On the estimation of the damping loss factor of highly damped structures
Mathias HINZ(1) , Guido SANTOS(1) , José Vitor MONTEIRO(1) , Júlio CORDIOLI(1) , Israel PEREIRA(2) , Sideto
FUTATSUGI(2)(1) Vibration and Acoustics Laboratory, Federal University of Santa Catarina, Brazil, julio.cordioli@ufsc.br(2) Smart Structures, Noise and Vibration R&D - EMBRAER, Brazil, israel.pereira@embraer.com.brAbstract
The damping loss factor (DLF) of a structure represents numerically its power dissipation capacity under agiven excitation ﬁeld, and it is an important input parameter for numerical models based on Statistical Energy Analysis (SEA). Different experimental approaches are described in the literature to obtain the DLF fromcomplex structures. Nevertheless, the application of such methods to estimate the DLF of highly damped structures, for example fuselage panels with viscoelastic materials, may be questionable. With its focus on the DLF
computation for highly damped structures, this paper intends to present a broad overview through a numericalexperimental approach on three of the main methods presented in literature: the Power Input Method (PIM),
the Periodic Structures Theory and the Inhomogeneous Wave Correlation (IWC). Numerical and experimental results are presented and discussed, exploring details, advantages and limitations according to each method
proposed.
Keywords: Viscoelasticity, Fuselage panels, Inhomogeneous Wave Correlation

1INTRODUCTION

The attenuation of structural vibration is a crucial step during the design of a high performance structure likean aircraft, as it directly impacts the cabin interior noise and vibration levels. As a result, passive dampingtreatments such as viscoelastic materials (VEM) are widely used due to its simplicity regarding manufacturing/assembly procedures as well as the cost-efﬁciency relation obtained with such a solution. The VEM application to resonant components, such as the skin of an aircraft fuselage, intensiﬁes the energy dissipation througha vibration cycle (excited, for example, by pressure ﬂuctuations on the turbulent boundary layer), reducing thetotal amount of energy transmitted vibroacoustically to the airplane interior.
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Figure 1. Viscoelastic layer applied to a fuselage panel. [1]The damping loss factor (DLF) is then the parameter used to quantify the amount of energy dissipated by agiven structure under an excitation (mechanical or acoustic) ﬁeld. Many are the techniques developed with thegoal of computing the overall damping loss factor of complex structures.

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This paper intends to evaluate three of the most widespread methodologies for damping loss factor estimation,
focusing on their main limitations for highly damped cases, recurrent in structures treated with VEM. Therefore,a brief review for the methods implemented is ﬁrst introduced. Then, numerical and experimental results arediscussed for the case of a simple ﬂat plate treated with VEM, seeking to illustrate the difference in the behavior
between light and heavily damped structures and how it affects each method. Finally, all the methods areimplemented and analyzed for a fuselage panel, also treated with VEM.

2DAMPING LOSS FACTOR COMPUTATION METHODS

2.1 Power Input Method (PIM)
The deﬁnition for a structure’s loss factor is given by the ratio between the dissipated energy per oscillation
radian (Ediss /2π) and the total reversible energy of the system (Evib ) [2]:η=Ediss
Pdiss=.
2πEvibωEvib(1)For the case of steady state conditions, the total power injected (Pin ) in the system is equal to the amount ofpower being dissipated. For a random excitation signal, this is given by:
Pin = Pdiss =12π

 ∞

−∞

Re{Y f f (ω)}SFf Ff (ω)dω(2)where Re{Y f f } is the real part of the driving point mobility function. SFf Ff is the force power spectral density
on the same point.
Now, we recall that the vibrational energy is the sum of the kinetic and elastic energies.
Er = Ek + Eu

(3)

According to SEA hypothesis, when the structure is submitted to a stationary white noise force, as long as adiffuse ﬁeld is established, the expectations of kinetic and elastic energies are equal [3]< Ek >=< Eu >(4)

and the vibrational energy is then given by Equation 5
Er = 2Ek(5)

where the kinetic energy is chosen since it is easier to measure than the elastic energy experimentally andnumerically. One must notice that highly damped structures will likely fail to meet the diffuse ﬁeld condition,
as the waves will be dissipated before reaching the boundaries. The diffuse ﬁeld hypothesis leads to:
2Ek =

