Time-domain simulation of damped impacted plates. II. Numerical model and results

Time-domain simulation of damped impacted plates.

II. Numerical model and results

Christophe Lambourg,a)Antoine Chaigne,b)and Denis Matignon

Ecole Nationale Supe´rieure des Te´le´communications, De´partement TSI, CNRS URA 820, 46 Rue Barrault, 75634 Paris Cedex 13, France

A time-domain model for the flexural vibrations of damped plates was presented in a companion paper关Part I, J. Acoust. Soc. Am. 109, 1422-1432 共2001兲兴. In this paper 共Part II兲, the damped-plate model is extended to impact excitation, using Hertz’s law of contact, and is solved numerically in order to synthesize sounds. The numerical method is based on the use of a finite-difference scheme of second order in time and fourth order in space. As a consequence of the damping terms, the stability and dispersion properties of this scheme are modified, compared to the undamped case. The numerical model is used for the time-domain simulation of vibrations and sounds produced by impact on isotropic and orthotropic plates made of various materials共aluminum, glass, carbon fiber and wood兲. The efficiency of the method is validated by comparisons with analytical and experimental data. The sounds produced show a high degree of similarity with real sounds and allow a clear recognition of each constitutive material of the plate without ambiguity. © 2001 Acoustical

Society of America.

PACS numbers: 43.40.Dx关CBB兴

I. INTRODUCTION

A time-domain model of damped isotropic and orthotro-pic plates was derived in a companion paper共Part I兲 in order to investigate the relevance of material properties in the qual-ity of sounds produced by vibrating structures. This plate model includes damping terms expressed by means of a gen-eral differential operator similar to the one commonly used in viscoelasticity. In the previous paper it was shown under which conditions this operator is able to account for vis-coelastic, thermoelastic and radiation losses in materials.1

In this paper共Part II, Sec. II兲 the equations of motion for the damped plate are briefly reviewed in the general case. For convenience, these equations are written in the Laplace domain with complex rigidity factors. The system is comple-mented by initial and boundary conditions. The initial impact is modeled by Hertz’s law of contact. Most of the presented results are obtained in the case of freely suspended plates. However, the model is able to account for other simple boundary conditions, such as clamped or simply supported plates.

The main purpose of this paper is to validate a vibra-tional model for the impacted plate. No attempt is made to compute the complete acoustic field around the plate with great accuracy. However, in order to facilitate the compari-son with real sounds, rather than with vibrational quantities 共such as velocity or acceleration at a given point of the plate兲, a simple radiation model is to calculate radiation from the plate displacement, in order to obtain simulated wave-forms having the same dimension as recorded sounds.

In Sec. III, the equations of the model are put into a

numerical form using an explicit finite-difference scheme of second order in time and fourth order in space. Similar schemes were successfully applied in the past to the coupling between plate vibration and acoustic radiation by Frendi

et al.2 and by Schedin et al. for the time-domain simulation of undamped isotropic plates subjected to impacts.3 The nu-merical formulation is followed by a necessary analysis of stability and dispersion properties, since no previous refer-ences were found in the literature for similar problems which included damping.

The results presented in Sec. IV show the efficiency of the simulation. Following the mode of presentation previ-ously used for xylophones,4the method is first validated by comparisons between analytical and numerical results in simple situations and, second, by comparisons between mea-surements and simulations for damped rectangular plates made either of isotropic or orthotropic materials.

II. THE GOVERNING EQUATIONS A. The damped-plate equations

The flexural vibrations of a rectangular Kirchhoff–Love plate are considered here共see Fig. 1 for the geometry of the problem兲. The equations are written in the orthotropic case. The losses are expressed by means of a differential operator similar to the one used in viscoelasticity. For a plate of small thickness h, the transverse displacement W(x,y ,t), as a function of the coordinates x,y and time t, is governed by the following equations 共expressed in the Laplace domain with Laplace variable s):1,5

a兲_{Present address: 64 rue des Moines, 75017 Paris, France.}

b兲_{Present address: ENSTA-UME, Chemin de la Hunie`re 91761 Palaiseau}

冉

M˜ _x共x,y,s兲 M˜_y共x,y,s兲 M˜ xy共x,y,s兲

冊

⫽⫺h3

冉

D˜1共s兲 D˜2共s兲/2 0 D˜₂共s兲/2 D˜₃共s兲 0 0 0 D˜4共s兲/2

冊

⫻

冉

W˜_,xx共x,y,s兲 W˜_{,y y}共x,y,s兲 W˜,xy共x,y,s兲

冊

, 共1兲

␳h共s2⫹Rfs兲W˜共x,y,s兲⫽M˜x,xx共x,y,s兲⫹M˜ y , y y共x,y,s兲 ⫹2M˜

xy ,xy共x,y,s兲⫹ f˜z共x,y,s兲, where the subscripts ,xx, , y y and ,xy denote the partial de-rivatives of the variables. The first three equations corre-spond to the viscoelasticlike strain-stress relationships for thin orthotropic plates. M˜ x(x, y ,s), M˜y(x,y ,s) and M˜ xy(x, y ,s) are the Laplace transforms of the bending and twisting moments and D˜i(s) are the four complex rigidities. W˜ (x,y,s) is the Laplace transform of the displacement. The

fourth equation in Eq. 共1兲 derives from Newton’s second law, where ␳ is the density of the material, R_f a viscous damping coefficient, and f_z(x,y ,t) an excitation source term. The finite-difference formulation of the problem is fa-cilitated by expressing the strain–stress relationships in terms of differential equations. In the present model, the complex rigidities are written in the form:

D˜i共s兲⫽Di共1⫹d˜i共s兲兲⫽Di

1⫹兺_vN_⫽1svpiv

1⫹兺_wN_⫽1sw_q w

, 共2兲

where the coefficients piv and qw depend on the damping mechanisms 共see Table II兲. The Di correspond to the static rigidities of the plate. The d˜i are perturbation terms due to damping. In order to ensure that the model is dissipative the coefficients pivand qw must fulfill a number of constraints.1

B. The boundary conditions

For a plate of finite size, Eq.共1兲 must be completed by the boundary conditions. Only three ideal conditions for

edges parallel to the axes of symmetry of the material are considered here. For the edges located at x⫽0 and x⫽lx共see Fig. 1兲, these conditions are6

• for a free edge共F兲:

Mx⫽0 and

⳵Mx

⳵x ⫹2

⳵Mxy

⳵y ⫽0, 共3兲

• for a simply supported edge 共SS兲:

W⫽0 and Mx⫽0, 共4兲

• for a clamped edge 共C兲:

W⫽0 and ⳵W

⳵x ⫽0. 共5兲

Similar conditions are given for the edges located at y ⫽0 and y⫽lyby interchanging x and y in Eqs.共3兲–共5兲.

