# 平面库伦摩擦系统的动力学分析Dynamic Analysis of Planar Coulomb Friction System

DOI: 10.12677/AAM.2021.104143, PDF, HTML, XML, 下载: 8  浏览: 23

Abstract: The existence of Coulomb friction in mechanical system leads to the existence of discontinuity. In this paper, the global dynamic behavior of an object moving on a horizontal plane under Coulomb friction is studied. In this case, the points on the sliding mode domain of the discontinuous system are all pseudo-equilibrium points of the system. On this basis, by constructing the Poincaré mapping, the conclusion that the set of points in the sliding mode domain is the global attractor of the system is given. Finally, the conclusion that the attractor is convergent in finite time is proved by numerical simulation.

1. 引言

2. 系统数学模型的建立

Figure 1. Coulomb friction system

$m\stackrel{¨}{x}+p\left(\stackrel{˙}{x}\right)+kx=0$ (1.1)

$p\left(\stackrel{˙}{x}\right)=\left\{\begin{array}{l}{p}_{0},\text{}\stackrel{˙}{x}>0\hfill \\ -{p}_{0},\text{}\stackrel{˙}{x}<0\hfill \end{array}$ (1.2)

${p}_{0}\ne 0$ 时，由于系统(1.1)~(1.2)是一个右端不连续的微分系统，当考虑到系统(1.1)在Filippov意义下的解，不难知道系统(1.1)的解是存在的，且满足初值条件 $x\left(0\right)={x}_{0},y\left(0\right)={y}_{0},$ 存在区间为 $\left[0,+\infty \right)$，并且系统的解是右唯一的。为了后续的方便讨论，将系统(1.1)~(1.2)化为等价的平面微分系统，令 $\stackrel{˙}{x}=y$

$\left\{\begin{array}{l}\stackrel{˙}{x}=y\hfill \\ \stackrel{˙}{y}=-\frac{kx}{m}-\frac{p\left(y\right)}{m}\hfill \end{array}$ (1.3)

$p\left(y\right)=\left\{\begin{array}{l}{p}_{0},\text{}y>0\hfill \\ -{p}_{0},\text{}y<0\hfill \end{array}$ (1.4)

$\left\{\begin{array}{l}\stackrel{˙}{x}=y\hfill \\ \stackrel{˙}{y}=-\frac{kx}{m}-\frac{{p}_{0}}{m},\text{}y>0\hfill \end{array}$ (1.5)

$\left\{\begin{array}{l}\stackrel{˙}{x}=y\hfill \\ \stackrel{˙}{y}=-\frac{kx}{m}+\frac{{p}_{0}}{m},\text{}y<0\hfill \end{array}$ (1.6)

$m{y}^{2}+k{\left(x+\frac{{p}_{0}}{k}\right)}^{2}={c}_{1}$ (1.7)

$m{y}^{2}+k{\left(x-\frac{{p}_{0}}{k}\right)}^{2}={c}_{2}$ (1.8)

3. 切换线上滑模动力学行为

Figure 2. Vector field distribution

${\Sigma }_{{C}_{1}}=\left\{\left(x,y\right)|-{A}_{1}

${\Sigma }_{s}=\left\{\left(x,y\right)|-\frac{{p}_{0}}{k}

${\Sigma }_{{C}_{2}}=\left\{\left(x,y\right)|\frac{{p}_{0}}{k}

$\left\{\begin{array}{c}\stackrel{˙}{x}=y,\\ \stackrel{˙}{y}=0,\end{array}$ (2.1)

4. 系统的全局动力学行为

${P}_{L}\left({x}_{0}\right)=\left\{\begin{array}{l}{x}_{1}^{L},{x}_{0}\in \left(\frac{{p}_{0}}{k},{A}_{2}\right]\hfill \\ {x}_{0},{x}_{0}=\left[-{A}_{1},\frac{{p}_{0}}{k}\right]\hfill \end{array}$

${P}_{U}\left({x}_{0}\right)=\left\{\begin{array}{l}{x}_{1}^{U},{x}_{0}\in \left[-{A}_{1},-\frac{{p}_{0}}{k}\right)\hfill \\ {x}_{0},{x}_{0}=\left[\frac{{p}_{0}}{k},-{A}_{2}\right]\hfill \end{array}$

${x}_{1}^{U}={P}_{U}\left({x}_{0}\right)=-{x}_{0}-\frac{2{p}_{0}}{k},\text{}{x}_{0}\in \left[-{A}_{1},-\frac{{p}_{0}}{k}\right)$ (3.1)

${x}_{1}^{L}={P}_{L}\left({x}_{0}\right)=-{x}_{0}+\frac{2{p}_{0}}{k},\text{}{x}_{0}\in \left(\frac{{p}_{0}}{k},{A}_{2}\right]$ (3.2)

$P\left({x}_{0}\right)={P}_{U}\left({P}_{L}\left({x}_{0}\right)\right)={x}_{0}-\frac{4{p}_{0}}{k},\text{}{x}_{0}\in \left[\frac{3{p}_{0}}{k},{A}_{2}\right]$ (3.3)

$P\left({x}_{0}\right)\in {\Sigma }_{S},\text{}这里{P}^{N}=\underset{N}{\underset{︸}{P\circ P\circ \cdots \circ P}}$

5. 数值仿真及应用分析

$k=1,\text{}m=1,\text{}{p}_{0}=\frac{1}{2}$ 时，系统的滑模域区间为 $\left[-0.5,0.5\right]$，从图3中可以看出，在有限时间内从任何点出发的轨线，最后都收敛到 $\left[-0.5,0.5\right]$ 区间的滑模域上。

Figure 3. Trajectory phase diagram of plane Coulomb friction system

6. 结论

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