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In-plane dynamic stiffness matrix for a free orthotropic plate
Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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In-plane dynamic stiffness matrix for a free orthotropic plate O. Ghorbel a, J.B. Casimir b,n, L. Hammami a, I. Tawfiq b, M. Haddar a a Ecole Nationale d'Ingénieurs de Sfax, Université de Sfax, Laboratoire de Mécanique, Modélisation et Production, Route Soukra Km 3.5, BP 1173, 3038 Sfax, Tunisia b Institut Supérieur de Mécanique de Paris, LISMMA-Quartz, 3 rue Fernand Hainaut, 93407 Saint-Ouen, France
a r t i c l e i n f o
abstract
Article history: Received 3 June 2015 Received in revised form 16 November 2015 Accepted 18 November 2015 Handling Editor: S. Ilanko
The aim of this paper is to describe a procedure for computing the dynamic stiffness matrix relative to the in-plane effect for an orthotropic rectangular plate. The dynamic stiffness matrix is calculated for free edge boundary conditions. The formulation is based on strong solutions for the equations of motion for an orthotropic plate obtained with the Levy series and a Gorman decomposition of the free boundary conditions. The results obtained for the in-plane harmonic response are validated by the Finite Element Method. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Dynamic Stiffness Method Orthotropic plate In-plane Vibrations Harmonic response
1. Introduction Structural vibrations are one of the main problems investigated by mechanical engineering in the domain of transport. Much research has focused on the vibration analysis of plates [1,2]. In-plane vibration is very important for understanding the energy transmission between two coupled plates [3–5] and modeling sandwich composite plates [6]. Bardell et al. [7] presented a significant contribution to understanding in-plane vibrations by using the Rayleigh–Ritz method to calculate the in-plane vibrational frequencies for plates with simply supported edges. Farag and Pan [8] expanded a solution for the in-plane vibration of a rectangular plate when the opposing edges are clamped. Yufei Zhang et al. [9] used a double Fourier cosine series to present a series solution for the in-plane vibration analysis of an orthotropic rectangular plate with elastically restrained edges and compared the analytic results with those obtained using the finite element method. Gorman extended the problem of lateral vibrations when analytical types of solutions are used for the in-plane vibration of rectangular plates [10], making it possible to apply the superposition method [11]. Gorman also presented the analytical method of in-plane solutions for the natural frequencies and for the mode shapes of simply supported and clamped rectangular plates [12]. In addition, Xing et al. [13] presented the exact solution for the in-plane natural frequencies of a rectangular plate when its opposing edges are simply supported. Du et al. [14] applied the Fourier series method to analyze the in-plane vibration of a rectangular plate with elastically restrained edges while Seok et al. [15] analyzed the free in-plane vibration for a rectangular cantilever plate. They used a variational method and an equation of motion for analyzing in-plane vibrations of thin orthotropic plates. Woodcock et al. [16] expanded the Hamilton principle and the Rayleigh–Ritz method to study the effects of the ply orientation on in-plane vibrations. Park [17] used the separation of the variables to derive the n
Corresponding author. E-mail address: [email protected] (J.B. Casimir).
http://dx.doi.org/10.1016/j.jsv.2015.11.028 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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equations for the clamped circular plate. Moreover, Gorman presented the exact solution for a rectangular plate when its two opposing edges are simply supported and the others are clamped or free [18]. The Dynamic Stiffness Method has proved efficiency for analyzing the harmonic response for complex structures composed of simple elements [19–21]. This meshless method characterized by the absence of structural discretization is based on exact solutions of the harmonic motion equations for free boundary conditions. The method uses the dynamic stiffness matrix KðωÞ that links the boundary displacements denoted U and the external forces applied on the boundaries denoted F, according to KðωÞ U ¼ F
(1)
Dynamic stiffness matrices have been developed for many elements. Casimir et al. [22] and Banerjee et al. [23,24] developed the dynamic stiffness matrix of various beams, while Tounsi et al. studied circular rings [25]. Boscolo and Banerjee [26] focused on in-plane vibration of isotropic plates according to the assumption that two opposite sides are simply supported. Some researchers have also investigated the dynamic stiffness matrix of shells [27–29]. The main objective of this paper is to develop the dynamic stiffness matrix of an orthotropic plate for in-plane vibrations according to the assumption that all the edges of the plate are free. Natural boundary conditions for the four edges are required to deal with future assemblies. In order to achieve this formulation, we used Gorman decompositions of four symmetry contributions and Levy type solutions, as explained in a previous paper concerning flexural vibrations [30]. A such approach has been recently used by NefovskaDanilovic and Petronijevic for the problem of in-plane vibrations of isotropic plates [31]. The dynamic stiffness matrix KðωÞ is built by superposing four symmetry contributions. Following this, the validation of the formulation is achieved by comparisons of harmonic responses obtained by the Finite Element Method.
