International audienceSandwich honeycomb panels are widely used in aerospace applications because they are very light and stiff. Strong mechanical or acoustical excitations, associated to low damping properties of such panels can lead to high vibration levels, generating fatigue and reliability problems. We propose in this paper to investigate the capability of micro-perforations of honeycomb panels for reducing their vibration levels. Micro-perforations are mostly known and used in acoustics for increasing absorption. A model of the panel vibration damping induced by the acoustic motion in the micro-perforations is proposed here. For this purpose, a lumped element model, based on Maa's results is developped for estimating a viscous damping force at the micro-scale. The resultant force is then homogenized for a group of cells (meso-scale model) and allows us to express a damping term for the global structure (macro-scale model). A perturbation technique is then used to compute the modal damping coefficients of a micro-perforated plate in order to evaluate the performance of the treatment

Gautier, François

Pelat, Adrien

Pézerat, Charles

Régniez, Margaux

English

Archive Ouverte en Sciences de l'Information et de la Communication

HAL Id: hal-02293595
https://hal.archives-ouvertes.fr/hal-02293595
Submitted on 20 Sep 2019
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Vibration damping of honeycomb sandwich panels
induced by micro-perforations
Margaux Régniez, Adrien Pelat, Charles Pézerat, François Gautier
To cite this version:
Margaux Régniez, Adrien Pelat, Charles Pézerat, François Gautier. Vibration damping of honeycomb
sandwich panels induced by micro-perforations. Eurodyn 2014, 2014, Porto, Portugal. ￿hal-02293595￿
ABSTRACT:  Sandwich honeycomb panels are widely used in aerospace applications because they are very light  and stiff. 
Strong mechanical or acoustical excitations, associated to low damping properties of such panels can lead to high vibration 
levels, generating fatigue and reliability problems. We propose in this paper to investigate the capability of micro-perforations of 
honeycomb panels for reducing their vibration levels. Micro-perforations are mostly known and used in acoustics for increasing 
absorption. A model of the panel vibration damping induced by the acoustic motion in the micro-perforations is proposed here.  
For this purpose, a lumped element model, based on Maa's results is developped for estimating a viscous damping force at the 
micro-scale.  The resultant  force  is  then homogenized for  a  group of  cells  (meso-scale  model)  and allows us  to  express  a 
damping term for  the  global  structure  (macro-scale  model).  A perturbation  technique  is  then  used  to  compute  the  modal  
damping coefficients of a micro-perforated plate in order to evaluate the performance of the treatment. 
KEY WORDS: acoustic impedance, vibration damping, micro-perforations.
1 INTRODUCTION
Sandwich materials composed of two skins glued on both 
sides of honeycomb cells are widely used in several domains 
because they offer the possibility to make lightweight and stiff 
structures.  In  term  of  vibroacoustic  characteristics, 
honeycomb sandwich materials have however high acoustic 
radiation  efficiencies,  or  reciprocally,  they  are  sensitive  to 
acoustic  excitations.  The  aim  of  the  present  study  is  to 
investigate  the  capability  of  micro-perforations  to  reduce 
vibrations of honeycomb sandwich materials.
Micro-perforations  are  well  known  devices  used  for  the 
attenuation  of  acoustic  levels.  Indeed,  micro-perforations 
provide an acoustic dissipation on the surface of the treated 
structure  due  to  the  viscous  effects  inside  the  micro-holes 
