Numerical investigation of acoustic radiation damping in sandwich structures

Suhaib Koji Baydoun(1)∗ , Steffen Marburg(1)(1) Chair of Vibroacoustic of Vehicles and Machines, Department of Mechanical Engineering, Technical University of Munich,Boltzmannstraße 15, 88748 Garching, Germany* Email: suhaib.baydoun@tum.deAbstract
Acoustic radiation damping is the primary energy dissipating mechanism for lightweight structures with large
radiating surfaces. Attempts to reduce the vibrational response of those lightweight structures by additional
mechanical damping can only be successful if the extent of mechanical damping is comparable or larger thanthe extent of radiation damping. In this regard, engineers are in need of reliable and flexible methods for thequantification of radiation damping as well as for the modeling of its effect in an early stage of the designprocess. In this paper, we address the equations of time-harmonic elastodynamics and acoustics by means offinite and boundary element methods respectively. The acoustic radiation damping of a honeycomb sandwich
panel is compared to the energy loss due to hysteretic material damping. Moreover, the influence of radiation
damping on geometrical as well as material properties is studied.
Keywords: Structural acoustic interaction, Acoustic radiation damping, Honeycomb sandwich structures

1INTRODUCTION

Acoustic radiation damping refers to the energy dissipation of structures due to sound radiation to the acoustic
far-field. In many engineering applications, radiation damping is rather insignificant and negligible. However, it
is relevant in cases where other damping mechanisms are only weakly pronounced, such as sandwich structures
exhibiting a high stiffness-to-weight ratio [2]. The quantification of radiation damping relies on the physicalcoupling between the fluid and the dynamic behavior of the structure. Therefore, it is hardly amenable toanalytical solutions except for plates with uniform thickness and density [4, 6, 7]. Moreover, experimental
damping determination necessitates reference measurements inside vacuum chambers to deduce the effect ofradiation damping, which are therefore accompanied by an excessive effort. In this regard, numerical structure
acoustic interaction analyses are a suitable way to systematically study the extent of acoustic radiation damping
in sandwich structures with different material and geometric configurations.
In this paper, we use the finite element method (FEM) and the collocation boundary element method (BEM) for
modeling the structural acoustic interaction. The theory is applied to the example of an air-loaded honeycomb
sandwich panel.

2NUMERICAL MODELING OF STRUCTURAL ACOUSTIC INTERACTION

We consider the fully coupled structural acoustic interaction of sandwich structures using FEM [11] and BEM[8] for discretizing the equations of linear time-harmonic elastics and acoustics, respectively. The resulting linear
systems of equations for the structural and for the acoustic subdomain read
K (1 − iηh ) − ω 2 M u = fs + ff ,(1)
H(ω)p = G(ω)vs .

(2)

The column vectors u and p contain the unknown displacement and sound pressure degrees of freedom at
the nodes. The stiffness and mass matrices of the structure are denoted as K and M. Structural damping is7470

modeled using the hysteretic loss factor ηh . The boundary element matrices H(ω) and G(ω) are obtained bya collocation discretization of the Kirchhoff-Helmholtz integral equation, relating the structural particle velocity
vs to the sound pressure. These matrices are implicitly dependent on the angular frequency ω = 2π f , where fis the frequency in Hz. The structure is excited by nodal forces fs . Equations (1) and (2) are coupled on thesound radiating boundary. There, the structure is subject to normal tractions due to the acoustic sound pressure,and the particle velocity in (2) equals the time derivative of the normal displacement on the boundary. The
coupling conditions can be expressed asff = Csf p

andvs = −iωCfs u,(3)where Csf and Cfs are the mesh coupling matrices obtained by a Galerkin projection [10]. The force vector ff
can be interpreted as the acoustic loading on the structural nodes and i denotes the imaginary unit. Finally, theglobal system of equations containing the coupling conditions emerges as   fK (1 − iηh ) − ω 2 M −Csf u= s .
(4)0−iωG(ω)Cfs
H(ω) pWhen only the structural response is of interest, (4) can be reformulated by forming the Schur complement andthereby omitting the pressure degrees of freedom [9], i.e.
K (1 − iηh ) − ω 2 M + iωCsf H(ω)−1 G(ω)Cfs u = fs ,(5)
where the expression iωCsf H(ω)−1 G(ω)Cfs can be interpreted as the effect of acoustic loading.
Once (4) or (5) is solved, the harmonic radiation loss factor ηr can be evaluated, quantifying the extent ofenergy dissipation due to sound radiation. It is defined asηr = Re (P) /|−iωEp |,(6)

