Compressible fluid-structure interaction and modal representation/Spudaus skyscio ir konstrukciju saveika bei savuju virpesiu formu savybes.

Kargaudas, V. ; Zmuida, M.

1. Introduction

Vibrations of structures in civil engineering, aerospace, biomechanics are frequently connected with fluid influence. Fluid is a part of the mechanical system, and compressible gas or liquid. It is the significant component of the whole mechanical model. Four different dam--water reservoir models, the first rigid dam--incompressible water, the fourth flexible dam--compressible water, are presented by Tiliouine, Seghir [1]. Galerkin variational formulation is established for each model and earthquake response studies presented. A method to compute the vibration modes of an elastic shell or plate in contact with a compressible fluid is considered by Hernandez [2]. Presence of zero-frequency spurious modes with no physical meaning is indicated. Elastoacoustic vibration modes are investigated by Mellado and Rodriguez [3]. Interaction of compressible flow and deformable structures is solved by Gretarsson et al. [4]. Hydrodynamic pressure on underwater glide vehicle and surface stresses are investigated by Du et al. [5]. Vibrations in magnetorheological fluids are studied by Bansevicius et al. [6].

Forced vibrations of two plates in incompressible fluid are investigated in [7]. These two plates, not connected together, interact through an incompressible fluid. Interaction of the different eigenmodes of the same plate in vacuum is also presented.

2. Equations of plate motion

Deflections of a plate AB (Fig. 1), supported at opposite edges, can be approximated by the functions of distance y and time functions [q.sub.s] (t)

u(y,t) = [n.summation over (s=1)] [q.sub.s](t)[[sigma].sub.s](y) (1)

where n is any integer. The base functions [[sigma].sub.s] (y) satisfy the boundary conditions of the plates when y = [y.sub.1], y = [y.sub.2]. In Fig. 1 [y.sub.1] = 0, [y.sub.2] = h and [[sigma].sub.s] ([y.sub.j]) = 0, [[sigma].sup.n.sub.s] ([y.sub.j]) = [d.sup.2][[sigma].sub.s]/d[y.sup.2] = 0, j = 1,2 , but any other values of [y.sub.1], [y.sub.2] and boundary conditions can be applied. Solution (1) is presented in n-dimensional vector space and is complete in the functional space [L.sub.2] [0, h] if n [right arrow] [infinity]. We can define the space of investigation when n = const < [infinity].

A virtual deflection of the plate [delta][u.sub.r] = [delta][q.sub.r][[sigma].sub.r](y), 1 [less than or equal to] r [less than or equal to] n. The inertia forces are -A [rho][n.summation over (s=1)][[??].sub.s] [[sigma].sub.s](y), so the virtual work [delta][W.sub.i] = -([n.summation over(s=1)] [d.sub.rs][[??].sub.s])[delta][q.sub.r], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], A is the cross-section area, [rho]--density of the plate. Potential energy of the deformation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], therefore the virtual work of the plate deformation -[partial derivative][PI]/[partial derivative][q.sub.r] [delta][q.sub.r] = -([n.summation over (s=1)] [c.sub.sr][q.sub.s])[delta][q.sub.r]. Modulus of elasticity E = [E.sup.*]/(1 - [v.sup.2]), where v is Poison's ratio.

[FIGURE 1 OMITTED]

Sum of the virtual work for any r = 1,2, ..., n is zero if influence of the fluid is neglected

[n.summation over (s=1)] [d.sub.rs][[??].sub.s] + [n.summation over (s=1)] [c.sub.rs][q.sub.s] = 0 (2)

or

D[??] + C[bar.q] = 0 (2*)

where D = [parallel][d.sub.rs] [parallel], C = [parallel][c.sub.rs][parallel] are n-by-n matrices, [[bar.q].sup.T] = [[q.sub.1], [q.sub.2], ..., [q.sub.n]]. If [[sigma].sub.s](y) are orthogonal, the matrix D and may be the matrix C is diagonal.