C. The interaction between the plate and the impactor In Eq.共1兲, it is assumed that the load f (x,y,t) due to the interaction between the plate and the impactor is governed by Hertz’s law of contact.7This method has been used with success in the past for the time-domain modeling of im-pacted xylophone bars8 and for a mixed time-frequency in-vestigation on impacts on plates.9The motion Wi of the im-pactor is a solution of the following system:

f共x,y,t兲⫽g共x⫺x0, y⫺y0兲FH共t兲, FH共t兲⫽

再

关共W共x0,y0,t兲⫺Wi共t兲兲/kH兴3/2 if Wi共t兲⬍W共x0, y0,t兲, 0 otherwise 共6兲 d2Wi共t兲 dt2 ⫽ FH共t兲 mi ,

where g(x⫺x0, y⫺y0) is a normalized spatial window

cen-tered at the impact point, FH is the interaction force, W ⫺Wi is the relative displacement between the impactor and the plate at the impact point, mi is the mass of the impactor and k_H is Hertz’s constant which depends on the surface geometries and elastic properties of the two bodies in con-tact. In Eq.共6兲, the vibrations in the impactor are neglected and it is assumed that the contact surface remains constant during the impact. These assumptions are reasonable when the dimensions of the impactor are small compared to those of the plate.9

D. The radiation equation

For the purpose of simplicity, only the case of baffled plates will be considered here. Thus, for computing the radi-ated pressure from the plate displacement W, the following time-domain formulation of the Rayleigh integral was used:

p共r,t兲⫽⫺ ␳a 2␲

冕 冕

S₀ 1 兩r⫺r0兩 ⳵2_W ⳵t2

冉

r0,t⫺ 兩r⫺r0兩 c

冊

dS0, 共7兲 where p is the sound pressure at the position r in the sound field, ␳a is the air density, c is the speed of sound in air, S0

is the surface of the plate and r0refers to the position of the

source points on the plate surface.

FIG. 1. Geometry of the plate of dimensions lxand lyand of thickness h. The flexural displacement W(x, y ) is oriented along the z axis.

III. THE NUMERICAL MODEL

The present work is limited to rectangular plates for which the boundaries coincide with the nodes of a regular rectangular grid. For this class of geometries, it is known that the Finite-Difference Methods 共FDM兲 are convenient for solving vibration problems in the time domain.10,11

In this section, the use of a 2–4 finite-difference scheme for solving the impacted plate equations is studied in terms of stability and numerical dispersion. This scheme is of sec-ond order in time and fourth order in space. The numerical parameters to be determined are the time step⌬t and the two spatial steps ⌬x, ⌬y along the x and y axis, respectively. The appropriate selection of these parameters is imposed both by the numerical stability condition and by the order of accuracy required for the dispersion.

In what follows, the guideline for the dispersion criteria was given by the accuracy required in the context of audio applications. According to Moore,12 the minimum value of the relative difference limen for pitch of a pure tone is about 0.5% at 2 kHz, and becomes greater than 5% for frequencies above 5 kHz. Since there are no available data for the differ-ence limen relative to tones made of simultaneous inhar-monic frequencies, the pure tone difference limen was used as reference for the required accuracy of the numerical model.

A. Explicit finite-difference schemes

The transverse displacement of the plate is computed at the nodes (x⫽l⌬x, y⫽m⌬y, t⫽n⌬t) of a rectangular grid. The value of a field variablev(x,y ,t) expressed on this mesh

is denoted v_l,mn ⫽v(l⌬x,m⌬y,n⌬t).

The elastic and inertial terms, involving second order partial derivatives versus time or space, are discretized by means of centered finite difference operators共see Appendix A兲. The damping terms are approximated by decentered op-erators which has the advantage to keep the explicit character of the numerical formulation. This strategy is justified by the fact that the damping terms d˜idefined in Eq.共2兲 are assumed to be first order correction terms. Centered operators for the damping terms would have led to an implicit scheme. The time derivatives in the damping terms d˜i are approximated by backward difference operators which, in terms of the z-transform, amounts to replacing the continuous Laplace variable s by the discrete operator (1⫺z⫺1)/⌬t. Thus, the discrete approximation of d˜i, written in descending power of z, is written: 兺r⫽0 N _␹ irz⫺r 1⫹兺_rN_⫽1␺_rz⫺r⫽ ⌬tN_⫹兺 r⫽1 N _⌬tN_{⫺r共1⫺z⫺1兲}r_p ir ⌬tN_⫹兺 r⫽1 N _⌬tN⫺r_共1⫺z⫺1_兲r_q r . 共8兲

In Eq. 共8兲, the constants ␹ir and␺r are obtained by identi-fying term by term the two sides of Eq.共8兲.

The finite-difference approximation of Eq. 共1兲 is then given by: 共Mx兲l,m n _⫽⫺h3_D 1

兺

r⫽0 N ␹1r共DxxW兲l,m n⫺r ⫺h3D2 2 r

兺

⫽0 N ␹2r共Dy yW兲l,m n⫺r_⫺

兺

r⫽1 N ␺r共Mx兲l,m n⫺r 共My兲l,m n _⫽⫺h3_D 3

兺

r⫽0 N ␹3r共Dy yW兲l,m n⫺r ⫺h3D2 2 r

兺

⫽0 N ␹2r共DxxW兲l,m n⫺r_⫺

兺

r⫽1 N ␺r共My兲l,m n⫺r 共9兲 共Mxy兲l⫹1/2 ,m⫹1/2 n _⫽⫺h3D4 2 r

兺

_⫽0 N ␹4r共DxyW兲l⫹ 1/2,m⫹1/2 n_⫺r ⫺

兺

r_⫽1 N ␺r共Mxy兲l⫹1/2 ,m⫹1/2 n_⫺r W_l,mn⫹1⫽共2⫺Rf⌬t兲Wl,m n ⫹共Rf⌬t⫺1兲Wl,m n⫺1 ⫹⌬t 2 ␳h 共共DxxMx兲l,m n _⫹共D y yMy兲l,m n ⫹2共D˜ xyMxy兲l,m n ⫹共 fz兲l,m n _兲,

where the discrete spatial operators Dxx, Dy y, Dxy and D˜xy are fourth order approximations共see Appendix A兲. It can be seen from Eq. 共9兲 that the transverse displacement W is re-cursively determined at time n⫹1 from its values at previous time steps.

B. Boundary conditions

The numerical approximation of the boundary condi-tions is obtained by means of the image method. It consists of replacing the boundaries by virtual sources located at points external to the plate. In what follows, only the bound-ary conditions for a free straight edge located at x⫽0, for a semi-infinite plate, is discussed关see Eq. 共3兲兴. The results can be easily extended to simply supported and clamped edges.13 The numerical formulation of the boundary conditions de-scribed below is conducted so as to maintain the overall ac-curacy of the scheme and to limit the computational domain. It is assumed that the semi-infinite plate is located in x ⬎0 共see Fig. 1兲 and that the transverse displacement W, at time n⌬t and at every previous time step, is known at each point l⌬x,m⌬y such that l⭓⫺1. The values of W at time

n⫹1 are then obtained by using the following procedure:

• First, the bending and twisting moments are computed at time n⌬t. This computation is performed by applying the first three equations in Eq.共9兲 to each point of the grid for which x⬎0. For x⫽0, M_x is equal to zero and M_y is de-rived from the displacement by using the second equation in Eq.共9兲, after replacing Dxx

(4)

by the second order operator

D_xx(2)in order to limit the spatial extension of the boundary domain. ( Mxy)_⫺1/2,m

n

is given by the third equation in Eq. 共9兲 where Dxy

(4)

is replaced by the operator D_x(2)oD_y(4) de-fined as:

共Dx (2)_oD y (4)_W兲 ⫺1/2 ,m⫹1/2 n ⫽_{24⌬x⌬y 关共}1 W_0,mn _⫺2⫺27W_0,mn _⫺1 ⫹27W0,m n _⫺W 1,m⫹2 n _兲⫺共W ⫺1,m⫺2 n _⫺27W ⫺1,m⫺1 n ⫹27W_⫺1,mn _⫺W ⫺1,m⫹2 n _{兲兴. 共10兲}

