﻿ 车辆纵向非光滑多体动力学建模与数值算法研究
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 力学学报  2015, Vol. 47 Issue (2): 301-309  DOI: 10.6052/0459-1879-14-323 0

### 引用本文 [复制中英文]

[复制中文]
Fan Xinxiu, Wang Qi. RESEARCH ON MODELING AND SIMULATION OF LONGITUDINAL VEHICLE DYNAMICS BASED ON NON-SMOOTH DYNAMICS OF MULTIBODY SYSTEMS[J]. Chinese Journal of Ship Research, 2015, 47(2): 301-309. DOI: 10.6052/0459-1879-14-323.
[复制英文]

### 文章历史

2014-10-22 收稿
2014-11-28 录用
2015–01–06 网络版发表

1 系统的描述及其动力学方程

 图 1 车辆纵向多体系统力学模型 Fig. 1 Model of the longitudinal multi-body system of the vehicle

 ${{ \varPhi }}({{ q}}) = [f_1 ,f_2]^{\rm T} = 0$ (1)

 ${{ g}}_N ({{ q}}) = \left[{g_{N1} ,g_{N2} } \right]^{\rm T} \geqslant 0$ (2)

 $\left.\begin{array}{l} M\ddot{ q}= Q+\varPhi^{\rm T}_q\lambda+ W_N\lambda_N+ Q^{\rm f}\\ \varPhi( q)=[f_1,f_2]^{\rm T}= 0\\ g_N=[g_{N1},g_{N2}]^{\rm T}\geqslant 0\\ \end{array}\right\}$ (3)

 $F_{\rm fi} = \left\{ \begin{array}{ll} - \mu _i F_{Ni} {\rm Sgn}(v_{ri} ),v_{ri} \ne 0 \\ - \mu _{0i} F_{Ni} {\rm Sgn}(\dot{v}_{ri} ),v_{ri} = 0 \\ \end{array} \right.$ (4a)

 $M_{\rm fi} = \left\{ \begin{array}{ll} - \delta _i F_{Ni} {\rm Sgn}(\dot{\theta }_{ci} ),& \dot{\theta }_{ci} \ne 0 \\ - \delta _i F_{Ni} {\rm Sgn}(\ddot{\theta }_{ci} ),& \dot{\theta }_{ci} = 0 \\ \end{array} \right.$ (4b)

 ${\rm{Sgn}}(x) = \left\{ {\begin{array}{*{20}{l}} {1,}&{x > 0}\\ {[- 1,1],}&{x = 0}\\ { - 1,x}&{lt;0} \end{array}} \right.$

$\mu _i$和$\mu _{0i}$分别为车轮与路面之间的动、静摩擦系数; $\delta _i$为车轮与路面间的滚阻系数. $v_{ri}$与$\dot{v}_{ri}$ 分别为车轮与路面接触点的速度和切向加速度. $\dot{\theta }_{ci}$与$\ddot{\theta }_{ci}$分别为车轮的角速度和角加速度.

 $F_{Ni} = \lambda _{Ni}\ \ (i = 1,2)$ (5)

 $\left[{{\begin{array}{*{20}c} {Q_{x_{ci} }^{\rm f} } \\ {Q_{y_{ci} }^{\rm f} } \\ {Q_{\theta _{ci} }^{\rm f} } \\ \end{array} }} \right] = \left[{{\begin{array}{*{20}c} {\dfrac{\partial g_{Ni} }{\partial y_{ci} }} & 0 \\[3mm] { - \dfrac{\partial g_{Ni} }{\partial x_{ci} }} & 0 \\ {R_i } & 1 \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {F_{fi} } \\ {M_{fi} } \\ \end{array} }} \right] \ \ (i = 1,2)$

 $Q^{\rm f} = W_{{T}} {{ \lambda }}_{{T}}$ (6)

