579 sin120601)

minus

1

2

1205792(1198862

119901minus 1198872

119901) sin 2120579

minus

1206012[

1

2

(1198862

119901minus 1198872

119901) sin 2120579 minus 119887119901119887 cos 120579]

(45)

According to the Cramer rule the variables in (43) canbe expressed as follows while the matrix 119872 has a nonzerodeterminant

1199091 =1198721

119873

1199092 =1198722

119873

1199093 =1198723

119873

1199094 =1198724

119873

(46)

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

Δkk

Δj

j

120601Δt

120601Δt

rΔt

rΔt

Δi

i

Figure 7 The change of the 119909-119910-119911 coordinate system

Liquid-free surface

The fixedliquid part

The pendulumpart

fixed part liquid-Height of the

free surface

Figure 8 Schematic diagram for the division into the pendulumpart and the fixed liquid part

where119872119894 119894 = 1 2 3 4 is the matrix formed by replacing the119894th column of119872 by the column vector119873

Based on (46)MATLABrsquos ODE algorithm is used to solvethe differential equations of the vehiclersquos dynamic model

33 The Inertia Tensor of the Pendulum Mass and the FixedLiquid Mass Before solving the differential equations of thevehiclersquos dynamicmodel the inertia tensor for the fixed liquidmass and the pendulum mass are needed

It is known that the liquid-free surface can be approxi-mately described by a tilted straight linewhen transient liquidsloshing occurs This means that the inertia tensor for theliquid bulk can be obtained However the obtained valuesare the sum of the pendulum part and the fixed liquid partAs it is quite difficult to divide liquid bulk physically into thependulum part and the fixed liquid part the fixed liquid partis assumed to locate at the bottomof the liquid bulk (as shownin Figure 8) and the height of the fixed liquid-free surface canbe acquired using (3) and (4) while the density of the liquidcargo is already given Based on this assumption the inertiatensor for the fixed liquid part can be easily obtained Hencethe inertia tensor for the pendulum part can also be obtained

According to a market survey the cross-sectional area ofcylindrical tanks is usually close to 24m2 HT5250GYQ3Ca typical tank truck of the HongTu brand is chosen asthe simulation object in this paper The radius of the tank

cross section is 18m and the thickness of the tank walls isneglected The length of the tank is 9m The liquid cargo isassumed to be water whose density is 1000 kgm3 CAD isused to calculate the inertia tensors for the pendulum massand the fixed liquid mass as the liquid fill level changes from01 to 09 with a 01 step size and as the tilt angle of the liquid-free surface changes from 0 degrees to 90 degrees with a 10-degree step size The calculation results for liquid fill levels of02 05 and 08 are given in Tables 1 2 and 3

As the fixed liquid part is assumed to be static as thetank tilts the inertia tensor of the fixed liquid mass is alwaysconstant

Polynomials are fitted for the data points listed in Tables1 2 and 3 respectively to obtain equations that describe theinertia tensor as a function of the liquid fill percentage and theliquid-free surface tilt angle For arbitrary tilt angle of liquid-free surface under constant liquid fill level interpolation isused to obtain the inertia tensor of the pendulum mass

4 Results and Discussion

41 Tank Truckrsquos Driving Stability When the Liquid Fill LevelIs 05 A MATLAB simulation is carried out for the steeringangle step test of the HongTu tank truck with a drivingspeed of 90 kmh and a liquid fill level of 05 The simulationresults for different driving variables are plotted in Figure 9Simulation results for a normal truck that has the sameparameter values load situation (ie 50 laden) and drivingvelocity are plotted in Figure 10

It is quite obvious that the vehiclersquos roll stability is greatlyinfluenced by transient liquid sloshing in a partially filledtank While the curves of the driving variables for thenormal truck quickly return to the steady state those forthe tank truck fluctuate up and down for about 40 s beforebecoming steady and the overshoot is quite large Howeverthe overshoot dampens quickly with the passage of time andthe damping frequency is close to the pendulum oscillationfrequency It can also be seen that the bigger the steering angleis themore significant the effect on vehicle roll stability givenby transient liquid sloshing is As can be seen in Figure 9(d)the maximum roll rate of the tank truck is 02 rads whilethe steering angle is 002 rad this value increases to 07 radswhen the steering angle is 005 rad