2. In-plane orthotropic plate equations 2.1. Internal forces–displacements relationship Let us consider an orthotropic rectangular plate characterized by its lateral dimensions 2a and 2b, and its thickness h. It is assumed that the principal material axes are parallel with the edges of the plate and with the axes of the Cartesian coordinate system denoted Ox and Oy. The in-plane displacements of any point on the middle surface of the plate along the xaxis and the y-axis are denoted u and v, respectively (see Fig. 1). The relationship of the stress/displacements is given by the following equations: 8 ∂u ∂v > > σ x ¼ Dx þ D1 > > ∂x ∂y > > > < ∂u ∂v σ y ¼ D1 þ Dy (2) ∂x ∂y > > > > > ∂u ∂v > > : σ xy ¼ Dxy ∂x þ ∂y where σx, σy and σxy are the stress tensor components. Dx, Dy, D1 and Dxy are the material constants defined by the following equation: 8 Ex > > Dx ¼ > > 1 ν > xy νyx > > > > ν Ey νyx Ex xy > < D1 ¼ ¼ 1 νxy νyx 1 νxy νyx (3) > > Ey > > > Dy ¼ > > 1 νxy νyx > > > :D ¼G xy xy
z
y
2b O
x
h 2a Fig. 1. Orthotropic plate.
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where Ex and Ey are Young's moduli along directions x and y, respectively, νxy, νyx are Poisson's ratios and Gxy is the shear modulus. The internal forces are obtained by a simple integration of the stresses over the thickness according to the following equation: Z h=2 Z h=2 Z h=2 σ x dz; N y ¼ σ y dz; Nxy ¼ σ xy dz (4) Nx ¼ h=2
h=2
h=2
2.2. In-plane equation of motion The in-plane equations of motion can be obtained by applying Hamilton's principle. They are given by Eq. (5) in the case where no distributed loads are applied inside the plate: 8 ∂Nx ∂Nxy ∂2 u > > > < ∂x þ ∂y ¼ ρh ∂t 2 (5) > ∂Nxy ∂Ny ∂2 v > > þ ¼ ρh 2 : ∂x ∂y ∂t where ρ is the mass density of the material. Eqs. (3)–(5) allow obtaining the system of Eq. (6) satisfied by the displacement components alone: 8 ∂2 u ∂2 v ∂2 u ∂2 v ∂2 u > > þ Dxy 2 þ Dxy ¼ρ 2 > Dx 2 þD1 < ∂x∂y ∂x∂y ∂x ∂y ∂t (6) > ∂2 v ∂2 u ∂2 v ∂2 u ∂2 v > > þ Dxy 2 þ Dxy ¼ρ 2 : Dy 2 þD1 ∂x∂y ∂x∂y ∂x ∂y ∂t Time t can be eliminated for harmonic responses. Therefore the displacements U and V are written as follows: ( uðx; y; tÞ ¼ Uðx; yÞejωt vðx; y; tÞ ¼ Vðx; yÞejωt
(7)
where Uðx; yÞ and Vðx; yÞ are the amplitudes of the in-plane harmonic displacement. Therefore the system of Eq. (6) reduces to 8 ∂2 U ∂2 V ∂2 U ∂2 V > 2 > > < Dx ∂x2 þ D1 ∂x∂y þ Dxy ∂y2 þ Dxy ∂x∂y ¼ ρω U (8) > ∂2 V ∂2 U ∂2 V ∂2 U > > þ Dxy 2 þDxy ¼ ρω2 V : Dy 2 þD1 ∂x∂y ∂x∂y ∂x ∂y After successive derivations, the equations of the system (8) can be decoupled into two equations, resulting in a set of uncoupled equations of motion as follows: 8 4 4 4 2 2 > > > A1 ∂ U þ A2 ∂ U þ A3 ∂U þ A4 ∂ U þ A5 ∂ U þ A6 U ¼ 0 > < ∂x4 ∂y4 ∂x2 ∂y2 ∂x2 ∂y2 (9) > ∂4 V ∂4 V ∂4 V ∂2 V > ∂2 V > B þB þB þ B þ B þ B V ¼ 0 > 5 1 2 3 4 6 2 : ∂x4 ∂y ∂y4 ∂x2 ∂y2 ∂x2 where Ai and Bi for i¼ 1,2,6 are constants dependent on material properties, mass density and circular frequency ω: Ex 1 νxy νyx Ey A2 ¼ 1 νxy νyx
