# Improvement of Transmission Loss Using Active  Control with Virtual Modal Mass

Authors: V. Lhuillier; L. Gaudiller; C. Pezerat; S. Chesne
Journal: Advances in Acoustics and Vibration
 (2008)

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## Abstract

This paper deals with an alternative modal active control approach to reduce sound transmission
through a structure excited by an acoustic wave. Active control makes it possible to conserve lightness
while improving acoustic performances. “Modal mass damping control” is proposed for light and small
structures having slight modal overlap. The aim of this control is to modify the modal distribution of high
radiation efficiency modes with active modal virtual mass and active modal damping. The active virtual
mass effects lower eigen frequencies to less audible frequency range while reducing vibration amplitudes
in a broad frequency range. An application of this concept is presented in a simple smart structure. It
is harmonically excited on large bandwidth by a normal acoustic plane wave. Results obtained by active
modal virtual mass and damping control are compared to other modal control approaches.


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## Body

## 1. Introduction

The acoustic
transmission loss can be improved by passive methods such as mass addition
(Mass Law [1]),
double panel, and/or the use of sound absorbing materials. Most of these
techniques generally involve an increase in mass or volume to provide a good
insulation at low frequencies. However, this is not convenient in the transport
sector.Active control can complete the passive methods and
reduce the sound produced by vibrating structures at low frequencies. Various
active control strategies have been developed.In the 1930s, research focused on active noise control
ANC (Lueg's Patent [2]) in which a secondary field destructively interferes
with the primary disturbance. However, these feedforward controllers are tricky
to implement because of the difficulty in measuring reference signal and
determining the feedback effect of the secondary sources on the reference
sensors. ANC was only successfully implemented in applications, where
disturbance is tonal such as ducts and exhaust stacks [3].Active structural acoustic control (ASAC) controllers
were then developed to reduce sound radiation by modifying panel vibrations
with shakers or piezoelectric patches.The ASAC feedforward approach has the same drawbacks
as ANC. Moreover, for the low-frequency range, the minimization of sound
radiation achieved by “modal restructuring” can increase the
vibration levels [4].
This method uses coupling between modes below the critical frequency to create
destructive interferences in the radiated field.If the disturbance signal is not available, control
strategies are limited to feedback controllers. The type of controller differs
as a function of the availability of a model. Elliott and Johnson [5] showed that at low
frequencies, reducing the volume velocity of the panel leads to reduced sound
power.Networks of sensors [6, 7] or distributed sensors such as PVDF film [8] have been implemented to
evaluate the volume velocity and to supply an SISO controller. System stability
is guaranteed [9] when
the actuators and the sensors are collocated and dual. An alternative to SISO
controllers called “decentralized MIMO” consists in scattering
manifold independents control units on the structure [10]. The results are fairly
good but the technical resources needed are so huge that smart structures
cannot be easily introduced in everyday equipment such as double-glazed
windows, machinery shielding, vehicles. Moreover, active damping provides good
performances without a model, but it is only efficient in the vicinity of a
resonance frequency.The model-based control strategy presents several
advantages. It allows minimizing the number of sensors with modal
reconstruction [11]
and reducing control energy by nonlinear modal control [12, 13]. Complex and nonlinear
structures can also be controlled with modal adaptative algorithm [14].In the vibroacoustic field, accurate modeling is also
required to adjust the controller. Baumann et al. [15, 16] proposed to compute sound power by using radiation
filters. These filters are introduced in the state space model of the structure
and the acoustic energy is incorporated in the cost function. Experiments
carried out by Bingham et al. [17] and Dehandschutter et al. [18] demonstrated the validity
of this method, but this technique is limited by the number of radiation
filters that have to be introduced in the state space formulation.Below the critical frequency, transmission is mass
controlled [1].
Alujevic and Gardonio [19] showed
that a light panel controlled by an active virtual mass with decentralized MIMO
controllers behaves like a noncontrolled heavy panel. Transmission loss is
increased but the sound power peaks still remain due to the lack of active
damping.The aim of this paper is to describe the reduction of
sound transmission by modifying the modal distribution. When structures are
relatively small and light, the panels have low modal overlap, so in this case,
modal control appears to be adapted. It enables limiting the number of active
components and concentrating control energy on high radiation efficiency modes.
Active modal damping squeezes sound power peaks and controls low radiation
efficiency modes which may be excited and transfer vibrating energy to structure
bounds. Moreover, above the critical frequency, when the incidence angle is
close to the coincidence angle, the behavior of the panel is controlled by
damping. Consequently, transmission loss drops dramatically for a lightly
damped structure.  The effect of adding modal masses
shifts the resonance frequencies to a less audible frequency range. Also, the
mass control of one eigen mode can be considered as an addition of virtual
modal mass. Thus, the acoustic characteristics of the structure should be
improved in the upper frequency range.This paper presents a state feedback controller which
acts on modal mass and modal damping. After the state-of-the-art presented
previously, Section2 deals with modeling the structure and the method used to
compute the sound power. Section 3 introduces the vibroacoustic controller
driven by modal accelerations and modal velocities. The principle of this new
“mass damping vibroacoustic modal control” approach is shown in
Section 4 on a simple one-dimensional structure excited by a normal incident
plane wave. It is then compared to other vibroacoustic modal controls in
simulations.


## 2. Modeling

### 2.1. Structural Modeling

The modal
control enables concentrating control energy on high-radiation efficiency modes
and limiting the number of active components. Moreover, the effectiveness of
this model-based strategy depends on the accuracy of the modeling. The first
step consists in building a model that includes the mass and the stiffness of
the actuators. The structure is divided into a finite number of elements. The
equation of motion governing the dynamic of the controlled system
is(1)(Ms+Ma)δ¨+Csδ˙+(Ks+Ka)δ=Fd+Fc,
with Fd being the disturbance force, Fc the control force, δ the displacement, M,C,K the mass, damping, and stiffness matrices,
respectively. The subscripts a and s denote the actuators and the structure,
respectively. The control effort driven by the displacement, the velocity, and
the acceleration acts on the stiffness, the damping, and the mass of the
system, respectively. When the structure is lightly damped and the modes are
sufficiently decoupled, the linear system is described by a set of decoupled
modal equations after a change of variable:(2)δ=ϕq,(3)q¨i+2ξiωiq˙i+ωi2qi=ϕiT(Fd+Fc)=fd(i)+fc(i),with qi being the modal amplitudes, ξi the modal viscous damping ratio, ϕi the modal shapes, and ωi the frequency of the ith mode. The response v of the structure can be given as a linear
combination of the modes:(4)v(x,y,z,t)=∑n=1Nqn(t)ϕn(x,y,z).The corresponding modal state
variable form is given by(5)x˙=Ax+Bu+Ew,y=Cx+Du,x={qq˙},with x being the state vector, u the control vector, w the disturbance noise, y the output vector, A,B,C,D the state matrices and E the disturbance input matrix. From this
formulation, the transfer function of the modal velocities can be easily
computed and introduced into the following acoustic modeling.