When the steering angle is 005 rad the pendulumrsquosamplitude changes from its original value of 471 rad (270degrees) to 509 rad (2917 degrees) and stays steady withthe aid of lateral acceleration and gravity The movementof the pendulum mass ball increases the vehiclersquos rollovertorque and results in a bigger roll angle for the tank truck(Figure 9(c)) compared to the normal truck (Figure 10(c))

As the maximum roll rate of the vehicle is already quitelarge even though the steering angle is small it seems that thevehicle may encounter rollover However as seen in Figure 9the tank truck does not experience rollover but returns to asteady state after about 50 s of fluctuation which means thatwhether or not the vehicle loses roll stability mostly dependson the duration of its transient response

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

Table 1 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 02

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 1831 18574 0 597 0 215 4088 10110∘ 1926 18613 117 607 338 215 4088 10120∘ 2208 18711 230 637 684 215 4088 10130∘ 2670 18871 336 687 1044 215 4088 10140∘ 3298 19075 432 754 1420 215 4088 10150∘ 4071 19283 515 837 1806 215 4088 10160∘ 4967 19485 582 933 2192 215 4088 10170∘ 5959 19650 632 1039 2558 215 4088 10180∘ 7020 19770 662 1153 2883 215 4088 10190∘ 8101 19783 672 1268 3135 215 4088 101

Table 2 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 05

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 10837 45002 0 2285 0 3998 35642 121210∘ 11023 45002 228 2305 1063 3998 35642 121220∘ 11575 45002 449 2364 2094 3998 35642 121230∘ 12477 45002 656 2461 3062 3998 35642 121240∘ 13702 45002 843 2592 3936 3998 35642 121250∘ 15212 45002 1005 2754 4691 3998 35642 121260∘ 16960 45002 1136 2941 5303 3998 35642 121270∘ 18895 45002 1233 3149 5754 3998 35642 121280∘ 20957 45002 1292 3370 6031 3998 35642 121290∘ 23084 45002 1312 3598 6124 3998 35642 1212

0 10 20 30 40 50 60 70

0

0002

0004

0006

0008

001

Time (s)

Slip

angl

e (ra

d)

minus0002

(a) Curves of slip angle

00501

0

01502

02503

035

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

(b) Curves of yaw rate

0 10 20 30 40 50 60 70minus07

minus06

minus05

minus04

minus03

minus02

minus01

001

Roll

angl

e (ra

d)

Time (s)

(c) Curves of roll angle

002040608

Roll

rate

(rad

s)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

minus08

minus06

minus04

minus02

(d) Curves of roll rate

474849

55152535455

Pend

ulum

ampl

itude

(rad

)

0 10 20 30 40 50 60 70Time (s)

Steering angle = 002 radSteering angle = 005 rad

(e) Curves of pendulum amplitude

002040608

1

Pend

ulum

angu

lar v

eloc

ity (r

ads

)

minus08minus06minus04minus02

minus10 10 20 30 40 50 60 70

Time (s)

Steering angle = 002 radSteering angle = 005 rad

(f) Curves of pendulum angular velocity

Figure 9 Truck driving stability when liquid fill level is 05

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

Table 3 The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 08

119868119909119909119901 119868119911119911119901 119868119909119910119901 119868119909119911119901 119868119910119911119901 119868119909119909119891 119868119911119911119891 119868119909119911119891

0∘ 16269 34190 0 2570 0 23308 104440 500810∘ 16363 34150 117 2579 751 23308 104440 500820∘ 16644 34040 230 2610 1461 23308 104440 500830∘ 17107 33880 336 2659 2091 23308 104440 500840∘ 17732 33680 432 2726 2611 23308 104440 500850∘ 18506 33470 515 2809 2998 23308 104440 500860∘ 19400 33270 582 2905 3240 23308 104440 500870∘ 20387 33100 632 3010 3337 23308 104440 500880∘ 21438 32980 662 3122 3298 23308 104440 500890∘ 22539 32980 672 3241 3135 23308 104440 5008