A1 ¼
A3 ¼ Gxy þ
E x Ey 2
ðGxy ð1 νxy νyx ð þ νxy Ey Þ2
ð1 νxy νyx Þ ð1 νxy νyx Þ2 Gxy Ex ρω2 A4 ¼ ρω2 ð1 νxy νyx ÞGxy Ey ρω2 A5 ¼ ρω2 ð1 νxy νyx ÞGxy A6 ¼
ρ2 ω4 Gxy
B1 ¼ Gxy Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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B2 ¼ Gxy
Ey Ex
Ey B3 ¼ þ 1 νxy νyx
G2xy ð1 νxy νyx Þ Ex
B4 ¼ B5 ¼
νxy Ey þ Gxy 1 νxy νyx Ex
Gxy ρω2 ð1 νxy νyx Þ ρω2 EL
1 νxy νyx
Ey ρω2 Gxy ρω2 ð1 νxy νyx Þ Ex Ex B6 ¼
2
ρ2 ω4 ð1 νxy νyx Þ Ex
3. The in-plane dynamic stiffness matrix of an orthotropic plate The Dynamic Stiffness Method is based on the calculation of the dynamic stiffness matrix. This matrix is built with a strong solution of the harmonic problem. Free edge boundary conditions are necessary to allow assemblies between plates along their four edges. In order to obtain this strong solution, a decomposition approach is used, as was done recently for the flexural problem [30]. This approach is based on a decomposition of the problem into four symmetry contributions denoted SS (symmetric–symmetric), SA (symmetric–antisymmetric), AS (antisymmetric–symmetric) and AA (antisymmetric–antisymmetric). For each symmetry contribution, the displacement solutions U and V are obtained with two series. The main difference with Kirchhoff's flexural problem is that two components of the displacement solution have to be obtained simultaneously from the solution of the differential system of Eq. (8).
3.1. Symmetry contributions Any displacement solution U can be decomposed into four contributions with respect to symmetry considerations. It can be written as follows: Uðξ; ηÞ ¼ U SS ðξ; ηÞ þ U AA ðξ; ηÞ þ U SA ðξ; ηÞ þ U AS ðξ; ηÞ
(10)
where ξ and η are the dimensionless coordinates defined by the following equation: x ξ¼ ; a
η¼
y b
(11)
The first subscript, S or A, indicates if the contribution is symmetric or antisymmetric, respectively, around the coordinate ξ. The second subscript indicates the symmetry property about the coordinate η. For example, UAS is the contribution of the displacement U which is antisymmetric around ξ and symmetric around η, that is to say: U AS ðξ; ηÞ ¼ U AS ð ξ; ηÞ ¼ U AS ðξ; ηÞ ¼ U AS ð ξ; ηÞ
(12)
Similar considerations can be made for the V component of the displacement but U and V are coupled by Eq. (8). These equations show that if U is symmetric around a spatial coordinate, V should be antisymmetric around this coordinate, and if U is antisymmetric around a spatial coordinate, then V should be symmetric around this coordinate. Therefore the component displacement V is split into four contributions as is U with Eq. (10), but the notation is quite different. For example, the symmetric–symmetric contribution of V is denoted VAA. The subscripts always indicate the symmetry property of the corresponding U component of the displacement.
