基于高斯原理的非理想系统动力学建模$^{\bf 1)}$

图1 沿地面滑行的刚杆

Fig. 1 A rigid bar that slides along the ground

4.1.1 杆件运动学与动力学方程

杆的动力学方程可以表示为

(30)$\begin{eqnarray} \left. \begin{array}{l} m\ddot{{x}}=F_{\rm t} \\ m\ddot{{y}}=F_{\rm n} -mg \\ \dfrac{1}{3}ml^2\ddot{{\theta }}=-F_{\rm n} l\cos \theta +F_{\rm t} l\sin \theta \\ \end{array} \right\} \end{eqnarray}$

单边约束方程为

$g_{\rm N} =y_A =y-l\sin \theta \geqslant 0 $

根据运动学关系

$\ddot{{y}}=\ddot{{y}}_A +l\ddot{{\theta }}\cos \theta -l\dot{{\theta }}^2\sin \theta $

可以得到接触点法向加速度和法向接触力之间的关系

(31)$\begin{eqnarray}\ddot{{y}}_A =A(\theta ,\mu )F_{\rm n} +B(\theta ,\dot{{\theta }})\end{eqnarray}$

其中

$A=\frac{1}{m}[1+3\cos \theta (\cos \theta -\mu \sin \theta )]\\ B=l\dot{{\theta }}^2\sin \theta -g $

同时可以计算得到方程(29)中的矩阵

$M=\left[ {{\begin{array}{c} {m} & \\ & m & \\ && {\frac{1}{3}ml^2} \\ \end{array} }} \right], Q=\left[ {{\begin{array}{c} 0 \\ {-mg} \\ 0 \\ \end{array} }} \right]\\ {W}_{\rm N} =\left[ 0 \ 1\ {-l\cos\theta } \right] \ \ {W}_{\rm t} =\left[ 1 \ 0 \ {l\sin\theta } \right] \\ {s}=[1], \ \ {U}=\left[ \mu \right] $

则

${A}={W}_{\rm N} M^{-1}(W_{\rm N}^{\rm T} +{W}_{\rm t}^{\rm T} {sU})= [m^{-1}(1+3\cos^2\theta -3\mu\sin \theta \cos \theta )]=\left[A \right]\\ {B}={W}_{\rm N} M^{-1}{Q}+{\bar{{W}}}_{\rm N} =\left[ l\sin \theta \dot{\theta}^2-g\right]=\left[ B \right] $

可见, 通过矩阵计算的结果与通过运动学及动力学方程计算的结果相吻合.

下面将通过两种方法来分析本问题中含摩擦的单边约束问题.

4.1.2 接触模型及对比分析

(1)单边接触的线性互补模型 (linear complementary problem, LCP)

当$y=0$且$\dot{{y}}=0$时, 应有

(32)$\begin{eqnarray}F_{\rm n} \geqslant 0,\ \ \ddot{{y}}_A \geqslant 0,\ \ F_{\rm n} \cdot \ddot{{y}}_A =0\end{eqnarray}$

即

(33)$\begin{eqnarray}F_{\rm n} \geqslant 0,\ \ AF_{\rm n} +B\geqslant 0,\ \ F_{\rm n}(AF_{\rm n} +B)=0\end{eqnarray}$

由线性互补性问题的解的唯一性判别方法, 当$A>0$时, 不管$B$为何值, 上述关于法向约束力的LCP问题均有唯一解存在.

(2) 单边接触的高斯优化模型

由式(29)所给出的极值原理计算可知: 对于本问题, 当 $A>0$时, 总会有

(34)$F_{\rm n} =\mbox{arg}\mathop {\min }\limits_{\lambda } \left( \frac{1}{2}(\lambda)^{\rm T}A{\lambda}|{A\lambda +B\geqslant 0}\right) = \mbox{arg}\mathop {\min }\limits_{\lambda } \left( \frac{1}{2}A{\lambda }^2| A\lambda +B\geqslant 0\right) $

可见, 当$A>0$时, 关于法向约束力的线性互补模型及高斯优化模型同时对应系统的唯一解.