0 1 2 3 4 5 6 7 8 9 10

0

0005

001

0015

Time (s)

Slip

angl

e (ra

d)

minus0025

minus002

minus0015

minus001

minus0005

Steering angle = 002 radSteering angle = 005 rad

(a) Curves of slip angle

0

005

01

015

02

025

Yaw

rate

(rad

s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(b) Curves of yaw rate

0

Roll

angl

e (ra

d)

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(c) Curves of roll angle

0

02

04

Roll

rate

(rad

s)

minus1

minus08

minus06

minus04

minus02

0 1 2 3 4 5 6 7 8 9 10Time (s)

Steering angle = 002 radSteering angle = 005 rad

(d) Curves of roll rate

Figure 10 Driving stability of normal truck with 50 laden situation

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

0 30 60 90 120 150

0

001

Time (s)

Roll

angl

e (ra

d)

minus007

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0

005

01

minus025

minus02

minus015

minus01

minus005

0 30 60 90 120 150Time (s)

Roll

rate

(rad

s)

(b) Curves of roll rate

Figure 11 Tank truck driving stability when liquid fill level is 02

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus006

minus005

minus004

minus003

minus002

minus001

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

005

01

015

Roll

rate

(rad

s)

minus02

minus015

minus01

minus005

(b) Curves of roll rate

Figure 12 Driving stability of normal truck that is 20 laden

From the comparison of Figures 9 and 10 it can also beseen that although the overshoots of the driving variablesof the tank truck are much bigger than those of the normaltruck the difference between their steady values is not so bigwhich means that the transient response of the tank truck isgreatly affected by liquid sloshing while the steady responseis only slightly influenced by it

42 Tank Truckrsquos Driving Stability When the Liquid Fill LevelsAre 02 and 08 To investigate the impact of the liquid filllevel on the vehiclersquos roll stability tank vehicles with liquid filllevels of 02 and 08 with a steering angle of 002 rad andwitha driving speed of 90 kmh are studied in a steering angle step

test The simulation results for the roll angle and roll rate areplotted in Figures 11 and 13

At the same time the roll stability of normal trucks withthe same steering angle parameter values load situationsand driving speed are also investigated to determine theinfluence of transient liquid sloshing in a partially filled tankon vehicle roll stability The simulation results for the rollangle and roll rate for the normal trucks are presented inFigures 12 and 14

As can be seen from Figure 11 when the liquid fill levelis 02 the vehicle takes about 120 s to come to a steadystate The fluctuation times of the vehicle driving variablesare much bigger than that when the liquid fill level was 05as can be seen by comparing Figures 9 and 11 However

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 13

0 5 10 15 20 25 30 35 40 45 50

0

02

Time (s)

Roll

angl

e (ra

d)

minus12

minus1

minus08

minus06

minus04

minus02

(a) Curves of roll angle

0 5 10 15 20 25 30 35 40 45 50Time (s)

0

02

04

06

Roll

rate

(rad

s)

minus08

minus06

minus04

minus02

(b) Curves of roll rate

Figure 13 Tank truck driving stability when liquid fill level is 08

0 1 2 3 4 5 6 7 8 9 10

0

Time (s)

Roll

angl

e (ra

d)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

(a) Curves of roll angle

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

01

02

Roll

rate

(rad

s)

minus05

minus04

minus03

minus02

minus01

(b) Curves of roll rate

Figure 14 Driving stability of normal truck that is 80 laden

the oscillation amplitude is reduced when the liquid fill levelis lower This is because when the liquid fill level is lowerand the pendulum mass is much smaller even though thearm length of the pendulum is bigger It can be deduced thatthe oscillation amplitude is determined by the liquid sloshingmass to a large degree This conclusion can also be proved bycomparing Figures 9 and 13 Although the liquid fill level of08 is bigger than that of 05 shown in Figure 9 the sloshingliquid mass for the former is smaller than that for the latteras seen from (3) and (4)Therefore vehicle roll stability whenthe liquid fill level is 05 is worse than that when the liquid filllevel is 08