3.2. Strong solutions Two series are used for each symmetry contribution to obtain a strong solution of the harmonic problem. These series are defined as the Levy series for the flexural problem but the parameters in the trigonometric functions were chosen to obtain Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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a non-vanishing solution along the four edges of the plate. These series are given in the following equation: 8 1 1 X X > 4n 3 4n 3 > > πη þ πξ 1SS U n ðξÞ cos 2SS U n ðηÞ cos U SS ðξ; ηÞ ¼ > > > 4 4 > n¼1 n¼1 > > > 1 1 > > > V ðξ; ηÞ ¼ X 1 V ðξÞ sin 4n 3 πη þ X 2 V ðηÞ sin 4n 3 πξ > > SS SS n SS n > > 4 4 > n¼1 n¼1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ 1AA U n ðξÞ sin 2AA U n ðηÞ sin > > U AA ðξ; ηÞ ¼ > 4 4 > n ¼ 1 n ¼ 1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ V AA ðξ; ηÞ ¼ 1AA V n ðξÞ cos 2AA V n ðηÞ cos > > < 4 4 n¼1 n¼1 1 1 X X > 4n 3 4n 3 > > ð Þ ð Þ πη þ πξ U ξ; η ¼ 1 U ξ sin 2SA U n ðηÞ cos > n SA SA > 4 4 > > n¼1 n¼1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ V SA ðξ; ηÞ ¼ 1SA V n ðξÞ cos 2SA V n ðηÞ sin > > 4 4 > > n ¼ 1 n ¼ 1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ U AS ðξ; ηÞ ¼ 1AS U n ðξÞ cos 2AS U n ðηÞ sin > > 4 4 > > n¼1 n¼1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ V AS ðξ; ηÞ ¼ 1AS V n ðξÞ sin 2AS V n ðηÞ cos > > 4 4 : n¼1 n¼1
5
(13)
The left-superscript 1 or 2 indicates whether the function depends on the coordinate ξ or the coordinate η. Introducing the series (13) into Eq. (9) gives ordinary differential equations satisfied by the functions kαβ U n and kαβ V n , where k is 1 or 2, αβ is SS, AA, SA or AS, see the following equation: 8 2k 4k > d U > A A d αβ U n 4 A2 2 2 A5 4 A6 k > > ϕ4 αβ n þ ϕ2 k21 3 þ ϕ2 b2 4 þ k k b þ b U ¼0 > 1 1 > A1 A1 A1 A1 A1 αβ n > dξ4 dξ2 > > > > 4k 2k > > d αβ U n A1 A4 2 A5 d αβ U n 4 2 2 4 A6 k > 2 2 A3 > þ k1 ϕ4 k1 b ϕ2 þ b U ¼0 > < dη4 þ ϕ k1 A2 þb A2 A4 A2 A2 αβ n dη2 (14) 2k 4k > d αβ V n d αβ V n > 4 B2 2 2 B5 4 B6 k > 4 2 2 B3 2 2 B4 > ϕ þ ϕ k1 þ ϕ b þ k1 k1 b þb V ¼0 > > B1 B1 B1 B1 B1 αβ n > dξ4 dξ2 > > > 4k 2k > > d αβ V n > B1 B4 2 B5 d αβ V n 4 2 2 4 B6 k > 2 2 B3 > þ k1 ϕ4 k1 b ϕ2 þ b V ¼0 > : dη4 þ ϕ k1 B2 þ b B2 B4 B1 B1 αβ n dη2 3Þπ with ϕ ¼ ba and k1 ¼ ð4n . The closed form solutions are given by the following equations: 4 8 1 1 1γ n ξ > 7e γ n ξ þ Bn e1κn ξ 7 e 1κn ξ > αβ U n αβ ðξÞ ¼ An e > > > > > 2 U ðηÞ ¼ C n e2γ n η 7 e 2γn η þDn e2κn η 7e 2κn η < αβ n αβ 1 > V ðξÞ ¼ An e1γ n ξ 7e 1γ n ξ þ Bn e1κn ξ 7 e 1κn ξ > > > αβ n αβ > > > : 2αβ V n ðηÞ ¼ C n e2γ n η 7e 2γ n η þ Dn e2κn η 7 e 2κn η αβ
(15)
