基于Padé模型降階法的車輛側(cè)傾動(dòng)力學(xué)研究

陳 翔，許 男，郭孔輝

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基于Padé模型降階法的車輛側(cè)傾動(dòng)力學(xué)研究

陳 翔，許 男※，郭孔輝

（吉林大學(xué)汽車仿真與控制國家重點(diǎn)實(shí)驗(yàn)室，長春130025）

為簡化對(duì)車輛側(cè)傾動(dòng)力學(xué)的研究特別是對(duì)車身側(cè)傾角的估算和控制，提出針對(duì)車身側(cè)傾角的Padé模型降階方法。該文首先建立了線性三自由度車輛模型，并在此基礎(chǔ)上推導(dǎo)出了車身側(cè)傾角的傳遞函數(shù)，由于此傳遞函數(shù)的高階特性難以用于車輛側(cè)傾動(dòng)力學(xué)的計(jì)算與控制中，所以接著對(duì)所推導(dǎo)出的側(cè)傾角傳遞函數(shù)進(jìn)行降階處理。又考慮到車身側(cè)傾的低頻特性以及Padé模型降階法在低頻區(qū)能有較好的擬合效果，該文采用Padé模型降階法對(duì)車身側(cè)傾角傳遞函數(shù)進(jìn)行降階處理。為了驗(yàn)證降階后模型的有效性，從時(shí)域、頻域和復(fù)域3個(gè)方面分別進(jìn)行了論證，并通過TruckSim和Matlab進(jìn)行了仿真對(duì)比。結(jié)果表明降階后的模型在低頻區(qū)具有較好的逼近效果，此降階方法可以簡化對(duì)車輛側(cè)傾動(dòng)力學(xué)的研究，并可以將此降階模型應(yīng)用于車身側(cè)傾角的估算和控制中。

車輛；模型；動(dòng)力學(xué)；側(cè)傾角；Padé降階法；時(shí)域；頻域；穩(wěn)定性

0 引 言

隨著農(nóng)業(yè)工程、建筑工程以及交通運(yùn)輸?shù)确矫娴男枰缆飞系霓r(nóng)用工程車、建筑工程車、重型卡車、大巴車以及箱式貨車等越來越多，這些車輛的重心一般都比較高，而由于重心高所導(dǎo)致的車輛側(cè)翻等安全事故也隨之增多[1-3]。所以側(cè)傾動(dòng)力學(xué)對(duì)于這些車輛的重要性是不言而喻的。國內(nèi)外學(xué)者針對(duì)這一問題已經(jīng)提出了一些側(cè)翻預(yù)警算法[4-5]以及防側(cè)翻的主動(dòng)控制策略[6-11]。而要進(jìn)行有效的控制，就需要對(duì)車輛的側(cè)傾角進(jìn)行精確的測量或估算[12]。Ryu等[13]采用GPS和INS組合導(dǎo)航系統(tǒng)配合測量獲得車輛側(cè)傾角和側(cè)偏角的實(shí)時(shí)數(shù)值。趙潤茂等[14]也采用了GPS（global positioning system）和INS（inertial navigation system）組合導(dǎo)航系統(tǒng)對(duì)田間插秧機(jī)進(jìn)行配合測量并預(yù)測了未來1～2 s時(shí)間內(nèi)的行駛姿態(tài)。但是更多的學(xué)者傾向于不增加傳感器，而只在車輛動(dòng)力學(xué)的基礎(chǔ)上運(yùn)用一些數(shù)學(xué)算法獲得車輛的側(cè)傾特性參數(shù)[15-21]。

由于側(cè)傾特性的表達(dá)式比較復(fù)雜，其傳遞函數(shù)屬于高階數(shù)學(xué)模型，這給估算和控制都帶來了困難。因此，本文在國內(nèi)外相關(guān)領(lǐng)域所提出的降階研究方法[22-25]的啟發(fā)下而提出對(duì)車輛側(cè)傾模型進(jìn)行降階計(jì)算的思想。同時(shí)又考慮到車輛側(cè)傾的低頻特性[26]以及Padé模型降階法在低頻區(qū)能有較好的逼近效果這一優(yōu)勢[27-28]。所以本文采用此降階法對(duì)側(cè)傾角傳遞函數(shù)進(jìn)行降階處理，并從時(shí)域、頻域以及復(fù)域等方面對(duì)其有效性進(jìn)行了驗(yàn)證。