 $\begin{array}{l} {Q^{\rm{f}}} = {[Q_{{x_1}}^{\rm{f}}\;Q_{{y_1}}^{\rm{f}}\;Q_{{\theta _1}}^{\rm{f}}\;Q_{{x_{c1}}}^{\rm{f}}\;Q_{{y_{c1}}}^{\rm{f}}\;Q_{{\theta _{c1}}}^{\rm{f}}\;Q_{{x_{c2}}}^{\rm{f}}\;Q_{{y_{c2}}}^{\rm{f}}\;Q_{{\theta _{c2}}}^{\rm{f}}\;Q_{{\theta _0}}^{\rm{f}}]^{\rm{T}}}\\ {W_T} = {\left[{\begin{array}{*{20}{c}} 0&0&0&1&0&{ - R}&0&0&0&0\\ 0&0&0&0&0&{ - 1}&0&0&0&0\\ 0&0&0&0&0&0&1&0&{ - R}&0\\ 0&0&0&0&0&0&0&0&{ - 1}&0 \end{array}} \right]^{\rm{T}}}\\ {\lambda _T} = {[\begin{array}{*{20}{c}} {{F_{{\rm{f1}}}}}&{{M_{{\rm{f1}}}}}&{{F_{{\rm{f2}}}}}&{{M_{{\rm{f2}}}}} \end{array}]^{\rm{T}}} \end{array}$

 $\label{eq3} M\ddot{ q} = Q + \varPhi _{q}^{\rm T} \lambda + W_{N} \lambda_{{N}} + {{ W}}_{T} {{ \lambda }}_{{T}}$ (7)

2 动力学方程的算法

2.1 互补条件

(1)与摩擦和滚阻相关物理量的互补条件: 静滑动摩擦力的正向与负向摩擦余量分别定义为[14]

 $\label{eq4} \left. {\begin{array}{l} F_{fi}^ + = \mu _{0i} F_{Ni} + F_{fi} \\ F_{fi}^ - = \mu _{0i} F_{Ni} - F_{fi} \\ \end{array}} \right\}$ (8)

 $\left. {\begin{array}{*{20}{l}} {\dot v_{ri}^ + = \frac{{(\left| {{{\dot v}_{ri}}} \right| + {{\dot v}_{ri}})}}{2}}\\ {\dot v_{ri}^ - = \frac{{(\left| {{{\dot v}_{ri}}} \right| - {{\dot v}_{ri}})}}{2}} \end{array}} \right\}$ (9)

 $\left. {\begin{array}{*{20}{l}} {M_{fi}^ + = {\delta _i}{F_{Ni}} + {M_{fi}}}\\ {M_{fi}^ - = {\delta _i}{F_{Ni}} - {M_{fi}}} \end{array}} \right\}$ (10)

 $\left. {\begin{array}{*{20}{l}} {\ddot \theta _{ci}^ + = \frac{{(\left| {{{\ddot \theta }_{ci}}} \right| + {{\ddot \theta }_{ci}})}}{2}}\\ {\ddot \theta _{ci}^ - = \frac{{(\left| {{{\ddot \theta }_{ci}}} \right| - {{\ddot \theta }_{ci}})}}{2}} \end{array}} \right\}$ (11)

 $\left. {\begin{array}{*{20}{l}} {\lambda _{Hi}^ + = \mu _i^ * {\lambda _{Ni}} + {\lambda _{Hi}}}\\ {\lambda _{Hi}^ - = \mu _i^ * {\lambda _{Ni}} - {\lambda _{Hi}}} \end{array}} \right\}$ (12)

 $\label{eq5} \left. {\begin{array}{l} \ddot{g}_{Hi}^ + = \dfrac{(\left| {\ddot{g}_{Hi} } \right| + \ddot{g}_{Hi} )}{2} \\[3mm] \ddot{g}_{Hi}^ - = \dfrac{(\left| {\ddot{g}_{Hi} } \right| - \ddot{g}_{Hi} )}{2} \\ \end{array}} \right\}$ (13)