V

ρ

12π

 ∞

−∞

Svi vi (ω)dωdV(6)where ρ is the material density, V its volume and Svi vi is the velocity power spectral density function for eachpoint i being measured.
Combining Equations 2 e 6 into Equation 1, and performing some manipulations [1], one can ﬁnd the ﬁnal
expression for the loss factor (η) computation, given by:η(ω) =Re{Y f f (ω)}ω ∑Ni=1 mi |Yi f (ω)|2(7)The equation above is the one commonly used for experimental/numerical computation when implementing thePIM theory. The procedure is simply developed by segmenting the complete structure in discrete elements(representing the masses mi ) and measuring the driving and transfer mobility functions in Equation 7.

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2.2 Periodic Structures Theory
When discussing numerical models, the Periodic Structures theory presents an interesting alternative to the conventional full FEM approach, when taking into account space periodicity of a given structure, represented by
the repetition of a cell [4].
Given this, one can propose a relationship between the structure’s degrees of freedom (DOF) as a function of
the wavenumbers (kx e ky ) and nodes’ distances given by the relation:q = P(kx , ky )q(8)where the periodicity matrix P summarizes this relationships between the original DOF vector q and a reducedarray q . With this considered, one can write the equation of motion for the given cell (considering as well theperiodicity for the internal forces) as:PH (K − ω 2 M)P q = PH F = 0(9)where PH is the hermitian of P. M and K are the original mass and stiffness matrices for the cell. The ﬁnal
equation is then written as:(10)(K̄(kx , ky ) − ω 2 M̄(kx , ky )) q = 0
For structural damping, the dissipated power is proportional to the local strain energy on the element (Ur ).
Considering local loss factors (ηr ) for the different regions on the structure being analyzed, the total dissipatedpower is given by(11)Pdiss = ∑ 2ωηrUrr

The total and strain energies can, ﬁnally, be expressed in terms of stiffness and mass matrices and the wave
mode, given by the eigenvector (φn ) obtained as a solution to Equation 10. Therefore, the loss factor for agiven wave n is given by:∑ 2ηr φnH K̄r φn(12)ηn = Hr
φn [K̄ + ω 2 M̄]φn
The ﬁnal step is then the computation of an average loss factor for a given frequency band. Assuming areverberant diffuse ﬁeld, the structure loss factor can be computed as an average of Equation 12 over all thewavenumber values whose natural frequencies lie within the band of interest.
2.3 Inhomogeneous Wave Correlation (IWC)
The IWC method [5] is based on the measurement of the correlation between harmonic displacement ﬁelds. Fora ﬁxed frequency (ω), a 2D harmonic displacement ﬁeld (ŵ) obtained from the desired structure is selected to
be a reference, compared with the harmonic ﬁeld generated by a inhomogeneous wave (Equation 13).
ϕ̂k,γ,θ (x, y) = eik(1+iγ)(xcosθ +ysinθ ) = eik̃(xcosθ +ysinθ )(13)First, the heading angle (θ ) of the inhomogeneous wave is ﬁxed, and multiple values of the wavenumber k andthe attenuation factor γ are iterated, creating several harmonic ﬁelds depending on the selected pair (kit , γit ).
Then, the Modal Assurance Criterion [6] (Equation 14) is applied to quantitatively compare how similar is theshape of each inhomogeneous wave displacement ﬁeld (ϕ̂(kit , γit )) and the reference one (ŵ).
∗(xi , yi )ΔSi  ∑N ŵ(xi , yi ).ϕ̂k,γ,θ(14)IWC(k, γ, θ ) = ∑N |ŵ(xi , yi )|2 ΔSi . ∑N |ϕ̂k,γ,θ (xi , yi )|2 ΔSi
where ˆ represents ω dependence and ∗ the complex conjugate. Si is the equivalent area for the measurement
point and N is total measured points.

