尚 进
论文题目：利用试验模态参数对轮胎侧偏特性的建模研究
作者简介：尚 进，男，1972年01月出生，1996年09月师从于清华大学管迪华教授，于2000年01月获博士学位。
摘 要
轮胎力学特性的建模，是车辆动力学研究的重点和难点。这是由于轮胎与地面相互的作用力，几乎是控制汽车运动唯一（除作用于车身的空气动力）因素，而轮胎的结构、材质和工况极其复杂。八十年代中期著名的轮胎力学大师们认为很难预测轮胎力学状态的突破。自三十年代轮胎力学产生以来，近半个多世纪在应用上占主导地位的模型，包括近来的著名的魔方模型（Magic Formula），当属经验的。为预测轮胎力学特性和深入了解其产生机理，基于不同假设基础上的物理和分析模型，如弦和梁模型等对轮胎偏离特性的描述，虽然在一定程度上促进了对轮胎特性形成的了解，但由于这些模型的参数缺乏物理基础，仍然是在大量的与工况相关的试验基础上提炼的，难以得到长足的发展和广泛应用。近一、二十年来应现代汽车产品模拟开发技术的要求发展了建立在模态分析基础上的弹性圈模型。本文对轮胎动力学建模研究的方法为直接利用试验模态参数的建模。它们均属当前轮胎模型发展的前沿。直接模态参数模型在参数测取上最大限度地减少了人为因素，更具科学的严谨性，也不受弹性圈只描述面内特性的限制，在增加侧向激振模态试验参数的基础上就可直接建立侧偏特性模型，这在国际上尚属首次，成为本论文的选题。轮胎的偏离特性与整车参数的匹配，是控制汽车操稳性的决定性因素，成为轮胎力学主要组成部分。利用模态参数直接建模方法的体系适应于轮胎所有特性的建模，使得本研究不仅可以给出好的建模结果，而且其中所探讨的方法对发展本系统的建模，具有直接参考价值，甚至对圈模型的发展也有参考价值。
轮胎侧偏特性的建模，除径、切激振所获取模态参数外，还需侧向激振提取的模态参数。第二章的试验除提取了200Hz以下的自由悬置轮胎的侧向模态参数外，通过试验和理论分析重点讨论了轮辋约束系统的特性对轮胎模态参数的影响。发现并指出此前的所有轮胎模型（包括理论模型）均无法分离约束系统（试验台）对它们带来的影响，这是因为它们建模所依赖的参数均是来自于试验台的经验数据。第二届（1997）国际轮胎力学会议已发现不同试验台测取的同一轮胎特性会有相当大的差异，这应当说与此不无关系，本研究提出了科学有效的解决此问题的方法。
论文的第三章利用轮胎稳态侧偏印迹变形方程及轮胎模态参数（传递矩阵），推导出稳态侧偏特性模型。该模型不仅可预测不同载荷下轮胎的侧偏特性，侧向力到偏离角，回正力矩以及汽胎拖距到偏离角的特性曲线，而且可以计算出不同工况下，印迹上的侧向力分布和相应的胎体、胎面的侧向变形，从而深入了解特性形成的机理，这是以往任何模型所不及的。此外,在Pacejka著名弦理论模型及其它分析模型中，描述轮胎迟滞特性的松弛长度是由轮胎试验得出，作为模型参数出现的，而本模型可以直接将其计算得出，也显示了本模型的科学性和优越性。对以上特性随载荷、侧偏角、地面摩擦系数变化的计算分析与以往大量的试验研究结果相符，并纯理论地首次发现松弛长度随载荷和偏离角变化的规律与近期发表的基于试验的研究结果相同（1997），而不是以往一直沿用的它是印迹半长的3倍。特别应当指出的是如以计算结果为试验数据，用著名的稳态经验模型Magic Formula 拟合，有很好的符合。建立偏离特性需要的基本参数有印迹长度及垂直压力分布，这些参数应从利用本方法建立的滚动模型计算得出，但本研究两者是并行进行的。在建偏离模型初期，滚动模型尚无最后结果的情况下，印迹长度取自垂直模型，而压力分布参考了郭孔辉院士的四次多项式，但最终直接运用了第五章滚动模型所得印迹参数结果，使模型达到了高度的解析化，并使结果与Magic Formula符合更好。
在第四章对轮胎非稳态建模中，考虑到轮胎旋转过程的加载，为了便于模型的推导，作者首次提出了导出传递函数的概念，并在非旋转轮胎模型参数的基础上，推导出导出传递函数的计算式。基于轮胎非稳态侧偏印迹变形方程及导出传递函数矩阵，推导出了轮胎非稳态侧偏模型。对京轮6.50R16轮胎的计算结果与试验结果有很好的一致性。由于试验条件所限，对非稳态偏离特性随载荷变化影响的分析是在两种同一量级的不同轮胎上进行比较的，两者完全定性一致。以上结果说明了建模的成功。计算还给出了不同速度下，高频带内的频响特性（速度高达150km/h，频率达到100Hz）。4.6节中专门考察了速度对动特性的影响，特别是引入导出传递函数带来的影响。分析结果表明高速下，引入导出传递函数是必须的。分析结果与仅能找到的高速试验的文献结果相比，定性一致，应当说本研究所提出的导出函数对求旋转体的加载响应均有参考价值。由于所建模型为非线性模型，又要进行离散化的数值计算，由所得到的频域模型，以FFT逆变换得到时域模拟模型是不现实的。本文在4.5节，直接从求胎体在侧向位移和摆动角输入下的时域响应，推导出了轮胎侧偏时域仿真模型，对京轮6.50R16轮胎的一例计算与试验结果有很好的符合，这也说明了建模方法的正确性。本文的建模研究虽然本着先易后难的原则，对稳态和非稳态先后分别进行的，但在4.8节中专门对本方法对稳态和非稳态建模和模型的统一性进行了论证，即建模的步骤和方法的统一、参数测取的统一、结构形式的统一，最后表现在稳态模型即是动态模型在零频的结果，无须对它们分开进行建模。这是以往建模方法不具备的，当然,非稳态模型在零频时应具有稳态数值是所有动态模型都应具备的。稳态和非稳态模型高度的统一，是本建模方法研究的重要优越性，具有发展前景。
滚动轮胎的印迹参数（印迹长度及压力分布）是侧偏特性建模的基础。以往建模方法，它们均出自经验规律和数据。本研究的第三、四章，虽然印迹长度是出自垂直模型的计算，但压力分布还是借助了郭孔辉教授的E指数模型的经验公式及数据。为使所研究模型完全的解析化，第五章利用模态参数建立了稳态滚动模型，利用粘性阻尼和结构阻尼分别计算了滚动阻力特性。结构阻尼下的滚动阻力特性与实际符合较好，与文献公认的轮胎具结构阻尼的结论也一致，但建立在圈模型基础上的滚动模型（1997，Savkoor）没能给出合理的滚动阻力特性。它在结构阻尼时，滚动阻力不随速度变化，这是圈模型阻尼环节设置不合理的原因。以滚动模型计算出的印迹参数作为侧偏特性计算的基础，所得结果与Magic Formula经验公式有更好的符合性。第六章对全文的工作作了全面的总结，指出进一步应研究发展的工作和本研究所探讨方法使用的一般性。
关键词：轮胎侧偏模型，试验模态参数，导出传递函数，滚动速度效应，
轮辋约束
Study
on Tire Cornering Properties by Using
Experimental Modal Parameters
ABSTRACT
The modeling of
tire mechanical properties is of importance in field of vehicle dynamics, but
very difficult. That’s because the interaction between tire and ground seizes
nearly unique control of vehicle motion (besides the air force applied on
vehicle body), at the same time, the tire material property, structure, and
operation condition etc is quite complex. In such a case, it isn’t strange that
the middle of 1980’s sees an unpredictable period in the progress of the art of
tire mechanics even among the well-known researchers.