• The application of the image method also involves the computation of the values of Mx at points x⫽⫺⌬x. This is performed by considering the second condition in Eq. 共3兲, for which a third order decentered approximation in ⌬x and fourth order in ⌬y leads to:

共Mx兲_⫺1,mn ⫽3共Mx兲1,m n _⫺1 2共Mx兲2 n_{⫺3⌬x共共D˜} y (4)_M xy兲_{⫺1/2 ,m}n ⫹共D˜ y (4)_M xy兲1/2 ,m n _{兲, 共11兲} where 共D˜ y (4) Mxy兲l⫹1/2 ,m n _⫽ 9 8⌬y 共共Mxy兲l⫹1/2 ,m⫹1/2 n ⫺共Mxy兲l⫹1/2 ,m⫺1/2 n _兲 ⫺ 1 24⌬y 共共Mxy兲l⫹1/2 ,m⫹3/2 n ⫺共Mxy兲l⫹1/2 ,m⫺3/2 n _{兲. 共12兲}

• The final task consists in computing W at time (n ⫹1)⌬t. This is performed by using the fourth equation in Eq. 共9兲 for the points l⬎0, and by replacing the fourth order operators in ⌬x by second order ones in this equa-tion at points l⫽0. The displacement at the image points

l⫽⫺1 is derived from the free edge condition M_x⫽0 at x⫽0, for which a second order in ⌬x, and fourth order in

⌬y approximation leads to the following expression of

W_⫺1,mn⫹1 : W_⫺1,mn⫹1⫽⫺⌬x 2_D 2 2D1 共Dy y (4)_W兲 0,m n⫹1_⫹2W 0,m n⫹1_⫺W 1,m n⫹1_{. 共13兲}

The influence of the damping terms is neglected in the dis-crete approximation of the boundary conditions. The schemes for the three other edges of the plate obey the same principles as the one presented above for x⫽0.

C. Stability

The numerical parameters of explicit difference schemes must fulfill a stability condition which guarantees the con-vergence of the numerical solution.14 In this paragraph, the stability condition for the 2–4 scheme applied to the un-damped plate model is presented. The results are then ex-tended to the damped model. A distinction is made in what follows between two definitions of stability:14

• The numerical scheme is called weakly stable if the solu-tion at a fixed time T⫽n⌬t is bounded for all ⌬t with respect to the Euclidean norm储Wn储⫽(兺l,m兩Wl,m

n _兩2₎1/2_.

• The scheme is called strongly stable if the solution remains bounded for all T.

A strong stability condition can be obtained for the un-damped model. In the un-damped case, paradoxically, only a weak stability condition can be given.

1. Undamped model

In this paragraph, the fluid damping constant Rf and the damping coefficients pir and qr are set equal to zero. The Fourier method is used for determining the stability criterion.14 It consists of studying the modulus of one par-ticular solution W_l,mn ⫽wnexp(jkxl⌬x⫹jkym⌬y) of the homo-geneous equation. Injecting W_l,mn in Eq.共9兲 for f_l,mn ⫽0 leads to the following equation for wn:

wn⫹1⫹共b共kx,ky兲2⌬t2⫺2兲wn⫹wn⫺1⫽0, 共14兲 with: b共kx,ky兲2⫽16

冉

␰1 2 ⌬x4

冉

X 2_⫹1 3X 4

冊

2 ⫹ ␰3 2 ⌬y4

冉

Y 2_⫹1 3Y 4

冊

2 ⫹ ␰2 2 ⌬x2_⌬y2

冉

X 2_⫹1 3X 4

冊冉

_Y2_⫹1 3Y 4

冊

⫹ ␰4 2 ⌬x2_⌬y2

冉

X⫹ 1 6X 3

冊

2

冉

Y⫹1 6Y 3

冊

2

冊

, 共15兲 and where ␰i⫽

冑

h2Di ␳ , X⫽sin kx⌬x 2 , Y⫽sin ky⌬y 2 . 共16兲

The stability criterion is satisfied if the discriminant of the characteristic polynomial associated with Eq. 共14兲 is nega-tive for all kxand ky. This leads to the following condition:

⌬t⭐ 1 2

冑

冉

4 3

冊

2

冉

␰1 2_⫹␰2 2 r₁2⫹ ␰3 2 r₁4

冊

⫹

冉

7 6

冊

4_␰ 4 2 r₁2 , 共17兲 where r1⫽⌬y/⌬x.

FIG. 2. Numerical isotropy I(␪,F,r) for a reduced frequency F equal to 0.2 and for different values of the degree r of anisotropy: r

2. Damped model

A model including the damping terms is now consid-ered. To facilitate the presentation, the case of a damped bending bar 共1D problem兲 is first presented and the results are then extended to plates共2D兲.

Similarly to Eq.共1兲, the equations governing the trans-verse displacement W(x,t) of a thin bar are written:

M˜ x⫽⫺EI 1⫹兺_rN_⫽1p_rsr 1⫹兺_rN_⫽1qrsr W ˜ ,xx 共18兲 ␳S共s2⫹Rfs兲W˜⫽M˜x,xx⫹ f˜z.

The corresponding FD scheme is written:

共Mx兲l n_⫽⫺IE

兺

r⫽0 N ␹r共DxxW兲l n⫺r_⫺

兺

r⫽1 N ␺r共Mx兲l n⫺r , 共19兲 W_ln⫹1⫽共2⫺R_f⌬t兲W_ln⫹共R_f⌬t⫺1兲W_ln⫺1 ⫺⌬t 2 ␳S 共共DxxMx兲l n ⫹共 fz兲l n_兲,

where E is Young’s modulus, I is the geometrical moment and S is the cross-section of the bar. The term fzdenotes the external force density by unit length. Dxx is a fourth order discrete operator.

A necessary condition for weak stability is obtained by studying the amplification matrix14 共see Appendix B兲. This condition is written:

⌬x2_⭓2⌬t

冑

IE pN

␳Sq_N. 共20兲

It is found that the condition expressed in Eq. 共20兲 is similar to the one obtained with the Fourier method in the undamped case, except that E is now replaced by E pN/qN. This term corresponds to the high frequencies’ asymptotic value of the complex Young’s modulus.

The stability condition for the 2D system Eq.共9兲 is de-termined in the same way as for the 1D case 共see Appendix B兲. It is found that the weak stability condition of the damped 2–4 scheme is similar to Eq. 共17兲, after replacing each D_i by D_ip_iN/q_N. Since p_iN/q_N is always greater than unity for a dissipative model, this result shows that the sta-bility condition is more restrictive when damping terms are taken into account than for the undamped plate. However, for the investigated materials 共aluminum, steel, glass and wood兲, it has been found that the consecutive increase in size of the minimum space step remains below a few percent, compared to the undamped case.

D. Accuracy

One method for estimating the deviation between exact and approximate solutions consists of calculating the differ-ence between continuous and discrete frequencies.10,15 For simplicity, only the undamped model will be discussed here.

