Study On The Analysis Method Of Potential Energy For Vehicle Cornering Stability

The transportation is the backbone of the social development, and its basic functionis to realize the spatial motion of passengers and goods. Since auto is invented,which are considered as the most important vehicles, cars with their specialsuperiority have remarkably boosted the development of modern society and theimprovement of the social condition.Since 1970s, the car factories and the relevant experts all over the world have paidmore attention and funds to improve the active and passive safety. With thecontinuously in-depth study, people are gradually aware of the fact that thehandling stability of automobiles is one of the most important factors that affectvehicles active safety, and the cornering stability is the absolutely necessary part.Nowadays, there have been some theories and methods about the vehicle corneringstability, but most of them were obtained by using experimentation or by undulysimplifying the vehicle model, so these methods can not explain the essentialreasons of losing control, meanwhile these experimental formulas and rulesobtained from experimentation may not be rational. If we want to look for the realreason of losing control in critical cornering, it is very important to analyze steeringsystem and deduce model based on the characteristic of vehicle-tire system, andthen it can provide more reasonable and efficient method for us to avoid the vehicleinstability and guarantee the safety operation of vehicle.Based on the theory of the Synergetic Theory and Nonlinear Dynamics, this paperdoes a little positive research on the analysis method of the vehicle corneringstability.Learned from the analysis of the vehicle steering system using the relative thoughtin the Synergetic Theory, it is illogical that we usually think that the point wherethe slip angle is equal to zero is only the vehicle stable point. Moreover, the resultdifferent from the traditional academic analysis and experiment can be receivedthrough the simulation experiments in the paper. The result is that there aremulti-stable points (including controllable and uncontrollable points), however thezero-stable point is involved in a stable potential energy limit cycle that iscomposed of steering angle of steering wheel, the vehicle slip angle and yawangular velocity .Through the compare of some primary tire models, this paper considers thestandard tire model of the magic formula as the basic tire model to do the nonlinearcharacteristic analysis. Learned from the analysis, there is the coupling of the stateparameters, the vehicle model presents strong nonlinear character, and thenonlinear character is much more evident under the situation of high speed andlarge steering angle. Furthermore when the vehicle is running, the tire also presentsnonlinear slip character. The side slip stiffnessC has the evident nonlinear trendchanged with the wheel slip angleα . The side slip force F only has theapproximate linear relationship with the wheel slip angleα in small area, however,with the increase of the wheel slip angle, when the wheel slip angle reaches acertain value, the side slip force does not increase with the slip angle, it appears ashort-time keeping state, at this time the side slip force reaches saturation, namelythe tire reaches its side adhering limit on common sense. When the slip anglecontinues to increase, the tire presents soft spring character, its sideslip stiffness Cdecreases gradually, making the side slip fore F degrease, at this time the tire isin the nonlinear state.For analyzing conveniently, the tire magic formula is fitted in the paper. Themethods contain the polynomial fitting, the piecewise linear fitting and thepointwise Taylor expansion fitting. We found in the fitting process, even the tenrank multinomial fitting is used, the error is a little big, however, the result is betterwhen using the subsection linear linearization method, and when the subsectionincreases infinitely, continuous smooth curve of the tire force is reached, at thesame time, which is equal to replace the nonlinear expression hard to resolved withthe infinite segment linear expression easy to resolved, and the quantity level of theerror is pretty small. Therefore the original formula is replaced by the Taylorexpansion of the magic formula, which becomes the basic tire model.Based on the vehicle dynamic equation and the Taylor formula of the magicformula, we build the potential energy function of vehicle steering system, andanalyze the stability of this system using the function. As a result, we find the factthat the stable region is a limit cycle, and it is composed of two state parameters -β andψ . When the two parameters are in the cycle, the vehicle state is stable, orelse it is unstable. Outside the stable region there are four small areas whosepotential energy is less then it of stable region, when vehicle leave the stable region,it is easy to enter into these region which represent unstable region. The stableregion vary with the variation of steering angle, when the input of steering angle issmall, the change of the limit cycle is unconspicuous, and the point (0,0) is thestable convergence point. If the input is large enough, the cycle disappear and thepoint is the unstable point.At first, we validate the point (0,0) using the method of singular point judgment ofnonlinear dynamics. The result indicates that, in normal condition, this point is thestable convergence point, and in certain condition is no longer the focus. Secondly,we construct the Lyapunov function to validate the variation tendency of energy ofthe steering system. Thirdly, in the control experiment of stability control systembased on the limit cycle, the control effect of energy was used to validate therationality of the limit cycle principle. All these experiment educe the result thatwas consistent with the conclusion of this paper.