PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Theoretical and experimental analysis of a skyhook damper for
active control of sound transmission
Neven ALUJEVIĆ1; Steven CLAES2; Martina ŠIMAG1; Paul SAS2
1

University of Zagreb, Croatia
2

KU Leuven, Belgium

ABSTRACTIn this paper active control of sound transmission through a double panel is considered. The active control
system consists of one accelerometer mounted on the radiating panel whose time-integrated and amplified
output is used as the error signal to drive a voice-coil actuator that reacts between the source and the radiating
panel. Effects of one such active damper in the centre of the double panel system are considered, although,in principle, it could be repeated in the x-y plane and then used in the centralised, distributed or a decentralisedmanner. Theoretical and an experimental results are presented and compared. In the theoretical part, themobility matrix model is used to investigate the stability of the feedback loop and to calculate the soundtransmission ratio with and without control. It is shown that large reductions of the sound transmission ratio
are possible, however only if the natural frequency of the fundamental source panel mode is higher than thenatural frequency of the fundamental radiating panel mode. The experimental part of the study is carried out
on the prototype active panel system and the measured result fully corroborate theoretical predictions.
Keywords: Sound, Vibration, Active control of sound transmission

1. INTRODUCTION
The construction of transportation vehicles such as cars, aircraft or helicopters typically involves
a considerable number of thin-walled and lightweight structural elements. From a vibroacoustic pointof view, such structural elements are efficient sound radiators and transmitters, particularly in the low
frequency range (1). There are several reasons why the low-frequency sound radiation andtransmission pose a particularly challenging problem in lightweight, thin -walled structures. First, thesound transmission through panels and similar structures is characterised by the so-called mass law.
The mass law imposes a decrease of the sound transmission ratio with the increase in frequency (2).
At low frequencies, however, the sound transmission ratio remains relatively high. In fact, the soundtransmission ratio through panel-like structures at low frequencies is largely controlled by structural
resonances so that both vibrations and sound transmission are high. The radiation of sound isparticularly high due to the lightly damped odd-odd structural modes characterised by high volume
velocity (3). Second, the mass law also imposes a decrease of the sound transmission ratio with theincrease of mass per unit area of the partition (2), imposing fundamental physical limitations tocontrolling the transmission of sound using lightweight structures. The combination of the two effects
two therefore causes problems in terms of high structural vibrations and the consequent high soundtransmission ratio of thin-walled lightweight structures at low frequencies.
These vibration and noise problems are traditionally solved by passive means (4). For example,
the resonance effect can be reduced by increasing structural damping using dissipative treatments.
These treatments are typically consisted of constrained damping layers of viscoelastic materials thatare glued to the structure. The problem with this technique is that it is typically targets only higher order modes of the structure and significantly increases the weight and cost of the product.
Alternatively, thin-walled structures can be equipped with single or multiple layers of sound absorbing
1

neven.alujevic@fsb.hr

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materials (4). These materials are typically open cell rubber foams or melamine sponges which absorb
noise by friction within the cell structure. Porous open cell foams are highly effective noise absorbers
across a broad range of medium to high frequencies (5). At low frequencies their performance
deteriorates. This is because of long acoustic waves at low frequencies. When the acoustic wavelength
becomes much larger than the thickness of the sound absorbing layer, the waves efficiently penetrate
through the layer. In conclusion, the control of sound transmission through lightweight, thin -walled
structures at low frequencies requires a special attention.
In this paper active control of sound transmission is considered through a double panel. A direct
velocity feedback control scheme with non-collocated sensor-actuator arrangement is used. It is shownthat active damping can be generated which reduces the sound transmission due to low-order structuralmodes. In Section 2 the mathematical model for the theoretical analysis is formulated whereas in
Section 3 the setup for the experimental measurement campaign is described. The results of thetheoretical and experimental analyses are contrasted and discussed in Section 4.