Inserting the solution W(x,y ,t)⫽(Aej␻t

⫹Be⫺ j␻t_)exp(jk

xx⫹jkyy) in Eq.共1兲, without damping terms and for fz⫽0, yields the continuous dispersion equation:

k⫽

冑

␻ ␰␪ with k⫽

冑

kx 2_⫹k y 2 , ␪⫽arctanky kx , ␰_␪⫽h 2

␳ 共D1cos4␪⫹共D2⫹D4兲cos2␪sin2␪⫹D3sin4␪兲. 共21兲

FIG. 3. 共a兲 Infinite isotropic plate profile at successive instants of time t

⫽兵0,0.05,0.10,0.15,0.20,0.25其ms. Top: analytical solution. Bottom: nu-merical solution.共b兲 Relative error 共in %兲 between analytical and numerical solutions versus time for four different sampling frequencies fe

Similarly, provided that the stability condition is verified, the introduction of the component W_l,mn ⫽(Aej␻_numn⌬t ⫹Be⫺ j␻numn⌬t_)exp(jk

xl⌬x⫹jkym⌬y) in Eq. 共9兲 leads to the numerical dispersion equation:

␻num⫽

2 ⌬tarcsin

b共kx,ky兲⌬t

2 , 共22兲

where b(kx,ky) is defined in Eq.共15兲.

In order to simplify the mathematical derivations, it is assumed here that␰2⫽0 and that␰4⫽

冑

2␰1␰3 which

consti-tutes a reasonable approximation for a large class of ortho-tropic materials.16 This allows us to characterize the anisot-ropy of the material with only one parameter r⫽␰3/␰1.

Under these conditions Eqs. 共21兲 and 共22兲, combined with Eq. 共17兲, yield: ⍀num⫽2 arcsin

冋

36

冑

9410

冉冉

X 2_⫹X 4 3

冊

2 ⫹

冉

Y2⫹Y 4 3

冊

2 ⫹2

冉

X⫹X 3 6

冊

2

冉

Y⫹Y 3 6

冊

2

冊

1/2

册

X⫽sin

冉

冉

4705 2592

冊

1/4

冑

_{⍀ cos␪}

冑

cos2␪⫹r2sin2␪

冊

共23兲 Y⫽sin

冉

冉

4705 2592

冊

1/4 _r

冑

_⍀sin␪

冑

cos2␪⫹r2sin2␪

冊

,

where ⍀⫽␻⌬t⫽2␲F and ⍀num⫽␻num⌬t are the reduced

angular frequencies.

Finally, the relative error in frequency is given by:

e共⍀,␪兲⫽⍀⫺⍀num

⍀ . 共24兲

It has been observed that about 15 time steps per period are required with the 2–4 scheme in order to keep the error in frequency smaller than 5% at 20 kHz, whereas more than 40 time-steps by period would have been needed with the 2–2 scheme. In the audio range 0–20 kHz, this degree of accuracy requires a sampling rate approximately equal to 300 kHz for the 2–4 scheme, whereas 800 kHz would have been needed with the 2–2 scheme.

For a material with a given anisotropy degree r, Eq.共23兲 shows that the numerical error not only depends on fre-quency, but also on the propagation angle ␪ in the plate. A convenient method for representing this error is to define an isotropy index:15

I共␪,F,r兲⫽cnum

c , 共25兲

where cnum is the numerical phase velocity and c the exact

phase velocity. Figure 2 represents the variations of I with␪, for a given reduced frequency F⫽0.2, and for different val-ues of the degree r of anisotropy. Notice that the angle ␪_m corresponding to the maximum index I_M moves from␲/4 to zero as the anisotropy degree r of the material decreases. However, the ratio IM/Im between the maximum and the minimum values of the isotropy index remains unchanged as

r varies.

In our applications, the plate equation has been solved with a 2–4 scheme and with a sampling rate of 192 kHz. This leads to an error of 8% in the estimation of the phase velocity at 20 kHz and of 2.5% at 10 kHz.

FIG. 4. Bending vibrations of a L-shaped isotropic plate, a few ms after the impact, with different bound-ary conditions.共a兲 Geometry of the plate at rest. 共b兲 Simply supported plate. 共c兲 Clamped plate. 共d兲 Free plate.

E. Computation of the plate-impactor interaction The discrete formulation of Eq. 共6兲 is given by

f_l,mn ⫽⫺gl,mFh n W_in⫹1⫽2W_in⫺W_in⫺1⫹⌬t 2 ms F_hn 共26兲 F_hn⫹1⫽

再

关共Wp n⫹1_⫺W i n⫹1_兲/k H兴3/2 if Wi n⫹1_⬍W p n⫹1 0 otherwise.

Wp is the displacement of the plate at impact point (x0,y0).

The spatial window gl,m is a triangular function of width 2⌬x and 2⌬y, centered at point (x0,y0).

F. Computation of the sound pressure

The semi-discrete formulation of the Rayleigh integral presented in Eq.共7兲 is obtained by using a standard trapezoi-dal rule: p共r,t兲⫽⫺ ␳a 2␲ l

兺

⫽0 Nx⫺1

兺

m⫽0 Ny⫺1 ⌬x⌬y 4 d2 dt2

冉

Wl,m共t⫺Rl,m/c兲 Rl,m ⫹Wl⫹1,m共t⫺Rl⫹1,m/c兲 Rl_⫹1,m ⫹ W_l,m⫹1共t⫺R_l,m⫹1/c兲 Rl,m_⫹1 ⫹Wl⫹1,m⫹1共t⫺Rl⫹1,m⫹1/c兲 Rl⫹1,m⫹1

冊

. 共27兲

In this equation, Nx and Ny are the nearest integers smaller than or equal to lx/⌬x and ly/⌬y, respectively, where 0⬍x⬍lxand 0⬍y⬍lyis the plate domain. Rl,mis the distance between the listening point and each mesh point of the plate and R_l,m/c is the corresponding time delay. After time-domain discretization, this quantity is approximated by the nearest multiple of⌬t such that:

Wl,m共n⌬t⫺Rl,m/c兲⯝W_l,m n⫺il,m with il,m⫽int

冉

Rl,m c⌬t

冊

. 共28兲 Finally, the discrete formulation of Eq. 共27兲 is written:

p共r,n⌬t兲⫽pn_⫽D tt

兺

i⫽i_min i_max

兺

l⫽0 N_x

兺

m⫽0 N_y ␦ii_l,mAl,mWl,m n⫺i , 共29兲 where

imin⫽min共il,m兲, imax⫽max共il,m兲

Dtt discrete backward time-derivative operator 共30兲 Al,m⫽⫺ ⌬x⌬y␳a 2␲Rl,m Bl,m Bl,m⫽

冦

1 if 0⬍l⬍Nx and 0⬍m⬍Ny 1/2 if 共共l⫽0 or Nx兲 and 共0⬍m⬍Ny兲兲 or 共共m⫽0 or Ny兲 and 共0⬍l⬍Nx兲兲 1/4 otherwise.

IV. RESULTS OF SIMULATIONS

In order to assess the validity of the method, this section starts with a comparison between the numerical result and the theoretical solution obtained in a simple case where the plate displacement is given analytically. This presentation is followed by the simulation of the plate for three different boundary conditions. The force pulse obtained with our nu-merical scheme is then compared to the force pulse calcu-lated for the same plate impacted with the same impactor using the method developed by McMillan.9 Finally, simu-lated waveforms, and their corresponding spectra, are com-pared to measurements for four different materials 共alumi-num, glass, carbon fiber and wood兲.