When the fluid is compressible and inviscid, the classical Helmholtz equation [DELTA][phi] = 1/[c.sup.2.sub.o] [[partial derivative].sup.2][phi]/[partial derivative][t.sup.2] for the potential function [phi](x,y,t) holds true, where [c.sup.o] is the sound speed in the fluid. By using the separation of variables method the velocity potential can be expressed

[[phi].sub.j] = [A.sub.j](sinh [[psi].sub.j]x - cosh[[psi].sub.j]x/tanh[[psi].sub.j]L) cos[X.sub.j]y[[??].sub.s](t) (3)

where [[??].sub.s] =sin[omega]t, [X.sub.j] = {j - 0.5)[pi]/h,

h[[psi].sub.j] = [pi]/2[square root of [(2j - 1).sup.2] - [[theta].sup.2.sub.o]], [[theta].sub.o] = 2[omega]h/[pi][c.sub.o] = 4 fh/[c.sub.o] (4)

Particular solution (3) depends on the frequency [omega] = 2[pi]f and this changes the whole solution of the fluid-structure interaction problem. If the sound speed [c.sub.o] [right arrow] [infinity], then [[theta].sub.o] [right arrow] 0 and [[psi].sub.j] [right arrow] [X.sub.j,] solution coincides with [7].

The boundary condition on the line x = 0 is [partial derivative][phi]/[partial derivative]x = [partial derivative][u.sub.s]/[partial derivative]t. If [u.sub.s] = [q.sub.s](t)[[sigma].sub.s](y), then from Eq. (3)

[[phi].sub.s] = [[??].sub.s][summation][A.sub.js](sinh[[psi].sub.j]x - cosh[[psi].sub.j]x/tanh[[psi].sub.j]L)cos[X.sub.j]y and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [[sigma].sub.s](y) = sin [[pi]s(y - [y.sub.1])]/[[y.sub.2] - [y.sub.1]], then

[D.sup.*.sub.js] = 2s[[??].sub.i]/[[(j - 0.5).sup.2] - [(s[[??].sub.1]).sup.2]][D.sub.js]/[pi], if s[[??].sub.1] [not equal to] j - 0.5;

[D.sup.*.sub.js] = sin[X.sub.j][y.sub.1]/2[[??].sub.1] - [D.sub.js]/[2[pi](2j - 1)] if s[[??].sub.1] = j - 0.5,

where [D.sub.js] = cos[X.sub.j][y.sub.1] - cos [pi]s cos [X.sub.j][y.sub.2], [[??].sub.1] = h/l. On the plate surface x = 0, applying relation p = [[rho].sub.o][??], [[rho].sub.o]--fluid density, pressure is expressed [p.sub.s] = [[rho].sub.o]h[[??].sub.s][[infinity].summation over(j=1)] 2[D.sup.*.sub.js]/h[[psi].sub.j]tanh[[psi].sub.j]L cos[X.sub.j]y. The virtual work of the fluid pressure, when virtual deflection is [delta][u.sub.r] = [[sigma].sub.r](y)[delta][q.sub.r], can be expressed [delta]W = -[[rho].sub.o][h.sup.2] [[alpha].sub.rs][[??].sub.s][delta][q.sub.r],

[[alpha].sub.rs] = [[infinity].summation over (j=1)] 2[D.sup.*.sub.js][D.sup.*.sub.jr]/h[[psi].sub.j]tanh[[psi].sub.j]L (5)

When the virtual work of the fluid pressure is added to the virtual work of plate deformation and inertia forces, the linear system of equations [n.summation over (s=1)]([d.sub.rs] + [[rho].sub.o][h.sup.2]d [[alpha].sub.rs])[[??].sub.s] + [n.summation over (s=1)] [c.sub.rs][q.sub.s] = 0 follows. In matrix notation

(D + [[rho].sub.o][h.sup.2]dH)[??] + C[bar.q] = 0 (6)

where H = [parallel][[alpha].sub.rs][parallel], and d is width of the plate, parallel to the axis z, perpendicular to the x, y plane.