2. MATHEMATICAL MODEL
The mathematical model is based on the mobility-impedance approach (4,6). To this end, it isassumed that the system can be divided into three components: the source panel, the radiating panel
and the acoustical transmission path. The acoustical path is due to the sound transmission through the
air confined between the radiating and the source panel. Each of these components is modelled using
point and transfer mobility or impedance functions. The acoustical transmis sion path is modelledusing transfer impedances between a finite number of cavity elements that are adjacent to the surfacesof the panels. The excitation of the source panel by the incident acoustic wave and the radiated soundpower from the radiating panel are also calculated by assuming that the two panels are divided into
the same number of elements. This number is obtained by choosing element dimensions to belx,e=lx/(4M) and ly,e=ly/(4N) where M and N are the highest modal orders used in calculations. The
mobility model scheme is given in Figure 1.
zx
receiving panel
fr,vr

fr,1 vr,1

fr,2 vr,2

ft,vt

fs,vs

fr,k vr,k...

ftr,k vtr,kfcr,1 vcr,1

acoustic transmission pathcontrolforce

ftr,1 vtr,11 ftr,2 vtr,2...

fts,1 vts,1f v,1 ts,2 ts,2...

fts,k vts,kfcs,1 vcs,1

fs,1 vs,1 fs,2 vs,2...

fs,k vs,kprimary excitation
force vector fp

source panel

Figure 1 – The mobility model scheme
The model considers only out of plane displacements/velocities and forces at the various types ofjunctions and at the centres of the plates and cavity elements. The time harmonic displacement orforce are given by w(t)=Re{w(ω)e jωt } or f(t)=Re{f(ω)e jωt }, where ω is the circular frequency in [rad/s]
and j=-1 1/2 . Thus w(t) and f(t) are the time-harmonic displacement and force functions while w(ω) andf(ω) are the complex frequency-dependent displacement and force phasors. In order to simplify themobility formulation used in this study, the time harmonic dependence is implicitly assumed in themathematical expressions which are therefore formulated in terms of the frequency-dependent phasors.
Also, the first derivative of the time-harmonic functions, for example the linear out of plane velocity,
ẇ(t)=Re{jωw(ω)e jωt} is represented by velocity frequency dependent phasors ẇ(ω)=jωw(ω). The
velocity and force phasors at the centres of the elements are grouped in the following two column
vectors:v { ^w1 , w2 ,...w j ,...wk 1 , wk ` ,T3246

f { ^ N z1 , N z 2 ,...N zj , N zk 1 , N zk `T(1)

where ẇj is the complex amplitude of the linear velocity along the z axis, and Nzj is the complex
amplitude of the force in the z direction, at the j-th element. The two panels are also excited by meansof p control forces. The velocity and control force phasors at the control positions in the source andradiating panels are grouped in the following two column vectors:v c { ^w1 , w2 ,...w j ,...wp 1 , wp ` ,Tfc { ^ N z1 , N z 2 ,...N zj , N zp 1 , N zp `T

(2)

It is thus in principle possible to model a number of control forces acting on the source and/or on
the radiating panel. In the remaining of the paper, however, a single reactive control force, located at
the x,y centre of the double panel is considered. The surface boundaries that the air cavity shares withthe source and the radiating plate are modelled using a finite number k of small elements, such thatthe element dimensions are considerably smaller than the shortest acoustic wavelength in the cavity.
The lateral surfaces of the air cavity are assumed to be rigid walls. Each of the top and bottom surface
elements can only vibrate in the direction normal to the surfaces themselves and their velocities andforces are defined at the geometrical centres of the elements. These junction vectors are grouped
together to form four combined vector pairs. These four groups are: the source velocity vector vs andthe source force vector f s, the radiating velocity vector vr and the radiating force vector f s ; thetransmission system velocity vector vt and the transmission system force vector f t; and finally, thecontrol velocity vector v c and the control force vector f c. The four groups of vectors are given by:v s { ^vs ,1 vs ,2

vs ,k ` ,fs { ^ f s,1

f s,2

f s,k `T(3)

vr { ^vr ,1 vr ,2

vr ,k ` ,fr { ^ f r ,1

f r ,2f r ,k `

(4)

