Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 26

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	cv,cvc	ζ	η	Q	Δω	τ
Viscous damping	c v	1	ζcvc
Critical damping ratio	ζ	cv/cvc	1	η/2	12Q	Δω/2ω0	1/ω0τ
Critical damping	c vc	4mks	2ζmω0
Damping loss	η	2cv/cvc	2ζ	1	1/Q	Δω/ω0	2/ω0τ
Qualtity factor	Q	ccv2cv	12ζ	1/η	1	ω0Δω	ω0τ2
3dB bandwidth	Δω		2ζω0	ηω0	ω0/Q	1	12τ
Decay time	τ		1/ζω0	2/ω0η	2Q/ω0	2/Δω0	1

In tools and software for vibroacoustic simulations many different quantities are used. The overview of all those different criteria shall help to avoid mistakes and confusion.