### 2.2. Acoustic Modeling

To predict the
acoustic performances, it is necessary to model the sound radiation of the
vibrating structure. The acoustic powerW can be calculated from the modal amplitudes
and from the frequency dependent radiation resistance matrix of structural
modes M [5]. The diagonal terms of M are the self-radiation
resistance and the off-diagonal terms are the mutual radiation resistance. The
global sound power W can be obtained by(6)W=q˙HMq˙,where the subscript H denotes the Hermitian. For complex geometries, M can be calculated from the radiation
resistance matrix of elemental radiators and modal shapes. Each element acts as
an elementary radiator, where the specific acoustic transfer impedance at
position y on an infinite plane to an observation point x is given by(7)z(x,y)=p(y)v(x)=jωρSe2πre−jkr,with r being the distance
between x and y, ρ the fluid density, k the acoustic wave number,
and Se the radiator area. It is assumed that the
elementary radiators radiate into free space and are small compared to the
acoustic wave length. The sound power radiated by only one single element We is equal to(8)We=Se2ℜ{ve*pe},where pe and ve are, respectively, the pressure and the
complex velocity of the elemental radiator. ℜ denotes the real part and * the conjugate. The global acoustic power W radiated by a set of elemental sources is
calculated with(9)W=S2ℜ{vHp},where v and p are, respectively, the velocity vectors and
pressure vectors of the radiators. Next, the matrix of acoustic transfer
impedances Z linking the pressure at each element to the
velocity at each element is introduced. rij denotes the distance between elemental
sources. Rij is the radiation resistance matrix of the
elemental radiators. W becomes(10)W=Se2ℜ{vHZv}=Se4vH(Z+ZH)v=vHRv,with (11)Rij=Se2ℜ{Zij}=ω2ρS24πcsin⁢⁡krijkrij,with c being the speed of sound
in the medium, ω the frequency, and ρ the mass density of air. The acoustic power
can be obtained from the radiation resistance matrix of the elemental radiators
and the velocity of structural modes after a change of variable
(2),(12)W(ω)=vHR(ω)v=q˙HϕHR(ω)ϕq˙,M(ω)=ϕHR(ω)ϕ.Then, for any structures and for any set of
frequencies, the radiation resistance matrix of structural modesM can be calculated and approximated using a
Laplace-domain multi-input, multiple output transfer function M(s). M(s) is then factorized into a stable causal
radiation filter G(s) (13) and W(s) is given by (14)(13)M(s)=GT(−s)G(s),(14)W(s)=Q˙(−s)TM(s)Q˙(s),with Q˙(s) being the modal velocities in the Laplace
domain.


## 2.1. Structural Modeling

The modal
control enables concentrating control energy on high-radiation efficiency modes
and limiting the number of active components. Moreover, the effectiveness of
this model-based strategy depends on the accuracy of the modeling. The first
step consists in building a model that includes the mass and the stiffness of
the actuators. The structure is divided into a finite number of elements. The
equation of motion governing the dynamic of the controlled system
is(1)(Ms+Ma)δ¨+Csδ˙+(Ks+Ka)δ=Fd+Fc,
with Fd being the disturbance force, Fc the control force, δ the displacement, M,C,K the mass, damping, and stiffness matrices,
respectively. The subscripts a and s denote the actuators and the structure,
respectively. The control effort driven by the displacement, the velocity, and
the acceleration acts on the stiffness, the damping, and the mass of the
system, respectively. When the structure is lightly damped and the modes are
sufficiently decoupled, the linear system is described by a set of decoupled
modal equations after a change of variable:(2)δ=ϕq,(3)q¨i+2ξiωiq˙i+ωi2qi=ϕiT(Fd+Fc)=fd(i)+fc(i),with qi being the modal amplitudes, ξi the modal viscous damping ratio, ϕi the modal shapes, and ωi the frequency of the ith mode. The response v of the structure can be given as a linear
combination of the modes:(4)v(x,y,z,t)=∑n=1Nqn(t)ϕn(x,y,z).The corresponding modal state
variable form is given by(5)x˙=Ax+Bu+Ew,y=Cx+Du,x={qq˙},with x being the state vector, u the control vector, w the disturbance noise, y the output vector, A,B,C,D the state matrices and E the disturbance input matrix. From this
formulation, the transfer function of the modal velocities can be easily
computed and introduced into the following acoustic modeling.