 $\label{eq6} \left. {\begin{array}{l} \ddot{g}_{Hi}^ + \geqslant 0,\ \ \lambda _{Hi}^ + \geqslant 0,\ \ \ddot{g}_{Hi}^ + \lambda _{Hi}^ + = 0 \\ \ddot{g}_{Hi}^ - \geqslant 0,\ \ \lambda _{Hi}^ - \geqslant 0,\ \ \ddot{g}_{Hi}^ - \lambda _{Hi}^ - = 0 \\ \end{array}} \right\}$ (14)

(2)接触定律的互补条件

 $\label{eq7} \ddot{g}_{Ni} \geqslant 0,\lambda _{Ni} \geqslant 0,\ \ \ddot{g}_{Ni} \lambda _{Ni} = 0$ (15)

2.2 动力学方程算法

 $\begin{array}{l} {Q^{\rm{f}}} = {W_T}{\lambda _T} = \\ \;\;\;\;{W_T}S\left[ {\begin{array}{*{20}{c}} { - {\mu _{01}}{\rm{Sgn}}({{\ddot g}_{T1}})}&{amp;}\\ { - {\delta _1}{\rm{Sgn}}({{\ddot g}_{T2}})}&{amp;}\\ {}&{amp; - {\mu _{02}}{\rm{Sgn}}({{\ddot g}_{T3}})}\\ {}&{amp; - {\delta _2}{\rm{Sgn}}({{\ddot g}_{T4}})} \end{array}} \right]{\lambda _N} + \\ \;\;\;{W_T}(E - S)\left[ {\begin{array}{*{20}{c}} { - {\mu _1}{\rm{Sgn}}({{\dot g}_{T1}})}&{amp;}\\ { - {\delta _1}{\rm{Sgn}}({{\dot g}_{T2}})}&{amp;}\\ {}&{amp; - {\mu _2}{\rm{Sgn}}({{\dot g}_{T3}})}\\ {}&{amp; - {\delta _2}{\rm{Sgn}}({{\dot g}_{T4}})} \end{array}} \right]{\lambda _N} \end{array}$

 $\begin{array}{l} S = {\rm{diag}}[1 - \left| {{\rm{Sgn}}\left( {{{\dot g}_{T1}}} \right)} \right|,1 - \left| {{\rm{Sgn}}\left( {{{\dot g}_{T2}}} \right)} \right|,1 - \left| {{\rm{Sgn}}\left( {{{\dot g}_{T3}}} \right)} \right|,\\ \quad 1 - \left| {{\rm{Sgn}}\left( {{{\dot g}_{T4}}} \right)} \right|] \end{array}$

 $\begin{array}{l} {Q^{\rm{f}}} = {W_T}S\left[ {\begin{array}{*{20}{c}} {\lambda _{H1}^ + - {\mu _{01}} \cdot {\lambda _{N1}}}\\ {\lambda _{H2}^ + - {\delta _1} \cdot {\lambda _{N1}}}\\ {\lambda _{H3}^ + - {\mu _{02}} \cdot {\lambda _{N2}}}\\ {\lambda _{H4}^ + - {\delta _2} \cdot {\lambda _{N2}}} \end{array}} \right] + \\ \quad {W_T}(E - S)\left[ {\begin{array}{*{20}{c}} { - {\mu _1}{\rm{Sgn}}({{\dot g}_{T1}})}&{amp;0}\\ { - {\delta _1}{\rm{Sgn}}({{\dot g}_{T2}})}&{amp;0}\\ 0&{amp; - {\mu _2}{\rm{Sgn}}({{\dot g}_{T3}})}\\ 0&{amp; - {\delta _2}{\rm{Sgn}}({{\dot g}_{T4}})} \end{array}} \right]\\ {\lambda _N} = \quad {W_H}\lambda _H^ + + {W_T}\left\{ {S\left[ {\begin{array}{*{20}{c}} { - {\mu _{01}}}&{amp;{\rm{0}}}\\ { - {\delta _1}}&{amp;{\rm{0}}}\\ {\rm{0}}&{amp; - {\mu _{02}}}\\ {\rm{0}}&{amp; - {\delta _2}} \end{array}} \right] + } \right.\\ \left. {\quad (E - S)\left[ {\begin{array}{*{20}{c}} { - {\mu _1}{\rm{Sgn}}({{\dot g}_{T1}})}&{amp;0}\\ { - {\delta _1}{\rm{Sgn}}({{\dot g}_{T2}})}&{amp;0}\\ 0&{amp; - {\mu _2}{\rm{Sgn}}({{\dot g}_{T3}})}\\ 0&{amp; - {\delta _2}{\rm{Sgn}}({{\dot g}_{T4}})} \end{array}} \right]} \right\}{\lambda _N} = \\ \quad {W_H}\lambda _H^ + - {W_T}\mu {\lambda _N} \end{array}$