Above all the tank vehiclersquos roll stability is greatly influ-enced when the liquid fill level is near to 05 in whichcase the sloshing liquid mass occupies a large proportion

of the liquid bulk and the mass of the liquid bulk is quitelarge In this situation the tank vehiclersquos roll stability is muchworse than that of normal cars However when the liquidfill level is either fairly small or fairly large the tank vehiclersquosroll stability is only slightly influenced by transient liquidsloshing Therefore a liquid fill level of 04ndash06 is the worstload situation for tankers and should be avoided if possible

5 Conclusions

To investigate the driving stability of tank trucks the paperused an equivalent trammel pendulum for liquid sloshing ina partially filled tank The trammel pendulum oscillated inthe tank and was described in reference to tank-fixed coordi-nates To couple the motion of the trammel pendulum with

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Discrete Dynamics in Nature and Society

that of the tank truck the motion equation of the pendulumunder a noninertial coordinate system was analyzed using aLagrangian function Based on this a dynamic model for atank truck was established using Newtonrsquos first law and theangular momentum A typical tank truck was selected andits driving stability is studied under a steering angle step testStudying tankersrsquo driving stability and the main influencingfactors is of great importance for evaluating tankersrsquo drivingsafety as well as for developing activepassive roll controlsystems for them

The following important discoveries were found from thedriving stability simulations

(1) The curves of the tank truck driving variables fluctu-ate until they return to the steady state The fluctua-tion frequency descends with an increase in the liquidfill level while holding the testing situation constantThe fluctuation frequency is close to the pendulumrsquososcillation frequency The overshoots of the drivingvariables depend on the pendulum mass which isequal to the liquid mass that participates in transientsloshing

(2) The pendulum mass occupies a large proportion ofthe mass of liquid bulk when the liquid fill level isin the range of 03ndash07 Also the mass of liquid bulkis quite large in such cases the overshooting of thetanker driving variables is quite large and the tankerrsquosdriving stability is greatly impacted by transient liquidsloshing Therefore overshoot control for tankersrsquodriving variables must be considered to support theirdriving safety Moreover the situation of a liquid filllevel near to 05 is the worst state in terms of load andshould be avoided

(3) The driving stability of a tanker depends on the dura-tion of the driving variables to a large degree

Since we made the assumption in deriving the inertiatensors for the pendulum mass and the fixed liquid massthat the fixed mass is located at the bottom of the tank andis always static when the tank tilts the inertia tensor valuediffers from the real situation and the simulation resultsfor the tank truckrsquos driving stability will also have beenslightly influenced Thus a theory and method for physicallydistinguishing the pendulumrsquos mass and the fixed liquid massfrom the liquid bulk will be proposed in a future study

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

This research is supported by Research on Evaluation andDetection Technology of Handling and Driving Stability forCommercial Vehicles (no 2009BAG13A04) and the NationalNatural Science Foundation of China (Grant nos 51375200and 51208225)

References

[1] J Woodrooffe ldquoEvaluation of dangerous goods vehicle safetyperformancerdquo Report TP 13678-E Transport Canada 2000

[2] C Kwon ldquoConditional value-at-riskmodel for hazardousmate-rials transportationrdquo in Proceedings of the Winter SimulationConference (WSC rsquo11) pp 1703ndash1709 December 2011

[3] T T Treichel J P Hughes C P L Barkan et al ldquoSafetyperformance of tank cars in accidents probabilities of ladinglossrdquo RSI-AAR Railroad Tank Car Safety Research and TestProject Association of American Railroads Washington DCUSA 2006

[4] M I Salem H Victor M Mucino et al ldquoReview of parametersaffecting stability of partially filled heavy-duty tankersrdquo SAETechnical Paper 1999-01-3709 1999

[5] X Kang S Rakheja and I Stiharu ldquoCargo load shift and itsinfluence on tank vehicle dynamics under braking and turningrdquoHeavy Vehicle Systems vol 9 no 3 pp 173ndash203 2002

