Page 285 - Physical Principles of Sedimentary Basin Analysis

8.4 The slider-block model of stick–slip motion 267

Exercise 8.2 This exercise treats the special case when μ s = μ d .
(a) Show that x(t) = vt − (v/λ) sin(λt).
(b) Show that dx/dt = 0for t n = 2nπ/λ for any integer n > 0.
(c) Show that the block does not stick, because the force is F s = kx s at the times t n .
The block therefore oscillates around position vt with an amplitude A 0 = v/λ and a period
t 0 = 2π/λ.

Exercise 8.3 Let 2λ(x s − x d )/v
π.
(a) Show that t 1
t 2 .
(b) Show that t 1 ≈ π/λ.
(c) Show that t 2 /t 1 ≈ 2λ(x s − x d )/(πv).

Note 8.2 The solution of equation

2
d x 2 2
+ λ x = f (t) where f (t) = λ (vt + x s − x d ) (8.20)
dt 2
is the sum of the homogeneous solution and a particular solution. The homogeneous solu-
tion solves the equation for f (t) = 0, and the particular solution is any solution of the
equation with the given f (t) = 0. The homogeneous solution is u(t) = A sin(λt) +
B cos(λt), and a particular solution is v(t) = vt + x s − x d , and both solutions are verified
by inserting them into equation (8.20). The solution is therefore x(t) = u(t) + v(t), and it
has two coefficients A and B that are given by the boundary conditions. The first bound-
ary condition at x = 0 implies that B =−(x s − x d ), and the second boundary condition
dx/dt = 0at x = 0 gives that A =−v/λ, which together give the position (8.12).

Note 8.3 The block comes to rest after the time span t 1 , which is a solution of dx/dt = 0.
We have
dx
= v (1 − cos(λt)) + λ (x s − x d ) sin(λt). (8.21)
dt
In order to solve equation dx/dt = 0 it is advantageous to rewrite expression (8.21)using

the trigonometric formulas 1−cos(λt) = 2sin 2 1 λt and sin(λt) = 2sin 1 λt cos 1 λt .
2 2 2
It then follows that
1 λ(x s − x d )

tan λt 1 =− (8.22)
2 v
or
1 −1 λ(x s − x d )
λt 1 = Nπ − tan (8.23)
2 v
where N is any integer. Adding (or subtracting) any number of π’s to an angle does not
change the tan-value. We therefore add π to the solution (N = 1) since we want the
smallest positive solution for the time span.