3. Eigen frequencies and modal representation

When vibrations are harmonic [bar.q] = [bar.g][e.sup.i[omega]t,] then from Eq. (2) (D - [lambda]C) [bar.g] = 0, where [lambda] = [[omega].sup.-2] and [bar.g] does not depend on time. If [bar.r] = [C.sup.1/2] [bar.g] , then the matrix equation is (B - [lambda]I) [bar.r] = 0, where B = [C.sup.-1/2]D[C.sup.-1/2] is symmetric matrix and I is the unit matrix. If the base functions [[sigma].sub.s](y) are orthogonal, then the matrix D is diagonal. When [y.sub.1] = 0, [y.sub.2] = h (Fig. 1)[d.sub.ss] = m/2, [c.sub.ss] = [s.sup.4]/[[GAMMA].sup.2], where [[GAMMA].sup.2] = 2[h.sup.3]/[[pi].sup.4] EI, m is mass of the plate.

If D is replaced by [D.sub.H] = D + [[rho].sub.o][h.sup.2]dH = = m([??] + [epsilon]H) = m[[??].sub.H], then vibrations Eq. (6) of the plate in fluid are presented

(I - [[omega].sup.2]m[[??].sub.H])[bar.r] = 0 (7)

where [[??].sub.H] = [C.sup.-1/2][[??].sub.H][C.sup.-1/2] is symmetric matrix, [[??].sub.H] = [??] + [epsilon]H, [epsilon] = [[rho].sub.o][h.sup.2]d/m . Really, from (5) the equality [[alpha].sub.rs] = [[alpha].sub.sr] follows, and the matrix H is symmetric, but every entry of the matrix [[alpha].sub.sr] = [[alpha].sub.sr] ([omega]) depends on the vibration frequency [omega] . So, the entries [[??].sub.sr] of the matrix [[??].sub.H] ([omega]) depend on the vibration frequency, the eigenvalues [[bar.[lambda]].sub.s] and eigenvectors [r.sub.s] = [r.sub.s] ([omega]) also depend on [omega]. The eigenvectors of the matrix [[??].sub.H] do not represent all vibration modes of the plate in compressible fluid. Alternatively, as the matrix [[??].sub.H] is symmetric, the real eigenvectors and eigenmodes can be determined by iterations.

For the steel plate [rho] = 7.8 kg/[dm.sup.3], E = 2.1 x [10.sup.7] N/[cm.sup.2], [delta] = 2.5 cm, l = h = 1 m, the first free frequency in vacuum [f.sub.1] = 6.54 Hz. If fluid is assumed incompressible water [[rho].sub.o] = 1 kg/[dm.sup.3], dimensionless parameter [epsilon] = 15.4, the first frequency is [f.sub.o1] = 0.634 Hz. The main parameter for compressible fluid is [[theta].sub.o] (4), and it depends on the distance h and sound speed [c.sub.o] = 1470 m/s for the water. The parameter [[theta].sub.o] does not depend on fluid density, and this is important when influences of water and air are compared.

The first approximation of the first eigenfrequency [f.sub.o1] = 0.634 Hz is applied and the new values of [[alpha].sub.sr] from (5) give new matrix H. After that the first eigenvalue [[??].sub.1] of the matrix is deduced from (7) and eigenfrequency [f.sub.1] = [square root of [[??].sub.1]/m]/2[pi] practically coincides with the value [f.sub.o1]. The set of eigenvectors [[bar.r].sub.s] ([f.sub.1]), s = 1,2, ..., 7 of the matrix [[??].sub.H] ([f.sub.1]) is complete and orthogonal in the n-dimensional vector space, but only the first eigenvalue and the first eigenvector have physical meaning. In the second line s = 1, [[theta].sub.o] = 0.0052 of the Table 1 are presented all eigenvalues of the matrix [[??].sub.H]([f.sub.1]).