## 2.2. Acoustic Modeling

To predict the
acoustic performances, it is necessary to model the sound radiation of the
vibrating structure. The acoustic powerW can be calculated from the modal amplitudes
and from the frequency dependent radiation resistance matrix of structural
modes M [5]. The diagonal terms of M are the self-radiation
resistance and the off-diagonal terms are the mutual radiation resistance. The
global sound power W can be obtained by(6)W=q˙HMq˙,where the subscript H denotes the Hermitian. For complex geometries, M can be calculated from the radiation
resistance matrix of elemental radiators and modal shapes. Each element acts as
an elementary radiator, where the specific acoustic transfer impedance at
position y on an infinite plane to an observation point x is given by(7)z(x,y)=p(y)v(x)=jωρSe2πre−jkr,with r being the distance
between x and y, ρ the fluid density, k the acoustic wave number,
and Se the radiator area. It is assumed that the
elementary radiators radiate into free space and are small compared to the
acoustic wave length. The sound power radiated by only one single element We is equal to(8)We=Se2ℜ{ve*pe},where pe and ve are, respectively, the pressure and the
complex velocity of the elemental radiator. ℜ denotes the real part and * the conjugate. The global acoustic power W radiated by a set of elemental sources is
calculated with(9)W=S2ℜ{vHp},where v and p are, respectively, the velocity vectors and
pressure vectors of the radiators. Next, the matrix of acoustic transfer
impedances Z linking the pressure at each element to the
velocity at each element is introduced. rij denotes the distance between elemental
sources. Rij is the radiation resistance matrix of the
elemental radiators. W becomes(10)W=Se2ℜ{vHZv}=Se4vH(Z+ZH)v=vHRv,with (11)Rij=Se2ℜ{Zij}=ω2ρS24πcsin⁢⁡krijkrij,with c being the speed of sound
in the medium, ω the frequency, and ρ the mass density of air. The acoustic power
can be obtained from the radiation resistance matrix of the elemental radiators
and the velocity of structural modes after a change of variable
(2),(12)W(ω)=vHR(ω)v=q˙HϕHR(ω)ϕq˙,M(ω)=ϕHR(ω)ϕ.Then, for any structures and for any set of
frequencies, the radiation resistance matrix of structural modesM can be calculated and approximated using a
Laplace-domain multi-input, multiple output transfer function M(s). M(s) is then factorized into a stable causal
radiation filter G(s) (13) and W(s) is given by (14)(13)M(s)=GT(−s)G(s),(14)W(s)=Q˙(−s)TM(s)Q˙(s),with Q˙(s) being the modal velocities in the Laplace
domain.


## 3. Modal Mass Damping Vibroacoustic Control

### 3.1. Principle

When a
structure is excited by a diffuse field below its critical frequency, the
transmission is “mass controlled” [1]. The effect of mass addition at low frequency slides
the resonance frequencies to a less audible frequency range, while reducing the
vibration amplitudes. Therefore, the overall transmission loss is raised.
Removing the stiffness scales down the eigen frequencies but increases the
level of vibration. Reducing eigen frequencies with stiffness control cannot be
considered due to the risk of instability.If the frequency is slightly greater than the critical
frequency, the transmission will be “damping controlled” [1]. In case of lightly damped
structures, active damping becomes essential to reduce transmission. At higher
frequencies, the good sound insulation does not justify extending the control
bandwidth.In the framework of small and lightly damped
structures, the modal control can be effective due to low modal overlap. When
the panel is excited by an acoustic plane wave, the low-order modes (odd-odd)
are highly excited. Controlling only these high-radiation modes may lead to a
considerable reduction of sound power.The aim of themodal mass—damping vibroacoustic
control— is to modify the modal distribution of the panel. The natural
frequencies of high-radiation modes can be lowered by the addition of active
modal mass and vibration amplitudes can be reduced at resonances frequencies by
modal damping. The active modal mass can be considered as a virtual mass concentrated on controlled modes. Consequently, the amplitudes of mass
controlled modes are reduced after resonances due to frequency shifts. Active
modal damping limits sound power peaks and the vibrations of low-radiation modes.
The control gain matrix is computed with an optimal control algorithm. If the
cost function considers exclusively sound power, certain nonradiating modes may
be highly excited leading to risks of failure and structure-born transmission.
The control is driven by modal acceleration and modal velocities from the
derivative state. In practice, modal displacements and velocities are
calculated from sensors signals with a Luenberger observer [20]. Then, the reconstructed
state feeds the controller whose dynamics is nearly independant of the
controlled structure. For simplicity, observation problems are not treated in
this article:(15)u=−Kx˙c=−∑i=1NcKv(i)q˙i+Ka(i)q¨i.Then, the active mass and
damping can be expressed in the modal equation:(16)[1+Ka(i)⋅fp(i)︸Active⁢modal⁢mass]q¨i+[2ξiωi+Kv(i)⋅fp(i)︸Active⁢modal⁢damping]q˙i+ωi2qi=fd(i)−fp(i)∑j⁢≠iKv(j)q˙j+Ka(j)q¨j︸Excitation⁢of⁢controlled⁢modes,with fp(i) being the control effort at the ith mode, xc the state vector of controlled modes, and Ka and Kv the gains relative to modal accelerations and
velocities. The state of the controlled system can be calculated from (5) and
(15) with the following transfer function:(17)Y(s)=(C−DKs)((I+BK)s−A)−1EW(s).Next, the modal velocities enables computing the sound
power.q¨ is a highpass filter. Consequently, when the
frequency is higher than the eigen frequencies of the mass controlled modes and
gains relative to the accelerations are high, the frequency response function
(FRF) of the control will be constant. The problem of unwanted excitation
(spillover) of modes especially above the control bandwidth has to be
considered. In the control bandwidth, spillover can be attenuated with good
actuator positions. At high frequency, the passive methods are sufficiently
efficient. For instance, in the framework of reducing the sound transmission of
the double panel (DP) the consequences of spillover on the higher modes are
limited due to their good natural transmission loss (TL) in midrange and high
frequencies. The performances of DP can be increased at low frequencies without
excessive alteration of the TL at higher frequencies by adding modal masses
exclusively to high radiation modes.


### 3.2. Optimization and Adjustment

When the
structure is assumed to be linear and its model is available, the optimization
method used can be the linear quadratic (LQ) algorithm. The constant gains are
obtained after minimization of a quadratic cost function. The optimal control
seeks to minimize both the state function of structural systemx,
which is usually vibration energy or acoustic energy, and the control cost u.
The control gains are computed by solving Ricatti's equation but using the
derivative of state vector x˙ does not allow computing gains in this way. If
the cost function considers the control energy, the optimal gain may favor
damping rather than mass control due to the acceleration FRF. During
experiments, the maximal acceptable voltage of the transducers is a factor that
limits controller performances. Therefore, the cost function computes the
acoustic energy and forbids a control voltage higher than the voltage limit.
This can be defined as follows:(18)if⁢V≤Vmax,J=∫ωinferiorωsuperiorQ˙(ω)HM(ω)Q˙(ω)dω,if⁢V>Vmax,J=∞.