Tire mechanics has been in study since 30’s. But what’s dominant in the application of tire model for nearly half century is the empirical tire model, including the famous Magic Formula model. Based on various of hypothesis, there also come forth other physically analytical models to predict tire mechanical property in certain aspect and to explore the corresponding generation mechanism, one of such models as string model or beam model for description of tire cornering property. Although they help to promote understanding of tire mechanical properties to some extent, there is no more space left for their further development and application since the model parameters have to be obtained from plenty of experiment data which depends on operation condition, but not directly from physical property of material and tire structure geometry. In the last 10-20 years, driven by the pressing requirement of vehicle dynamics simulation technique, REF (Ring on Elastic Foundation) model based on modal tests has aroused more and more attention and developed quickly. While in this dissertation, tire structure is modeled directly with modal parameters. The two methods just mentioned stand abreast at the front edge of tire mechanics research. However, the latter modeling scheme transcends the limitation of REF method in describing only tire in-plane properties, thus it can be easily applied to model tire cornering property provided lateral modal parameters under lateral excitation tests, which embodies superiority of the modeling approach to other ones. The main content of the dissertation concerns modeling tire cornering property by the direct modal approach, which is carried out for the first time. As is known, matching level between tire lateral properties and vehicle parameters influence the handling of vehicle determinately, and consequentially modeling tire lateral properties is the most pivotal subject of tire mechanics. The direct modal parameters modeling method still abates the artificial influence to a great extent, so as to provide more accurate and convincible parameters. It’s still advantageous that the method can be adapted to descript all kinds of tire properties. Based on the present research, it’s affirmed that the method not only works out good results up to the facts, but also is valuable to systemize the tire modeling approach, even can provide reference for the further development of REF model.
In modeling of tire lateral properties, it’s required to extract modal parameters through lateral excitation test, besides what obtained by radial and tangential excitation tests. In chapter 2, the lateral modal parameters below 200Hz are thus extracted by the lateral excitation tests of a free-suspended tire. Then, based on the results, primary effect of wheel constraints on tire modal parameters has been discussed. It’s discovered and indicated definitely that it’s the fact that all of the previous tire models, including the theoretic ones, are built on parameters from empirical data on rig tests makes them unavailable to get rid of the influence of constrain of test rig on themselves. In the second international tire mechanics conference, there points out the great difference among tire characteristics of the same tire obtained under different test rigs, which must present a certain relation to constrain effect. An effective resolution is provided hereafter for the problem about the influence of tire constrains.