(3) 两种模型之间的对比分析

① 由高斯优化模型可以推导得出LCP模型

根据式(34)所代表的约束最优化问题, 其可行域为$\{\lambda|A\lambda +B\geqslant 0\}$, 设其非空, 则实际法向约束力$F_{\rm n} $为函数$f(\lambda )=(\lambda)^{\rm T}A{\lambda}/2$极小值点的K-T条件是

(35)$\begin{eqnarray} \lambda \geqslant 0,A\lambda +B\geqslant 0, \lambda (A\lambda +B)=0, \lambda(A\lambda +B)=0\ \ \ \end{eqnarray}$

式(35)与直接建立的LCP模型(33)吻合.

② 由LCP模型可推出高斯优化模型

LCP模型与二次型优化问题紧密相连, 式(33)所代表的LCP模型可转化为二次型优化问题

(36)$\begin{eqnarray}\mathop{\mbox{min}}\limits_{F_{\rm n} } \frac{1}{2}AF_{\rm n}^2 +BF_{\rm n},\ \ F_{\rm n} \geqslant 0\end{eqnarray}$

最优化问题(36)的对偶问题为

(37)$\begin{eqnarray}\mathop {\mbox{min}}\limits_{F_{\rm n} } \frac{1}{2}AF_{\rm n}^2,\ \ AF_{\rm n} +B\geqslant 0\end{eqnarray}$

将符号$F_{\rm n} $由$\lambda $代替, 则得到了与高斯优化模型式(34)吻合的优化模型.

由此可见, 本文中得到的高斯优化模型与LCP模型是分别通过不同的方法得到的, 但同时又可以根据最优化理论相互推导得到, 因此可以有效证明两种方法的可靠性.

4.2 双杆系统的多点接触问题

如图2所示两根相同的均质杆$AB$和$BC$, 长为$l$, 质量为$m$, 在端点$B$用光滑铰链连接, 且受到外力$F$作用, $A$和$C$两端放在非光滑的水平面上, 杆与地面间动滑动摩擦系数为$\mu $.

图2

图2 沿地面滑行的双刚杆

Fig. 2 Two-bar system that slides along the ground

4.2.1 多点接触系统高斯优化模型

系统的广义坐标矢量可取为${q}=[x\ \ y \ \ \theta _1\ \ \theta _2]^{\rm T}$, 其中$(x,y)$为$AB$杆的质心坐标, $\theta_{\rm 1}$为$AB$杆与水平线的夹角, $\theta _2 $为$BC$杆与铅直线的夹角, 则$A$点与$C$点的单边接触方程为

(38)$\begin{eqnarray} \left.\begin{array}{l} g_{{\rm N}A} =y_A =y-\dfrac{l}{2}\sin \theta _1 \geqslant 0\\ g_{{\rm N}C} =y_C =y+\dfrac{l}{2}\sin \theta _1 -l\cos \theta _2 \geqslant 0 \\ \end{array}\right\} \end{eqnarray}$

则可以计算得到优化模型(29)中的各矩阵项

$M= {{\begin{bmatrix} {2m} & 0 & {-\dfrac{1}{\mbox{2}}ml\sin \theta _1 } & {\dfrac{1}{\mbox{2}}ml\cos \theta _2 }\\ 0 & {2m} & {\dfrac{1}{\mbox{2}}ml\cos \theta _1 } & {\dfrac{1}{\mbox{2}}ml\sin \theta _2 }\\ {-\dfrac{1}{\mbox{2}}ml\sin \theta _1 } & {\dfrac{1}{\mbox{2}}ml\cos \theta _1 } & {\dfrac{1}{3}ml^2} & {-\dfrac{1}{\mbox{4}}ml^2\sin (\theta _1 -\theta _2 )}\\ {\dfrac{1}{\mbox{2}}ml\cos \theta _2 } & {\dfrac{1}{\mbox{2}}ml\sin \theta _2 } & {-\dfrac{1}{\mbox{4}}ml^2\sin (\theta _1 -\theta _2 )} & {\dfrac{1}{3}ml^2} \\ \end{bmatrix} }} $