## 3.1. Principle

When a
structure is excited by a diffuse field below its critical frequency, the
transmission is “mass controlled” [1]. The effect of mass addition at low frequency slides
the resonance frequencies to a less audible frequency range, while reducing the
vibration amplitudes. Therefore, the overall transmission loss is raised.
Removing the stiffness scales down the eigen frequencies but increases the
level of vibration. Reducing eigen frequencies with stiffness control cannot be
considered due to the risk of instability.If the frequency is slightly greater than the critical
frequency, the transmission will be “damping controlled” [1]. In case of lightly damped
structures, active damping becomes essential to reduce transmission. At higher
frequencies, the good sound insulation does not justify extending the control
bandwidth.In the framework of small and lightly damped
structures, the modal control can be effective due to low modal overlap. When
the panel is excited by an acoustic plane wave, the low-order modes (odd-odd)
are highly excited. Controlling only these high-radiation modes may lead to a
considerable reduction of sound power.The aim of themodal mass—damping vibroacoustic
control— is to modify the modal distribution of the panel. The natural
frequencies of high-radiation modes can be lowered by the addition of active
modal mass and vibration amplitudes can be reduced at resonances frequencies by
modal damping. The active modal mass can be considered as a virtual mass concentrated on controlled modes. Consequently, the amplitudes of mass
controlled modes are reduced after resonances due to frequency shifts. Active
modal damping limits sound power peaks and the vibrations of low-radiation modes.
The control gain matrix is computed with an optimal control algorithm. If the
cost function considers exclusively sound power, certain nonradiating modes may
be highly excited leading to risks of failure and structure-born transmission.
The control is driven by modal acceleration and modal velocities from the
derivative state. In practice, modal displacements and velocities are
calculated from sensors signals with a Luenberger observer [20]. Then, the reconstructed
state feeds the controller whose dynamics is nearly independant of the
controlled structure. For simplicity, observation problems are not treated in
this article:(15)u=−Kx˙c=−∑i=1NcKv(i)q˙i+Ka(i)q¨i.Then, the active mass and
damping can be expressed in the modal equation:(16)[1+Ka(i)⋅fp(i)︸Active⁢modal⁢mass]q¨i+[2ξiωi+Kv(i)⋅fp(i)︸Active⁢modal⁢damping]q˙i+ωi2qi=fd(i)−fp(i)∑j⁢≠iKv(j)q˙j+Ka(j)q¨j︸Excitation⁢of⁢controlled⁢modes,with fp(i) being the control effort at the ith mode, xc the state vector of controlled modes, and Ka and Kv the gains relative to modal accelerations and
velocities. The state of the controlled system can be calculated from (5) and
(15) with the following transfer function:(17)Y(s)=(C−DKs)((I+BK)s−A)−1EW(s).Next, the modal velocities enables computing the sound
power.q¨ is a highpass filter. Consequently, when the
frequency is higher than the eigen frequencies of the mass controlled modes and
gains relative to the accelerations are high, the frequency response function
(FRF) of the control will be constant. The problem of unwanted excitation
(spillover) of modes especially above the control bandwidth has to be
considered. In the control bandwidth, spillover can be attenuated with good
actuator positions. At high frequency, the passive methods are sufficiently
efficient. For instance, in the framework of reducing the sound transmission of
the double panel (DP) the consequences of spillover on the higher modes are
limited due to their good natural transmission loss (TL) in midrange and high
frequencies. The performances of DP can be increased at low frequencies without
excessive alteration of the TL at higher frequencies by adding modal masses
exclusively to high radiation modes.


## 3.2. Optimization and Adjustment

When the
structure is assumed to be linear and its model is available, the optimization
method used can be the linear quadratic (LQ) algorithm. The constant gains are
obtained after minimization of a quadratic cost function. The optimal control
seeks to minimize both the state function of structural systemx,
which is usually vibration energy or acoustic energy, and the control cost u.
The control gains are computed by solving Ricatti's equation but using the
derivative of state vector x˙ does not allow computing gains in this way. If
the cost function considers the control energy, the optimal gain may favor
damping rather than mass control due to the acceleration FRF. During
experiments, the maximal acceptable voltage of the transducers is a factor that
limits controller performances. Therefore, the cost function computes the
acoustic energy and forbids a control voltage higher than the voltage limit.
This can be defined as follows:(18)if⁢V≤Vmax,J=∫ωinferiorωsuperiorQ˙(ω)HM(ω)Q˙(ω)dω,if⁢V>Vmax,J=∞.


## 4. Application to a One-Dimensional Structure

The aim of this
section is to present modal adjustment possibilities. Constraints such as
number of components, dimensions, saturation, and measurements are not
discussed in this first study. Simulations are performed on a one-dimensional
structure using the previously described control approach. The structure is
excited by a normal acoustic plane wave to emphasize active modal mass control
within the framework of sound transmission. This modal method enables
concentrating control energy on high-radiation efficiency modes and limits the
number of actuators. Consequently, the actuators are mainly used to act on the
first odd modes. The first step of this control strategy requires using system
model.


