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Vibroacoustic Simulation. Alexander Peiffer

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Vibroacoustic Simulation - Alexander Peiffer

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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

normal upper Delta upper E Subscript cycle Baseline equals integral Subscript 0 Superscript upper T Baseline c Subscript v Baseline ModifyingAbove x With dot StartFraction d x Over d t EndFraction d t equals c Subscript v Baseline ModifyingAbove u With caret squared omega squared integral Subscript 0 Superscript 2 pi slash omega Baseline s i n squared left-parenthesis omega t plus phi right-parenthesis d t equals pi c Subscript v Baseline omega ModifyingAbove u With caret squared

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

normal upper Delta upper E Subscript cycle Baseline equals pi c Subscript v eq Baseline omega ModifyingAbove u With caret squared (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):

normal upper Delta upper E Subscript cycle Baseline equals alpha Subscript x Baseline ModifyingAbove u With caret squared (1.57)

Comparing this with Equation (1.56) gives:

c Subscript v normal e normal q Baseline equals StartFraction alpha Subscript x Baseline Over pi omega EndFraction (1.58)

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

minus m omega squared bold-italic u e Superscript j omega t Baseline plus j StartFraction alpha Subscript x Baseline Over pi omega EndFraction omega bold-italic u e Superscript j omega t Baseline plus k Subscript s Baseline bold-italic u e Superscript j omega t Baseline equals bold-italic upper F Subscript x Baseline e Superscript j omega t (1.59)

and rewriting (1.59) leads to

minus m omega squared bold-italic u e Superscript j omega t Baseline plus k Subscript s Baseline left-parenthesis 1 plus j eta right-parenthesis bold-italic u e Superscript j omega t Baseline equals bold-italic upper F Subscript x Baseline e Superscript j omega t (1.60)

with the structural loss factor

eta equals StartFraction alpha Subscript x Baseline Over pi k Subscript s Baseline EndFraction (1.61)

and

bold-italic k Subscript s Baseline equals k Subscript s Baseline left-parenthesis 1 plus j eta right-parenthesis (1.62)

In this case the displacement response reads:

bold-italic u equals StartFraction bold-italic upper F Subscript x Baseline Over k Subscript s Baseline left-parenthesis 1 plus j eta right-parenthesis minus m omega squared EndFraction equals StartFraction bold-italic upper F Subscript x Baseline Over m left-parenthesis omega 0 squared left-parenthesis 1 plus eta right-parenthesis minus omega squared right-parenthesis EndFraction (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:

StartFraction ModifyingAbove u With caret Over ModifyingAbove u With caret Subscript 0 Baseline EndFraction equals StartFraction 1 Over StartRoot left-bracket 1 minus left-parenthesis omega slash omega 0 right-parenthesis squared right-bracket squared plus eta squared EndRoot EndFraction (1.64)

and

phi 0 equals arc tangent StartFraction negative eta Over 1 minus left-parenthesis omega slash omega 0 right-parenthesis squared EndFraction (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

eta equals 2 zeta equals StartFraction 1 Over upper Q EndFraction (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:

normal upper Pi Subscript diss Baseline equals eta omega one-half k Subscript s Baseline ModifyingAbove upper X With caret squared equals eta omega upper E (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:

eta equals StartFraction 1 Over 2 pi EndFraction StartFraction normal upper Delta upper E Subscript cycle Baseline Over upper E EndFraction (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

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