$Q= {{\begin{bmatrix} 0 & F & {\dfrac{1}{2}Fl\cos \theta _1 } & 0 \\ \end{bmatrix} }} ^{\rm T} \\ {\bar{{W}}}_{\rm N} = {{\begin{bmatrix} {\dfrac{l}{2}\cos \theta _1 \dot{{\theta }}_1^2 +\dfrac{l}{2}\sin \theta _2 \dot{{\theta }}_2^2 } \\ {-\dfrac{l}{2}\sin \theta _1 \dot{{\theta }}_1^2 +\dfrac{l}{2}\cos \theta _2 \dot{{\theta }}_2^2 } \\ \end{bmatrix} }} \\ {W}_{\rm t} = {{\begin{bmatrix} 1 & 0 & {\dfrac{l}{2}\mbox{sin}\theta _1 } & 0 \\ 1 & 0 & {-\dfrac{l}{2}\mbox{sin}\theta _1 } & {l\mbox{cos}\theta _2 } \end{bmatrix} }} \\ {W}_{\rm N} = {{\begin{bmatrix} 0 & l & {-\dfrac{l}{2}\cos \theta _1 } & 0 \\ 1 & 0 & {\dfrac{l}{2}\mbox{sin}\theta _1 } & {l\sin \theta _2 } \\ \end{bmatrix} }} \\ {U}= {{\begin{bmatrix}{c} \mu & \\ & \mu \\ \end{bmatrix} }} ,\ \ \ {S}= {{\begin{bmatrix}{c} {sgn(\dot{{x}}_A )} & \\ & {sgn(\dot{{x}}_{\rm C} )} \\ \end{bmatrix} }} $

则将上述矩阵代入, 便可以确定关于法向约束力矢量的优化模型

(39)$\begin{eqnarray}{\lambda}=\mbox{arg}\min \left[ \frac{1}{2}(\lambda^*)^{\rm T}{A\lambda }^* | A\lambda ^* +B\geqslant 0 \right]\end{eqnarray}$

其中, ${A}={W}_{\rm N} M^{-1}(W_{\rm N}^{\rm T} +{W}_{\rm t}^{\rm T} {sU})$; ${B}={W}_{\rm N} M^{-1}{Q}+{\bar{{W}}}_{\rm N}$.

4.2.2 基于优化模型的动力学的计算过程及数值模拟

算法流程如下:

(1)输入初始条件${q}_0$, ${\dot{{q}}}_0 $, 积分区间为$[0,T]$, 步长$\Delta t$以及系统的物理、几何参数;

(2)计算优化模型(39)中的各系数矩阵;

(3)计算法向约束力矢量${\lambda }$:

解决上述优化问题(39)的计算方法很多, 本例选择了比较成熟的传统优化算法: 牛顿法来计算法向约束力矢量^[18].

(4) 计算广义加速度矢量${\ddot{{q}}}$:

采用下列公式计算广义加速度矢量

(40)\begin{eqnarray}{\ddot{{q}}}=M^{-1}{Q}+M^{-1}{W}_{\rm N}^{\rm T} {\lambda }+M^{-1}{W}_{\rm t}^{\rm T} {SU\lambda }\end{eqnarray}

(5)根据步骤4所得到的广义加速度, 积分得到下一时刻广义坐标的速度和位移: $\dot{{q}}_{t+\Delta t} $, $q_{t+\Delta t} $;

(6)判断此时$t$是否满足终止条件$t\leqslant T$:

(a)如果满足, 程序终止并输出;

(b)如果不满足, 令$t=t+\Delta t$ 转到步骤2继续运行.

依据上述算法流程, 选取下列参数及初始条件对系统进行动力学数值模拟

$m=2 \mbox{kg}, \ \ l=2 \mbox{m},\ \ \mu =0.2;\\ \theta _1 =\theta _2 =\frac{\pi}{4},\ \ x=y=l\sin \frac{\pi }{4} \mbox{m},\\ \dot{{x}}=-1 \mbox{m}/\mbox{s},\ \ \dot{{y}}=1 \mbox{m}/\mbox{s},\ \ \dot{{\theta }}_2 =0.1 {\mbox{rad}}/\mbox{s},\\ \dot{{\theta }}_1 =-0.1 {\mbox{rad}}/\mbox{s}. $

图3$\sim\!$图5的动力学的数值模拟结果所显示出系统的运动特性及约束力特点与物理规律相吻合, 可以展示出采用本文中的高斯优化模型并结合优化计算方法来解决含摩擦的刚体系统的动力学问题的可行性及有效性.

