# Vibroacoustic - precise formula for determining the EDPC from stored energies

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Wer are calculating some vibroacoustics problems, requiring precise control of energy lost (dissipated) due to damping. The inquiry concerns the ANSYS internal algorithms allowing the determination of energy-dissipated-per-cycle (EDPC) in harmonic analyses.

The benchmark is very simple. It consists in rectangular plate, loaded symetrically in in-plane axial ("X") direction. The boundary conditions eliminate any possible movement but that along X asis. Therefore, it is a quasi-1D situation. On the other hand, we did not want to use beam elements, as the real problem at stake requires shells.

The main part of the benchmark macro is as follows:

XA=3.0

YA=1.0

THICK=0.005

/prep7

RECT,0,XA,0,YA

ET,1,SHELL181

MP,EX,1,2e11

MP,NUXY,1,0.3

MP,DENS,1,7800

SECTYPE,1,SHELL

SECDATA,THICK

ESIZE,YA/2

AMESH,all

NSEL,S,LOC,X,0

D,all,all,0

ALLS

D,all,UZ,0

D,all,ROTX,0

D,all,ROTY,0

D,all,UY,0

....

/solu

ANTYP,HARMIC

....

DMPRAT,0.05

KBC,1

freqstart=400

freqend=500

freqstep=5

*DO,given_frequency,freqstart,freqend,freqstep

HARF,given_frequency

N1=NODE(XA,YA/2,0)

F,N1,FX,-F1AMP

ALLS

SOLVE

..... postprocess of energies ....:

SET, (imaginary or real)

ETABLE, (kind of energy)

SSUM

*GET .... SSUM, ITEM,....

*ENDDO

We have determined the natural frequency of the plate associated with its "spring-like" behaviour (~450 Hz), and studied its response around that resonance. At each frequency, the various energies (TENE, SENE, KENE, AENE) are stored (ETABLE,... and summed in all elements. Both the real and imaginary parts are taken separately into account.

The plot of energies is as follows:

Now, we are struggling to understand, how the energy dissipated per cycle (EDPC) can be deduced from these energies.

The ANSYS conference paper (Modeling of Material Damping Properties in ANSYS, C. Cai, H. Zheng, M. S. Khan and K. C. Hung) seems to define the dissipated energy in the following way (page 5):

Let us call this approach "Energy-based" approach.

Further on, Cai et co-authors declare:

So it seems rational to sum up maximum strain energy (SENE or TENE) and multiply it by 2 * Pi * 2 * DMPRAT. So we do, yet problems remain:

1) Is this simple formula really sufficient, when at least three energy types are present (SENE=TENE, KENE and AENE)?

2) Which component (real or imaginary or some combination of both) should be taken into account, for proper representation of Um ?

Doubts were further backed up by our assessment of the dissipated energy in a different way - let us call it "driving-point-based" one. Namely, we calculated the real (active) part of work exerted by the force in the application point, storing its displacement amplitude and phase. We claim that the following formula is correct:

EDPC(drivinpointapproach) = |u| |F| cos (Δθ)

where  Δθ is the difference of phases between the force and the resulting displacement at the application node.

In this case, it is unnecessary to think of a vector product (as F and u are made to be aligned).

As one may notice on the plot, EDPCdrivinpointapproach is different from EDPCenergybasedapproach. They are close to each other only at the resonance peak, but differ largely at frequencies far from the resonance.

Consequently, we are looking for a precise formula for determining the EDPC from stored energies.

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