T

T

T

vt { ^vts

vtr ` { ^vts,1 vts,2vts,kvtr ,1,1 vtr ,2,2

vtr , k `T

(5)ft { ^fts

ftr ` { ^ fts,1

fts,2

fts, kftr ,,1

ftr ,2,

ftrr , k `T(6)v c { ^v cs

v cr ` { ^vcs ,1

vcs ,2

vcs , pvcr ,11vcr ,22

vcr , p `T(7)

fc { ^fcs

fcr ` { ^ f cs ,1

f cs ,2f cs , pf cr ,,11f cr ,22f cr , p `T(8)where vs,j and fs,j represent the complex velocities and forces for the j-th source panel element, vr,j andfr,j represent the complex velocities and forces at the j-th radiating panel element, v ts,j and fts,j represent
the complex velocities and forces for the j-th acoustic element on the source panel junction, v tr,j andftr,j represent the complex velocities and forces for the j-th acoustic element on the radiating panel
junction, vcs,j and fcs,j , and finally v cr,j and fcr,j represent the control system complex velocities andforces for the j-th control force at the j-th control point either on the source or on the radiating panel
footprint of the control force actuators.
The dynamics of the bare source and radiating panels are modelled using a mobility matrix
formulation, so that velocity and force vectors can be expressed in the form:v s Yss f s  Ysp f p  Ysc fc ,v r Yrr fr  Yrf f f  Yrc fc(9)where Yss, Ysp , Ysc and Y rr, Yrf, Yrc are mobility matrices of the source and the radiating panel, andf p , f c , and f f are the primary excitation vector, the control force vector, and the flanking excitation
vector, respectively. The details on how the mobility matrices used in Eqs. (9) are populated, and also
on how the the mobility and impedance matrices introduced in the forthcoming part of the formulation
are populated, are given in (6). Furthermore, details on how to calculate the resonance frequencies
ω m,n and mode shapes ϕm,n for rectangular panels having various boundary conditions can be found in(4). The primary and flanking excitation vector are given by:
f p { ^ f p ,1

f p ,2

f p,k ` ,T

f f { ^ f f ,1

f f ,2f f ,k `T(10)

The flanking excitation vector acting on the radiating panel could be caused by a subsystem
connected to it or by an additional flanking path connecting the source panel to the radiating panel.
The flanking excitation has not been considered throughout this study covered, so that the flanking
excitation vector f f is assumed to be a zero vector. If the source plate is excited by a plane acoustic
wave, then the components of the primary excitation vector are determined by pressure field generated

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by the plane wave over the surface of the source panel:lx l y j kx x j  k y y j(11)kwhere P is the amplitude of the plane wave which has an acoustic wave number in the x direction
given by k x=k sin(θ)cos(ϕ) and in the y direction given by k y=k sin(θ)sin(ϕ), where k is the wave
number, θ and ϕ are azimuthal and elevation angles, whereas x j and y j are coordinates of thef p, j ( x j , y j , Z)Pe

 j k x k ygeometrical centre of the corresponding element of the source panel. The term Pe x j y j in Eq.
(11) is the pressure at the geometrical centre of an element while the term lxly /k is the area of theelement. Therefore the excitation is modelled by assuming that the pressure field over the surface ofthe element can be approximated by the pressure at the centre of the element. The dynamics of thetransmission system is expressed using the following impedance matrix expression:ft Z t v t(12)where Zt is an impedance matrix of the transmission system. The elements in the Zt matrix, due to thenature of the acoustical coupling, are fully populated. Thus the velocity at one element will generate
a force, which is caused by pressure fluctuations at the centres of all the other elements, on both source
and radiating plates. The source and radiating panel mobility matrices, Eqs.(9), can be grouped
together in one equation:v sr Ysr , sr f sr  Ysr , pf f pf  Ysr ,c fc(13)where the mobility matrices and the excitation vectors have the form:
0 ºª Ys , pf
0 ºTTª Ys , srª Ys ,c º
Ysr , pf { «f pf { ^f p f f ` { ^f p 0`,,,
Ysr , sr { «Ysr ,c { «
(14)
»
»
»
Yr , pf ¼
Yr , sr ¼¬ 0¬ 0¬ Yr ,c ¼
and the velocity and force vectors are given by:TT(15)
v sr { ^v s v r ` ,f sr { ^f s fr `where vsr and f sr are respectively the source-radiating velocity vector and the source-radiating force
vector. The source-radiating vectors are related to the corresponding coupling system vectors so as tosatisfy the continuity (for the velocity vectors) and equilibrium (for the force vectors) principles at
each junction:
vt v sr ,ft fsr(16)
If Eqs.(16) are substituted into Eq.(12) the source-radiating force vector and force radiating
velocity vector can be related through the following impedance expression:
fsr Zt v sr
(17)
Substitution of Eq.(17) into Eq. (13) yields:Ysr , sr Zt v sr  Ysr , pf f pf  Ysr ,c fc ,

v sr

vsr

(I  Ysr , sr Zt )1 Ysr , pf f pf  Ysr ,c fc

v sr(I  Ysr , sr Zt )1 Ysr , pf f pf  (I  Ysr ,sr Zt )1 Ysr ,c fc ,v sr

Qtp f pf  Qtc fc

(18)
(19)

where the matrices Qtp and Qtc are given by:(I  Ysr , sr Zt )1 Ysr , pf ,(I  Ysr , sr Zt )1 Ysr ,c(20)