The sign 7 changes with the contribution and the 1 γ n , 2 γ n , 1 κ n and 2 κ n are the complex roots of the characteristic equations. They are obtained by the following equations: 8 2 > A1 A2 A3 ðA1 b ρω2 Þ > 2 2 > k B þ þ b ρω2 > 1 1 > pffiffiffiffiffiffi > B B B 1 1 1 > 1 > þ Δ1 =ϕ4 > γn ¼ 2 > > ϕ > > > > > 2 > > A1 A2 A3 ðA1 b ρω2 Þ 2 2 > > k1 B1 þ þ b ρω2 pffiffiffiffiffiffi > > B B B > 1 1 1 2 < γ ¼ Δ1 =ϕ4 n 2 ϕ (16) > > A1 A2 A3 ðA1 a2 ρω2 Þ > 2 2 2 > > k B þ þ a ρω 1 > 1 pffiffiffiffiffiffi >1 B1 B1 B1 > > κn ¼ þ Δ2 =2 > > 2 > > > ϕ > > A1 A2 A3 ðA1 a2 ρω2 Þ : > 2 > k B þ þ a2 ρω2 pffiffiffiffiffiffi > 1 1 > >2 B B B 1 1 1 > > Δ2 =2 : κn ¼ ϕ2 Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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where Δ1 and Δ2 are defined by Eqs. (17) and (19), respectively: 2
Δ1 ¼ d 4ce where
(17)
8 c ¼ A1 ϕ4 > > < 2 2 d ¼ ϕ2 k1 A3 þ ϕ2 b A4 > > : e ¼ k4 A b2 k4 A þA b4 1 5
1 2
(18)
6
Δ2 ¼ o2 4pq
(19)
where 8 o ¼ A2 > < 2 2 p ¼ ϕ2 k1 A3 þ ϕ2 b A5 > : q ¼ k4 A b2 k4 A þ A b4 1 1
1 4
(20)
6
3.3. Edge in-plane displacements The procedure for calculating the dynamic stiffness matrix (DSM) consists of a sequence of symbolic computations of equations that link coefficients An ; Bn ; C n and Dn to the projections of the edge displacement and external force on a functional basis. First, the edge displacements are obtained from the U and V solutions inside the plate given by Eqs. (13) and (15). These displacements are linked to the coefficients according to a symbolic matrix equation. For example, the edge displacements for the symmetric–symmetric contribution are such that 8 pffiffiffi 1 1 X X > 4n 3 2 > 1 2 > πη þ ð 1Þn þ 1 U ð 1; η Þ ¼ U ð 1 Þ cos U ð η Þ > SS SS n SS n > 4 2 > > n¼1 n¼1 > p ffiffiffi > 1 1 > X X > 2 4n 3 1 2 > > ð 1Þn þ 1 πη þ > V SS ð1; ηÞ ¼ SS V n ð1Þ sin SS V n ðηÞ < 2 4 n¼1 n ¼ 1 pffiffiffi (21) 1 1 X X > 4n 3 2 1 2 > nþ1 > πξ ð 1Þ U ð ξ; 1 Þ ¼ U ð ξ Þ þ U ð 1 Þ cos > SS SS n SS n > 4 2 > > n¼1 n¼1 > pffiffiffi : > 1 1 > X X > 4n 3 2 > 1 2 nþ1 > ð Þ ð Þ ð Þ πξ ð 1Þ V ξ; 1 ¼ V ξ þ V 1 sin > SS SS n SS n : 4 2 n¼1
n¼1
and after limiting the Levy series to N terms, we obtain 0
A1
1
B B1 C B C B C B C1 C B C 0 1 B D1 C U SS ð1; ηÞ B C B C B C B V SS ð1; ηÞ C B ⋮ C B C ¼ ∂HSS B C B U SS ðξ; 1Þ C BA C @ A B NC B C V SS ðξ; 1Þ B BN C B C BC C B NC B C @ DN A where
0
∂H 11 ðηÞ
B ∂H ðηÞ B 21 ∂HSS ¼ B B ∂H 31 ðξÞ @ ∂H 41 ðξÞ
⋯ ⋯ ⋯ ⋯
∂H 1;4ðN 1Þ þ 4 ðηÞ
(22)
1
∂H 2;4ðN 1Þ þ 4 ðηÞ C C C ∂H 3;4ðN 1Þ þ 4 ðξÞ C A
(23)
∂H 4;4ðN 1Þ þ 4 ðξÞ
The functions ∂Hij are obtained by a symbolic computation. 3.4. External forces A similar symbolic computation is achieved for the external forces applied on each edge of the plate. First, the force/ displacement relationships given by Eqs. (2) and (4) are used to evaluate internal forces from the displacement solutions U and V. Then, free boundary conditions are used to obtain the relations between the external forces and the coefficients An , Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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Table 1 Functional basis. SS Uð1; ηÞ V ð1; ηÞ Uðξ; 1Þ V ðξ; 1Þ
AA
cos ðmπηÞ
ð2m 1Þπη 2 cos ðmπξÞ
ð2m 1Þπη 2 cos ðmπηÞ
sin
ð2m 1Þπξ 2
ð2m 1Þπξ 2 cos ðmπξÞ
ð2m 1Þπη 2 cos ðmπηÞ
sin
sin
ð2m 1Þπξ 2
cos ðmπηÞ
ð2m 1Þπη 2 ð2m 1Þπξ sin 2 cos ðmπξÞ
sin
cos ðmπξÞ
sin
AS
sin
sin
SA
Bn , C n , Dn . These relations are defined by 0