A. Comparison with analytical displacement

An analytical solution to the plate equation can be ob-tained only for a limited number of cases. One example of such an analytical solution can be found in textbooks for an infinite isotropic plate subjected to the initial Gaussian shape:6

FIG. 5. Interaction force between an orthotropic plate and an impactor dur-ing the contact time, for two different initial velocities of the impactor:共a兲 1.45 m/s,共b兲 0.6 m/s. Left: Simulation with the present Finite-Differences Method共FDM兲. Right: Simulation made by McMillan 共Ref. 9兲.

TABLE I. Elastic and geometrical parameters of plates and impactors. Aluminum plates ␳⫽2660 kg m⫺3 _D 1⫽6160 MPa D4⫽8600 MPa plate a1: lx⫽304 mm ly⫽192 mm h⫽2 mm fc⫽6 kHz plate a3: lx⫽419.5 mm ly⫽400 mm h⫽4 mm fc⫽3 kHz Glass plate ␳⫽2550 kg m⫺3 _D 1⫽6700 MPa D4⫽10270 MPa platev1: lx⫽229.5 mm ly⫽220.5 mm h⫽2 mm fc⫽5.8 kHz Carbon plate ␳⫽1540 kg m⫺3 _D

1⫽8437 MPa D2⫽463 MPa D3⫽852 MPa D4⫽2267 MPa

plate c1: lx⫽399 mm ly⫽200 mm h⫽2.2 mm critical zone⫽关3.6,11.3兴 kHz

Carbon–Epoxy plate

␳⫽1581 kg m⫺3 _D

1⫽6912 MPa D2⫽51.5 MPa D3⫽4750 MPa D4⫽1015 MPa

plate m1: lx⫽70.7 mm ly⫽70.7 mm h⫽2 mm

Wooden plate

␳⫽388 kg m⫺3 _D

1⫽1013 MPa D2⫽27.5 MPa D3⫽53.7 MPa D4⫽221 MPa

plate b1: lx⫽515 mm ly⫽412.5 mm h⫽4.8 mm critical zone⫽关2.4,10.3兴 kHz

Titanium impactor radius R⫽10 mm mi⫽1.12 kg kH⫽4.0⫻10⫺6m N⫺2/3 Rubber impactor radius R⫽10 mm mi⫽23.6 g kH⫽9.0⫻10⫺6m N⫺2/3 TABLE II. Damping parameters. Thermoelastic damping共aluminum兲

D

˜_{1共s兲⫽D1}

冉

_1⫹ sR1 s⫹ 共c1/h2兲

冊

, D˜4(s)⫽D4

R1⫽8.45⫻10⫺3 c1⫽8.0⫻10⫺4rad m2s⫺1 Viscoelastic damping共glass兲

D ˜_{i共s兲⫽Di}

冉

1⫹ sR1 s⫹s1⫹ sR2 s⫹s2

冊

, i⫽关1, 4兴 R1⫽1.63⫻10⫺3 s1⫽5180 rad s⫺1 R2⫽1.962⫻10⫺3 s2⫽55 100 rad s⫺1 Viscoelastic damping共carbon兲

D ˜_{i共s兲⫽Di}

冉

_1⫹ sRi1 s⫹si1⫹ sRi2 s⫹si2

冊

, i⫽关1, 3, 4兴 D˜2(s)⫽D2 R11⫽1.32⫻10⫺3 s11⫽10.1⫻103rad s⫺1 R12⫽5.0⫻10⫺3 s12⫽94.0⫻103rad s⫺1 R31⫽8.8⫻10⫺3 s31⫽2.5⫻10 3_{rad s}⫺1 _R 32⫽44.0⫻10⫺3 s32⫽70.0 rad s⫺1 R41⫽10.4⫻10⫺3 s41⫽2.27⫻103rad s⫺1 R42⫽14.4⫻10⫺3 s42⫽40.0⫻103rad s⫺1 Viscoelastic damping共wood兲

D ˜_{i共s兲⫽Di}

冉

_1⫹ sRi1 s⫹si1⫹ sRi2 s⫹si2

冊

, i⫽关1, 3, 4兴 D˜2(s)⫽D2 R11⫽8.18⫻10⫺3 s11⫽3.2⫻103rad s⫺1 R12⫽10.0 10⫺3 s12⫽50.2⫻103rad s⫺1 R31⫽16.7⫻10⫺3 s31⫽1.1⫻103rad s⫺1 R32⫽70.0⫻10⫺3 s32⫽50.2 rad s⫺1 R41⫽15.2⫻10⫺3 s41⫽1.75⫻103rad s⫺1 R42⫽35.0⫻10⫺3 s42⫽50.2⫻103rad s⫺1 Radiation damping共isotropic plates兲

D ˜_{i共s兲⫽Di}

冉

_1⫹2␳ac ␳h␻c 兺1 3_b m共s/␻c兲 m 兺0 3 an共s/␻c兲n

冊

i⫽关1, 4兴 _␻c⫽c2

冑

␳ h2_D 1 ␳a⫽1.2 kg m⫺3 c⫽344 m s⫺1 a0⫽1.1669 a1⫽1.6574 a2⫽1.5528 a3⫽1 b1⫽0.0620 b2⫽0.5950 b3⫽1.0272 Fluid damping Rf in s⫺1

W共0,r兲⫽W0e⫺r

2_/a2

. 共31兲

In this case, the displacement at time t is given by:

W共t,r兲⫽ W0 1⫹␶2e ⫺ r2_/a2_(1⫹␶2₎ ⫻

冋

cos r 2_␶ a2共1⫹␶2兲 ⫹␶sin r2_␶ a2共1⫹␶2兲

册

, 共32兲 where ␶⫽4t a2

冑

h 2_D 1␳. 共33兲

Figure 3共a兲 shows the shape of the plate at successive instants of time 共between 0 and 0.25 ms兲 for an aluminum plate with thickness h⫽2 mm, and for an initial Gaussian shape of width a⫽2.5 cm. The other dimensions of the plate are sufficiently large (1 m⫻1 m) so that no reflexions from the boundaries are observed during the selected time scale. No differences can be easily seen between the exact solution 关top of Fig. 3共a兲兴 and the simulated solution obtained with

FIG. 6. Aluminum plate. FFT analysis of the sound pressure during the first 20 ms.共a兲 Simulated vs measured frequencies in the range 0–5 kHz 共‘‘䊊’’兲. The solid line indicates perfect matching between the two frequency sets

共slope equal to 1兲. 共b兲 Comparison between simulated 共‘‘⫻’’兲 and measured 共‘‘䊊’’兲 magnitudes of the main FFT peaks.

FIG. 7. Glass plate. FFT analysis of the sound pressure during the first 20 ms.共a兲 Simulated vs measured frequencies in the range 0–5 kHz 共‘‘䊊’’兲. The solid line indicates perfect matching between the two frequency sets

共slope equal to 1兲. 共b兲 Comparison between simulated 共‘‘⫻’’兲 and measured 共‘‘䊊’’兲 magnitudes of the main FFT peaks.

FIG. 8. Carbon plate. FFT analysis of the sound pressure during the first 20 ms.共a兲 Simulated vs measured frequencies in the range 0–1.5 kHz 共‘‘䊊’’兲. The solid line indicates perfect matching between the two frequency sets

共slope equal to 1兲. 共b兲 Comparison between simulated 共‘‘⫻’’兲 and measured 共‘‘䊊’’兲 magnitudes of the main FFT peaks.

