
130 Friction-Induced Vibration and Self-Organization
where
=
are the components of displacement in the
direction
is the time
G is the shear modulus
Λ is the Lame’s constant
Ρ is the density
Letters with hats on top are used for dimensional quantities. Equation
(6.41) pertains to the lower body, whereas the corresponding quantities for
the upper body are denoted with a prime (′). Equation (6.41) is transformed
into a dimensionless coordinate system moving to the right with constant
speed
(
not necessarily equal to
, which yields, Adams 1995)
β−
∂
∂
+
∂
∂
+β−
∂
∂∂
=
∂
∂
−
∂
∂∂
β
∂
∂
+−
∂
∂
+β−
∂
∂∂
=
∂
∂
−
∂
V
u
x
u
x
u
xx
u
t
V
u
xt
u
V
u
u
u
V
u
() (1
(1 ...