NOISE REDUCTION DUE TO ACTIVE VIBRATION DAMPING

(1)

HAL Id: jpa-00230483

https://hal.archives-ouvertes.fr/jpa-00230483

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

NOISE REDUCTION DUE TO ACTIVE VIBRATION DAMPING

A. Adobes

To cite this version:

A. Adobes. NOISE REDUCTION DUE TO ACTIVE VIBRATION DAMPING. Journal de Physique

Colloques, 1990, 51 (C2), pp.C2-769-C2-772. �10.1051/jphyscol:19902179�. �jpa-00230483�

(2)

COLLOQUE DE PHYSIQUE

Colloque C2, suppl6ment au n 0 2 , Tome 51, F6vrier 1990 ler Congr&s F r a n ~ a i s dfAcoustique 1990

NOISE REDUCTION DUE TO ACTIVE VIBRATION DAMPING

A. ADOBES

Direction des Etudes et Recherches, EDF, DBpartement ~coustique et MBcanique Vibratoire, 1 Avenue du G6nbral de Gaulle, F-92141 Clamart.

France

-

On discute la pertinence d'un critbre de minimisation, generalement utilise en absorption vibratoire active, sur la reduction du niveau sonore dans un local de reference. La simulation numerique modblise la structure par Bl6ments finis et le local par differences finies.

Abstract -

The relevancy of a minimization criterion, often used in active vibration damping, is discussed from the poin! of vue of noise reduction in a test room, The numerical simulation models the structure using a finite element method and the room using a finite difference method.

The mechanical resistance of structures is often the main purpose of classical active vibration damping studies. Sometimes, however, the mechanical resistance of structures is not problematical, and the real question is the decrease of the sound radiated by them. In this case, the minimization test should be chosen according to acoustic criteria, We use a finite element method to discretize the structure and a finite difference method to discretize the fluid. The efficiency of a classical minimization criterion to reduce the sound pressure level in a test room is evaluated. The mathematical model defined by Luzzato and Jean, 111, is applied in this new perspective.

LL DFFlNlTlON OF THF PROW FM

The configuration under study is described in figure 1.

Figure 1 : A plate undergoing a mechanical excitation radiates sound into a test room.

One of the walls of a room, which is called the test room; is modelled as a plate. None of the other walls is allowed to vibrate. The plate is submitted to an external load, the primary perturbating load, and therefore generates noise in the room. We want to apply to the plate a secondary antagonist load such that the sound pressure level in the room is reduced.

W. THEORFTICAL PROBLFM

We suppose that the fluid in the room is so light that its effect on the vibration of the plate can be neglected.

This assumption allows us to split the problem into two steps. The first step is the computation of the dynamic response of the plate to the primary and the secondary loads. This step includes the derivation of a feedback system based on an acoustic criterion. The second step is the computation of the sound pressure level in the test room.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19902179

(3)

UI.1. First s m : active vlb-

. . . -

studv of the P[ate;

The configuration of the mechanical system is described by an element u of a vector space U. The loads are represented by an element f of the algebraic dual F o f U. Let L be a linear mapping from U into 7.

L u = f or U = L - I f ( 1 )

In our case, f can be splitted into a vector fe describing the primary load and a vector f, describing the secondary load. Mathematically, the feedback system is a linear mapping which to every continuous velocity field u associates a suitable continuous load fc. In practice, u is measured with q transducers and fc is applied to the plate with r exciters. Let $ denote the set of complex numbers. Let B denote a linear mapping from U into $4, associating to vector u E U, a Set of q complex numbers. 8 describes the behavior of the transducers.

Let D denote a linear mapping from

Gr

into F, associating to a set of r complex numbers, a vector fc E U. D describes the behavior of the exciters. Let S denote a linear mapping from $q into

Gr. g

is the amplification system. This active vibration damping system is represented in figure 2.

Plate L b

Figure 2 : diagrammatical representation of the active vibration damping system The configuration under study leads us to rewrite equation (1).