TABLE III. General simulation parameters. Typical time and spatial steps

fe⫽1/⌬t⫽192 kHz ⌬x⫽⌬y⫽9.5 mm

Impact position

xo⫽lx/2 yo⫽ly/2

Listening point

our finite-difference scheme关bottom of Fig. 3共a兲兴. Therefore, in order to better show the discrepancies between these two solutions, Fig. 3共b兲 shows the relative error between the two waveforms as a function of time for four different sampling frequencies. For a sampling frequency equal to 200 kHz, for example, it can be seen that the relative error remains always smaller than 5% and tends rapidly to less than 1%.

B. Boundary conditions

Figure 4 illustrates the ability of the numerical method to calculate the flexural vibrations of the plate under various boundary conditions: simply supported 关Fig. 4共a兲兴, clamped 关Fig. 4共b兲兴 and free plate 关Fig. 4共c兲兴. This figure shows, in addition, that the scheme is able to simulate L-shaped plates, provided that the sides are parallel to the axes. Thus, the model is not restricted to simple rectangular plates.

C. Impact simulation: Comparison with another method

In her thesis, McMillan tackles similar problems of im-pacted plates, using a method based on Green functions of the vibrating structure at the impact point.9Figure 5 shows the simulated forces obtained with the two methods, using the same geometric and elastic parameters, in the case of a Carbon–Epoxy plate struck by a Titanium impactor 共see Table I兲. The force histories obtained with the present finite-difference method are on the left-hand side of the figure, whereas the force histories obtained with the McMillan method are on the right-hand side. Figure 5共a兲 corresponds to an impact with initial velocity 1 m/s, whereas Fig. 5共b兲

cor-responds to an impact velocity of nearly 0.6 m/s. It can be seen that the two methods yield identical waveforms. The ripples in the force pulses are due to the fact that the inverse of the lowest eigenfrequency of the plate here is about 20 times lower than the interaction time, a consequence of the particular selected set of parameters.

D. Comparison between measured and simulated pressure

The simulated sounds are now compared with measured sounds. Due to the large number of excited modes in each case, no quantitative information can be easily derived from the visual comparison between measured and simulated waveforms. In this respect, comparisons in the spectral do-main are more informative. All spectra are calculated by FFT 共fast Fourier transform兲 on the first 20 ms on the sound and normalized 共in dB兲 with respect to the magnitude of the highest peak. In order to allow quantitative comparison be-tween measured and simulated spectra, only the most signifi-cant peaks are displayed in the figures. These data are ex-tracted from the FFT analysis by means of a simple peak detection algorithm. The values of the parameters used for the simulations can be found in Tables I–III.

Figure 6 shows the results obtained for an aluminum plate excited with a xylophone rubber mallet with initial ve-locity 1.45 m/s. Figure 6共a兲 shows that the number and fre-quencies of the spectral peaks, between 0 and 5 kHz, are very well reproduced. The spectral envelopes of both real and simulated sounds are comparable. The displayed spectral domain is limited here to 5 kHz for reasons of clarity, and also because the magnitude of the peaks above 5 kHz are relatively small 共less than ⫺40 dB below the maximum兲, a consequence of both excitation spectrum and damping. The damping here is mainly due to thermoelastic and radiation losses.1

Figure 7 shows the results obtained for a glass plate. Here again, the measured and simulated frequencies and magnitudes of the spectral peaks look very similar, which is confirmed by listening to the corresponding sounds. The losses in this material are mainly due to viscoelasticity and radiation.1

A first example of impact against an orthotropic plate is given in Fig. 8 which compares measurements with simula-tions in the case of a plate made of carbon fibers. It can be seen on the spectra that the high frequencies are rapidly damped: the magnitude of the peaks are less than ⫺40 dB below the maximum for frequencies above 1.2 kHz. Here, the impact produces a duller sound than with aluminum and glass. The damping is almost entirely due to viscoelastic losses in the material, and the radiation losses can be neglected.1 Similar conclusions can be drawn from Fig. 9 which shows the results obtained in the case of wood 共Spruce兲. Here, the damping of the vibrations in the plate is mainly due to viscoelastic losses. However, due to the di-mensions of the plate, the radiation losses cannot be totally neglected. The discrepancies between measurements and simulations for some of the peaks are probably due to the approximations made in the radiation model.

FIG. 9. Wooden plate. FFT analysis of the sound pressure during the first 20 ms.共a兲 Simulated vs measured frequencies in the range 0–1.5 kHz 共‘‘䊊’’兲. The solid line indicates perfect matching between the two frequency sets

共slope equal to 1兲. 共b兲 Comparison between simulated 共‘‘⫻’’兲 and measured 共‘‘䊊’’兲 magnitudes of the main FFT peaks.

In a previous paper, measured decay factors were com-pared to the values predicted by the continuous plate model solved in the frequency domain.1 To completely assess the validity of the method, Fig. 10 shows, in addition, the com-parison between measured and simulated decay factors, where the simulated damping factors are now obtained from a frequency analysis of the simulated waveforms. The analy-sis method used here is the Matrix Pencil method.17 Figure 10共a兲 shows that the thermoelastic model of losses accounts for the apparent erratic distribution of the decay factors in the low-frequency range for metallic plates. This distribution es-sentially follows from the fact that the decay time of each eigenfrequency depends on its modal shape. In this fre-quency range 共below 2 kHz, for the plate shown on this figure兲 the radiation damping is negligible. The low values of the decay factors below the critical frequency play a major role in duration and tone quality of the sound generated by the free vibrations of the plate. Figure 10共b兲 shows the rel-evance of the asymptotic radiation model which accounts for

the losses above the critical frequency of the plate. For this plate 共aluminum a3, see parameters in Tables I and II兲, the

average decay factor above 4 kHz is about 100 times the average decay factor below 1 kHz.

Figures 10共c兲 and 共d兲 show the relevance of the vis-coelastic model for orthotropic plates. It can be seen that the viscoelastic losses fully account for the measured decay fac-tors, except for the wooden plate above 2 kHz where the radiation losses are not negligible. Here again, the model is able to reproduce the apparent erratic distribution of decay factors with frequency. This distribution is not due to error in the measurements 共a commonly erroneous conclusion兲 but rather to the fact that there is a specific damping model for each complex rigidity 共see Table II兲. Notice further that 13 damping parameters only are needed here for reproducing adequately the temporal evolution of more than 40 modes. This is an interesting feature of the method in terms of data reduction.

FIG. 10. Decay factors.共a兲 Aluminum plate. Range 0–1.2 kHz. 共‘‘䊊’’兲 Experiments; 共‘‘⫻’’兲 simulations. 共b兲 Aluminum plate. Range 0–12 kHz. Top: simulations; bottom: measurements.共c兲 Carbon plate. Range 0–5 kHz. Top: simulations; bottom: measurements. 共d兲 Wooden plate. Range 0–3 kHz. Top: simulations; bottom: measurements.

V. CONCLUSION

A time-domain model for the simulation of flexural vi-brations and sounds radiated by damped isotropic and ortho-tropic plates has been presented. In order to demonstrate the particular interest of the method for the synthesis of tran-sients, the study was made for plates excited by an impact.

The key point of the model is that the losses in the plates are modeled in the time domain with the help of a general differential operator applied to the rigidities of the material. This formulation gives great flexibility for modeling the damping of plates. As shown in a companion paper, this formulation allows the modeling of the three main mecha-nisms of damping in plates: viscoelasticity, thermoelasticity, and radiation.1 Therefore, it makes it possible to reproduce adequately the complicated frequency dependence of the de-cay times observed in a large variety of materials with only a small number of damping parameters.