where P and Ep denote the radiated sound power and the potential energy of the vibrating structure, respectively.
Only the real part Re (·) of the sound power corresponds to the far-field sound radiation and hence to energy
dissipation. Recent results [1] show that spurious numerical damping could lead to an overestimation of damping
phenomena when studying them with BEM. However, the occurrence of numerical damping does not seem tobe an issue in exterior acoustics.
The radiated sound power and the potential energy can be determined from the acoustic and the structural
responses, i.e.
111Ep = uT Ku∗ − fTs u,
P = pT Θv∗s ,222
where Θ denotes the boundary mass matrix that is obtained from numerical integration of the boundary element
interpolation functions. In the case that structural damping is considered, the total loss factor is the sum ofradiation and structural loss factors. The latter is defined as the relation between the power loss due to structural
damping and the power corresponding to the total potential energy. It is given as
Re uT Ku∗
1.
(7)
ηs = ωηh2|−iωEp |3

RADIATION DAMPING OF A HONEYCOMB SANDWICH PANEL

The acoustic radiation damping of a honeycomb sandwich panel (a = 0.696 m, b = 0.464 m) in air is studied
using coupled structural acoustic analyses. The material properties of the baseline panel are given in Tab. 1.
Besides radiation damping, additional mechanical damping resulting from a hysteretic loss factor of ηh = 0.01 is
considered in all simulations. The panel is free of any structural boundary conditions and modeled without an7471

Table 1. Baseline properties of the honeycomb sandwich panelAluminum face sheets
Thickness
t0.2 mmDensityρa2780 kg/m3
Young’s modulus
E73 GPa
Poisson’s ratio
νa0.34
Aluminum honeycomb core
Thickness
h20 mmDensityρc44.8 kg/m3
Young’s modulus Ex , Ey
18.9 MPa
Young’s modulusEz
1.89 GPa
Shear modulus
Gxy
3 MPa
Shear modulus
Gyz
200 MPa
Shear modulus
Gxz
200 MPa
Poisson’s ratio
νc0.1

acoustic baffle. The structural mesh consists of shell and solid elements to model the face sheets and the core,respectively. The discretized model has a total of 16071 displacement and 3392 pressure degrees of freedom.
The radiation loss factor of the panel resulting from a point-force excitation at a corner node is displayedin Figure 1 along with the structural loss factor. At low frequencies, radiation damping is rather small andinsignificant but it increases gradually until reaching its maximal value in the coincidence region. At higher
frequencies, the radiation loss factor shows an asymptotic behavior. The comparison to the structural loss factor
clearly demonstrates the significance of radiation damping for this type of sandwich structures.
Figure 2 shows the radiation loss factors of the panel with different face sheet thicknesses. All other material
and geometric parameters are left unchanged as given in Tab. 1. The results indicate that the frequency range
in which coincidence occurs is not affected by the face sheet thickness. This is a reasonable result when considering that the critical frequency is proportional to the mass and inversely proportional to the flexural rigidityof the panel (recall that the flexural rigidity in turn is approximately proportional to the thickness of the facesheet). However, it can also be seen that the extent of radiation damping significantly decreases with increasing
thickness of the face sheets. Above coincidence, this is mainly due to the effect of added mass of the panel perunit area.
Many authors note that the acoustic properties of sandwich structures are highly sensitive to the shear stiffnessof the core [3, 5]. However, the results in Fig. 3 indicate that the radiation damping of the panel is not significantly affected by the shear stiffness of the core - neither below nor above coincidence.
Lastly, the influence of the core thickness is displayed in Fig. 4. Below coincidence, the resonance effects dominate the radiation damping properties, and a clear classification of the different panels is hardly possible. Above
coincidence, again the effect of added mass leads to a decrease of the radiation loss factor with increasing corethickness.