图3

图3 杆$AB$的质心坐标的时间历程图

Fig. 3 Time history of barycentric coordinates of bar $AB$

图4

图4 杆$AB$的转角的时间历程图

Fig. 4 Time history of and rotation angles of bar $AB$

图5

图5 接触点的法向约束力的时间历程图

Fig. 5 Time history diagram of normal binding force of contact point

5 结语

本文在从基本的动力学原理——高斯原理出发, 研究了高斯原理在非理想系统中的应用, 推导并证明了非理想系统中高斯拘束的函数表达式, 从而得到了以函数极值形式表达的非理想多体系统的优化模型. 该工作的意义在于:

(1) 借助于高斯原理, 非光滑的多体系统动力学的求解过程可以从求解含互补条件的微分代数方程变为自然寻优的过程, 其寻优的过程非借助于互补性及最优化理论之间的关系, 而是使得目标函数的选择更加符合自然哲学的思想;

(2)因为原理本身所具有的极值形式可以将非理想系统的动力学问题引入求解优化问题的框架内, 从而可以尝试探索现代智能优化算法在非理想多体系统中的应用; 同时对于带控制的多体系统, 可以根据技术需要将动力学分析与系统的优化结合进行.

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广义坐标形式的高斯最小拘束原理及其推广

力学与实践, 2014,36(6):779-785

当采用广义坐标描述系统的运动时,相比质点形式的高斯最小拘束原理,通过广义坐标形式的高斯最小拘束原理来建立动力学优化模型,计算效率更高. 从高斯原理的变分形式出发推导了广义坐标形式的高斯最小拘束原理,并研究了非理想约束、单边约束及刚体碰撞情形下的高斯最小拘束原理的形式. 研究认为:对刚体碰撞情形下,高斯最小拘束原理不能取代碰撞恢复定律,碰撞恢复定律以碰撞后广义速度的约束方程形式起作用.

( Yao

Wenli

, Dai

Baoqing

Gauss principle of least constraint in generalized coordinates and its generalization

Mechanics in Engineering, 2014,36(6):779-785 (in Chinese))

[29]

姚文莉

基于广义坐标形式的高斯最小拘束原理的多刚体系统动力学建模

北京大学学报(自然科学版), 2016,52(4):708-712

 By variational Gauss principle in the form of kinetic energy and explicit express, the meaning of items in Gauss constraints is confirmed. The optimization model of dynamics of multi-rigid-body system in the form of Descartes generalized coordinates is established based on Gauss principle of least constraint in generalized coordinates. The method to write the Gauss constraints in other coordinate system according to the above model is studied. The dynamical problem of multi-rigid-body system can be turned into a problem for a minimum, and provided by the relation between the Descartes generalized coordinates and other coordinates system, the Gauss constraint in this kind of coordinates system is easy to be obtained. The modeling method is more convenient and is of universality. The dynamical models are established for planar motion and fixed axis rotation by Descartes generalized coordinate and Lagrange coordinate, and the validity of this method is proved.

( Yao

Wenli

Dynamical modeling of multi-rigid-body system based on Gauss principle of least constraint in generalized coordinates

Acta Scientiarum Naturalium Universitatis Pekinensis, 2016,52(4):708-712 (in Chinese))

[30]

杨流松, 姚文莉, 岳嵘.

优化形式的刚体系统动力学模拟

力学与实践, 2015,37(4):488-491

基于广义坐标形式的高斯最小拘束原理来研究刚体系统的动力学问题的优化方法. 相比目前动力学建模普遍采用的质点形式的高斯最小拘束原理,广义坐标形式的高斯最小拘束原理因对所选择的广义坐标没有要求,而使得建模过程更简单及具有更强的通用性. 本文分别建立了有约束和无约束条件下问题的优化动力学模型,对问题进行了动力学数值模拟,并与用拉格朗日微分方程处理的模型进行了对比分析,从而验证了所提方法的有效性.

( Yang

Liusong

, Yao

Wenli

, Yue

Rong

Simulation of body dynamics with optimization

Mechanics in Engineering, 2015,37(4):488-491 (in Chinese))