1 A1 BB C B 1 C B C B C1 C 0 1 B C Nx ð1; ηÞ BD C B N ð1; ηÞ C B 1C B xy C B C B C B C B Ny ðξ; 1Þ C ¼ ∂GSS B ⋮ C @ A BA C B NC Nxy ðξ; 1Þ B C B BN C B C BC C @ NA DN
(24)
where 0
∂G11 ðηÞ B ∂G ðηÞ B 21 ∂GSS ¼ B B ∂G31 ðξÞ @ ∂G41 ðξÞ
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
∂G1;4ðN 1Þ þ 4 ðηÞ
1
∂G2;4ðN 1Þ þ 4 ðηÞ C C C ∂G3;4ðN 1Þ þ 4 ðξÞ C A ∂G4;4ðN 1Þ þ 4 ðξÞ
(25)
Functions ∂Gij are obtained by a symbolic computation and stored in a matrix. 3.5. Functional basis The external forces and edge displacement functions are projected on a functional basis. The DSM gives the exact relations between these projections. To do this, a functional basis is defined on each edge of the plate and for each contribution. For example, in the case of the symmetric–symmetric contribution, the functional basis on the edge ξ ¼ 1 is given by the functions cos mπη for m A f1 ,.., Ng. Table 1 gives all the functional bases used. The edge displacement projections on these bases are computed by symbolic computational software for each symmetry contribution. These projections still depend on the unknown parameters Ai ; Bi ; C i and Di . They are stored in four matrices denoted HSS ; HAA ; HSA and HAS . For example, the symmetric–symmetric matrix is such that 0
1 1 SS U 0 B 2 C B SS U 0 C B C B 1 C 0 1 B SS U 1 C A1 B C B 1V C BB C B SS 1 C B 1C B 2 C B C B C U B C1 C B SS 1 C B C B 2 C BD C B SS V 1 C B 1C B C B C B C C ⋮ B C ¼ HSS B B ⋮ C B1 C BA C B SS U N 1 C B NC B C B C B1 C B BN C B SS V N 1 C B C B C B C B2 C @ CN A B SS U N 1 C B C DN B2 V C B SS N 1 C B 1 C B C @ SS V N A 2 SS V N
(26)
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and is given by the following equation: 0 R 1 1
1 H 11 ðηÞ R 1 1 2 1 H 21 ðξÞ
2
dη
B B dξ B B R1 B H ðηÞ cos ðπηÞ dη B 1 31 πη B R1 B dη B 1 H 41 ðηÞ sin 2 B B ⋮ B B R1 B 1 H 11 ðηÞ cos ððN 1ÞπηÞ dη B B R ð2N 3Þ B 1 πη dη HSS ¼ B 1 H 21 ðηÞ sin B 2 B R1 B H ðξÞ cos ððN 1ÞπξÞ dξ B 1 11 B B R1 ð2N 3Þ B πξ dξ B 1 H 21 ðξÞ sin 2 B B B R1 ð2N 1Þ B πη dη H ð η Þ sin B 1 31 2 B BR ð2N 1Þ @ 1 πη dη 1 H 41 ðξÞðηÞ sin 2
⋯ ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
1 2
R1
1 R 1 1 2 1
H 1;4ðN 1Þ þ 4 ðηÞ dη
1
C C C C R1 C H ðηÞ cos ðπηÞ dη C 1 3;4ðN 1Þ þ 4 πη C R1 C dη H ð η Þ sin C 4;4ðN 1Þ þ 4 1 2 C C ⋮ C R1 C C 1 H 1;4ðN 1Þ þ 4 ðηÞ cos ððN 1ÞπηÞ dη C C R1 ð2N 3Þ C πη dη C 1 H 2;4ðN 1Þ þ 4 ðηÞ sin C 2 C R1 C H ðξÞ cos ððN 1ÞπξÞ dξ C 1 1;4ðN 1Þ þ 4 C C R1 ð2N 3Þ πξ dξ C C 1 H 2;4ðN 1Þ þ 4 ðξÞ sin 2 C C R1 C ð2N 1Þ πη dη C 1 H 3;4ðN 1Þ þ 4 ðηÞ sin C 2 C C R1 ð2N 1Þ A πη dη 1 H 4;4ðN 1Þ þ 4 ðηÞ sin 2 H 2;4ðN 1Þ þ 4 ðξÞ dξ
(27)
The projections of the forces applied are computed in the same way and stored in four matrices GSS ; GAA ; GSA and GAS . 3.6. Dynamic stiffness matrix The last step consists in calculating the DSM for each contribution by eliminating the parameters Ai ; Bi ; C i and Di using a numerical inversion of the matrices Gαβ . Therefore the DSM of the αβ-symmetry contribution is given by 1 Kαβ ¼ Hαβ Gαβ
(28)