 ${{ \mu }} = {{ S}}\left[{{\begin{array}{*{20}c} {\mu _{01} } & {\rm 0} \\ {\delta _1 } & {\rm 0} \\ {\rm 0} & {\mu _{02} } \\ {\rm 0} & {\delta _2 } \\ \end{array} }} \right]{\rm + }({{ E - S}})\left[ {{\begin{array}{*{20}c} {\mu _1 {\rm sgn}(\dot{g}_{T1} )} & 0 \\ {\delta _1 {\rm sgn}(\dot{g}_{T2} )} & 0 \\ 0 & {\mu _2 {\rm sgn}(\dot{g}_{T3} )} \\ 0 & {\delta _2 {\rm sgn}(\dot{g}_{T4} )} \\ \end{array} }} \right]$

 $\label{eq8} M\ddot{ q} = Q + \varPhi_{q}^{\rm T} \lambda + W_{N} {{ \lambda }}_{N} + {{ W}}_{H} {{ \lambda }}_{H}^{+} - {{ W}}_{T} {{ \mu \lambda }}_{N}$ (16)

 $\label{eq9} {{ \varPhi }}_q {{\ddot{ q}}} + {{\dot{ \varPhi }}}_q {{\dot{ q}}} + \alpha {{ \varPhi }}_q {{\dot{ q}}} + \beta{{ \varPhi }} = 0$ (17)

 $\begin{array}{l} \ddot q = {M^{ - 1}}Q + {M^{ - 1}}\Phi _q^{\rm{T}}\lambda + {M^{ - 1}}({W_N} - {W_T}\mu ){\lambda _N} + \\ \quad {M^{ - 1}}{W_H}\lambda _H^ + \end{array}$ (18)

 $\begin{array}{l} {{ \lambda }} = {{ D \varPhi }}_q {{ M}}^{ - 1}({{ W}}_{N} - {{ W}}_T {{ \mu }}){{ \lambda }}_{N} + {{ D \varPhi }}_q {{ M}}^{ - 1}{{ W}}_H {{ \lambda }}_{H}^+ + {{ A}} \\ {{ D}} = [- {{ \varPhi}}_q {{ M}}^{ - 1}{{ \varPhi }}_{q}^{\rm T}]^{ - 1} \\ {{ A}}^\ast = {{ \varPhi }}_q {{ M}}^{ - 1} Q + \dot{ \varPhi}_q {{\dot{ q}}} + \alpha {{ \varPhi }}_q {{\dot{ q}}} + \beta {{ \varPhi }} \\ {{ A}} = {{ D A}}^\ast \\ \end{array}$

 $\label{eq10} \ddot{ q} = M^{ - 1}{{ Q}} + {{ M}}^{ - 1}{{\tilde{ W}}}_N {{ \lambda }}_{N} + M^{ -1}{{\tilde{ W}}}_H {{ \lambda }}_{H}^{+} + {{ M}}^{ - 1}{{ \varPhi }}_{q}^{\rm T} {{ A}}$ (19)

 $\begin{array}{l} {{\tilde{ W}}}_N = {{ \varPhi }}_{q}^{\rm T} {{ D \varPhi }}_q {{ M}}^{ - 1}({{ W}}_{N} - {{ W}}_T {{ \mu }}) +{{ W}}_{N} - {{ W}}_T {{ \mu }} \\ {{\tilde{ W}}}_H = {{ \varPhi }}_{q}^{\rm T} {{ D \varPhi }}_q {{ M}}^{ - 1}{{ W}}_H + {{ W}}_H\\ \end{array}$

