Vehicle Dyn

- M02MAE - Vehicle Refinement Supervisor: Damian HartyCoursework ReportCoursework 1 – Vehicle Dynamics2011 MSc Automotive EngineeringLiu YunchuanStudent ID: 3542130December 2011AbstractIn this report, a two-wheel model was developed to assist evaluating the general response of three cars with typical forms of layout: VW Beetle (RR), BMW M5 (FR) and VW Polo (FF). The eigenvalues of the characteristic equation can obtained by building an excel spreadsheet, which was used to determine the stability of the car at different speeds. In addition, the damping ratios and natural frequencies were worked out to analyse the transient response of the vehicle under a steering input.ContentsList if Equations ............................................................................................................. i i List if Figures ................................................................................................................. i i Nomenclature ................................................................................................................. i i Glossary ......................................................................................................................... i i1.Introduction (1)2.Modeling and Supporting Theories (1)2.1 Two-Wheel Model (2)2.2 Spreedsheet Model (2)2.3 Supporting Theories (3)2.3.1 Steering Characteristics of the Vehicle (3)2.3.2 Static Response of the System (4)2.3.2 Transient Response of the System (4)3.Vehicle Examination (6)3.1 Methodology (6)3.2 VW Beetle Examination (6)3.2.1 Critical Speed Examnations (6)3.2.2 Radial Tyre Examnations (6)3.2.3 Tyre Width Examnations (6)3.2.4 Summary (7)3.3 BMW M5 Examination (7)3.3.1 Original Car Examnations (7)3.3.2 Modified Car Examnations (7)3.3.3 Sammary (8)3.4 VW Polo Examination (8)3.4.1 Targeted Damping Ratio Examinations (8)3.4.2 Minimal Tyre Witdth Change Examnations (8)3.4.3 Sammary (8)4.Conclusions (8)References/Bobilography (9)Appendices (10)List of Equations2.1 (2)2.2 (2)2.3 (2)2.4 (3)2.5 (3)2.6 (3)List of figuresFigure 2.1 Vehicle Model with Three Degrees of Freedom (1)Figure 2.2 Two-Wheel Vehicle Model in Earth-fixed Co-ordinate System (2)Figure 2.3 Possibilities of car behaviour during cornering (4)Figure 2.4 Transient Response of the System (5)Figure 2.5 System Output with Different Damping Ratios (5)NomenclatureGlossary1. IntroductionIn recent years, much attention has been attached to the vehicles handling performance as the increasing consumer appeal for safer and easier driving cars. In response to that demand, the proportion of cars equipped with electronic stability control system such as Electronic Stability Program (ESP) has grew tremendously, and those systems are likely to become mandatory in the coming years with the tighten legislation.Generally, Cars with good handling performance are both fun and safe to drive, otherwise, they prone to lose control as the speed increases especially under emergency avoidance condition, which often involves acute lane departures. According to the National Highway Transportation Safety Administration (NHTSA), more than 39% of crash-related fatalities are caused by lane departures (Rajamani 2006:14).To help the understanding of the vehicle’s general response to the steering input, a simplified two-wheel model and an Excel spreadsheet was developed(Attached in CD),so the characteristic of the vehicle can be described by the eigenvalues (roots of the characteristic equation of the system), damping ratios and natural frequencies calculated from the characteristic equation.2. Modelling and Supporting Theories2.1 Two-Wheel ModelForecasting vehicle’s response is highly demanding for there are multiple factors affecting its behaviour and it is the interactions of all the effects rather than independent factors that add complexity to the analysis. (Blundell and Harty 2004:2). Before assessing the complex model (Figure 2.1), a classical two-wheel model can be used to analyze the dynamic characteristic of the vehicle.Figure 2.1 Vehicle Model with Three Degrees of Freedom: Lateral Yaw and Roll.(Pacejka 2002:8)Having only two degrees of freedom: Y (measures lateral position) and Ψ(measures yaw angle), the two-wheel model (Figure 2.2) is expressed in a so-called “body-centred” co-ordinate system based on several assumptions:1)The vehicle is assumed to have a planar motion: the movements from thesuspension are omitted.2)The differences between the inside and outside wheels when taking cornerswith much longer radiuses than the wheelbase was neglected; thus fourwheels are represented by two: a front one and a rear one.3)External factors are ignored, including aerodynamic force and roadconditions.Ac.gBFigure 2.2 Two-Wheel Vehicle Model in Earth-fixed Co-ordinate SystemIn the model, the steering angle of front wheel is represented by δand the slip angles of front and