L u = fe+fc = f e + D ( S . ( B ( u ) ) ) ( 2

Let

LI

denote the surface of the plate, u denote the velocity field normal to the plate and 9(u) be the weighted-quadratic form defined as :

The Helmholtz equation governing the room establishes a linear relationship between the velocity i i i l d normal to the radiating structure, u, and the pressure field in the room, p. The minimization of the functional J(u) should therefore lead to a reduction of the mean quadratic pressure in the room, if not to its minimum.

The minimization problem can be stated as follows I11 :

To find 9, a linear mapping from $q into

Gr,

V fe E

3

u ^EU, 3 G E

Gr,

such that,

L u = f e + D ( G ) , G = S ( B ( u ) ) and Inf E(R(N)) = E(R(G)) ( 4 ) where R(N) = ( L -1 D )(N)

+

L -1 D (fe).

We make a pure tone analysis of this problem so that all the variable fields such as the acoustic pressure and the velocity fields, are multiplied by e-Jwt, where w denotes the excitation pulsation. In our case, the linear operator, L, is the bending vibration operator :

2

L =(- Eh3 A A -M,o ) 2

12(1-v )

( 5 ) Where E denotes the Young modulus, v the Poisson's ratio, Ms the mass per unit area, h the thickness of the plate. It is assumed that the plate is thin enough so that both the stress normal to the mid-surface and the shearing stresses can be neglected. Kirchhoff's theory for plate bending is then used.

We now apply the classical mathematical model 121, which underlies all approximation processes, (modal analysis, finite element method...), to approximate the continuous solution, u, of equation (2). The basic linear bidimensionnal finite element, the Adini's rectangle 131, is used. u is expressed in terms of the corresponding set of basis functions, Ep, Ep= (el, ee, e3,

... :,

ep), where p denotes the order of truncation of the approximation process.

U Z E ~ U ~

( 6 ) Where UP is a vector containing the nodal velocities, the nodal derivatives of the velocities with respect to the x direction and the nodal derivatives with respect to the y direction. The principle of virtual powers leads to

(4)

the equation for the bending vibration of the plate, approximated by the finite element method :

11

accounts for the structural losses that take place in the plate as it vibrates. KPp and M P p denote respectively the stiffness matrix and the mass matrix. DPr is a rectangular matrix, with p rows and r columns, representing the linear mapping D. F P is a vector representing the primary load and GTis the unknown vector in our minimization process. Equation (7 ) can be rewritten as :

The approximation process leads to the following approximated matricial expression of flu).

J(u)-

up * z:up

( 9 ) U P * denotes the complex conjugate of _" UP. ZPpdenotes a matrix, proportional to IMPp, whose general term is :

~ j = J i e :

cj

dn

( 1 0 ) Replacing U P into equation (9) by its expression derived in equation (a), we obtain the expression of J(u) in terms of vector Gr. The minimum of 3(u) is reached for a value of GT, GrO, which cancels the derivative of Xu) with respect to GI. The calculus leads to the following expression of vector GTo.

Once Gro is determined, vector U P can be computed and the continuous velocity field, u, which mi&nizes J(u) is approximated using (6).

The room is governed by the Helmholtz equation : ( A + k 2 ) p = v

Where p denotes the acoustic pressure field, k denotes the wave number and v denotes the velocity field. As in the case of the plate, we apply the approximation process, 121, to solve equation (12) using now the finite difference method. Let Hn = [h,, h,,

...,

hn] be a sequence of n vectors of the vector space I 3 p. Let choose Hn compatible with a finite difference scheme. n is the order of truncation of the approximation process for the Helmholtz problem. Thus one can write :

n n

p ^E

C

hi Pi= HnP

i = 1

( 1 3 ) Where p n is a vector containing the nodal Dressures of the finite difference mesh.

Equations (12) and (13) lead to linear sysiem (14), representing the Helmholtz equation :

n n t n n

H n P = ( H n , v ) = V ( 1 4 )

where W n, denotes the matrix representation of the Helmholtz operator and (.,.) denotes the inner product, (see 121 and eq.15). Replacing v by the already calculated plate velocity field, u, into equation (14) allows one to compute . pn, after solving the linear system. The general term of vector Vn is given by :

v1=J

hi u dQ

n ( 1 5 )

Once p n is computed, an approximation of p is obtained in al! the room, using equation (13).