### 4.1. Modeling

The smart
structure considered and shown in Figure1 is a clamped-clamped beam to which
three piezoelectric patches (P1 89 ceramics) are fixed. The dimensions and
material characteristics are given in Table 1. An FE Model is developed using Ansys software, where the beam is modeled with
Solid45 elements (8 nodes with 3 mechanical dof/node) of 2.5×2.5×1.5 mm and the
piezoelectric patches are modeled with Solid5 elements (8 nodes with up to 6
dof/node including structural and electrical dof) of 2.5×2.5×0.7 mm. For the
sake of simplicity, 3 identical patches are bonded to the smart structure.Table 1
Simulations parameters.
Beam densityρbeam=7800Kg/m3Beam widthb=4.0*10−2mBeam
Young modulusE=2.1*1011PaBeam heighth=1.5*10−3mModal
damping ratioξi=0.01Beam lengthlb=6.0*10−1mCeramic
densityρc=7700Kg/m3Patch widthbp=3.0*10−2mCeramic
Young modulusY11=6.67*1010PaPatch heighteP=0.7*10−3mY33=5.26*1010PaPatch lengthlP=8*10−2mPermittivityϵ11s=1142Position 1A1=2.5*10−3mϵ33s=668Position 2A2=4.25*10−1mPiezoelectricityd31=−108pC/NPosition 3A3=1.625*10−1md33=240pC/NFigure 1
Smart structure.The first 14 modal shapes are extracted and the
optimal position of the actuators are sought by maximizing the coupling
coefficientk of one high-radiation efficiency mode
and minimization of coupling coefficient of other modes. The generalized
coupling coefficient kij of the ith mode for the jth piezoelectric patch bonded to the structure
is defined by Hagood and von Flotow [21] as follows:(19)kij2=(ωijD)2−(ωijE)2(ωijE)2,where ωijD and ωijE are, respectively, the natural frequencies of
the structure when the piezoelectric jth patch is open-circuited and
short-circuited. The modal coupling coefficient is defined by Badel et al.  [22]:(20)|αij|=C0PKPDikij2withC0P=2ϵ33slPbeP,where KPDi is the equivalent stiffness of the ith mode and C0P is the null strain capacitance of the
piezoelectric patch. lP,b,andeP are, respectively, the length, width, and
thickness of the patches. Next, the modal coupling is introduced in
(2):(21)q¨i+2ξiωiq˙i+ωi2qi=fd(i)+∑j=1NActuatorsαijVj,where Vj is the control voltage of the jth patch. The results of optimal placement are
presented in Figure 1.Once the structural modeling has been completed, the
radiation resistance matrixM(ω) is calculated in a frequency band from 10 to
900 Hz and approximated using the Laplace-domain multi-input, multiple output
transfer function M(s) (14). The eigen frequencies of the smart
structure are presented in Table 2.Table 2
Eigen frequencies.
Mode12345678Frequency (Hz)2469130215325451617794


### 4.2. Control Gain Matrix

#### 4.2.1. Transferring Control Energy from Damping to Mass

In order to
present the advantages of a derivative state feedback control, the control
effort is initially concentrated on modal damping before being transferred to
modal mass. The initial terms of the gain matrix relative to damping can be
calculated by solving the Ricatti equation stemming from the minimization of
the frequency weighted energy cost function (22). Then, the terms relative to
modal mass are introduced and those such as maximal voltage actuators inputs
remain constant. The modal mass control is applied here exclusively on the
first mode because of its high contribution to the sound power. Indeed, at low
frequency and for this type of structure, the sound power is mainly radiated by
volumetric modes [5].In Figure2, exclusive modal damping control is
represented in blue. The considered excitation is a normal acoustic plane wave
which is uniform on all the plate. The line color gets hotter when the first
mode acceleration gain increases. The left-hand plot shows the evolution of the
sound power of the controlled system computed with (14) and (17). When the
modal acceleration gain increases, the first eigen frequency scales down and
the overall sound power is reduced by the active mass. However, limiting the
control effort does not allow still introducing as much active modal damping.
Consequently, the sound power peak of the mass controlled mode is not reduced
as in the exclusive active damping control.Transfer of control energy from modal damping to modal mass: (a) sound power,
(b) control.
(a)(b)The performances of the controlled systems differ
drastically according to the weighting on the mass and damping. If the aim is
to reduce the sound power peaks, the control will focus its energy on damping.
A good tradeoff between mass and damping can be found when reducing sound
power.The right-hand side plot shows the control frequency
response of actuatorA1. This frequency response becomes constant when the
gains applied to acceleration are high compared to those applied to damping.
With full damping, the command is like a pass band filter, whereas it is like a
highpass filter in mass control. Consequently, the control energy of the
vibroacoustic modal mass control is slightly higher than that of the regular
state feedback controller. Note that in Figure 2(b), the control voltage is
calculated at the initial state which leads to the highest control voltage. So,
the first peak of the control appears at 24 Hz.


#### 4.2.2. Minimization of Acoustic Energy

In Figure3(a), the cost function computed with (18) between 10 and 900 Hz is drawn with
respect to the acceleration gain and velocity gain of the first eigen mode.
When the control voltage is higher than the maximal acceptable voltage of the
transducers, the cost function reaches infinity. The bold line indicates the
border beyond which the cost function tends to infinity. The acoustic energy
reaches a minimum for a given gain couple. When the gain applied on acceleration
is high compared to that on the velocity, the acoustic energy increases because
of the peak on the first mode due to the limitation of control voltage. The
right-hand side plot shows the sound power of the mass damping controlled
system computed with an optimal gain matrix.Evolution of acoustic energy with respect to the acceleration gain and the
velocity gain on first eigen mode: (a) sound power with optimal gains (b).
(a)(b)The optimal gain matrix is obtained from the results
presented in Figure3. Then, as shown in Figure 4(a) time response simulation
of the structure excited by a plane shock normal wave is performed with the
optimal gain matrix. The sound power is presented in the left-hand side plot.
The high-frequency components of the sound power vanish after a few
oscillations due to the active damping on these modes. Consequently, the sound
power is mainly created by the first mode which is initially more disturbed. In
the right-hand side plot, the first mode velocity shows that the mass effect
increases the oscillation period and the amplitude is reduced with active
damping.(a) Sound power, (b) first mode velocity.
(a)(b)


### 4.3. Effects of the Modal Mass Control Compared to Other Modal Controls

#### 4.3.1. Other Approaches

In this
section, the modal mass control is compared with two other modal control
approaches. The first is a pure vibration control using a regular state feedback
control with gains computed by minimizing the frequency weighted (FW) energy
cost function (22):(22)J=∫0∞(xtQx+ρutu)dt.The second control strategy considers vibration and
acoustics. In Baumann et al.' [15] development, the radiation filters (RF) G(s) are converted to state space form (23) and
introduced in the augmented state space of the system (24):(23)r˙=AGr+BGq˙,z=CGr+DGq˙,(24)[{q˙q¨}r˙]=[A00BGAG][{qq˙}r]+[B0]u,with z being the result of the passing velocity
components through the radiation filter. z2 is the sound power. In the case of a normal
incident plane wave, the general force decays with increasing frequency, which
is why an integrator term is added in (26). If the optimal gain matrix is
computed by a minimization of z2,
low-frequency modes will not be controlled because the M(ω) terms are small when the modes do not radiate
independently. Consequently, the cost function that considers the disturbance
can be written as follows:(25)JRad=∫0∞(z2Tz2+ρuTu)dt,(26)z2=ℒ−1(s−1G(s)Q˙(s)),where z22 is the weighted sound power.


