Modulated Impulsive Forces

Consider the rod sliding on a rough surface in Fig. 7. In case of Coulomb friction, the friction force F_f depends on the normal force F_N by a constant coefficient $\mu$, i.e., $F_f = \mu \vert F_N\vert$ [6], active in the direction opposite to v_A,x, the velocity of the contact point at the surface. F_y is the kinetic force exerted by the center of mass in the vertical direction. Combined with the gravitational force, F_g, this yields the normal force F_N = F_y + F_g

  

Figure 7: Coulomb friction.

Let the variables be defined as in Fig. 7, and v_x be the horizontal velocity of the center of mass, M, v_y the vertical velocity, and $\omega$ the angular velocity. Fig. 7 shows that the linear velocities relate to the angular velocity as

$$\begin{eqnarray} \html{eqn88}v_x = -l \omega sin \theta \\ v_y = l \omega cos \theta\end{eqnarray}$$

The system has three inertial, energy storing, components, i.e., the linear inertias m_x and m_y and the rotational inertia J. In this example, their values are all unity.

In case of sliding motion, point A is in contact with the floor, so v_A,y = 0, which requires $v_y = -l \omega cos \theta$.This poses an algebraic constraint on v_y and $\omega$, which may cause impulses governed by conservation principles between the two when initially this constraint is not satisfied. As a result the friction force that is related to $\dot{v}_y$ by $F_f = \mu \vert F_N\vert$may be of an impulsive nature. This impulse propagates into $\omega$ because of the $\dot{\omega} = -l sin \theta F_f + l cos \theta F_N$ constraint, and, therefore, causes exogeneous impulsive forces to interact with impulses due to conserved quantities, and these forces are modulated by the change of value of these quantities. Refer to [7] for a detailed analysis.

According to the previous discussion, the system equations can be compiled as $${\small \left.\begin{array} {c} \left[\begin{array} {cccccc}... ... \\ 0 \\ 0 \\ -{F}_g \\ 0\end{array}\right]\end{array}\right. }$$
In this system of equations, $$\dot{\bar{x}} = \left[\begin{array} {c} \dot{v}_{y} \\ \dot{... ...r{x} = \left[\begin{array} {c} v_y \\ \omega\end{array}\right]$$
$$\dot{\hat{x}} = \left[\begin{array} {c} \dot{v}_{x}\end{arra... ...ight]; \hat{x} = \left[\begin{array} {c} v_x\end{array}\right]$$
The remaining equations in h yield $$\hat{y} = \left[\begin{array} {c} F_y \\ F_f \\ F_N\end{array}\right]$$

$$F' = \left[\begin{array} {ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 ... ...ay} {ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$$
and $$\hat{H}' = \left[\begin{array} {ccc} 0 & 1 & -\mu \\ -1 & 0... ...& 0 & 0 \\ 0 & 0 & 0 \\ 0 & -1 & -l cos\theta\end{array}\right]$$
The pseude-inverse of F' is $$F'^+ = \left[\begin{array} {ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ ... ...eta}{l cos\theta} & 0 & \frac{1}{l cos\theta}\end{array}\right]$$
Therefore, $$\hat{H} F'^+ F'' = \left[\begin{array} {ccc} 1 - \mu \frac{... ...ta} & -1 & \frac{1}{l cos\theta} \\ 0 & 0 & 0\end{array}\right]$$
results in

$$\begin{eqnarray} \html{eqn106}(1 - \mu \frac{sin\theta}{cos\theta})(v_x - v_x^0)... ...x^0) -(v_y - v_y^0) + \frac{1}{l cos\theta}(\omega - \omega^0) = 0\end{eqnarray}$$

which can be solved in combination with $\bar{H} \bar{x} = 0$,$$v_y + l cos\theta \omega = 0$$
to give the state projection

$$\begin{eqnarray} \html{eqn110}\omega = \frac{-l(cos\theta - \mu sin \theta) v_y^... ..._y^0 + \omega^0)} {1 + l^2 cos\theta ( cos\theta - \mu sin\theta)}\end{eqnarray}$$

This is conform the topological analysis in [7]. Note that the change of $\omega$ and v_y due to the modulated impulsive Coulomb friction force, $F_y = \mu \vert F_N\vert$, factors in correctly.