Using now Eq.(17) with Eq.(19) gives the source-radiating force vectors:
f sr Zt Qtp f pf  Zt Qtc fc ,f sr R tp f pf  R tc fc

(21)

Qtp

Qtc

where Rtp and R tc are given by:(22)
Rtc Zt Qtc
Rtp Zt Qtp ,Similar to Eq.(13), the control velocity vector can also be expressed using the mobility method:
vc Yc, sr fsr  Yc , pf f pf  Yc ,c fc(23)
where the mobility matrices have the form:ª Yc , pfs
0 ºª Yc , s
Yc , pf «,Yc , sr «»¬ 0¬ 0 Yc , r ¼

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0 º
,
Yc , pfr »¼

Yc ,cª Yc ,cs« 0¬

0 º
Yc ,cr »¼(24)

Yc,s , Yc,pfs , Yc,cs and Yc,r , Y c,pfr , Yc,cr are mobility matrices of the source and the radiating panel, at thecontrol locations. Substitution of the right term of Eq.(21) into Eq.(23) yields:
vc

Yc, sr Rtp f pf  Yc , sr Rtc fc  Yc , pf f pf  Yc ,c fc ,vc

vc

Yc, sr R tp  Yc, pf f pf  Yc , sr R tc  Yc ,c fc

Tcp f pf  Tcc fc

(25)
(26)

where Tcp and Tcc are given by:
Tcp

Yc , sr R tp  Yc , fp ,Tcc

Yc , sr R tc  Yc ,c(27)

With feedback control, the control force vector f c is related to the control velocity vector v c bymeans of the control law matrix H:
fc Hvc(28)
so that the control velocities can be calculated using Eq.(26) as follows:vc

Tcp f pf  Tcc Hvc ,vc

I  Tcc H

1

Tcp f pf

(29)

whereas the source and radiating panel forces are determined by the right term of Eq.(21). Details on
how the control matrix H is populated can be found in (7). Finally, the source and radiating velocities
are then given by Eq.(13). The sound power radiated by the radiating panel can then be evaluatedusing the velocities of the radiating elements which are a subset of vsr, the right term of Eq.(4), as:WZ

v reH Rv re(30)

where R is the radiation resistance matrix and ( )H denotes the Hermitian transpose (the complex
conjugate). The sound power of the incident plane wave can be calculated using the following
expression
lx l y(31)
Wi P 2cos T
2U cwhere U and c are the air density and the speed of sound. The sound power transmission ratio canbe calculated as T W Wi .

3. EXPERIMENTAL SETUP
The experimental study is carried out on a prototype double panel which is shown in Figure 2.

Figure 2 – Experimental setup

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The unconstrained x,y - dimensions of the panels are 0.3×0.4 m. They are made of a fiberglass
reinforced epoxy used for PCB boards. The width of the air cavity between them is 0.036 m. The
prototype panel is mounted on top of the open side of wooden box with thick and heavy walls with
the inner dimensions of 0.3×0.4×0.5 m. The box has a loudspeaker with a 30 cm membrane diameterto produce high levels of sound pressure inside the box at low frequencies. The thick and heavy box
side walls ensure that the flanking sound power radiated through the walls is at least 20 dB lower thanthat radiated through the panel. White noise signal is used to drive the loudspeaker. A reference
microphone is located in a corner inside the box, whereas the measurement microphone is locatedoutside, 10 centimetres above the radiating panel centre. The sound transmission ratio is approximatedby the FRF between the sound pressure measured by the outside microphone and the reference
microphone inside. The double panel system is equipped with a miniature voice coil actuator, (8),mounted at its centre, in the air cavity between the two panels. Its mechanical terminals are attachedto the two panels so that it can react against them. One miniature MEMS accelerometer, (9), is attachedat the actuator footprint at the radiating panel. Its time-integrated output, proportional to the radiating
panel velocity at the actuator junction point, is amplified and fed back to the actuator as the error
signal. The time integration is performed with an analogue integrator with a 1 Hz cut -off frequency.
In such a way a skyhook damper is realised (10,11). In order to ensure the stability of the feedback
loop, it is necessary that the uncoupled source panel fundamental natural frequency is higher than thatof the uncoupled radiating panel. Analytical stability analyses carried out previously with simple
lumped parameter 2 DOF models providing more details on why this is necessary to ensure stability
can be found in references (12–14). In this study the prototype panels were fabricated with slightly
different thicknesses: 2 mm for the source panel and 1.5 mm for the radiating panel so as to ensurethat the fundamental natural frequency of the source panel is higher than that of the radiating panel.