#### 4.3.2. Transmission Loss

Considering
weak fluid-structure coupling, the parietal pressure can be described with the
so-calledblocked pressure [1]. The disturbance load corresponds to the incident
pressure is given by(27)p(x)=−2pin,where pin is the amplitude of the wave of the acoustic
plane wave.  The transmission loss can be defined as follows:(28)TL=10⁢log10(WinW),with W being the radiated sound power and Win the incident sound power defined by [23, 24](29)Win=|pin|2lb⋅b⁢⋅cos(θi)2ρc,where ρ is the fluid density, c the speed of sound in the medium, and θi the incidence angle. For a normal incident
acoustic plane wave, θi=0.


### 4.4. Comparison

Figure5
presents the transmission losses of the uncontrolled and controlled structure
with the three different approaches. The structure is still excited by a normal
incident acoustic plane wave that exclusively disturbs odd modes. The three
controllers are tuned so that the maximal voltage inputs are the same. The
modal mass damping controller uses the optimal gain matrix determined
previously. The transmission losses of the FW and the RF controllers have the
same characteristics. They give good isolation at resonance frequencies with damping.
Consequently, the peaks disappear but the general behavior of transmission loss
does not change. Note that the optimal gain matrix of the RF controller is
found without searching weighting factors Q as in the frequency weight function (22). The
modal mass damping control reduces the peaks at the resonance frequency with
damping in the same way as FW and RF controllers, and it also improves the
general behavior of the transmission loss due to the mass effect. This active
modal mass is above all a mass addition. The reduction of the first eigen
frequency from 24 Hz to 21.6 Hz leads to the diminution of modal velocity after
the first resonance. Contrary to damping, the mass effect is not limited to the
vicinity of the resonance frequency of the controlled mode.Figure 5
Transmission loss.Also, the three controllers can generate spillover.
This phenomenon can be limited by considering residual modes in the patch
placement optimization and by introducing them in the cost function [9]. Additional passive
components are frequently introduced to reduce the spillover induced by the
controllers. For future applications such as double panel control, the
spillover problem can be limited by the good natural properties of the controllers
in midrange and high frequency.


## 4.1. Modeling

The smart
structure considered and shown in Figure1 is a clamped-clamped beam to which
three piezoelectric patches (P1 89 ceramics) are fixed. The dimensions and
material characteristics are given in Table 1. An FE Model is developed using Ansys software, where the beam is modeled with
Solid45 elements (8 nodes with 3 mechanical dof/node) of 2.5×2.5×1.5 mm and the
piezoelectric patches are modeled with Solid5 elements (8 nodes with up to 6
dof/node including structural and electrical dof) of 2.5×2.5×0.7 mm. For the
sake of simplicity, 3 identical patches are bonded to the smart structure.Table 1
Simulations parameters.
Beam densityρbeam=7800Kg/m3Beam widthb=4.0*10−2mBeam
Young modulusE=2.1*1011PaBeam heighth=1.5*10−3mModal
damping ratioξi=0.01Beam lengthlb=6.0*10−1mCeramic
densityρc=7700Kg/m3Patch widthbp=3.0*10−2mCeramic
Young modulusY11=6.67*1010PaPatch heighteP=0.7*10−3mY33=5.26*1010PaPatch lengthlP=8*10−2mPermittivityϵ11s=1142Position 1A1=2.5*10−3mϵ33s=668Position 2A2=4.25*10−1mPiezoelectricityd31=−108pC/NPosition 3A3=1.625*10−1md33=240pC/NFigure 1
Smart structure.The first 14 modal shapes are extracted and the
optimal position of the actuators are sought by maximizing the coupling
coefficientk of one high-radiation efficiency mode
and minimization of coupling coefficient of other modes. The generalized
coupling coefficient kij of the ith mode for the jth piezoelectric patch bonded to the structure
is defined by Hagood and von Flotow [21] as follows:(19)kij2=(ωijD)2−(ωijE)2(ωijE)2,where ωijD and ωijE are, respectively, the natural frequencies of
the structure when the piezoelectric jth patch is open-circuited and
short-circuited. The modal coupling coefficient is defined by Badel et al.  [22]:(20)|αij|=C0PKPDikij2withC0P=2ϵ33slPbeP,where KPDi is the equivalent stiffness of the ith mode and C0P is the null strain capacitance of the
piezoelectric patch. lP,b,andeP are, respectively, the length, width, and
thickness of the patches. Next, the modal coupling is introduced in
(2):(21)q¨i+2ξiωiq˙i+ωi2qi=fd(i)+∑j=1NActuatorsαijVj,where Vj is the control voltage of the jth patch. The results of optimal placement are
presented in Figure 1.Once the structural modeling has been completed, the
radiation resistance matrixM(ω) is calculated in a frequency band from 10 to
900 Hz and approximated using the Laplace-domain multi-input, multiple output
transfer function M(s) (14). The eigen frequencies of the smart
structure are presented in Table 2.Table 2
Eigen frequencies.
Mode12345678Frequency (Hz)2469130215325451617794