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Figure 2. Example of IWC(k,γ) mapping for a ﬁxed θ .
The processes is then repeated ﬁxing now different values θ , selecting for each heading angle the best correlation found by the IWC equation.
9090150

12075150301800.0753018033002103002406015002100.1512060

330300240270270(k) - Wavenumber(γ) - Attenuation Factor

Figure 3. Example of pairs of k and γ selected for each θ of a frequency ω.
Once the previous steps were performed for all the 2D harmonic displacement ﬁelds (ŵ) (one for each ω), thedamping loss factor can be then computed from Equation 15. [7] Im(k̃4 ) (15)
η(ω) =  Re(k̃4 ) 3IMPLEMENTATION

3.1 Case 1 - Numerical ﬂat plate
The ﬁrst case consists of retrieving the harmonic displacement ﬁelds from a numerical model developed in theRsoftware VA One , for a simple ﬂat plate considering three different cases of damping.
• Dimensions → L ×W × t = 0, 5 × 0, 35 × 0, 0012m• Density → ρ = 2700 mkg3

• Young and Shear moduli → E  = 71GPa & G = 26, 7GPa
• Frequency range → f = 50 − 1 − 5600Hz• Damping Loss factors → η1 = 0, 01; η2 = 0, 1; η3 = 0, 34244

Figure 4. PIM - 2 Forces / 64
Acquisition PointsFigure 5. IWC - 1 Force / 1836
Acquisition Points (Full mesh)

3.2 Case 2 - Experimental ﬂat plate with periodic VEM
The second case is again an aluminum ﬂat plate, now considering two different setups of a periodically attached
VEM. The ﬁrst one presents a 25% VEM coverage, while the second, 50%. The mobilities in this case wereR
measured using a laser vibrometer (PolytecPSV-500).
• Dimensions → L ×W × t = 0, 5 × 0, 35 × 0, 0012m• Frequency range → f = 50 − 2 − 12800Hz• Acquisition Points → 493Figure 6. 25% coverage.

Figure 7. 50% coverage.

Beyond the experimental procedure, results for the mobilities were also calculated through numerical models
of such conﬁgurations, using equivalent properties obtained experimentally [8] to characterize the areas withR, while the periodic structure method
VEM application. The FEM model was implemented through Nastran
R
was implemented through VA One , taking into account the geometry pattern used in the specimen measured.
3.3 Case 3 - Fuselage Panel with VEM
The third case is a fuselage panel with stringers and frames conﬁgured to create a 4x6 cell structure. Thefuselage panel skin is covered with VEM in constrained form, covering approximately 50% of the total innerRarea. For this case, as in the last one, the mobilities were measured using a laser vibrometer (Polytec
PSV500).
• Dimensions → L ×W = 1.8 × 1.13m• Frequency range → f = 50 − 10000Hz• Acquisition Points → 451

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Figure 8. VEM disposition on the Panel.

Figure 9. Excitation (Beam - blue , Stringer green, Skin - red) and Acquisition nodes.

As in the ﬂat plate with periodic VEM, the properties of the applied VEM area were obtained experimentally toR
supply the numerical models. A FEM model was then built on Nastran
software, while the periodic structureR
method was applied in VA One , taking into account one cell of the fuselage panel.

4RESULTS AND FURTHER DISCUSSIONS

4.1 Case 1Figure 10 shows the damping loss factor estimations for three scenarios of DLF (η1 = 0, 01, η2 = 0, 1 andR
η3 = 0, 3, constant in frequency) used as input for the ﬂat plate FEM model in VA One(K) - Damping Loss Factor

10.1

0.01
PIM - DLF 0,3 - 64 points / 2 forces
IWC - DLF 0,3 - full grid / 1 force
PIM - DLF 0,1 - 64 pontos / 2 forças
IWC - DLF 0,1 - full grid / 1 force
PIM - DLF 0,01 - 64 points / 2 forces
IWC - DLF 0,01 - full grid / 1 force

0.001

1001000(f) - Frequency [Hz]Figure 10. DLF estimation for η1 = 0, 01, η2 = 0, 1 and η3 = 0, 3.
It becomes clear that for cases categorized as low and moderate damping (η1 e η2 ), results for the two methods
converge quite well to the input result. Nevertheless, for the high damping scenario (η3 ), the results for thePIM start to diverge with a frequency increase. One observes that this behavior is based on the fact previously
stated that the reverberant ﬁeld gradually disappears for higher frequencies as all the energy is dissipated before
the wave can reach the boundaries, as illustrated in Figures 11 and 12, where reverberant and direct ﬁelds arerespectively dominants. One can ﬁnally observe that the IWC was the method with the closest results to theexpectations for the highly damped model.