1 整車動(dòng)力學(xué)模型

1.1 建模說明

1.2 整車動(dòng)力學(xué)方程

根據(jù)牛頓力學(xué)定律對(duì)整車進(jìn)行動(dòng)力學(xué)建模，并可得到如下數(shù)學(xué)方程式[29-30]

整車側(cè)向方向

整車橫擺方向

車身側(cè)傾方向

前后軸側(cè)向力

式（1）－（4）中m為簧上質(zhì)量，kg；m、m為前后軸簧下質(zhì)量，kg；為車輛行駛速度，m/s；l、l為前后軸到質(zhì)心的距離，m；h為車身質(zhì)心距車輛側(cè)傾中心高度，m；為重力加速度，m/s2；為前軸中心處當(dāng)量轉(zhuǎn)角，(°)；為車輛橫擺率，(°)/s；為橫擺角加速度，(°)/s2；為車輛質(zhì)心側(cè)偏角，(°)；為側(cè)偏角速度，(°)/s；為車身側(cè)傾角，(°)；為側(cè)傾角速度，(°)/s；為側(cè)傾角加速度，(°)/s2；F、F為前后軸側(cè)向力，N；I為簧上質(zhì)量繞軸的轉(zhuǎn)動(dòng)慣量，kg·m2；I為簧上質(zhì)量繞軸的轉(zhuǎn)動(dòng)慣量，kg·m2；I、I為前后軸簧下質(zhì)量繞軸的轉(zhuǎn)動(dòng)慣量，kg·m2；I為簧上質(zhì)量對(duì)平面的慣性積，kg·m2；K、K為前后輪側(cè)偏剛度，N/rad；為車身側(cè)傾角剛度，(N·m/rad)；為車身側(cè)傾角阻尼，N·m·s/rad；下標(biāo)為過自身質(zhì)心的軸且與車輛行駛速度方向同向；下標(biāo)為過自身質(zhì)心的軸且與豎直向上方向同向。

注：lf為前軸到質(zhì)心的距離，m；lr為后軸到質(zhì)心的距離，m；δ為前軸中心處當(dāng)量轉(zhuǎn)角，(°)；δL為前軸左輪轉(zhuǎn)角，(°)；δR為前軸右輪轉(zhuǎn)角，(°)；αf為前軸中心處當(dāng)量側(cè)偏角，(°)；αfL為前軸左輪側(cè)偏角，(°)；αfR為前軸右輪側(cè)偏角，(°)；αr為后軸中心處當(dāng)量側(cè)偏角，(°)；αrL為后軸左輪側(cè)偏角，(°)；αrR為后軸右輪側(cè)偏角，(°)；Bw為輪距，m；ho為車輛側(cè)傾中心距地面高度，m；hb為車身質(zhì)心距車輛側(cè)傾中心高度，m；分別為車輛橫擺率、橫擺角加速度，(°)·s-1、(°)·s-2；分別為車輛質(zhì)心側(cè)偏角、質(zhì)心側(cè)偏角速度，(°)、(°)·s-1；分別為車身側(cè)傾角、側(cè)傾角速度、側(cè)傾角加速度，(°)、(°)·s-1、(°)·s-2；CG(O)為整車質(zhì)心；O'為車輛轉(zhuǎn)向瞬心。

1.3 整車動(dòng)力學(xué)狀態(tài)方程

且有

1.4 側(cè)傾角

由式（5）進(jìn)行拉普拉斯變換可得

進(jìn)而可以求得側(cè)傾角的傳遞函數(shù)表達(dá)式如下

其中

由式（7）可求得穩(wěn)態(tài)側(cè)傾角增益

由式（10）可知，當(dāng)側(cè)傾影響系數(shù)為0時(shí)，穩(wěn)態(tài)側(cè)傾角增益只和車速、側(cè)傾角剛度、車身質(zhì)量、車身質(zhì)心至側(cè)傾中心高度以及軸距相關(guān)，并與車速的平方成正比。