whose  dimensions  are  less  than  the  boundary  layer  of  the 
fluid.  Micro-perforations  were  particularly  studied  for  rigid 
structures  [1-3],  [8-9]  and  more  recently  for  vibrating 
structures  [10-11],  where  the  objectives  are  always  the 
improvement of the acoustic absorption. Of course, when the 
structure is excited by an acoustic field, the decrease of the 
wall pressure due to the presence of micro-holes implies that 
the vibration level is also reduced. The reduction of the wall 
pressure  can  be  quantified  by  the  well  known  acoustic 
absorption  coefficient  quantifying  the  ratio  between  the 
absorbed energy and the incidence energy.  This treatment is 
also  particularly  effective  for  high  levels,  where  the 
dissipation phenomenon becomes non linear [13]. Otherwise, 
whatever  the  excitation  (airborne  or  structure-borne 
excitations), the dynamic comportment of the structure can be 
changed  due  to  fluid  motion  (generated  by  the  vibration) 
inside  micro-holes.  Viscous  effects  can  then  introduce 
dissipation  leading  to  a  non-negligible  damping  when 
numerous  micro-perforations  are  considered.  This 
phenomenon  was  shown  on  micro-perforated  membranes 
[12]. The goal of this paper is to see if this damping effect is 
also interesting for flexural  motion of honeycomb sandwich 
panels.
The  proposed  modeling  is  divided  in  three  parts, 
corresponding to models at three different scales: the micro-
scale,  the  mesoscale  and  the  macro-scale.  The  micro-scale 
corresponds  to  the  scale  of  the  honeycomb cell,  where  the 
physical  phenomena are described and modeled to compute 
the dissipation delivered by one hole. The mesoscale contains 
several  perforated  cells,  where  the  damping  effect  can  be 
homogenized. The macro-scale is the scale of the structure, 
where the effects on modal properties can be obtained.
2 DYNAMIC  MODELING  OF  THE  MICRO-
PERFORATED STRUCTURE 
All  equations  of  the  paper  are  written  at  the  harmonic 
regime.  The  convention  chosen  is e j ωt , ω being  the 
circular frequency.
2.1 At the micro-scale 
The honeycomb sandwich plate is supposed to be modeled 
by  the  Kirchhoff's  plate  theory.  As  a 
consequence,  at  the  scale  of  a  single 
honeycomb  cell,  the  motions  of  the 
wall  cells  (or  plate  cross-section)  can 
be described by a superposition of two 
elementary  vibratory  motions:  a 
translation and a rotation.
Both structural  motions are  coupled 
to the internal  fluid motions, inducing 
the damping effect to be modeled.
Translation of the cell induces an air 
flow in the hole which is supposed to 
be be done on one side of the cell. Air 
Vibration damping of honeycomb sandwich panels induced by micro-perforations
Margaux Regniez1,2, Adrien Pelat2, Charles Pézerat2, François Gautier2
1CNES, 18 Avenue Edouard Belin, 31400 Toulouse, France
2Université du Maine, CNRS UMR 6613, LAUM, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France 
emails: margaux.regniez@univ-lemans.fr, adrien.pelat@univ-lemans.fr, nicolas.joly@univ-lemans.fr, charles.pezerat@univ-
lemans.fr, francois.gautier@univ-lemans.fr
Figure  1:  Single  
microperforated  cell  and  
resultant viscous force.
1
flow is also induced by the cell rotation, but is supposed to be 
very weak. Rotation effects are such ignored in the modeling. 
To  calculate  the  equivalent  viscous  force f⃗ v resulting 
from  the  microperforated  honeycomb  cell  translation  (see 
figure  1),  the  fluid  motion  is  described  by  a  mass-spring-
damping system as shown on figure 2, for which the equation 
of motion is:
jωM f x˙ f=
−K f
jω ( x˙ f− w˙ )−C f ( x˙ f −w˙ )          (1)
