About Damping
Phenomenological damping models are typically invoked to model the intrinsic frictional damping present in most materials (material damping). These models are easiest to understand in the context of a system with a single degree of freedom. The following equation of motion describes the dynamics of such a system with viscous damping:
(2-12)
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:
(2-13)
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 2-12.
Table 2-12: Relationships Between Measures of Damping.
Damping Parameter
Definition
Relation to damping Ratio
Damping ratio
–
Logarithmic decrement
where t0 is a reference time and τ is the period of vibration for a decaying, unforced degree of freedom.
Quality factor
where Δω is the bandwidth of the amplitude resonance measured at of its peak.
Loss factor
where Qh is the energy lost per cycle and Wh is the maximum potential energy stored in the cycle. The variables Qh and Wh are available as solid.Qh and solid.Wh.
At the resonant frequency:
In the frequency domain, the time dependence of the force and the displacement can be represented by introducing a complex force term and assuming a similar time dependence for the displacement. The equations
and
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 2-12 takes the following form in the frequency domain:
(2-14)
where the time dependence has canceled out on both sides. Alternatively, this equation can be written as
(2-15)
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.