
155Friction-Induced Instabilities and Vibrations
for the lower body, where m = 1 for a mode with a period of 2π and m = 2 for
an anti-symmetric mode v(x + 2π,y) = –v(x,y) with a period of 4π. The terms
that represent the translation of the system as a rigid body can be set to zero,
(A
0
= 0 and B
0
= 0), without loss of generality. Note that the subscript k has
been omitted for conciseness.
In order to satisfy the Navier equation (6.113)—s
1
and s
2
in Equation
(6.126)—the latter must be roots of the characteristic equation
β+ β+ +
β−
sQ Q
n
m
sQQ(1
1
24
11
2
2
2
1
22
12
(6.128)
The solution of Equation (6.128) is
=± −=±− βsQ
12
1
2
(6.129) ...