rear wheels are denoted by αf and αr respectively, while a andb represents the length of the front and rear axle to the vehicle’s centre of gravity(c.g.).The mass and the moment of inertia of the vehicle are represented by m andI zz.2.2 Spreadsheet ModelWith the model built above, following basic equations can be used to represent the vehicle on pneumatic tyres:∑F x=mv̇x(2.1)∑F y=m(rv x+v̇y)(2.2)∑M z=I zz ṙ(2.3) To facilitate the calculation, it is assumed that the vehicle is having a static state cornering where the longitudinal speed of the vehicle varies so slowly that longitudinal acceleration V x can be treated as 0 and the cornering stiffness C af and C ar of the two axles are linear in this case.Based on (2.1), (2.2), (2.3), the characteristic equation can be developed as: λ2+(−a11−a22)λ+(a11∙a22−a21∙a12)=0(2.4) Where:a11=Cαf+CαrmV x,a21=Cαf a−Cαr bmV x−v x,a12=Cαf a−Cαr bI zz V x,a22=Cαf a2+Cαr b2I zz V x,And the roots of the equation can be calculated:λ=−B±√B2−4AC2A(2.5) Where: A=1,B=(−a11−a12),C=(a11∙a22−a21∙a12).Since there is no guarantee that B2−4AC>0, an “if” function was used to determine whether the eigenvalue is a complex number; by judging the sign of the value of B2−4AC, the roots could be worked out with the help of the “complex “ function of Excel. The formula used in the spreadsheet is shown below:Where: ∆=B2−4ACThe damping ratio and the natural frequency of the system could be obtained by calculating the cosine of the angle between real axis and the root point and the magnitude (length) of the vector from the origin to the root point respectively.2.3 Supporting TheoriesThere are two major concerns with regard to the vehicle handling in this report: The stability of the vehicle and the response characteristic of the vehicle with a steering input at different longitudinal speeds (V x). From the control engineering’s perspective, it is the static and transient response of the system that needs to be examined.2.3.1 Steering Characteristic of the VehicleAs shown in Figure 2.3, three scenarios could happen if the driver keeps a constant steering input at certain speed: understeer, neutral steer and oversteer, In order to determine the steering characteristics of the vehicle, stability factor K is introduced where:K=(m(a+b)2)∗(aC ar−bC af)(2.6)If K>0, the car understeers; the radius of its path decreases at a constant speed with a settled steering angle.If K<0, the car oversteers; the radius of its path decreases at a constant speed with a settled steering angle.On the critical point of the two scenarios, when K<0, the vehicle will always follow the path therefore it is theoretically neutral steered.Figure 2.3: Possibilities of car behaviour during cornering (Vlcek 2011)2.3.2 Static Response of the SystemThe vehicle is steady with a constant steering when the speed is slow and the real part of the eigenvalue is negative. As the speed builds up, the root approaches to the right part of the axis simultaneously, until it surpasses the critical point where λ=0: the car will spin so the system is unstable.2.3.3 Transient Response of the SystemHowever, a statically stable system is not necessarily safe, there is a potential risk that the car will wobble from one side to another under an abrupt steering input when there is an emergency happens; therefore, it is necessary to investigate the system’s transient response.If B2−4AC>0, the system is a first-order system, whose real roots lies on the X axis , the damping ratio of the system will be a constant: ζ=1, indicating the system is critically damped. In this case, the vehicle will not oscillate under any circumstances. Besides, if the root lies on the negative part of the axis, the amplitude decays exponentially, so the system is stable. Otherwise, a positive real root will give an exponential increase output, meaning the car will slip straightaway.If B2−4AC<0, the system is second-ordered, whose complex roots do not lie on the X axis. If the root lies on the left side of the Y axis, it is convergent; the amplitude will decay so the car will eventually rebalance itself after oscillation. Otherwise, the root is divergent that the amplitude of the oscillation keeps growing until the car fails, which is the least desired result of all (Figure 2.4).Figure 2.4 Transient Response of the System (Roymech 2010)The damping ratio measures how fast oscillations in a system decay after a disturbance. It can be expressed by the ratio between the length of the real root and the length from root point to the origin, so the roots far away above the X axis has a bigger oscillation. A well damped vehicle system will have a good directional convergence without the driver’s correction (Mimuro, et al. 1990). However, the rise time of the overdamped (ζ > 1) system will becomes longer. (Figure 2.5).Figure 2.5 System Output with Different Damping Ratios (Nogueira 2007) The natural