Nonsmooth spatial frictional contact dynamics of multibody systems

Abstract

Nonsmooth dynamics algorithms have been widely used to solve the problems of frictional contact dynamics of multibody systems. The linear complementary problems (LCP) based algorithms have been proved to be very effective for the planar problems of frictional contact dynamics. For the spatial problems of frictional contact dynamics, however, the nonlinear complementary problems (NCP) based algorithms usually achieve more accurate results even though the LCP based algorithms can evaluate the friction force and the relative tangential velocity approximately. In this paper, a new computation methodology is proposed to simulate the nonsmooth spatial frictional contact dynamics of multibody systems. Without approximating the friction cone, the cone complementary problems (CCP) theory is used to describe the spatial frictional continuous contact problems such that the spatial friction force can be evaluated accurately. A prediction term is introduced to make the established CCP model be applicable to the cases at high sliding speed. To improve the convergence rate of Newton iterations, the velocity variation of the nonsmooth dynamics equations is decomposed into the smooth velocities and nonsmooth (jump) velocities. The smooth velocities are computed by using the generalized-$\mathbf{a}$ algorithm, and the nonsmooth velocities are integrated via the implicit Euler algorithm. The accelerated projected gradient descend (APGD) algorithm is used to solve the CCP. Finally, four numerical examples are given to validate the proposed computation methodology.

Appendix

1.1 A.1 Dynamic equation of the rigid sliding rod

As shown in Fig. 31, a rigid rod is subject to oblique sliding under the action of gravity. The length of the rod is $l$, $\mu$ denotes the friction coefficient. According to the normal and tangential contact forces shown in Fig. 31, it is straightforward to establish the dynamic equations of the rod as follows:

$$\begin{aligned} &F_{1} + \mu F_{2} - mg - m\ddot{y} = 0, \\ \end{aligned}$$

(38)

$$\begin{aligned} &F_{2} - \mu F_{1} - m\ddot{x} = 0, \\ \end{aligned}$$

(39)

$$\begin{aligned} &\frac{l}{2} \bigl[ ( \mu F_{1} + F_{2} ) \sin \theta + ( \mu F_{2} - F_{1} )\cos \theta \bigr] + r \bigl[ ( \mu F_{1} - F_{2} )\cos \theta + ( \mu F_{2} + F_{1} )\sin \theta \bigr] = J\ddot{\theta }, \end{aligned}$$

(40)

where $r$ denotes the cross-sectional radius of the rod and $J$ is the moment of inertia of the rod, satisfying

$$ J = \frac{ml^{2}}{12} + \frac{mr^{2}}{4}. $$

(41)

The coordinates of the rod mass center yield

$$ \left \{ \textstyle\begin{array}{l} x = \frac{l}{2}\cos \theta + r\sin \theta \\ y = \frac{l}{2}\sin \theta + r\cos \theta \end{array}\displaystyle \right . $$

(42)

Fig. 31

Sliding oblique rigid rod model

Substituting Eqs. (41), (42), (38) and (39) into Eq. (40) gives the dynamic equation of the sliding rod as follows:

$$\begin{aligned} &mg \biggl[ \mu l\sin \theta + \frac{1}{2} \bigl( \mu ^{2} - 1 \bigr)l\cos \theta + \bigl( \mu ^{2} + 1 \bigr)r\sin \theta \biggr] \\ &\qquad{}- m \biggl[ \frac{1}{2}\mu l^{2} - rl\cos ( 2 \theta ) + \mu rl\sin ( 2\theta ) \biggr]\dot{\theta }^{2} \\ &\quad= \biggl[ \bigl( 1 + \mu ^{2} \bigr)J + \frac{1}{4} \bigl( 1 - \mu ^{2} \bigr)ml^{2} - mrl\sin ( 2\theta ) + \bigl( \mu ^{2} + 1 \bigr)mr^{2} \biggr]\ddot{\theta } \end{aligned}$$

(43)

Given the initial condition of the rod as $\theta = \pi /4, \dot{\theta } = 0$, Fig. 13 and Fig. 14, show the numerical solutions of Eq. (43). According to the results, the normal contact force $F_{2}$ becomes negative after the moment 0.271 s when the rod gets separated from the vertical wall.