4. RESULTS AND DISCUSSION
The stability analysis is carried out using the Nyquist criterion by studying the properties of the
open-loop sensor-actuator frequency response function (FRF). The corresponding Bode and Nyquist
plots determined theoretically and experimentally are shown in Figure 3.

Figure 3 – Bode and Nyquist plots of the open loop sensor-actuator FRF. Up: simulated, bottom: measured

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The simulated and measured results agree reasonably well. The FRF behaves as if the sensoractuator pair were collocated up to approximately 400 Hz. Above this frequency phase lags occur due
to higher order modes so that the feedback loop is conditionally stable. The frequencies at which the
Nyquist locus crosses the negative real axis are shown in Bode plots using the red circles. The
amplitudes of the FRF at the crossover frequencies are much smaller that the FRF amplitudes at thestructural resonances. Therefore, a relatively large gain margin enables implementing large feedback
gains and generating large active damping effects. This is illustrated in Figure 4, which shows thesound power transmission ratio with and without control, both simulated and measured. The two plotssuggest that large sound transmission occurs due to the two fundamental resonances of the double
panel system occurring at 135 Hz and 181 Hz.

Figure 4– The sound power transmission ratio. Left: simulated, right: measured. Solid, blue lines: nocontrol, dashed red lines: with control.
This is because the two corresponding structural modes are characterized by large volumetriccomponents of the radiating panel vibration, as shown in Figure 5. Above the second resonance (themass-air-mass mode shown in Figure 5) the sound transmission ratio rolls off steeply with frequency
since the vibration of the two panels and the sound transmission become governed by the mass law,
Figure 4. Therefore the active reduction of the sound transmission at frequencies below the mass -airmass resonance is important. As shown by the red dashed lines, reductions of about 20 dB are both
anticipated theoretically (Figure 4, left plot) and determined experimentally (Figure 4, right plot). The
simulation and experimental results agree well, except that the (0,0,1) acoustical mode of the wooden
box air cavity causes a resonance at about 340 Hz in the experimentally determined sound powertransmission ratio. Note that the theoretical model does not include the acoustics of the rectangular
air cavity beneath the source panel.

Figure 5– The two fundamental modes of the double panel. Left: first mode, right: mass-air-mass mode3251

CONCLUSIONS
The active control of sound transmission through a double panel using a skyhook damper locatedin the air cavity between the source and the radiating panel is investigated theoretically andexperimentally. The stability analysis indicates that the feedback loop is surprisingly robust withrespect to the non-collocated sensor-actuator arrangement. The arrangement is non-collocated because
the reactive force actuator is collocated (and dual) to the sensor measuring rel ative velocity across its
terminals. However in the present scheme an absolute velocity sensor at the radiating panel actuatorterminal is used to provide the error signal. Although the theoretical foundation for this type of active
vibration control have been laid for lumped parameter systems in the past, in this study it is shown
both experimentally and theoretically that this kind of control scheme can be successfully used in
distributed parameter systems. In particular, it is shown that the sound power transmission ratio
through double panels can be significantly reduced in the low frequency range, below the mass -airmass resonance. This is important because at those frequencies the acoustic and structural wavelengths
are large so that the effectiveness of traditional passive treatments such as sound absorbing foams or
viscoelastic constrained damping layers is unsatisfactory. Therefore active control can be used as a
viable alternative to passive sound transmission control methods in order to reduce the very low
frequency sound transmission.

ACKNOWLEDGEMENTS
This project has received funding from the European Union's Horizon 2020 research andinnovation programme under the Marie Sklodowska-Curie grant agreement no. 657539.

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