2 Padé模型降階

2.1 Padé模型降階法介紹

降階后的傳遞函數(shù)()的分子分母系數(shù)算法如式（10）～（11）所示

式（12）中的0、1…c為中間過度系數(shù)，它們的值可由式（11）進(jìn)行計(jì)算獲得，=1,2,…,。

2.2 側(cè)傾角降階模型

由式（12）～（14）可知，穩(wěn)態(tài)時(shí)

對(duì)比式（8）和（16），可以發(fā)現(xiàn)降階前后的穩(wěn)態(tài)表達(dá)式并沒有發(fā)生改變，所以降階后的穩(wěn)態(tài)特性完全不變，這也說明了Padé降階法在低頻區(qū)有很好的逼近效果。

3 降階效果驗(yàn)證

Padé降階法已經(jīng)在很多工程領(lǐng)域得到了應(yīng)用并取得了不錯(cuò)的效果。但在車輛側(cè)傾動(dòng)力學(xué)中的應(yīng)用尚屬首次，因此其降階效果還需要進(jìn)一步驗(yàn)證。接下來，本文將從時(shí)域、頻域和復(fù)域等3個(gè)角度對(duì)降階效果做全面的說明。

3.1 時(shí)域驗(yàn)證

本文將通過成熟仿真軟件Trucksim對(duì)所得到的降階模型進(jìn)行驗(yàn)證說明。其中，降階模型的計(jì)算在Matlab軟件中完成。驗(yàn)證所采用的車輛為一前軸轉(zhuǎn)向的兩軸多功能載重箱式貨車。如圖2所示即為具有一定代表性的目標(biāo)仿真車輛，其轉(zhuǎn)向比例系數(shù)=1/25，即為前輪轉(zhuǎn)角與方向盤轉(zhuǎn)角之比。此車輛的其他相關(guān)參數(shù)如表1所示。

表1 車輛仿真參數(shù)

圖2 仿真車輛

時(shí)域驗(yàn)證中選取了典型的角階躍工況和正弦工況進(jìn)行仿真。角階躍工況為：a. 車速=40 km/h，轉(zhuǎn)向盤轉(zhuǎn)角δ=180°，起躍時(shí)間=0.66 s；b. 車速=80 km/h，轉(zhuǎn)向盤轉(zhuǎn)角δ=90°，起躍時(shí)間=0.66 s。正弦工況為：a. 車速=40 km/h，轉(zhuǎn)向盤轉(zhuǎn)角幅值δ=90°，頻率=0.4 Hz；b. 車速=80 km/h，轉(zhuǎn)向盤轉(zhuǎn)角幅值δ=60°，頻率=0.2 Hz。

如圖3所示分別為上述a、b 2種工況的角階躍對(duì)比曲線，如圖4所示分別為上述a、b 2種工況的正弦對(duì)比曲線。從這4幅圖可以看出，車輛在高速和低速以及瞬態(tài)和穩(wěn)態(tài)情況下都能有較好的逼近效果。其中角階躍的穩(wěn)態(tài)時(shí)段的吻合度非常接近，著表明其穩(wěn)態(tài)特性擬合的效果很明顯。對(duì)于瞬態(tài)而言，從4幅圖中可以看出，曲線的走勢也是完全吻合，降階后的系統(tǒng)基本沒有出現(xiàn)延時(shí)或者超前的現(xiàn)象，這說明降階后的系統(tǒng)與原系統(tǒng)具有時(shí)間的一致性。而從圖4可以發(fā)現(xiàn)，工況剛開始時(shí)的側(cè)傾角擬合度稍差，但一個(gè)周期以后其擬合度就會(huì)變穩(wěn)定，這是因?yàn)閯傞_始有靜止突然啟動(dòng)，相當(dāng)于從高頻逐漸過渡到穩(wěn)定的低頻，所以前面時(shí)段的擬合度會(huì)稍差一些。并且隨著車速的提高，其擬合精度也會(huì)有所降低。

注：圖中Trucksim表示軟件仿真，RM表示降階模型，下同。

Note:Trucksim represents simulation by Trucksim, RM denotes reduced model，same as below.