The sounds synthesized with this model allow a clear recognition of the materials without ambiguity. This accurate reproduction would not be possible with a simple description of the damping, as with a single loss factor, for example. All measured and simulated sounds presented in this paper can be heard at the following Web address: http:// wwwy.ensta.fr/⬃chaigne/plaque/

compar–exp–sim.html.

The equations are solved numerically by means of a finite-difference scheme of second order in time and fourth order in space 共2–4 scheme兲. The presence of complex ri-gidities makes it necessary to investigate thoroughly the sta-bility of the scheme, which contributes to filling a gap in the literature. It has been shown that a weak stability condition can be obtained from the study of the amplification matrix, following the method by Richtmyer and Morton.14

The accuracy of the method is studied in terms of nu-merical dispersion. A comparison with another scheme of lower order in space 共2–2 scheme兲 shows that, for a given accuracy, the required computing time with the selected 2–4 scheme is about 4 times lower than with the 2–2 scheme.

In order to show the validity of the method, a compari-son is made first between an exact analytical solution and the numerical result obtained for an infinite undamped isotropic plate. The impact model is further validated by comparing the calculated force pulses imparted to the plate with the force pulses obtained with another method.9 Finally, simu-lated sounds are systematically compared to measurements for four different materials. These comparisons show a par-ticularly good agreement in terms of frequencies and damp-ing factors despite the relatively low number of dampdamp-ing parameters inserted in the model.

The present model is limited to rectangular plates and to plates with rectangular corners共see Fig. 4兲, for which the use of finite differences is particularly well-suited. Extension of this model to more complicated shapes makes it necessary to use other numerical methods, such as finite elements based on a variational formulation of the problem.18

APPENDIX A: FINITE-DIFFERENCE OPERATORS The discrete operators given below are obtained by lin-ear combinations of Taylor expansions of the field variable

v(x,y ,t). In this paper, (D_tp

(i)

v)l,m

n _{denotes the approximation} of the pth time derivative of v at point (l⌬x,m⌬y,n⌬t),

where the subscripts (i) is the order of the truncation error in ⌬t.

1. First order decentered operators

The following operators are used to approximate the time-derivatives in the damping terms.

共Dt (1) v兲n_⫽ 1 ⌬t 共vn⫺vn⫺1兲⫽共vl,m n _兲 ,t⫹O共⌬t兲 共Dtt (1) v兲n⫽共Dt (1)_␱D t (1) v兲n ⫽_⌬t12共vn⫺2vn⫺1⫹vn⫺2兲⫽共vl,m n _兲 ,tt⫹O共⌬t兲 共A1兲 共Dt3 (1) v兲n⫽共Dt (1)_␱D t (1)_␱D t (1) v兲n ⫽_⌬t13共vn⫺3vn⫺1⫹3vn⫺2⫺vn⫺3兲 ⫽共vl,m n _兲 ,ttt⫹O共⌬t兲. ⯗

2. Second order centered operators

共Dtt (2) v兲l,m n _⫽ 1 ⌬t2共vl,m n_⫹1⫹v l,m n_⫺1⫺2v l,m n _兲 ⫽共vl,m n _兲 ,tt⫹O共⌬t2兲, 共A2兲 共Dxx (2) v兲l,m n _⫽ 1 ⌬x2共vl⫹1,m n _⫹v l⫺1,m n _⫺2v l,m n _兲 ⫽共vl,m n _兲 ,xx⫹O共⌬x2兲, 共A3兲 共Dy y (2) v兲l,m n _⫽ 1 ⌬y2共vl,m_⫹1 n _⫹v l,m_⫺1 n _⫺2v l,m n _兲 ⫽共vl,m n _兲 , y y⫹O共⌬y2兲, 共A4兲 共Dxy (2) v兲_ln_{⫹1/2 ,m⫹1/2}⫽ 1 ⌬x⌬y 共vl⫹1,m⫹1 n _⫹v l,m n ⫺vl_⫹1,m n _⫺v l,m_⫹1 n _兲, ⫽共vl⫹1/2 ,m⫹1/2 n _兲

,xy⫹O共⌬x2兲⫹O共⌬y2兲,

共A5兲 共D˜ xy (2) v兲_l,mn ⫽ 1 ⌬x⌬y 共vl⫹1/2 ,m⫹1/2 n _⫹v l⫺1/2 ,m⫺1/2 n ⫺vl⫹1/2 ,m⫺1/2 n _⫺v l⫺1/2 ,m⫹1/2 n _兲 ⫽共vl,m n _兲

3. Fourth order centered operators 共Dxx (4) v兲l,m n _⫽ 1 12⌬x2关⫺共vl⫹2,m n _⫹v l⫺2,m n _兲 ⫹16共vl⫹1,m n _⫹v l⫺1,m n _兲⫺30v l,m n 兴 ⫽共vl,m n _兲 ,xx⫹O共⌬x4兲, 共A7兲 共Dy y (4) v兲_l,mn ⫽ 1 12⌬y2关⫺共vl,m⫹2 n _⫹v l,m⫺2 n _兲 ⫹16共vl,m_⫹1 n _⫹v l,m_⫺1 n _兲⫺30v l,m n _兴 ⫽共vl,m n _兲 , y y⫹O共⌬y4兲 共A8兲 共Dxy (4) v兲l⫹1/2 ,m⫹1/2 n _⫽ 1 242⌬x⌬y 关共vl⫺1,m⫺1 n _⫹v l⫹2,m⫹2 n _⫺v l⫺1,m⫹2 n _⫺v l⫹1,m⫺1 n _兲⫹27共v l⫺1,m⫹1 n _⫹v l,m⫹2 n _⫹v l⫹1,m⫺1 n _⫹v l⫹2,m n _兲 ⫺27共vl_⫺1,m n _⫹v l,m_⫺1 n _⫹v l_⫹1,m⫹2 n _⫹v l_⫹2,m⫹1 n _兲⫹272_共v l,m n _⫹v l_⫹1,m⫹1 n _⫺v l_⫹1,m n _⫺v l,m_⫹1 n _兲兴 ⫽共vl⫹1/2 ,m⫹1/2 n _兲

,xy⫹O共⌬x4兲⫹O共⌬y4兲 共A9兲

共D˜ xy (4) v兲l,m n _⫽ 1 242_{⌬x⌬y 关共}vl⫺3/2 ,m⫺3/2 n _⫹v l⫹3/2 ,m⫹3/2 n _⫺v l⫹3/2 ,m⫺3/2 n _⫺v l⫺3/2 ,m⫹3/2 n _兲⫹27共v l⫺3/2 ,m⫹1/2 n _⫹v l⫺1/2 ,m⫹3/2 n ⫹vl⫹1/2 ,m⫺3/2 n _⫹v l⫹3/2 ,m⫺1/2 n _兲⫺27共v l⫺3/2 ,m⫺1/2 n _⫹v l⫺1/2 ,m⫺3/2 n _⫹v l⫹1/2 ,m⫹ 3/2 n _⫹v l⫹3/2 ,m⫹1/2 n _兲 ⫹272_共v l⫺1/2 ,m⫺1/2 n _⫹v l⫹1/2 ,m⫹1/2 n _⫺v l⫹1/2 ,m⫺1/2 n _⫺v l⫺1/2 ,m⫹1/2 n _{兲兴⫽共v} l,m n _兲