4CONCLUSION

The numerical framework based on FEM and BEM enables flexible and systematic structural acoustic analyses
without any limitations on the geometry, material and boundary conditions. Using the numerical framework,
the extent of acoustic radiation damping of a honeycomb sandwich panel has been studied. The comparison
to structural loss factors indicates the significance of radiation damping for honeycomb sandwich structures.
Regarding the influence of material configurations, it has been found that the radiation damping exhibits ahigh sensitivity to the face sheet and core thicknesses. In fact, the radiation loss factor above coincidence is7472

10−1
Radiation loss factor

10−1

Loss factor

10−2
10−3
10−4

10−2
10−3t = 0.1 mm
t = 0.2 mm
t = 0.4 mm
t = 0.8 mm
t = 1.6 mm

10−4Radiation loss factorStructural loss factor
10−50

1,000

2,000

3,000

10−54,000

0Frequency in Hz

3,000

4,000

Figure 2. Radiation loss factor of the honeycomb
sandwich panels with different face sheet thicknesses.

10−1

10−1

10−2
10−3
Gyz , Gxz = 50 MPa
Gyz , Gxz = 100 MPa
Gyz , Gxz = 200 MPa
Gyz , Gxz = 400 MPa
Gyz , Gxz = 800 MPa

10−40

1,000

2,000

3,000

4,000

Frequency in Hz
Figure 3. Radiation loss factor of the honeycomb
sandwich panels with different shear stiffnesses ofthe core.

Radiation loss factor

Radiation loss factor

2,000Frequency in Hz

Figure 1. Comparison between the radiation loss
factor and the structural loss factor (ηh = 0.01) of
the honeycomb sandwich panel.

10−5

1,000

10−2
10−3
h = 15 mm
h = 20 mm
h = 25 mm
h = 30 mm

10−4
10−5

0

1,000

2,000

3,000

4,000

Frequency in Hz
Figure 4. Radiation loss factor of the honeycomb
sandwich panels with different shear thicknesses of
the core.

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approximately inversely proportional to the surface density of the panel. Further investigations are required tofully understand the effect of the stiffness properties of the core and the face sheets on radiation damping.

ACKNOWLEDGEMENTS
This work was conducted with the support of the German Research Foundation (DFG) in the context of thepriority program 1897 “Calm, Smooth and Smart - Novel Approaches for Influencing Vibrations by Means ofDeliberately Introduced Dissipation.”

REFERENCES[1] S. K. Baydoun and S. Marburg. Quantification of numerical damping in the acoustic boundary element method for two-dimensional duct problems. Journal of Theoretical and Computational Acoustics,
26(3):1850022, 2018.
[2] B. L. Clarkson and K. T. Brown. Acoustic radiation damping. Journal of Vibration, Acoustics, Stress, andReliability in Design, 107:357–360, 1985.
[3] H. Denli and J. Q. Sun. Structural-acoustic optimization of composite sandwich structures: A review. The
Shock and Vibration Digest, 39(3):189–200, 2007.
[4] F. Fahy and P. Gardonio. Sound and Structural Vibration. Academic Press, Oxford, United Kingdom, 2007.
[5] Y.-J. Kim and J.-H. Han. Identification of acoustic characteristics of honeycomb sandwich composite panels
using hybrid analytical/finite element method. Journal of Vibration and Acoustics, 135(1):11006, 2013.
[6] G. A. Kriegsmann. Assessment of a new radiation damping model for structural acoustic interaction.
Journal of the Acoustical Society of America, 86(2):788–794, 1989.
[7] R. A. Mangiarotty. Acoustic radiation damping of vibrating structures. Journal of the Acoustical Society ofAmerica, 35(3):369–377, 1963.
[8] S. Marburg and B. Nolte. A unified approach to finite and boundary element discretization in linear timeharmonic acoustics. In S. Marburg and B. Nolte, editors, Computational Acoustics of Noise Propagation inFluids. Finite and Boundary Element Methods, pages 1–34. Springer, Berlin, Germany, 2008.
[9] H. Peters, N. Kessissoglou, and S. Marburg. Modal decomposition of exterior acoustic-structure interaction.
Journal of the Acoustical Society of America, 133(5):2668–2677, 2013.
[10] H. Peters, S. Marburg, and N. Kessissoglou. Structural-acoustic coupling on non-conforming meshes withquadratic shape functions. International Journal for Numerical Methods in Engineering, 91(1):27–38, 2012.
[11] O. C. Zienkiewicz, R. L. Taylor, and D. Fox. The Finite Element Method for Solid and Structural Mechanics.
Butterworth-Heinemann, Oxford, United Kingdom, 2014.

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