 $\label{eq11} {{\ddot{\pmb g}}}_N = {{ W}}_N^{\rm T} {{\ddot{ q}}}$ (20)

 $\label{eq12} {{\ddot{\pmb g}}}_N = {{ W}}_N^{\rm T} {{ M}}^{ - 1}{{\tilde{ W}}}_N {{ \lambda }}_{N} + {{ W}}_N^{\rm T} {{ M}}^{ - 1}{{\tilde{ W}}}_H {{ \lambda }}_{H}^{+} + {{ C}}$ (21)

 $\label{eq13} {{\ddot{\pmb g}}}_T = {{ W}}_T^{\rm T} {{\ddot{ q}}}$ (22)

 $\label{eq14} {{\ddot{\pmb g}}}_H = {{ W}}_H^{\rm T} {{\ddot{ q}}}$ (23)

 $\label{eq15} {{\ddot{\pmb g}}}_{H }^ + - {{\ddot{\pmb g}}}_{H }^ - = {{ W}}_H^{\rm T} {{ M}}^{ - 1}{{\tilde{ W}}}_N {{ \lambda }}_{N} + {{ W}}_H^{\rm T} {{ M}}^{ - 1}{{\tilde{ W}}}_H {{ \lambda }}_{H}^{+} + {{ L}}$ (24)

 $\label{eq16} {{ \lambda }}_{H}^{{ + }} + {{ \lambda }}_{H}^ - = 2{{ \mu }}_H {{ \lambda }}_{N}$ (25)

 ${{ \mu }}_{H} = ({{ \mu }}_0 {{ B}})^{\rm T} \,,\ \ \ \ {{ \mu }}_0 = \left[ {{\begin{array}{*{20}c} {\mu _{01} } & {\delta _1 } & { 0} \\ { 0} & {\mu _{02} } & {\delta _2 } \\ \end{array} }} \right]$

 $\label{eq17} \left[{{\begin{array}{*{20}c} {\ddot{\pmb g}_N } \\ {\ddot{\pmb g}_{H}^{{ + }} } \\ {{{ \lambda }}_H^ - } \\ \end{array} }} \right] = \left[{{\begin{array}{*{20}c} {{{ W}}_N^{\rm T} {{ M}}^{ - 1}{{\tilde{ W}}}_N } & {{{ W}}_N^{\rm T} {{ M}}^{ - 1}{{\tilde{ W}}}_H } & 0 \\ {{{ W}}_H^{\rm T} {{ M}}^{ - 1}{{\tilde{ W}}}_N } & {{{ W}}_H^{\rm T} {{ M}}^{ - 1}{{\tilde{ W}}}_H } & E \\ {2{{ \mu }}_H } & { - E} & 0 \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {{{ \lambda }}_N } \\ {{{ \lambda }}_{H}^{{ + }} } \\ {\ddot{\pmb g}_H^ - } \\ \end{array} }} \right] + \left[{{\begin{array}{*{20}c} C \\ L \\ 0 \\ \end{array} }} \right]$ (26)

 $\begin{array}{l} [\begin{array}{*{20}c} {\ddot{\pmb g}_N } & {\ddot{\pmb g}_H^ + } & {{{ \lambda }}_{H}^{{ - }} } \\ \end{array}]^{\rm T} \geqslant 0,\ \ {}[{{\begin{array}{*{20}c} {{{ \lambda }}_N } & {{{ \lambda }}_{H}^{{ + }} } & {\ddot{\pmb g}_H^ - } \\ \end{array} }}]^{\rm T} \geqslant 0 \\ {} [\begin{array}{*{20}c} {\ddot{\pmb g}_N } & {\ddot{\pmb g}_H^ + } & {{{ \lambda }}_{H}^{{ - }} } \\ \end{array}][\begin{array}{*{20}c} {{{ \lambda }}_N } & {{{ \lambda }}_{H}^{{ + }} } & {\ddot{\pmb g}_H^ - } \\ \end{array}]^{\rm T} = 0\\ \end{array}$