## 4.2. Control Gain Matrix

### 4.2.1. Transferring Control Energy from Damping to Mass

In order to
present the advantages of a derivative state feedback control, the control
effort is initially concentrated on modal damping before being transferred to
modal mass. The initial terms of the gain matrix relative to damping can be
calculated by solving the Ricatti equation stemming from the minimization of
the frequency weighted energy cost function (22). Then, the terms relative to
modal mass are introduced and those such as maximal voltage actuators inputs
remain constant. The modal mass control is applied here exclusively on the
first mode because of its high contribution to the sound power. Indeed, at low
frequency and for this type of structure, the sound power is mainly radiated by
volumetric modes [5].In Figure2, exclusive modal damping control is
represented in blue. The considered excitation is a normal acoustic plane wave
which is uniform on all the plate. The line color gets hotter when the first
mode acceleration gain increases. The left-hand plot shows the evolution of the
sound power of the controlled system computed with (14) and (17). When the
modal acceleration gain increases, the first eigen frequency scales down and
the overall sound power is reduced by the active mass. However, limiting the
control effort does not allow still introducing as much active modal damping.
Consequently, the sound power peak of the mass controlled mode is not reduced
as in the exclusive active damping control.Transfer of control energy from modal damping to modal mass: (a) sound power,
(b) control.
(a)(b)The performances of the controlled systems differ
drastically according to the weighting on the mass and damping. If the aim is
to reduce the sound power peaks, the control will focus its energy on damping.
A good tradeoff between mass and damping can be found when reducing sound
power.The right-hand side plot shows the control frequency
response of actuatorA1. This frequency response becomes constant when the
gains applied to acceleration are high compared to those applied to damping.
With full damping, the command is like a pass band filter, whereas it is like a
highpass filter in mass control. Consequently, the control energy of the
vibroacoustic modal mass control is slightly higher than that of the regular
state feedback controller. Note that in Figure 2(b), the control voltage is
calculated at the initial state which leads to the highest control voltage. So,
the first peak of the control appears at 24 Hz.


### 4.2.2. Minimization of Acoustic Energy

In Figure3(a), the cost function computed with (18) between 10 and 900 Hz is drawn with
respect to the acceleration gain and velocity gain of the first eigen mode.
When the control voltage is higher than the maximal acceptable voltage of the
transducers, the cost function reaches infinity. The bold line indicates the
border beyond which the cost function tends to infinity. The acoustic energy
reaches a minimum for a given gain couple. When the gain applied on acceleration
is high compared to that on the velocity, the acoustic energy increases because
of the peak on the first mode due to the limitation of control voltage. The
right-hand side plot shows the sound power of the mass damping controlled
system computed with an optimal gain matrix.Evolution of acoustic energy with respect to the acceleration gain and the
velocity gain on first eigen mode: (a) sound power with optimal gains (b).
(a)(b)The optimal gain matrix is obtained from the results
presented in Figure3. Then, as shown in Figure 4(a) time response simulation
of the structure excited by a plane shock normal wave is performed with the
optimal gain matrix. The sound power is presented in the left-hand side plot.
The high-frequency components of the sound power vanish after a few
oscillations due to the active damping on these modes. Consequently, the sound
power is mainly created by the first mode which is initially more disturbed. In
the right-hand side plot, the first mode velocity shows that the mass effect
increases the oscillation period and the amplitude is reduced with active
damping.(a) Sound power, (b) first mode velocity.
(a)(b)


## 4.2.1. Transferring Control Energy from Damping to Mass

In order to
present the advantages of a derivative state feedback control, the control
effort is initially concentrated on modal damping before being transferred to
modal mass. The initial terms of the gain matrix relative to damping can be
calculated by solving the Ricatti equation stemming from the minimization of
the frequency weighted energy cost function (22). Then, the terms relative to
modal mass are introduced and those such as maximal voltage actuators inputs
remain constant. The modal mass control is applied here exclusively on the
first mode because of its high contribution to the sound power. Indeed, at low
frequency and for this type of structure, the sound power is mainly radiated by
volumetric modes [5].In Figure2, exclusive modal damping control is
represented in blue. The considered excitation is a normal acoustic plane wave
which is uniform on all the plate. The line color gets hotter when the first
mode acceleration gain increases. The left-hand plot shows the evolution of the
sound power of the controlled system computed with (14) and (17). When the
modal acceleration gain increases, the first eigen frequency scales down and
the overall sound power is reduced by the active mass. However, limiting the
control effort does not allow still introducing as much active modal damping.
Consequently, the sound power peak of the mass controlled mode is not reduced
as in the exclusive active damping control.Transfer of control energy from modal damping to modal mass: (a) sound power,
(b) control.
(a)(b)The performances of the controlled systems differ
drastically according to the weighting on the mass and damping. If the aim is
to reduce the sound power peaks, the control will focus its energy on damping.
A good tradeoff between mass and damping can be found when reducing sound
power.The right-hand side plot shows the control frequency
response of actuatorA1. This frequency response becomes constant when the
gains applied to acceleration are high compared to those applied to damping.
With full damping, the command is like a pass band filter, whereas it is like a
highpass filter in mass control. Consequently, the control energy of the
vibroacoustic modal mass control is slightly higher than that of the regular
state feedback controller. Note that in Figure 2(b), the control voltage is
calculated at the initial state which leads to the highest control voltage. So,
the first peak of the control appears at 24 Hz.


## 4.2.2. Minimization of Acoustic Energy

In Figure3(a), the cost function computed with (18) between 10 and 900 Hz is drawn with
respect to the acceleration gain and velocity gain of the first eigen mode.
When the control voltage is higher than the maximal acceptable voltage of the
transducers, the cost function reaches infinity. The bold line indicates the
border beyond which the cost function tends to infinity. The acoustic energy
reaches a minimum for a given gain couple. When the gain applied on acceleration
is high compared to that on the velocity, the acoustic energy increases because
of the peak on the first mode due to the limitation of control voltage. The
right-hand side plot shows the sound power of the mass damping controlled
system computed with an optimal gain matrix.Evolution of acoustic energy with respect to the acceleration gain and the
velocity gain on first eigen mode: (a) sound power with optimal gains (b).
(a)(b)The optimal gain matrix is obtained from the results
presented in Figure3. Then, as shown in Figure 4(a) time response simulation
of the structure excited by a plane shock normal wave is performed with the
optimal gain matrix. The sound power is presented in the left-hand side plot.
The high-frequency components of the sound power vanish after a few
oscillations due to the active damping on these modes. Consequently, the sound
power is mainly created by the first mode which is initially more disturbed. In
the right-hand side plot, the first mode velocity shows that the mass effect
increases the oscillation period and the amplitude is reduced with active
damping.(a) Sound power, (b) first mode velocity.
(a)(b)


