
160 Friction-Induced Vibration and Self-Organization
governed by Equation (6.153), whereas the particular form of the function
k(c(x – ξ)) is different for each problem.
6.4.3.7 Special Solutions for Limiting and Resonance Cases
6.4.3.7.1 Frictionless Case
Simplications can be made for a number of special cases. Let us rst con-
sider the frictionless static case (μ = 0, V = 0). It is possible to show that the
roots are purely imaginary (no energy is dissipated; Λ = iλ where l is real) for
this case and that the eigenvalue problem is self-adjoint. For purely imagi-
nary roots, Equations (6.129–6.135) simplify to
=λ −
β=λ−
Q
n
m
Q
n
m
,
1
2
2
1