where K f=
P0γ S f
2
V cav
corresponds to the effect of the cell
cavity, M f=S f
ℑ(Z Maa )
ω and C f=S f ℜ(Z Maa ) are
respectively the mass and dissipation of the fluid inside the 
micro-perforation, computed using respectively the imaginary 
and real parts of the acoustical impedance given by Maa [1-3]: 
Z Maa=
32νρ0t
d 2 (√1+ x
2
32+√2 x
d
8 t)+ jωρ0t(1+ 1√32+ x22 +
8
3π
d
t )
(2)
with, x=√ωρ0ν d2 , d the  hole's  diameter, t the
hole's  thickness, ν the  kinematic  viscosity, ρ0 the  air 
density and ω the circular frequency.
The applied force on the honeycomb cell due to the fluid 
dissipation is written as:
f⃗ v=[C f ( x˙ f− w˙ )+ K fjω ( x˙ f −w˙ )]. z⃗ .          (3)
Expression  (3)  is  valid  only  if  we  consider  that  the  one 
degree of freedom model described in figure 2 is valid. For an 
air piston inside a perforation, this is not the case since the 
stiffness term in (3) is not applied to the plate. This point is 
not considered in the paper.
It  is important to note that the observed reaction is local, 
which means that  each elementary force f⃗ v associated to 
one  cell  is  independent  from  the  forces  associated  to  the 
neighboring  cells,  No  interaction  between  adjacent  cell  is 
considered.
2.2 At the mesoscale 
At the  mesoscale the considered surface is noted dS , it 
is sized  to be higher than the cell dimension but small enough 
compared to the acoustic wavelength.
The  relationship  between  the  viscous  forces f⃗ v (in  N) 
and the resultant  force distribution f⃗ Rv (in N/m²) on dS
is:
f⃗ Rv dS=N f⃗ v=
σ
S f
f⃗ v dS          (4)
where N=nt /c nc is  the  number  of  holes  in  the  surface
dS .  The  number nh /c is  the  number  of  holes  per  cell 
and nc the  number  of  honeycomb  cell  in dS ,
σ=
nt / hS f
S c
is the perforation rate and S f the surface of a
hole. In this study, the number of holes per cell nh /c is fixed 
to one.
The motion of the elementary surface dS  is supposed to 
be described by the classical bending equation:
−ω2ρh w( x , y ,ω)dS+DΔ2 w( x , y ,ω)dS=Δ p dS+ f Rv dS (5)
where w( x , y ,ω) corresponds  to  the  transverse  flexural 
displacement, ρh is  its  surface  density, D the  flexural 
rigidity and Δ p the difference of pressures at each side of 
the panel. The resultant force distribution f⃗ Rv given by
f⃗ Rv dS= σS f (C f +K fjω)( x˙ f−w˙ ). z⃗ dS=−C v w˙ . z⃗ dS    (6)
allows us to define the damping coefficient C v which will 
have sense, as expressed, if it is real and positive.
2.3 At the macro-scale 
 The macro-scale corresponds to the scale of the complete 
structure. In the following, a rectangular honeycomb sandwich 
panel is considered . It is considered to be simply supported 
on its edges. The dimensions are expressed on the figure 4. 
Figure 2: One degree of freedom representation of the fluid movement inside  
the micro-perforated honeycomb cell.
Figure 3: Assembly of microperforated honeycomb cells to form the surface  
element dS.
2
Equation of motion (5) is considered  on the whole the surface 
of the structure and established for the harmonic regime: 
−ω2ρh w( x , y ,ω)+ j ωC (x , y)w( x , y ,ω)+DΔ2 w( x , y ,ω)=Δ p
(7)
where C (x , y)= I (x , y)C v          (8)
is  the  damping  operator,  taking  into  account  the  micro-
perforation  positions,  specified  thanks  to  the  indicator 
function I (x , y) ( I (x , y)=1 if there is a hole at point 
(x,y), and I (x , y)=0 otherwise).
The damping properties  of  the micro-perforated  plate  are 
investigated  by  looking  at  the  modes  of  the  damped  plate 
described by equation (7). Since the damping for the micro-
perforated plate is expected to be low, modes of the damped 
plate are supposed to be closed to the ones of the undamped 