4. Results and discussion 4.1. Symmetric–symmetric contribution We consider a carbon-epoxy plate with Young's moduli Ex ¼ 18:1 GPa and Ey ¼ 50:9 GPa, shear modulus Gxy ¼ 11 GPa, Poisson's ratio νxy ¼ 0:8 and mass density ρ ¼ 1526 kg=m3 . Its dimensions are a ¼ 0:5 m; b ¼ 0:25 m and h ¼ 0:002 m. Free– free–free–free (FFFF) boundary conditions are considered. The sole symmetric–symmetric contribution is isolated by using symmetric–symmetric in-plane loads. Uniform and identical harmonic loads are defined on the edges located on ξ ¼ 1 and ξ ¼ 1, that is to say: F x ð1; ηÞ ¼ F x ð 1; ηÞ ¼ 1
(29)
Fig. 2 shows this loading. The harmonic response along the x-direction is evaluated at point A located on coordinates (1,0). Fig. 3 shows this response over the frequency range [0, 10 000 Hz] and gives a comparison with the responses obtained by finite element models. Four-node Discrete Kirchhoff Quadrilateral (DKQ) elements are used in the FEM for two meshing refinements (40 40 and 80 80 elements).
Fig. 2. Symmetric–symmetric loads.
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−20 FEM 40x40 FEM 80x80 1 DSM
−40 −60
20 log10|U|
−80 −100 −120 −140 −160 −180 −200 −220
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
Frequency (Hz) Fig. 3. Harmonic response to a symmetric–symmetric load.
−120 FEM 40x40 FEM 80x80 FEM 100x100 1 DSM
−140
20 log10|U|
−160
−180
−200
−220
−240 8000
8100
8200
8300
8400
8500
8600
8700
8800
8900
9000
Frequency (Hz) Fig. 4. Harmonic response on [8000 Hz, 9000 Hz].
Very good convergence can be seen between the two approaches in the [0, 6000 Hz] frequency range, but beyond 6000 Hz, the precision of the FEM responses decreases. Fig. 4 shows more detailed responses in the [8000 Hz, 9000 Hz] frequency ranges. A third FEM using 100 100 DKQ elements is used to prove the convergence. Table 2 gives the computation times for calculating the response for one frequency and shows the performances of the present formulation compared to FEM.
4.2. Distributed force The next case studied is a plate subjected to a uniform distributed force on its right edge only, see Fig. 5. This force is defined by F x ð1; ηÞ ¼ 1
(30)
The material properties and the dimensions of the plate are the same as the previous one. Fig. 6 shows the harmonic response in the direction x of point A located at (1,0) . Very good convergence between the FEM results and DSM results can be observed. For frequencies higher than 5000 Hz, the FEM mesh must be fine enough to reach the precision of the DSM. Fig. 7 gives a more detailed representation of the curves. Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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Table 2 Computation times. Model
1 frequency calculation time
1 CEM 20 20 FEM 40 40 FEM 80 80 FEM 100 100 FEM
0.014 0.015 s 0.069 s 0.45 s 1.2 s
F y
z
x
Fig. 5. Distributed force.
0 FEM 20x20 FEM 40x40 1 DSM
20 log10|U|
−50
−100
−150
−200
−250
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Frequency (Hz)
Fig. 6. Harmonic response for a distributed load.
