Chapter 5.4, Problem 35E

Sticky Friction. An alternative for the damping friction model $F=-b{y}^{\prime }$ discussed in Section 4.1 is the “sticky friction” model. For a mass sliding on a surface as depicted in Figure 5.18, the contact friction is more complicated than simply $-b{y}^{\prime }$. We observe, for example, that even if the mass is displaced slightly off the equilibrium location $y=0$, it may nonetheless remain stationary due to the fact that the spring force $-ky$ is insufficient to break the static friction’s grip. If the maximum force that the friction can exert is denoted by $\mu$, then a feasible model is given by

Figure $5.18$Mass-spring with friction

$F_{\text{friction}}=\{\begin{array}{l}ky,\text{if}\left|ky\right|<\mu \text{and }{y}^{\prime }=0,\\ \mu \text{sign}\left(y\right)\text{, }\text{if}\left|ky\right|\ge \mu \text{and }{y}^{\prime }=0,\\ -\mu \text{sign}\left(y^{\prime }\right)\text{, if }{y}^{\prime }\ne 0.\end{array}$

(The function sign(s) is $+1$ when $s>0,-1$ when $s<0$, and $0$ when $s=0$.) The motion is governed by the equation

$m\frac{d^{2}y}{d{t}^2}=-ky+F_{\text{friction}}(16)$

Thus, if the mass is at rest, friction balances the spring force if $\left|y\right|<\mu /k$ but simply opposes it with intensity $\mu$ if $\left|y\right|\ge \mu /k$. If the mass is moving, friction opposes the velocity with the same intensity $\mu$.

a. Taking $m=\mu =k=1$, convert (16) into the first-order system

$y^{\prime }=v(17)$,

$v^{\prime }=\{\begin{array}{l}0,\text{ if}\left|y\right|<1\text{and }v=0,\\ -y+\text{sign}\left(y\right)\text{, if}\left|y\right|\ge 1\text{and }v=0,\\ -y-\text{sign}\left(v\right)\text{, if }v\ne 0.\end{array}$

b. Form the phase plane equation for (17) when $v\ne 0$ and solve it to derive the solutions

$v^2+{\left(y\pm 1\right)}^2=c,$

where the plus sign prevails for $v>0$ and the minus sign for $v<0$.

c. Identify the trajectories in the phase plane as two families of concentric semicircles. What is the center of the semicircles in the upper half-plane? The lower half-plane?

d. What are the critical points for (17)?

e. Sketch the trajectory in the phase plane of the mass released from rest at $y=7.5$. At what value for y does the mass come to rest?