Calculations of the matrix H([omega]s) and the matrix [[??].sub.H] ([f.sub.s]), s = 2,3, ..., 7 were performed, the eigenvalues in the lower lines s = 2,3, ... ,7 of the Table 1. Every eigenvalue [f.sub.s] of the corresponding matrix [[??].sub.H] ([f.sub.s])is almost the same as in the line with [[theta].sub.o] = 0 of the Table 1. But the first eigenvalue of the matrix [[??].sub.H] ([f.sub.7]) [f.sub.1]([f.sub.7]) = 0.486 < 0.634 = [f.sub.1] ([f.sub.1]) of the matrix [[??].sub.H] ([f.sub.l]).

There are the set of eigenvectors [[bar.r].sub.s] ([f.sub.j]), s = 1,2, ..., for every frequency [f.sub.j], s = 1, 2, ..., but only the eigenvectors [[bar.r].sub.s] ([f.sub.s]) have the physical meanings of the eigenmodes of the plate. All the vectors [[bar.r].sub.s] ([f.sub.j]), j = const, s = 1, 2, ..., are orthogonal and complete in the vector space of investigation. Only the vector s = j have physical sense. The eigenmodes [[bar.r].sub.s] ([f.sub.s]), s = 1, 2, ..., n are not orthogonal and may be not complete in the vector space of investigation.

Another example presents vibrations of wood plate in air, when h = 1 m, plate thickness [delta] = 0.4 cm, density [rho] = 0.4 kg/[dm.sup.3], E = 12 x [10.sup.5] N/[cm.sup.2]. Density of air [[rho].sub.o] = 1.2 g/[dm.sup.3], therefore [epsilon] = 0.75. Density of the air is much less then the density of water, and diminution of frequency in the fluid is not so significant (Table 2). The speed of sound in air [c.sub.o] = 340 m/s, and therefore parameter [[theta].sub.o] is higher and exceeds critical value [[theta].sub.o] = 1 when eigenvibration number s > 3 . If [[theta].sub.o] > 1, then h[[psi].sub.j] in (4) has an imaginary value and some terms in [[alpha].sub.rs] (5) are negative with product h[[psi].sup.*.sub.j]tanL[[psi].sup.*.sub.j] in denominator, where h[[psi].sup.*.sub.j] = [pi]/2[square root of [[theta].sup.2.sub.o] - [(2j - 1).sup.2]].

The matrix of hydrodynamic interaction H does not depend on fluid density [[rho].sub.o,] but depends on compressibility. If [c.sub.o] [right arrow] [infinity] then the matrix H (f) coincides with the matrix H (0) in compressible fluid when f = 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The matrices H (6.07), H (230) are presented for the eigenfrequencies s = 1, s = 3, s = 5 (Table 2). The latter matrix corresponds to the parameter [[theta].sub.o] = 2.71 > 1 and some entries are negative. Nevertheless, the matrix H (230) and the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are symmetric, therefore all eigenvalues and eigenvectors are real and can be defined positive (Table 2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Only the first column of the matrix [T.sub.H(6.07)] is the true first mode of the plate in compressible fluid. All other columns are the eigenvectors of the plate, and all these eigenvectors, with the first mode included, make a set of orthogonal vectors, complete in the functional space of investigation. This is true with the set of eigenvectors

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The last column is not only the eigenvector, but also can be assumed as eigenmode number 5. The product of the matrices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the eigenfrequencies. But if the set of eigenmodes forms the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

then [T.sub.MODE] [T'.sub.MODE] [not equal to] 1. Notice that absolute values of entries in diagonal of [T.sub.MODE] are much larger then all other absolute values of the same matrix, even though the matrix TH(230) has the diagonal values less then in [T.sub.MODE].

4. Discussion

When vibrations are forced by harmonic force F = [F.sub.z]sin[[omega].sub.z]t, [f.sub.z] = [[omega].sub.z]/2[pi], and the frequency [f.sub.z] coincides or is near the eigenfrequency [f.sub.j] of the plate in compressible fluid (underlined values in Tables 1, 2), the mode of vibration can be assumed equal to the eigenvector--the j-th column in the matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If the mode of the forced vibrations should be more precise, the other eigenvectors of the matrix [T.sub.H]([f.sub.s]) can be applied. If all n eigenvectors are necessary, any set of eigenvectors [T.sub.H]([f.sub.s]), s = 1,2, ... , n is acceptable. The set of eigenmodes can be unsuitable as base functions because the set can be not complete in the vector space of investigation. Moreover, the eigenmodes are not orthogonal. It may be indicated, that added masses are useful only when rigid bodies are in fluid. In some sense the coefficients [[alpha].sub.rs] = [[alpha].sub.rs] can be presented as substitute to the added mass.