## 4.3. Effects of the Modal Mass Control Compared to Other Modal Controls

### 4.3.1. Other Approaches

In this
section, the modal mass control is compared with two other modal control
approaches. The first is a pure vibration control using a regular state feedback
control with gains computed by minimizing the frequency weighted (FW) energy
cost function (22):(22)J=∫0∞(xtQx+ρutu)dt.The second control strategy considers vibration and
acoustics. In Baumann et al.' [15] development, the radiation filters (RF) G(s) are converted to state space form (23) and
introduced in the augmented state space of the system (24):(23)r˙=AGr+BGq˙,z=CGr+DGq˙,(24)[{q˙q¨}r˙]=[A00BGAG][{qq˙}r]+[B0]u,with z being the result of the passing velocity
components through the radiation filter. z2 is the sound power. In the case of a normal
incident plane wave, the general force decays with increasing frequency, which
is why an integrator term is added in (26). If the optimal gain matrix is
computed by a minimization of z2,
low-frequency modes will not be controlled because the M(ω) terms are small when the modes do not radiate
independently. Consequently, the cost function that considers the disturbance
can be written as follows:(25)JRad=∫0∞(z2Tz2+ρuTu)dt,(26)z2=ℒ−1(s−1G(s)Q˙(s)),where z22 is the weighted sound power.


### 4.3.2. Transmission Loss

Considering
weak fluid-structure coupling, the parietal pressure can be described with the
so-calledblocked pressure [1]. The disturbance load corresponds to the incident
pressure is given by(27)p(x)=−2pin,where pin is the amplitude of the wave of the acoustic
plane wave.  The transmission loss can be defined as follows:(28)TL=10⁢log10(WinW),with W being the radiated sound power and Win the incident sound power defined by [23, 24](29)Win=|pin|2lb⋅b⁢⋅cos(θi)2ρc,where ρ is the fluid density, c the speed of sound in the medium, and θi the incidence angle. For a normal incident
acoustic plane wave, θi=0.


## 4.3.1. Other Approaches

In this
section, the modal mass control is compared with two other modal control
approaches. The first is a pure vibration control using a regular state feedback
control with gains computed by minimizing the frequency weighted (FW) energy
cost function (22):(22)J=∫0∞(xtQx+ρutu)dt.The second control strategy considers vibration and
acoustics. In Baumann et al.' [15] development, the radiation filters (RF) G(s) are converted to state space form (23) and
introduced in the augmented state space of the system (24):(23)r˙=AGr+BGq˙,z=CGr+DGq˙,(24)[{q˙q¨}r˙]=[A00BGAG][{qq˙}r]+[B0]u,with z being the result of the passing velocity
components through the radiation filter. z2 is the sound power. In the case of a normal
incident plane wave, the general force decays with increasing frequency, which
is why an integrator term is added in (26). If the optimal gain matrix is
computed by a minimization of z2,
low-frequency modes will not be controlled because the M(ω) terms are small when the modes do not radiate
independently. Consequently, the cost function that considers the disturbance
can be written as follows:(25)JRad=∫0∞(z2Tz2+ρuTu)dt,(26)z2=ℒ−1(s−1G(s)Q˙(s)),where z22 is the weighted sound power.


## 4.3.2. Transmission Loss

Considering
weak fluid-structure coupling, the parietal pressure can be described with the
so-calledblocked pressure [1]. The disturbance load corresponds to the incident
pressure is given by(27)p(x)=−2pin,where pin is the amplitude of the wave of the acoustic
plane wave.  The transmission loss can be defined as follows:(28)TL=10⁢log10(WinW),with W being the radiated sound power and Win the incident sound power defined by [23, 24](29)Win=|pin|2lb⋅b⁢⋅cos(θi)2ρc,where ρ is the fluid density, c the speed of sound in the medium, and θi the incidence angle. For a normal incident
acoustic plane wave, θi=0.


## 4.4. Comparison

Figure5
presents the transmission losses of the uncontrolled and controlled structure
with the three different approaches. The structure is still excited by a normal
incident acoustic plane wave that exclusively disturbs odd modes. The three
controllers are tuned so that the maximal voltage inputs are the same. The
modal mass damping controller uses the optimal gain matrix determined
previously. The transmission losses of the FW and the RF controllers have the
same characteristics. They give good isolation at resonance frequencies with damping.
Consequently, the peaks disappear but the general behavior of transmission loss
does not change. Note that the optimal gain matrix of the RF controller is
found without searching weighting factors Q as in the frequency weight function (22). The
modal mass damping control reduces the peaks at the resonance frequency with
damping in the same way as FW and RF controllers, and it also improves the
general behavior of the transmission loss due to the mass effect. This active
modal mass is above all a mass addition. The reduction of the first eigen
frequency from 24 Hz to 21.6 Hz leads to the diminution of modal velocity after
the first resonance. Contrary to damping, the mass effect is not limited to the
vicinity of the resonance frequency of the controlled mode.Figure 5
Transmission loss.Also, the three controllers can generate spillover.
This phenomenon can be limited by considering residual modes in the patch
placement optimization and by introducing them in the cost function [9]. Additional passive
components are frequently introduced to reduce the spillover induced by the
controllers. For future applications such as double panel control, the
spillover problem can be limited by the good natural properties of the controllers
in midrange and high frequency.


## 5. Conclusion

A new modal
active control is proposed in this paper. It permits adjusting a frequency
response template of the controlled structure. This control has been developed
within an acoustic framework designed to reduce sound transmission through a
structure. For each mode, the active modal mass addition lowers the eigen
frequency of the mass controlled mode and reduces its velocity after resonance.
The use of this method on a simple smart structure equipped with piezoelectric
patches is described with an explanation of the concept by way of a simulation.
The good performances at resonance frequencies obtained with an active damping
controller can be completed with this approach which modifies the modal
distribution of the structure. The “mass damping vibroacoustic modal
control” is compared to other control approaches which mainly use damping
to reduce sound transmission.This preliminary study can be adapted to more complex
structures such as double panels. Indeed double panels provide good
transmission loss at midrange and high frequencies, making it possible to
concentrate control energy on the first modes. Thus, the method proposed is
well adapted to this kind of structure. Moreover, the downward shift of the
eigen frequencies can be considered as virtual transformations of structures
that could be used in the field of sound quality.



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*Source: 603084-2008-07-16.xml*