In this equation u is the displacement of the degree of freedom,
m is its mass,
c is the damping parameter, and
k is the stiffness of the system. The time (
t) dependent forcing term is
f(
t). This equation is often written in the form:
where ζ = c/(2mω0) and
ω02 = k/m. In this case
ζ is the
damping ratio (
ζ = 1 for critical damping) and
ω0 is the
undamped resonant frequency of the system. In the literature it is more common to give values of
ζ than
c. The damping ratio
ζ can also be readily related to many of the various measures of damping employed in different disciplines. These are summarized in
Table 2-10.
are written where ω is the angular frequency and the amplitude terms
U and
F can in general be complex (the arguments provide information on the relative phase of signals). Usually the real part is taken as implicit and is subsequently dropped.
Equation 2-20 takes the following form in the frequency domain:
There are three basic damping models available in the structural mechanics interfaces for explicit modeling of material damping — Rayleigh damping, viscous damping, and loss factor models based on introducing complex quantities into the equation system. There are also other phenomena which contribute to the damping. Some material models, such as viscoelasticity and plasticity are inherently dissipative. It is also possible to model damping in spring conditions.