frequency influences greatly on whether the driver has enough time to react to the oscillation and how the driver feels when in the car. An ideal vehicle system has a natural frequency between 0.6Hz to 1.2Hz and a damping ratio from 0.5 to 0.8 (Wei et al. 2006). It is also noteworthy that the natural frequency is the undamped frequency, and it is the damped frequency that really defines the oscillation characteristic of the system. However, the difference is ignored in this case using approximation.With all the parameters obtained above, further analysis can be carried out.3. Vehicle Examinations3.1 MethodologyAfter the spreadsheet been built, the eigenvalues of the characteristic equation can be obtained by trying different V x inputs.; after several attempts, the range of the value was narrowed down and the critical speed could be sought by approaching the point that λ=0, and same method can be used to find other critical values in the following examinations. The figures used for examine are attached in appendices. 3.2 VW Beetle Examinations3.2.1 Critical Speed ExaminationsBy applying the method illustrated above, it was observed that the critical speed of the beetle is 38.14m/s, which is roughly 137km/h, It is worth mentioning that the characteristic equation has two real roots, and one of them is always negative, put it differently, the car is always stable, which is not reasonable, thus it is ignored.The d amping ratio of the VW Beetle is a constant: ζ = 1, the car will gradually adjust itself to the new path until the speed surpasses approximately 38m/s, in which scenario, the car will spin unless the driver intervenes.3.2.2 Radial Tyre ExaminationsThe radial tyres offer twice as much as the cornering stiffness compared with the cross-ply tyres. So the cornering stiffness of both the front and rear axle has been shifted into -100000N/rad; apply the same technique and a critical speed of 53.94 m/s is sought.3.2.3 Tyre Width ExaminationsFrom the last case, it is obvious that the stability of the system is enhanced with the increased cornering stiffness. In this case, the target speed is set as 66m/s, which is greater than 38m/s, therefore the car is unstable because of the insufficient cornering stiffness. To increase the cornering stiffness, same proportion of the tyre width increment is required for they are highly proportional presuming that the cornering stiffness is linear.Since that when Cα2= -149738N/rad, the eigenvalue λ=0, so the cornering stiffness of the two axles needs to be increased to -149738N/rad, hence the increment is:a%=(w2w1−1)=(Cα2Cα−1)∗100%=199.48%Given the result above, if the Beetle keeps its cross-ply tyres, the width of the tyres needs to be tripled in order to retain the car stable under the speed of 66m/s. Therefore, the solution of changing the tyres into radial ones with 50% more width is more advisable.3.2.4 SummaryIn conclusion, VW Beetle is a stable car before reaching the critical point, otherwise, it spins without oscillating. The reason for the vehicle become easilyunstable is that the car is rearward weight biased; the heavy tail generates a great amount of moment within XY plane to turn over the car at high speed corners or during acute lane changes.The critical speed of the VW Beetle is around 38.14m/s and the speed can be increased to 53.94m/s by doubling the cornering stiffness if the old cross-ply tyres are changed into radial ones. The width of the tyre needs to be tripled in order to enable the Beetle reach the speed of 66m/s without compromising stability.3.3 BMW M5 Examinations3.3.1 Original Car ExaminationsTo understand the characteristic of the vehicle, first of all, it is essential to examine the steering characteristic of the car with coefficient K:K=(1800(1.37+1.39)2)∗( 1.37−110000− 1.39−110000)=4.2963×10−5>0.It is clear that the steering characteristic of M5 is understeer while the value of K is very close to 0; the car possesses a neutralized steering characteristic which gives a good performance in the corners.Furthermore, Set 70m/s to V x, the eigenvalue of the car is:λ=−1.8231±0.83204i.The eigenvalue is a complex number, so the vehicle oscillates, but the oscillation is minor, and for its real part is negative, the car is stable at the speed.The damping ratio and the natural frequency are 0.9097 and 0.3189 respectively, which suggests the car is well damped so the car does not oscillate very severely after the steering input and the frequency of 0.3189 gives the driver enough time to adjust. In a nutshell, the car processes very balanced handling performance.3.3.2 Modified Car ExaminationsAfter the modification, the cornering stiffness of the rear tyre has increased by 40% to -154000N/rad. therefore the characteristics of the system at the speed of 70m/s becomes:K′=(1800(1.37+1.39)2)∗( 1.37−154000− 1.39−110000)=8.838×10−4.It is obvious that the modified car has a better understeer tendency, which is more stable when taking corners but the dynamic performance may not be as good as before.