plates and are computed using a perturbation technique. 
The eigen values λ0k and the mode shapes ϕ0k of the 
undamped plate are the solutions of 
(ρh λ0k2 +DΔ2)ϕ0k=0 ,          (9)
associated to the imposed boundary conditions.
The  eigenvalues λk and  the  mode  shapes ϕk of  the 
damped plate are the solutions of 
(λk2ρh+λk C (x , y)+DΔ2)ϕk=0        (10)
and are supposed to be written as 
λk=λ0k+δ λ        (11)
ϕk=ϕ0k+δϕ        (12)
 where δλ and δϕ are  correction  terms,  supposed to 
be small. 
Reporting (11) and (12) into (10) leads to:
(DΔ2+λ0k2 ρh)ϕ0k+2 λ0kρhδ λϕ0k+λ0k C( x , y)ϕ0k+(δλ)2ρh ϕ0k
+(DΔ2+λ0k2 ρh)δ ϕ+C (x , y)δλ ϕ0k+2 λ0kδλρh δϕ+(δ λ)2ρhδ ϕ
+λ0kC (x , y)δϕ+δ λC (x , y)δϕ=0 .        (13)
The third line, the last three terms of the second line and the 
last  term of the first line of equation (13) are second order 
terms since they correspond to products of small values. The 
first term of the first line of the same equation is equal to zero 
because it corresponds to the equation (9) of the conservative 
associated system. Finally, equation (13) leads to:
(DΔ2+λ0k2 ρh)δ ϕ+(2 λ0kρh )δ λϕ0k+λ0k C (x , y)ϕ0k=0 .
(14)
Projecting  equation  (14)  on  the  mode ϕ0k gives  the 
correction terms δλ as
δλ=
−∫
S
ϕ0k
2 C (x , y)dS
2∫
S
ρhϕ0k
2 dS
       (15)
where S is  the  panel  surface.  This  leads  to  a  modal 
damping coefficient for the plate equal to
ξk=
−δλ
ω0k =
∫
S
ϕ0k
2 C( x , y)dS
2ω0k∫
S
ρhϕ0k
2 dS
.        (16)
If microperforations are uniformly spread over the structure, 
we get C (x , y)=C v and thus,
ξk=
C v
2ω0kρh
.        (17)
The  special  case  where  the  hole  distribution  is  uniform 
leads to the maximum value of  the modal  damping.  In  the 
general case, we have
ξk=
C v
2ω0kρh
−
∫̃
S
ϕ0k
2 C v d S̃
2ω0k∫̃
S
ρhϕ0k
2 d S̃
       (18)
where S̃ corresponds to the non-perforated surface.
Results  are  presented  in  the  next  paragraph  for  several 
perforation rates σ .
2.4 Parametric study
The  analysis  of  the  term C v (eq.  (6))  is  made  using  a 
parametric study. C v is written as
C v=
σ
S f (C f+
K f
jω)( jωM fK fjω+C f+ jω M f )=Rv+ jω I v  (19)
where Rv=
σ (C f+K f M f C f )
S f(C f2−(ωM f −K fjω )
2)
and
I v=
σ(M f C f2−K f M f2−K f2 M fω2 )
S f(C f2−(ωM f −K fjω )
2)
,  and plotted on the figure 5 for 
three  different  values  of  perforation  rate,  depending  on the 
following holes and cavity dimensions:
Perforation rate σ
(%)
Perforations diameter 
(m)
Honeycomb cell 
diameter (m)
0,1975 0,2.10-3 9.10-3
0,3968 0,2.10-3 6,35.10-3
0,8928 0,6.10-3 6,35.10-3
These curves show the classical effect of the resonance of 
the one-degree-of-freedom oscillator, whose natural frequency 
is f 0=
1
2π √ K fM f .
Figure 4: Plate's notations.
3
Using equation (17),  the modal damping of  a rectangular 
simply supported sandwich plate (dimensions 0,5m x 0,33m x 
0,02m and bending stiffness 150 kg.m².s-2) is computed for the 
first five modes (between 200 and 1000 Hz). Figure 6 plots 
the results as a function of perforation rate σ . 
Eigenfrequencies of the plates are found to be 210 Hz, 405 
Hz, 648 Hz, 729 Hz and 842 Hz.
The order of magnitude of the modal damping is rather low 
and  is  found  to  be  1.10-8  for  first  and  second  modes.  For 
modes of higher frequency and for higher perforation rates, 
the  modal  damping  is  found  to  be  2.10-6.  A  simple 
computation allows us to estimate the value that such modal 
damping should have to obtain a mobility difference of 1 dB 
between  the  non-perforated  and  the  perforated  panels.  This 
value  is  found  to  be  10-3.  Hence,  the  current  treatment  as 
dimensioned here, leads to a too low damping to reach this 
mobility difference.
3 CONCLUSION 
