Vibroacoustic Simulation. Alexander Peiffer

as would be the case with viscous damping. Examples for damping processes are:

       structural or hysteretic damping

       coulomb or dry-friction damping

       velocity-squared or aerodynamic drag damping.

As most systems are lightly damped so that damping can be neglected except near resonance, it is sufficient to approximate the non-viscous damping in terms of equivalent viscous damping. A good way to formulate a criteria that works for all types of damping is to consider the dissipated energy per cycle of vibration. For viscous damping this reads as

      With the dissipated energy per cycle ΔEcycle for any particular damping type the equivalent viscous damping cveq can be determined from

(1.56)

      1.2.5.1 Hysteretic Damping

      In many cases damping is caused by structural damping. If materials like aluminium or steel are cyclically stressed they form a hysteresis loop. Experimental observations show, that the energy dissipated per cycle is proportional to the square of the strain (displacement):

(1.57)

      Comparing this with Equation (1.56) gives:

       (1.58)

      Entering this into the equation of motion in complex form (1.27)

(1.59)

      and rewriting (1.59) leads to

       (1.60)

      with the structural loss factor

(1.61)

      and

       (1.62)

      In this case the displacement response reads:

(1.63)

The structural loss factor is of great importance for structural dynamics not only because of the frequent occurrence in practical systems but also for numerical methods. The equation of motion remains similar to the equation for undamped systems. Additionally, the frequency of structurally damped systems does not change. This can be seen if the equivalent viscous damping of structurally damped systems is introduced in the solution for the magnification factor and the phase:

       (1.64)

      and

       (1.65)

      Amplitude and phase resonance occur at the same frequency ω₀. At resonance the viscously damped system amplification is u^/u^0=1/2ζ. Hence

       (1.66)

      There is a further interpretation of the loss factor. ΔEcycle is the energy dissipated per cycle. The dissipated power given by Πdiss=ΔEd/T=ΔEcycleω/2π. Using equations (1.57) and (1.61) we get:

       (1.67)

      At the beginning of the cycle the total energy E is stored as potential energy in the spring. The dissipated power is a product of damping loss, frequency and the total energy of the system. This will be frequently used in the following sections, but particularly in Chapter 6 about statistical energy methods. The energy aspect leads to an equivalent definition of the loss factor:

       (1.68)

Thus, the damping loss factor can be seen as the criteria, defining the relative loss of energy per cycle divided by 2π. In this zoo of damping criteria we have related c_v, ζ, c_vc, η, Δω, τ and Q. Table 1.1 summarizes those quantities and puts them in relation to the others.

Table 1.1 Relation of important damping criteria

Name	Symbol