In chapter 3, steady tire cornering model is derived using footprint deformation equation together with tire modal parameters (transfer matrices). The model can predict tire lateral properties under different load and characteristic curve of side force, aligning torque or pneumatic trial vs. slip angle, and besides, work out side force distribution in the footprint and deformation of carcass and tread, also make a contribution to explore generation mechanism of these characteristics. Furthermore, the model derived here can work out the relaxation length, while within Pacejka’s famous string model or other tire models it is generally obtained from experiment and used as a known parameter for model input. The fact reveals again the superiority of the model. The dependent relation of tire properties on load, slip angle, or ground friction coefficient, calculated through the model, agrees well with the experiment results before. What’s more, it’s disclosed for the first time in theory that relationship of relaxation length following load and slip angle shouldn’t easily expressed by three times of half footprint length, though which is commonly used, but do appear consistent with the experiment results publicized in 1997. What makes the model especially attractive is that when substitute part of the tire characteristic parameters calculated from the direct modal model into the Magic Formula, the results from the direct modal model appear consistent well with the approximation results from Magic Formula.
In order to get the lateral properties, the cornering model must take footprint length and vertical pressure distribution as its basic inputs, which are just outputs from the tire rolling model also built on by the same method as for cornering model. However, the difficulty is, that these two models corresponding to different tire properties have to be performed simultaneously. Thus, in the beginning of modeling when no definite result from rolling model can be used, footprint length is temporarily gotten from vertical model, and the pressure distribution is approximated with Guo Konghui’s 4th order polynomials. Then for final modeling, footprint length and pressure distribution from rolling model in chapter 5 will be taken in. Thus final model becomes completely analytical, and its calculated results accord well with those from the Magic Formula.
In chapter 4, nonsteady tire model is established and studied. Considering that tire is loaded in process of its rotation, at the same time, in order to derive the model conveniently, the author defines first the derived transfer function and expresses it with static tire modal parameters in order to get the final cornering model. Then using footprint deformation equation in nonsteady state and the derived transfer function, the nonsteady cornering model can be obtained. The simulation result for Jinglun 6.50R16 with the above derived model accords well with test data. Due to limited experiment condition, the analysis of cornering property vs. load is just performed for two types of tires which are within the same weigh level, and their results are consistent qualitatively too. These facts illustrate the success of the modeling approach in the dissertation. Frequency response within high frequency band under different velocity is also worked out (wherein velocity reaches as high as 150Km/h, and frequency around 100Hz). Effect of velocity on dynamic performance is especially discussed in section 4.6, where effect of derived transfer function has to be included since analysis results indicate the necessity. The result is consistent with what’s found in literature for description of test result of tire cornering property in high velocity. It can be seen that it’s advantageous to use the derived transfer function to get response of a loaded rotation body. Due to nonlinear of the nonsteady model as well as the discretization for numerical computation, it’s not available to get corresponding time domain model just through inverse Fourier transformation of the frequency domain model. In section 4.5, the author resolves straightly the time domain response under inputs of lateral displacement and swing angle. Then the final cornering model in time domain is also applied to Jinglun 6.50R16 and its calculation result is compared with experiment results. Their perfect agreement validates the modeling approach once again.
Although the steady and nonsteady model are discussed separately, with relatively easy steady model established first, these two models can be unified together, which is discussed particularly in section 4.8, for the following aspects as similar modeling approach and process, similar parameters measurement, similar formula structure, and finally the equivalence of steady model to nonsteady model in state of zero frequency. Certainly, the above equivalence relation should hold for every dynamic model. Therefore, it’s not necessary to model under steady and nonsteady states. The high unitarity of the modeling approach embodies its advancement again.
When establishing cornering model, the footprint characteristic parameters (consist of footprint length and pressure distribution) must be input. Generally, they are resulted from empirical approach or test data. While in chapter 3 and 4, footprint length is gotten from the vertical model, and pressure distribution is approximated with reference to Guo’s empirical E-exponent model. In order to make the tire model completely analytical, the author also establishes the steady rolling model using modal parameters. Using the rolling model, the rolling resistance force corresponding to viscous damp and structural damp can be worked out respectively. The rolling resistance force corresponding to structural damp gives perfect up-to-fact answer, which agree well with accepted conclusion from literatures. However, it’s not the case if using rolling model based on REF model (1997, Savkoor), and that will result in resistance force independent of rolling speed. That’s because of the defect of damp element setting in the model. When substitute the footprint parameters calculated from rolling model as known inputs into the cornering model, similar results to magic formula will be obtained.
Chapter 6 outlines all work in the dissertation. In addition, continued progress is envisioned and general adaptation of the modeling method is specified.
Key words: tire cornering model, modal test parameters, derived transfer function
rolling speed effect, spindle constrain