The eigenmodes are important when resonant vibrations are induced and one or two eigenvectors of the corresponding matrix are required to present the forced vibration. Real fluid always is compressible, so any investigation of the fluid and structure raise the problem--what is the practical and general theoretic significance of the fluid compressibility.

http://dx.doi.org/ 10.5755/j01.mech.18.1.1284

References

[1.] Tiliouine, B.; Seghir, A. 1998. Fluid-structure models for dynamic studies of dam-water systems, 11th European Conf. on Earthquake Engineering, Paris, France.

[2.] Hernandez, E. 2006. Approximation of the vibration modes of a plate and shells coupled with a fluid, Journal of Applied Mechanics 73: 1005-1010. http://dx.doi.org/10.1115/1.2173675

[3.] Mellado, M.; Rodriguez, R. 2001. Efficient solution of fluid-structure vibration problems, Journal of Applied Numerical Mathematics 36: 389-400. http://dx.doi.org/10.1016/S0168-9274(00)00015-5

[4.] Gretarsson, J.T.; Kwatra, N.; Fedkiw, R. 2011. Numerically stable fluid-structure interactions between compressible flow and solid structures, Journal of Computational Physics 230: 3062-3084. http://dx.doi.org/10.1016/jjcp.2011.01.005

[5.] Xiao-xu, D.; Bao-wei, S.; Guang, P. 2011. Fluid dynamics and motion simulation of underwater glide vehicle, Mechanika 17(4): 363-367. http://dx.doi.org/10.5755/j01.mech.17.4.562

[6.] Bansevicius, R.; Zhurauski, M.; Dragasius, E.; Chodocinskas, S. 2008. Destruction of chains in magnetorheological fluids by high frequency oscillation, Mechanika 5(73): 23-26.

[7.] Kargaudas, V.; Zmuida, M. 2008. Forced vibrations of two plates in fluid and limit eigenmodes, Mechanika 2(70): 27-31.

V. Kargaudas, Kaunas University of Technology, Studentu 48, 51367 Kaunas, Lithuania, E-mail: vkargau@ktu.lt

M. Zmuida, Kaunas University of Technology, Studentu 48, 51367 Kaunas, Lithuania, E-mail: mykolas.zmuida@ktu.lt

M. Zmuida, Kaunas Technical College, Tvirtoves a. 35, 50155 Kaunas, Lithuania, E-mail: mykolas.zmuida@ktu.lt

Received March 10, 2011

Accepted February 02, 2012

Table 1
Vibrations of the steel plate

s [[theta].sub.0] [f.sub.1] [f.sub.2] [f.sub.3] [f.sub.7]

-- 0 6.54 26.1 58.8 320
1 0.0052 0.634 4.08 11.47 92.3
2 0.0333 0.634 4.08 11.47 92.3
3 0.0936 0.632 4.07 11.46 92.3
4 0.192 0.627 4.07 11.46 92.3
5 0.333 0.613 4.06 11.45 92.3
6 0.519 0.578 4.03 11.42 92.2
7 0.753 0.486 3.96 11.36 92.2

Table 2
Vibrations of the wood plate

s [[theta].sub.0] [f.sub.1] [f.sub.2] [f.sub.3] [f.sub.4]

-- 0 6.072 30.93 75.58 140.1
1 0.0714 6.067 30.93 75.58 140.1
2 0.363 5.914 30.82 75.49 140.0
3 0.879 4.142 29.94 74.77 139.4
4 1.65 7.47 31.1 75.7 140.3
5 2.71 10.26 35.9 81.0 145.8

s [f.sub.5]

-- 224.6
1 224.6
2 224.5
3 224.0
4 224.8
5 230.4