3 数值仿真

 图 2 车辆传动系统扭转振动力学模型 Fig. 2 Model of torsional vibration of the vehicle

 图 3 $\theta _{1}$的时间历程 Fig. 3 Time history of $\theta _{1}$
 图 4 $V_{\rm r1}$和$F_{\rm f1}$的时间历程 Fig. 4 Time history of $V_{\rm r1}$ and $F_{\rm f1}$
 图 5 $V_{\rm r2}$和$F_{\rm f2}$的时间历程 Fig. 5 Time history of $V_{\rm r2}$ and $F_{\rm f2}$
 图 6 $\ddot{ y}_{{\rm c1}}$和$F_{{\rm N1}}$的时间历程 Fig. 6 Time history of $\ddot{ y}_{\rm c1}$ and $F_{{\rm N1}}$
 图 7 $\ddot{ y}_{{\rm c2}}$和$F_{{\rm N2}}$的时间历程 Fig. 7 Time history of $\ddot{ y}_{{\rm c2}}$ and $F_{{\rm N2}}$

 图 8 $T_{\rm f}$的时间历程 Fig. 8 Time history of $T_{\rm f}$
 图 9 $V_{\rm r1}$和$F_{\rm f1}$的时间历程 Fig. 9 Time history of $V_{\rm r1}$ and $F_{\rm f1}$
 图 10 $V_{\rm r2}$和$F_{\rm f2}$的时间历程 Fig. 10 Time history of $V_{\rm r2}$ and $F_{\rm f2}$

 图 11 $T_{\rm f}$的时间历程 Fig. 11 Time history of $T_{\rm f}$
 图 12 $\ddot{\theta }_1$的时间历程 Fig. 12 Time history of $\ddot{\theta }_1$
 图 13 $F_{{\rm N2}}$的时间历程 Fig. 13 Time history of $F_{{\rm N2}}$
 图 14 $V_{\rm r2}$和$F_{\rm f2}$的时间历程 Fig. 14 Time history of $V_{\rm r2}$ and $F_{\rm f2}$

 图 15 约束方程二次范数$\left\| {{ \varPhi }} \right\|$的时间历程 Fig. 15 Time history of $\left\| {{ \varPhi }} \right\|$
4 结 论

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RESEARCH ON MODELING AND SIMULATION OF LONGITUDINAL VEHICLE DYNAMICS BASED ON NON-SMOOTH DYNAMICS OF MULTIBODY SYSTEMS
Fan Xinxiu, Wang Qi
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Fund: The project was supported by the National Natural Science Foundation of China (11372018)
Abstract: A method for modeling and simulation of longitudinal vehicle dynamics is presented based on non-smooth dynamics of multibody systems in this paper. The vehicle is the multi-rigid-body system which consists of a vehicle body, wheels and transmission system. It is assumed that wheels are linked with the vehicle body by shock absorbers and the transmission system is treated as the disk connecting with driving wheels by spring-damper system. The friction and the rolling resistance between wheels and the road are taken into account and the Coulomb's friction law is used to describe the frictional forces. Firstly, the lateral and bilateral constraint equations of the system are given in generalized coordinates of the system and the dynamical equations of the system are obtained by Lagrange's equations of the first kind. Secondly, the complementary formulations of friction law, rolling resistance law and contact law between wheels and road are given in order to determine state transitions from contact to separation and sticking to slipping. Based on the event-driven scheme, the problem of state transitions is formulated and solved as a horizontal linear complementarity problem (HLCP). The algorithm for solving these non-smooth DAEs is presented. The Baumgarte stabilization method is used to reduce the constraint drift. Finally, the multibody system of a vehicle is considered as an illustrative application example to analyse its dynamical behaviour affected by the engine output torque, the friction coeffcient and the rolling resistance coeffcient between wheels and road. The numerical results show that the phenomenon of the stick-slip between driving wheel and road occurs continually when the coeffcients have special values.
Key words: longitudinal dynamics of the vehicle    rolling resistance    Coulomb friction    linear complementarity    stick-slip