圖3 不同角階躍工況下的側(cè)傾角

Fig.3 Roll angle of different angle step condition

圖4 不同正弦工況下的側(cè)傾角

3.2 頻域驗(yàn)證

為保持分析的一致性，頻域分析中仍然選取低速40 km/h以及高速80 km/h來進(jìn)行說明。

如圖5所示，4幅圖分別是2種速度下的幅頻特性和相頻特性對(duì)比曲線圖，其中橫軸都表示角頻率。圖5a、5b中，當(dāng)角頻率≤6 rad/s時(shí)，降階后系統(tǒng)與原系統(tǒng)的幅頻特性和相頻特性基本重疊；圖5c、5d中，當(dāng)角頻率≤7 rad/s時(shí)，降階后系統(tǒng)與原系統(tǒng)的幅頻特性和相頻特性也基本重疊，這說明降階后的系統(tǒng)具有非常好的低頻逼近特性，基本符合了側(cè)傾動(dòng)力學(xué)的應(yīng)用范圍。而隨著頻率的不斷升高，其幅頻特性和相頻特性的逼近效果也會(huì)不斷降低，以至最后失真。所以此降階方法不適合急速轉(zhuǎn)向工況以及急速側(cè)傾運(yùn)動(dòng)等危險(xiǎn)工況，但是對(duì)于絕大部分的使用工況是完全適用的。

3.3 系統(tǒng)穩(wěn)定性驗(yàn)證

系統(tǒng)降階后是否具有與原系統(tǒng)一致的穩(wěn)定性也是非常重要的。而要說明系統(tǒng)的穩(wěn)定性，一般可在復(fù)域中進(jìn)行論證。

首先，開環(huán)系統(tǒng)的穩(wěn)定性一般由系統(tǒng)的特征根進(jìn)行說明，如圖6所示，1、2、3、4分別為原系統(tǒng)的4個(gè)特征根，1和2分別為降階系統(tǒng)的特征根。此復(fù)平面中的實(shí)軸表示特征根的實(shí)部，而虛軸表示特征根的虛部。可以看出降階后的系統(tǒng)特征根與原系統(tǒng)特征根都處在復(fù)平面的左半平面，即降階后仍然穩(wěn)定；并且降階后的特征根所在區(qū)域與原系統(tǒng)特征根所在區(qū)域基本相同，即降階系統(tǒng)與原系統(tǒng)的相對(duì)穩(wěn)定性也基本一致。

注：圖中OS表示原系統(tǒng)，RS表示降階后的系統(tǒng)，ω為角頻率(rad·s-1),下同。

注：圖中實(shí)軸、虛軸分別表示特征根是實(shí)部與虛部，下同；T1~T4為原系統(tǒng)特征根，R1、R2為降階后的特征根。

閉環(huán)系統(tǒng)的穩(wěn)定性也是考察系統(tǒng)穩(wěn)定的一部分[31]。如圖7所示，分別是車速為40以及80 km/h的Nyquist對(duì)比圖，從兩幅圖可以看出，它們都沒有包圍點(diǎn)（?1,0），即閉環(huán)穩(wěn)定性是一致的。兩圖的形狀也基本一致，并且在低頻區(qū)（趨近于0）達(dá)到了幾乎完全吻合的情況，這不僅說明了降階系統(tǒng)與原系統(tǒng)具有一致的閉環(huán)穩(wěn)定性，還進(jìn)一步驗(yàn)證了系統(tǒng)在低頻區(qū)具有良好的逼近效果。

圖7 不同車速下的Nyquist圖

4 結(jié) 論

1）為簡化對(duì)車輛側(cè)傾特性的研究，首先建立了線性三自由度車輛模型，在此基礎(chǔ)上得出了側(cè)傾角傳遞函數(shù)的表達(dá)式以及穩(wěn)態(tài)側(cè)傾角增益的表達(dá)式。

2）結(jié)合車身側(cè)傾特性一般處于低頻區(qū)以及Padé模型降階法在低頻區(qū)能有較好的擬合效果這兩個(gè)特點(diǎn)，本文采用Padé模型降階法對(duì)車身側(cè)傾角傳遞函數(shù)進(jìn)行了降階計(jì)算。