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Figure 11. Displacement ﬁeld for f = 200Hz.

Figure 12. Displacement ﬁeld for f = 5000Hz.

4.2 Case 2
Figures 13 and 14 illustrate, respectively, loss factor estimates for the two different VEM coverage cases (25%and 50%). All the results present a good agreement, which indicates that the added damping caused by theVEM treatment was not enough to cause a direct ﬁeld predominance, which would affect the PIM results.

(K) - Damping Loss Factor1IWC - Experimental
PIM - Experimental
Periodic - Numerical
PIM - Numerical

0.1

0.01

0.001
1001000(f) - Frequency [Hz]10000

Figure 13. DLF estimation for 25% coverage.

(K) - Damping Loss Factor

1

IWC - Experimental
PIM - Experimental
Periodic - Numerical
PIM - Numerical

0.1

0.01

0.001
1001000(f) - Frequency [Hz]10000

Figure 14. DLF estimation for 50% coverage.
4.3 Case 3
Figure 15 shows the fuselage panel loss factor estimation for both numerical and experimental analyses. Theresults show a very good convergence between themselves, for both numerical and experimental data, withPIM and IWC capturing the same behavior inside a small margin of variation. Again, the application o VEM
covering nearly 50% of the skin of a fuselage panel still didn’t reach a state of dominance of the direct ﬁeld,
which would lead the PIM to failure.

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0(η) − Damping Loss Factor

10−110−210

PIM Numerical
PIM Experimental
IWC Numerical
IWC Experimental
Periodic Numerical−310310(f) − Frequency [Hz]410Figure 15. DLF estimation for fuselage panel.

5CONCLUSIONSThe purpose of this paper was to evaluate and compare different methodologies for loss factor estimation incomplex damped structures. While the PIM showed to be a very simple and fast way of computing the DLF, it
might fail in critical cases where the dissipation is so high that there is no reﬂection of waves in the structure,
failing this way the diffuse ﬁeld assumption.
In the other hand, the IWC presented good results when compared to those obtained by the traditional methods,
not presenting any limitation for high values of damping. However, although the IWC method may present
advantages, it has also its own limitations. Among them, it is possible to emphasize the necessity to use a relatively complex and expensive system of measurement as the laser scanning, as well as the high computational
cost to handle the calculation and analyze the results when compared to other methods.

REFERENCES[1] Hinz, M. Numerical evaluation of damping loss factor estimation techniques for sheet metal structures.
Trabalho de Conclusão de Curso, Universidade Federal de Santa Catarina, Florianópolis, 2018.
[2] Cremer, L.; Heckl, M.; Petersson, B. Structure-Borne Sound: Structural Vibrations and Sound Radiation at
Audio Frequencies, Springer Verlag, Berlin, 2005.
[3] Le Boat, A. Foundation of Statistical Energy Analysis in Vibroacoustics, OUP Oxford, Lyon, 2015.
[4] Cotoni, V.; Langley, R.; Shorter P. A statistical energy analysis subsystem formulation using ﬁnite element
and periodic structure theory. Journal of Sound and Vibration, 318:1077-1108, 2008.
[5] Berhaut, J. Contribution a l’identiﬁcation large bande des structures anisotropes: application aux tables
d’harmonie des pianos. Thèse de Doctorat, Ecole Centrale de Lyon, Lyon, 2004.
[6] Ewins, D. Modal Testing: Theory and Practice, Research Studies Press Letchworth, London, 1984.
[7] Cherif, R.; Chazot, J.; Atalla, N. Damping loss factor estimation of two-dimensional orthotropic structuresfrom a displacement ﬁeld measurement. Journal of Sound and Vibration, 356:61-71,2015.
[8] Santos, G. Determinação do amortecimento de painéis aeronáuticos a partir da caracterização dinâmica demateriais viscoelásticos via ajuste de modelo. Dissertação de Mestrado, Universidade Federal de Santa Catarina, Florianópolis, 2019.

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