In this paper, we propose a methodology for computing the 
modal damping induced by micro-perforations performed on 
one  side  of  a  honeycomb  sandwich  panel.  This  method  is 
based on the Maa model for the fluid motion inside each hole, 
which  allows  us  to  estimate  an  equivalent  viscous  force 
(micro-scale  model).  The  resultant  force  is  then  computed 
considering a group of cells (meso-scale model) and allow us 
to express a term, whose real part expresses apparent damping 
for the global structure (macro-scale model).
A  perturbation  technique  is  used  to  estimate  the  modal 
damping coefficient.  The model can take into account a non-
uniform distribution of microperforations and the upper bound 
result  for  the  modal  damping  has  been  found.  Indeed,  it 
corresponds  to  the  case  where  the  distribution  of 
microperforations is homogeneous on the structure.
The order of magnitude observed for such modal damping 
is found to be rather low for the proposed geometry. 
ACKNOWLEDGMENTS
The authors  would  like  to  thank  Didier  Gangloff  from the 
Centre  National  des  Etudes  Spatiales  (CNES)  and  Eric 
Labiole from Thales Alenia Space, for the financial support of 
the thesis work.
REFERENCES
[1] D-Y.  Maa,  Theory  and  design  of  microperforated  panel  sound-
absorbing constructions, Scientia Sinica Vol. XVIII No. 1, Jan. - Feb. 
1975.
[2] D-Y.  Maa,  Wide-band  sound  absorber  based  on  microperforated  
panels, Chinese Journal of Acoustics, Vol. 4 No. 3, 1985. 
[3] D-Y. Maa, Potential of microperforated panel absorber, J. Acoust. Soc. 
Am. 104(5), November 1998.
[4] M.  Géradin,  D.  Rixen,  Théorie  des  vibrations  –  Application  à  la  
dynamique des structures, Elsevier Masson, Ch. III, 1993.
[5] J-L. Guyader, Vibration in continuous media, ISTE 2006.
[6] R.V. Waterhouse,  Interference Patterns in Reverberant Sound Fields, 
The Journal of the Acoustical Society of America, 27 (2), March 1955.
[7] Rayleigh,  Theory of sound, Dover Publications, New York, Vol. 1, p. 
42, 1945.
[8] T.  Dupont,  Transparence  et  absorption  acoustiques  des  structures  
microperforées, Thèse de l'Institut National des Sciences Appliquées de 
Lyon, 2002.
[9] N. Atalla, F. Sgard,  Modeling of perforated plates and screens using  
rigid frame porous models, Journal of Sound and Vibration 303, 195-
208, 2007.
[10] Y.Y. Lee, H.Y. Sun and X. Guo,  Effects of the panel and Helmholtz  
resonances on a micro-perforated absorber, Technical Note, Int. J. of 
Appl. Math and Mech., Vol. 4, 49-54, 2005.
[11] T. Bravo, C. Maury and C. Pinhède, Sound absorption and transmission  
through flexible micro-perforated panels backed by an air layer and a  
thin plate, J. Acoust. Soc. Am. 131 (5), May 2012.
[12] D. Cackley and J.  S.  Bolton,  The modeling of unconventional  sound  
absorbing  materials:  microperforated  films  and  closed  cell  foams, 
Société des Ingénieurs de l'Automobile Conference: lightweighting and 
acoustical materials in vehicles, UTC, October 22, 2013.
[13] R. Tayong Boumda,  T. Dupont and P. Leclaire,  On the variations of  
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panels  at high level of excitation,  J.  Acoust.  Soc.  Am. 127 (5),  May 
2010.
[14] M. Bruneau, Manuel d'Acoustique Fondamentale, Hermes, 1998.
Figure 6: Modal damping as a function of perforation rates, plotted for the  
first 5 modes.
Figure 5: Real part of the viscous damping as a function of frequency, for  
different perforation rates.
4


Vibration damping of honeycomb sandwich panels induced by micro-perforations

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Vibration damping of honeycomb sandwich panels induced by micro-perforations

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