【正文】 gineers) concept as follows:aij188。FzR.External forces acting on the vehicle are gravitational forces,tireroad contact forces, and suspensions forces. The externalmoments are the moments due to tireroad forces and suspensions0246810012345678910Time (s)Time (s)1086420Yaw rate (deg/s)0,000,050,100,150,200,250,30(Rigid)(Liquid)Lateral rear load transfertFig. 10. Vehicle response to single0123456789106420246Roll angle (deg)Time (s)(Rigid)(Liquid)2468(Rigid)(Liquid)A/Solids 28 (2020) 1026–1034forces. In this case we assume that the vehicle travels at a fixedspeed, so the longitudinal force will be negligible.The nonlinear variation of tire cornering stiffness with verticalload is described using the quadratic equation to obtain the lateraltire force Fy:Fyi188。 FzL(16)If FzR188。 FzrC0DFzr(15)where R, L, f and r represent right, left, front and rear side respectively. Fzis the vertical wheel load。DFzfFzRr188。CrSrcos240。12189。254。 j is the yaw angle of vehicle。zs222。rs188。xu。0。0。u。 w C2 V 254。u。x。mjrjC2aj254。 z222。 msas。 kg/ms). The shift of the liquid load is evaluated accordingδ (deg)06100,00,51,01,52,0Time (s)24 8input.totheinstantaneousdisplacement mass centre coordinatesandtheequivalent force exerted by the liquid on the tank. These quantitiesare calculated by using the numerical integration which covers thefield of the liquid.For a steadystate turning as shown in Fig. 5, we obtain a goodcorrelation between the two models. In the early stages, there isa small difference due to free surface oscillation, which is neglectedin the analytic model. Once liquid oscillation starts, the value isstabilized and the mean value approaches the analytical model.The correlation is also good between the twomodels fora singlemanoeuvre illustrated by Fig. 6. However, the amplitude is largerfor the numerical model pared to the analytical model. A small5. Nonlinear vehicle modelSeveral methods can be exploited to develop a vehicle model,such as Virtual Work, Lagrange and Newton. Alternativeapproaches to dynamic vehicle models use simple models witha reasonable puting time. Our model is developed using theNewton method, based on the conservation of the linear andangular momentum of a solid body.The dynamic equations of motion are derived for a unit vehiclethat possesses both front and rear steering capabilities as well asadapted to simpler vehicles, such as vehicles with only front wheel0 20 40 60 80 100 120 140 160 180 200012345678910Y (m)X (m)0481021012δ (deg)Time (s)62Fig. 4. Single lane change manoeuvre.M. Toumi et al. / European Journal of Mechanics A/Solids 28 (2020) 1026–1034 1029delay was also noted in the response of the numerical modelpared to the analytical model. The small difference betweenthe two models is probably due to the assumption of linearity andthe iterative calculation of the free surface for the analytical modelpared to the numerical model.0,10,00,10,20,30,40,5NumericAnalyticYL (m)0100,2Time (s)10,010,511,011,512,012,5NumericAnalyticIxx (kg m2)103246801Time (s)2468Fig. 5. Response to constant radiussteering, by the assignment of a constant control input of zero tothe appropriate (unavailable) control effectors. The free bodydiagram (FBD) for the vehicle under consideration is shown inFig. 7. The body fixed reference frame is labelled on the FBD with itsorigin atthe vehicle’s centre of gravity. The z axis is pointing up, the0,540,570,600,630,660,69 NumericAnalyticZL (m)048100,51Time (s)01530456075NumericAnalyticLateral pressure force (KN)2604810Time (s)26cornering manoeuvre.0 100,40,30,20,10,00,10,20,30,40,5NumericAnalyticTime (s)YL (m)ZL (m)0,510,540,570,600,630,660,69 NumericAnalytic14Numeric1032 468 010Time (s)2468laneM. Toumi et al. / European Journal of Mechanics A/Solids 28 (2020) 1026–10341030x axis is pointing towards the front of the vehicle, and the y axis ispointing towards the vehicle’s right side. This model assumes thatleft and right steer angles are the same for front and rear axles.8910111213AnalyticIxx (kg m2)010Time (s)2468Fig. 6. Response to singleThis model divides the vehicle’s mass into unsprung mass, muand sprung mass, ms. The sprung mass is the body of the vehiclesupported by its suspension. This sprung mass is assumed to pivotabout an imaginary roll axis. Neglecting the pitch and verticalmovement, the equations are derived from the vehicle’s linear andangular movements:Xi188。 m, b188。240。 kh240。Xliquidc188。Xliquidc188。 VanLoon et al., 2020).C15 Averypopularmethodforadvectinganinterfaceisthevolumeoffluid method (VOF) originally developed by Hirt and Nichols(1981). This method makes use of a fluid volume fraction aq,with values between zero and one, indicating the fraction offunction F is then advected according to the local flow velocity.Ineachcontrolvolume,thevolumefractionsofallphasessumtophases and represent volumeaveraged values, as long as thevolume fraction of each phase is known at each location.VOF is probably the most successful technique because of itssimplicity and its robustness. In this study we use the Fluent software v which uses this technique. The properties appearing inthe transport equations are determined by the presence of theponent phases in each control volume. There is a twophasesystem. The phases are represented by the subscripts 1 for the gasand 2 for the liquid. If the volume fraction of the liquid is beingtracked, the density in each cell is given by:0 20 40