[6] W H Wang Q Cao K Ikeuchi and H Bubb ldquoReliability andsafety analysis methodology for identification of driversrsquo erro-neous actionsrdquo International Journal of Automotive Technologyvol 11 no 6 pp 873ndash881 2010

[7] WWangW Zhang H GuoH Bubb andK Ikeuchi ldquoA safety-based approaching behavioural model with various drivingcharacteristicsrdquo Transportation Research C vol 19 no 6 pp1202ndash1214 2011

[8] F Solaas and O M Faltinsen ldquoCombined numerical andanalytical solution for sloshing in two-dimensional tanks ofgeneral shaperdquo Journal of Ship Research vol 41 no 2 pp 118ndash129 1997

[9] K Modaressi-Tehrani S Rakheja and I Stiharu ldquoThree-dimensional analysis of transient slosh within a partly-filledtank equipped with bafflesrdquo Vehicle System Dynamics vol 45no 6 pp 525ndash548 2007

[10] L Strandberg ldquoLateral stability of road tankersrdquo VIT Report138A National Road and Traffic Research Institute LinkopingSweden 1978

[11] R Ranganathan S Rakheja and S Sankar ldquoSteady turningstability of partially filled tank vehicles with arbitrary tankgeometryrdquo Journal of Dynamic Systems Measurement andControl vol 111 no 3 pp 481ndash489 1989

[12] R Ranganathan ldquoRollover threshold of partially filled tankvehicles with arbitrary tank geometryrdquo Proceedings of theInstitution of Mechanical Engineers D vol 207 no 3 pp 241ndash244 1993

[13] S Sankar and S Surial ldquoSensitivity analysis approach for fastestimation of rollover stability of heavy articulated vehiclesduring steady state turningrdquo Heavy Vehicle Systems vol 1 no3 pp 282ndash303 1994

[14] C B Winkle ldquoRollover of heavy commercial vehiclesrdquo ReportUMTRI-99019 2001

[15] X Li -S X Zheng -L andH Liu -F ldquoImproved algorithmon rollstability evaluation of partially filled tractor-tank semitrailerrdquoJournal of Jilin University vol 42 no 5 pp 1089ndash1094 2012(Chinese)

[16] KModaressi-Tehrani S Rakheja andR Sedaghati ldquoAnalysis ofthe overturning moment caused by transient liquid slosh insidea partly filled moving tankrdquo Proceedings of the Institution ofMechanical Engineers D vol 220 no 3 pp 289ndash301 2006

[17] X L Zheng X S Li and Y Y Ren ldquoAnalysis of rollover thresh-old for partially-filled tank vehicles impacted by transient liquid

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 15

sloshingrdquo Journal of Beijing Institute of Technology vol 21supplement 2 pp 169ndash174 2012

[18] F T Dodge ldquoAnalytical representation of lateral sloshing byequivalentmechanicalmodelsrdquo inTheDynamic Behavior of Liq-uids in Moving Containers H N Abramson and S SilvermanEds chapter 6 National Aeronautics and Space AdministratorWashington DC USA 1966

[19] H N Abramson W H Chu and D D Kana ldquoSome studies ofnonlinear lateral sloshing in rigid containersrdquo Journal of AppliedMechanics vol 33 pp 777ndash784 1966

[20] M I Salem Rollover Stability of Partially Filled Heavy-DutyElliptical Tankers Using Trammel Pendulums to Simulate FluidSloshing [PhD thesis] West Virginia University Department ofMechanical and Aerospace Engineering 2000

[21] X-L Zheng X-S Li and Y-Y Ren ldquoEquivalent mechanicalmodel for lateral liquid sloshing in partially filled tank vehiclesrdquoMathematical Problems in Engineering vol 2012 Article ID162825 22 pages 2012

[22] L Dai L Xu and B Setiawan ldquoA new non-linear approachto analysing the dynamic behaviour of tank vehicles subjectedto liquid sloshingrdquo Proceedings of the Institution of MechanicalEngineers K vol 219 no 1 pp 75ndash86 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of