Chapter 7, Problem 50RE

Coulomb Friction Revlslted In Problem 27 in Chapter 5 in Review we examined a spring/mass system in which a mass m slides over a dry horizontal surface whose coefficient of kinetic friction is a constant μ. The constant retarding force f_k = μmg of the dry surface that acts opposite to the direction of motion is called Coulomb friction after the French physicist Charles-Augustin de Coulomb (1736–1806). You were asked to show that the piecewise-linear differential equation for the displacement x(t) of the mass is given by

$m\frac{d^{2}x}{d{t}^2}+kx=\{\begin{array}{l}{f}_k\text{, }{x}^{\prime }<0(motiontoleft)\\ -f_k\text{, }{x}^{\prime }>0(motiontoright).\end{array}$

1. (a) Suppose that the mass is released from rest from a point x(0) = x₀ > 0 and that there are no other external forces. Then the differential equations describing the motion of the mass m are

  x″ + ω²x = F, 0 < t < T/2

  x″ + ω²x = −F, T/2 < t < T

  x″ + ω²x = F, T < t < 3T/2,

and so on, where ω² = k/m, F = f_k/m = μg, g = 32, and T = 2π/ω. Show that the times 0, T/2, T, 3T/2, ... correspond to x′(t) = 0.

1. (b) Explain why, in general, the initial displacement must satisfy ω² |x₀| > F.
2. (c) Explain why the interval −F/ω² ≤ x ≤ F/ω² is appropriately called the “dead zone” of the system.
3. (d) Use the Laplace transform and the concept of the meander function to solve for the displacement x(t) for t ≥ 0.
4. (e) Show that in the case m = 1, k = 1, f_k = 1, and x₀ = 5.5 that on the interval [0, 2π) your solution agrees with parts (a) and (b) of Problem 28 in Chapter 5 in Review.
5. (f) Show that each successive oscillation is 2F/ω² shorter than the preceding one.
6. (g) Predict the long-term behavior of the system.