3）最后系統(tǒng)的從時(shí)域、頻域和復(fù)域3個(gè)方面分析并證明了降階模型與原模型具有良好的逼近效果和一致的穩(wěn)定性。所以此降階法可以用來研究車輛的側(cè)傾特性并可使車輛側(cè)傾角的計(jì)算和控制更加簡便。

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Research on vehicle roll dynamics based on Padé techniques for model reduction

Chen Xiang, Xu Nan※, Guo Konghui

(130025,)

With the continuous development of agricultural engineering, architectural engineering and transportation, more and more technical vehicles for agriculture and architecture, heavy trucks, buses, vans, and so on have run on the road. However, the vehicle rollover accidents caused by high center of gravity have also increased in recent years. In order to study the vehicle roll dynamics and prevent rollover accidents by effective control, it is necessary to calculate the roll angle of vehicle body accurately. In this paper, a linear 3-DOF (degree of freedom) vehicle model was built firstly, then the transfer function of the vehicle body roll angle was deduced from it and the steady-state gain of the roll angle was also obtained. The denominator of the roll angle transfer function is a fourth order expression, and the numerator is a second order expression. High order system is difficult to be calculated and controlled for transient-state, so this paper proposes main effort to reduce the order of the vehicle roll model inspired by the thought of reducing system order from some researchers. Considering that the frequency of the roll dynamics is low generally, and the Padé technique for model reduction has good approximation in low frequency domain, the Padé reduction technique was adopted in this paper. After reducing the order of the roll model, the denominator and numerator of the transfer function came to be a second order expression and a first order expression respectively. Subsequently, the validity of this reduction method was illustrated from 3 aspects, i.e. time domain, frequency domain and complex domain respectively. In the time domain, the original model was simulated by Trucksim, and was also simulated by Matlab. Then, the roll angle of reduced model was compared with that of the original model in the vehicle driving conditions of angle step and sine wave. Simulation results show that, the steady-state gain and the changing tendency both fit well. With the vehicle velocity increasing, the approximation result will get a little bad. It also confirms that the approximation result is good in low frequency and this method is suitable for vehicle roll dynamics. In the frequency domain, the frequency characteristics of amplitude and phase were expatiated. As the Bode diagram shows, the frequency characteristics of the reduced system and the original system coincide almost completely when the angular frequency is not exceeding 6 rad/s. Lastly, the open loop stability and closed loop stability were also analyzed. The root locus diagram and the Nyquist stability criterion were used to elaborate the open loop stability and closed loop stability respectively. The results reveal that, the open loop stability and closed loop stability of the reduced system are consistent with that of the original system, and the relative consistency of the 2 systems also fits well. So the conclusion is that: This reduction method is suitable for calculating the vehicle roll angle and this reduction model approximates to the original model very well in low frequency domain which is the frequently-used working condition of vehicle. Additionally, in order to study the steady-state characteristics of the roll dynamics, the steady-state roll angle expression was also deduced and the roll influence coefficient was proposed in this paper. This coefficient is relative with the distance between front axle and mass center, the distance between rear axle and mass center, the distance from vehicle body mass center to roll axle, vehicle body mass, front axle mass, rear axle mass, front tire sideslip stiffness and rear tire sideslip stiffness. The special case was also analyzed when the roll influence coefficient is equal to 0. This provides a theoretical basis for studying the steady-state characteristics of the roll dynamics.

vehicles; models; kinetics; roll angle; Padé reduction techniques; time domain; frequency domain; stability

10.11975/j.issn.1002-6819.2017.17.012

U270.1+1

A

1002-6819(2017)-17-0091-07

2017-04-13

2017-08-30

中國汽車產(chǎn)業(yè)創(chuàng)新發(fā)展聯(lián)合基金資助（U1564213）

陳翔，主要從事車輛動(dòng)力學(xué)仿真與控制。長春吉林大學(xué)汽車仿真與控制國家重點(diǎn)實(shí)驗(yàn)室，130025。Email：jluchenxiang@163.com

許男，講師，主要從事車輛系統(tǒng)動(dòng)力學(xué)與控制。長春吉林大學(xué)汽車仿真與控制國家重點(diǎn)實(shí)驗(yàn)室，130025。Email：xunan@jlu.edu.cn