,xy⫹O共⌬x4兲⫹O共⌬y4兲. 共A10兲

APPENDIX B: STABILITY 1. 1D problem

By introducing the spatial Fourier transforms of both the displacement wnexp(jkl⌬x) and bending moment (mx)nexp(jkl⌬x), Eq. 共19兲 can be rewritten in the following matrix form:

vn⫹1⫽G˜共⌬t,k兲vn, 共B1兲

with

v_n⫹1⫽关w_n⫹1,w_n,¯,w_n⫺N⫹1,共n_x兲_n,¯,共n_x兲_n⫺N⫹1兴T_{, 共B2兲}

where (nx)n⫽⌬x2(mx)n. The amplification matrix G˜ (⌬t,k) is given by:

冢

A ⫺BC␹2 ¯ ¯ ⫺BC␹N C␺1 ¯ ¯ ¯ C␺N 1 0 ¯ ¯ 0 0 ¯ ¯ ¯ 0 0 1 0 ¯ 0 ⯗ ⯗ ⯗  ⯗ ⯗ ⯗ 0 ¯ 0 1 0 0 ¯ ¯ ¯ 0 B␹0 ¯ ¯ ¯ B␹N ⫺␺1 ¯ ¯ ¯ ⫺␺N 0 ¯ ¯ ¯ 0 1 0 ¯ ¯ 0 ⯗ ⯗ 0 1 0 ¯ 0 ⯗ ⯗ ⯗  ⯗ 0 ¯ ¯ ¯ 0 0 ¯ 0 1 0

冣

, 共B3兲 with P⫽2⫺Rf⌬t, B⫽4IEX2, C⫽ 4X2⌬t2 ␳S⌬x4 , F⫽Rf⌬t⫺1, X⫽sin k⌬x 2 , A⫽共P⫺BC␹0兲共F⫺BC␹1兲. 共B4兲

The discrete system in Eq. 共B1兲 is called weakly stable if there exists a real constant K(␶,T) such that:14

储G˜_共⌬t,k兲n_{储⭐K共␶,T兲, for all}

再

0⬍⌬t⬍␶ 0⭐n⌬t⭐T

k real

, 共B5兲

where 储G˜ (⌬t,k)储 denotes the spectral norm of the matrix

G˜ (⌬t,k). A condition for strong stability is obtained by

re-placing K(␶,T) by 1 in Eq.共B5兲. The following criterion has been applied here:14

‘‘If G˜ (⌬t,k) is uniformly Lipschitz continuous for ⌬t⫽0, so far as G˜ (⌬t,k)⫽G˜ (0,k) ⫹O(⌬t), as ⌬t→0, where the constant implied by the expression O(⌬t) does not depend on k, then the scheme is weakly stable if and only if

G˜ (0,k) is stable.’’

To apply this criterion, the first step consists of studying the dependence of G˜ (⌬t,k) on ⌬t. This parameter appears in the coefficients A, C and F defined in Eq.共B4兲 and in the expressions of the coefficients␹r and␺r:

␹r⫽共⫺1兲r pN qN⫹O共⌬t兲 共B6兲 ␺r⫽共⫺1兲 r_{⫹O共⌬t兲.}

All the coefficients of the amplification matrix can be expressed as g_{i j}⫹O(⌬t) and thus G˜ is uniformly Lipschitz continuous. The determination of a stability condition now consists of the study of G˜ (0,k). It must be checked whether the spectral radius of this matrix remains smaller than or equal to unity for all k or not. The eigenvalues␭iof G˜ (0,k) are such that the vector:

冉

共nx兲n wn

冊

⫽␭i

冉

共nx兲n⫺1

wn_⫺1

冊

共B7兲

is a solution of the discrete system in Eq. 共19兲 for ⌬t⫽0, given by:

冉

r

兺

⫽1 N 共⫺1兲r_␭ i ⫺r _⫺BpN qNr

兺

⫽0 N 共⫺1兲r_␭ i ⫺r C ␭_i⫺2⫹␭_i⫺1

冊

冉

共nx兲n wn

冊

⫽

冉

0 0

冊

. 共B8兲 The nontrivial solutions of this system are obtained by zero-ing the determinant of the matrix:

冉

r

兺

⫽0 N 共⫺1兲r_␭ i r

冊

冉

_␭ i 2_⫹

冉

X 4 ␰s IE pN ␳SqN ⫺2

冊

␭i⫹1

冊

⫽0 共B9兲 from which the following necessary condition for weak sta-bility is obtained:

⌬x2_⭓2⌬t

冑

IE pN

␳SqN

. 共B10兲

2. 2D problem

A necessary condition for the system Eq.共9兲 to be stable is now determined in the same way as for the 1D case. It is assumed that the dissipativity conditions on the coefficients

piNand qNare fulfilled共see Ref. 1兲. As ⌬t tends to zero,␹ir and␺ir can be expanded as follows:

␹ir⫽共⫺1兲r piN

q_N⫹O共⌬t兲, 1⭐i⭐4

共B11兲

␺ir⫽共⫺1兲r⫹O共⌬t兲, i⫽关1,4兴.

As for the 1D case, it can be shown that the amplification matrix is Lipschitz-continuous for⌬t⫽0. The eigenvalues ␭i of G˜ (0,kx,ky) are now solutions of the following system:

冉

Q共␭i兲 0 0 ⫺共P1共␭i兲⫺P21共␭i兲兲 0 Q共␭_i兲 0 ⫺共P₃共␭_i兲⫺P₂₃共␭_i兲兲 0 0 Q共␭_i兲 ⫺P₄共␭_i兲 C1 C3 C4 P共␭i兲

冊

•

冉

共nx兲n 共ny兲n 共nxy兲n wn

冊

⫽

冉

0 0 0 0

冊

, 共B12兲 with: Q共␭i兲⫽

兺

r⫽0 N 共⫺1兲r_␭ i ⫺r_{, P} 23共␭i兲⫽2h3D2X2 p2N qN Q共␭i兲, C1⫽ 4X2 ␳h␰_s2, P共␭i兲⫽␭i⫺2⫹␭i ⫺1_, P3共␭i兲⫽ 4h3D3Y2 r₁2 p3N qN Q共␭i兲, C3⫽ 4Y2 ␳h␰_s2r₁2, 共B13兲 P1共␭i兲⫽4h 3_D 1X 2p1N qN Q共␭i兲, P4共␭i兲⫽ 2h3_D 4XY r1 p_4N qN Q共␭i兲, C4⫽ 8XY ␳h␰_s2r1 , P21共␭i兲⫽ 2h3D2Y2 r₁2 p2N qN Q共␭i兲, ␰s⫽ ⌬x2 ⌬t , r1⫽ ⌬y ⌬x. As for the 1D case, the determination of the stability condi-tion is obtained by zeroing the determinant of the matrix Eq. 共B12兲 which amounts to studying the roots of the polyno-mial: P共␭兲⫹C1

冉

A1 p1N qN ⫹A21 p2N qN

冊

⫹C3

冉

A3 p3N qN ⫹A23 p2N qN

冊

⫹C4A4 p4N qN ⫽0, 共B14兲 with A1⫽h3D1X, Y⫽sin k⌬y 2 , A21⫽ h3D2Y2 2r₁2 , 共B15兲

A3⫽ h3_D 3Y2 r₁2 , A23⫽ h3_D 2X2 2 , A4⫽ h₃D₄XY 2r1 . This yields a stability condition similar to Eq. 共17兲, where the Diare replaced by DipiN/qN.

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