IV. MAIN RESULTS

The numerical package was run for

o

= 408 Hz, using the following data :

- Data for the test room ; Lx=3.5 m, Lz=3.5 m, Ly=5 m. 2, the specific complex impedance of the walls, including the plate, was taken equal to, Z = (1000, -1). The only source in the room was the vibrating plate.

Sata for the plate ; Simply Supported, E = 7.1 1012 Nlm, v = 0.33,

M s

= 3.429 kglm2, h = 5.25 10-3 m ,

11

= 0.1. It was square with an edge length a, a =3 m. The bottom left corner of the plate lay at Pl(x1, zl), such that, XI= 0.25 m and z1=.0.25 m. The primary load was a Dirac, FoG[(x-xo),(y-yo),(z-zo)] such that

,

xo = 1.75 m, yo = 0.0 m, z,= 1.75 m, and Fo = (1000.0). Its position is numbered 0 in figure 3.

The program was successively run for 9 configurations. Configuration nOO included no counter-exciter.

(5)

Configuration nOi included the i first counter-exciters of the list defined in figure 3. The computations results, in terms of J(u) and of the average Sound Pressure Level, ASPL, in the test room, are summarized in figure 4. ASPL is defined as :

ASPL = lologlo(;

i = l . ,

As one can see in figure 4, ~ ( u ) is, as expected, a decreasing function of the number of counter-exciters. Such is not the average sound pressure level in the test room. The ASPL increases slightly when one adds counter- exciter number 3 or counter-exciter number 8.

V. CONCLUSION

The minimization criterion that we chose proves to be satisfactory in so far as its application involves large reductions of the average Sound Pressure Level in the test room. Other criteria, such as the minimization of the weighted quadratic form a p ) , defined hereafter,

%PI = l p 2 ~ ) l du

U ( 1 7 )

with u denoting the volume of the test room, would certainly prove more efficient. One should note that minimazing N p ) , as one usually does in active noise control, is more difficult when the counter-exciters are shakers instead of loudspeakers. In our case, the best functional to minimize might nevertheless be the integral over the plate surface of the active intensity radiated by the plate. The practical application of such a criterion would require the use of intensimeters instead of the classical microphones and accelerometers.

Figure 3 : Coordinates of the primary and of the secondary Dirac distributions Counter-

exciter n o 1 2 3 4 5 6 7 8

Figure 4 : Unlike ~(u), the Average SPL is not a decreasing function of the number of counter-exciters coord. ( m ) (y=O.)

REFERENCFS

x 1.25 2.25 1.75 1.75 1.25 2.25 2.25 1.25

Number of counter-exciters

9 ( u ) Average SPL

111 E. LUZZATO, M. JEAN, 1983, Mechanical analysis of active vibration damping in continuous structures, Journal of Sound and Vibration 86(4), 455-473.

(21 E. LUZZATO, 1988, Analyse de I'interaction entre systemes dynamiques lineaires et representation des conditions aux limites non-locales. HE-24 1 88.1 1 Ddpartement Acoustique EDF

,

Clarnart FRANCE.

131 A.ADOBES, 1987, Methodes d'etudes des resonateurs. Premiers rbsultats et axes de recherche. HE 22187.13. Departernent Acoustique EDF, Clarnart FRANCE.

z 1.75

1.75 1.25 2.25 2.25 2.25 1.25 1.25

0 2 9 4 e - 7 ( d B ~ 1 5 8 . 9 8

1 1 0 0 e - 9 1 2 6 . 0 7

2 5 2 5 e - 1 1

109.1 6 3

109.75 4

5 0 6 e - 1 1 3 0 2 e - 1 1 2 8 0 e - 1 1 107.65

5

1 0 5 . 9 7 6

103.57 7

2 0 4 e - 1 ' 1 7 0 e - l 1 1 1 5 e - 1 1 97.1 5

8

99.61