λ’=−2.1905+4.4677iWith the eignevelue obtained, the system’s enhancement in stability can be confirmed and the larger imaginary part indicates the system oscillates more wildly with a steering input.The damping ratio and the natural frequency of the modified M5 is 0.4402 and 0.7191 respectively. The damping ratio drop from approximately 0.9097 to 0.4402 indicates the oscillation will last longer before damped out and the overshoot will increase with rise time become shorter. The higher natural frequency suggests that the driver needs to react faster to resume the control of the vehicle.3.3.3 SummaryTo sums up, the BMW M5 is a great example of both fun and safe to drive car, the balanced weight distribution enables the car to maintain static and dynamic stable at the same time. The modified car benefits from a wider rear tyre by a better grip and static stability, while the side effect is the dynamic stability is affected –the car will have a wilder oscillation when taking corners.3.4 VW Polo Examinations3.4.1 Targeted Damping Ratio ExaminationsMany companies requires their new model to have a damping ratio of 0.7 because at this point, a better compromise has been made between the rise time and overshoot, in other words, when ζ=0.7, the system is both fast and stable.Change the longitudinal speed in the spreadsheet until the damping ratio ζ reaches0.7, and the speed of the car when ζ=0.7 is approximately 40.28m/sWhen the VW hatchback is driving at the speed of 40.28m/s the eigenvalue is:λ=-3.657±3.732iWhere the root of the system is convergent due to the real part of the eigenvalue is negative, and the imaginary part is considerably large, it tends to oscillate quite heavily after the steering input, and the natural frequency is roughly 0.83,which is a figure, the driver’s response time is quite limited.3.4.2 Minimal Tyre Witdth Change ExamnationsIn order to reach the targeted value of damping ratio, the tyre width (cornering stiffness) is subjected to change. However, if the cornering stiffness of the two axles increases by the same value, the augmentation could be around 230% for both axles. However, since there is no requirement for polo that the size of the front and wheels needs to be same, it is proved that by varying the cornering stiffness of one axle can also meet the target of ζ=0.7, and the minimal change is to increase the width of the front wheel or reduce width of the rear wheel by 12.7%.3.4.3 SummaryLike the most family cars, the polo is a FF layout car which has a tendency to understeer in linear region; it is easy and safe to drive for most people while its transient response is not very ideal: the oscillation is severe and the natural frequency is high.By changing the distribution of the cornering stiffness, it is possible to adjust to the ideal damping ratio. Therefore, the final conclusions can be drawn.4. Conclusions1)The response of the vehicle under steering input can be described by theeigenvalues of the characteristic equation.2)By changing the tyres’ s type or increasing the tyre width, the system’s staticstability can be augmented while the transient stability could be sacrificed.3)The vehicle’s behaviour is determined by the distribution of the weight andcornering stiffness.Blundell, M and Harty, D. (2004) The multibody systems approach to vehicle dynamicsOxford: Butterworth HeinemannGuldner, J., Tan, H. and Patwardhan, S. (1996) ‘Analysis of automatic steering control for highway vehicle with look-down lateral reference systems’. Vehicle System Dynamics 26(4), 243-269Mimuro,T, Ohsaki, M, Yasuna, H, and Satoh,K. (1990) ‘Four Parameter Evaluation Method of Lateral Transient Response’ . Journal of Automobile Engineering 216(4), 297-305. Available From < /content/216/4/297.abstract> [3 Dec 2011]Nogueira, N. (2007) The List of Reference Illustrated [Online] available from /wiki/File:Damping_1.svg [1 Dec 2011]Pacejka,H. (2002) Tyre and vehicle dynamicsOxford: Butterworth-HeinemannRajamani, R. (2006) Vehicle dynamics and controlNew York: SpringerRoymech (2010) The List of Reference Illustrated [Online] available from /Related/Control/root_locus.html [3 Dec 2011]Vlcek , M. (2011) The List of Reference Illustrated [Online] available from http://old.fst.zcu.cz/_files_web_FST/_SP_FST%28SVOC%29/_2011/_sbornik/Papers Pdf/PhD/Vlcek_Martin.pdf [5 Dec 2011]Wei, C., Xie, M., Pan, C., Li, M., Liu Y., and Y. Zhou, K (2006) ‘Integrated Root Method and Stability Factor to Study Handling Stability of Automotive’. Chinese Journal of Mechanical Engineering 38(9), 19-22BibliographyMandal, A. (2006) Introduction to control engineering modeling, analysis and design New Delhi : New Age International P Ltd.,Gillespie, T. 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