−80 1 DSM FEM 50 x 50 FEM 100 x 100 FEM 200 x 200
−100 −120
20log10|U|
−140 −160 −180 −200 −220 −240 8000
8100
8200
8300
8400
8500
8600
8700
8800
8900
9000
Frequency (Hz) Fig. 7. Harmonic response in the frequency range [8000 Hz, 9000 Hz].
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4.3. Harmonic response for a concentrated force An important step of validation consists in evaluating the results when more than one term is necessary in the series development of the loads. This can be achieved using concentrated forces. The next load case implies such a load. A harmonic concentrated force located at ξ ¼ 1 and η ¼ 1 is used and the response is computed in the x-direction at point A located at coordinates (1,0). The projections of the concentrated force on the functional basis given in Table 1 are defined by the following equation: 8 1 > > > F0 ¼ > > 2b > > < ð 1Þn F Sm ¼ (31) > b > > n > > ð 1Þ A > > : Fm ¼ b These projections are such that F ð1; ηÞ ¼ F 0 þ
M X
F Sm cos mπη þ
m¼1
M X m¼1
π F Am sin ð2m 1Þ η 2
(32)
The response is evaluated at point Pðξ ¼ 1; η ¼ 1Þ. Fig. 8 shows this response and a comparison with finite element responses over 7000 Hz–8000 Hz range. −80 FEM 50 x 50 FEM 100 x 100 FEM 200 x 200 1 DSM
−100
20 log10|U|
−120 −140 −160 −180 −200 −220 −240 7000
7100
7200
7300
7400
7500
7600
7700
7800
7900
8000
Frequency (Hz) Fig. 8. Harmonic response for a concentrated load.
20 1 DSM
0
20log10|U|
-20
-40
-60
-80
-100
-120
0
50
100
150
200
250
300
Frequency (Hz) Fig. 9. Harmonic response of the Woodcock plate.
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Table 3 Eigenfrequencies. Model
F1
F2
F3
F4
F5
F6
F7
F8
Woodcock 1 DSM 20 20 4-node FEM 40 40 4-node FEM 10 10 8-node FEM 20 20 8-node FEM Symmetry properties
109.86 109.97 110.03 109.88 109.83 109.82 SS
126.23 126.15 126.26 126.17 126.14 126.13 AS
138.31 138.30 138.73 138.43 138.34 138.33 SA
205.00 204.99 206.43 205.39 205.08 205.04 AA
223.85 223.64 224.80 223.89 223.61 223.58 AS
225.19 225.20 226.39 225.44 225.18 225.12 SS
236.27 236.17 237.73 236.61 236.35 236.24 SA
252.38 252.31 253.13 252.35 252.14 252.09 AA
Very good convergence can be observed between the FE results and the DSM response obtained with 9 terms in the series development. 4.4. Natural frequencies This section gives a comparison with the available literature concerning the in-plane vibrations of orthotropic plates. No reference was found regarding the in-plane harmonic response for free boundary conditions, therefore the comparison is restricted to the calculation of natural frequencies. The in-plane natural frequencies of an orthotropic plate were given in the article of Woodcock et al. [16] cited in the Introduction of our paper. A modal analysis could be performed using the William–Wittrick algorithm [33]. Instead, the harmonic response curve is used because there is no damping in the model and the modal density is poor. A fine frequency sweep is used and the positions of the peaks are detected. The Woodcock plate is a 0:376 0:376 m2 square plate with a thickness of 1:077 mm. The constitutive material is called Material I in the paper. It is a material with Young's moduli Ex ¼ 13:8 GPa and Ey ¼ 0:9 GPa, shear modulus Gxy ¼ 0:71 GPa, Poisson's ratio νxy ¼ 0:2 and mass density ρ ¼ 100 kg=m3 . Free–free–free–free (FFFF) boundary conditions are considered. Two concentrated forces are applied in the corner located on coordinates (1,1). One in x direction and one in y direction are defined. The harmonic response is processed along direction x at the point located on coordinates (1,0). It is shown in Fig. 9. Eight natural frequencies are obtained on [0,300 Hz]. These frequencies are given on line 3 of Table 3. The other lines give Woodcock's results (Table 8 of his paper for a 0 angle ply) and the results of modal analysis performed with finite element models. Very good agreement between all the results is shown. The use of the Dynamic Stiffness Matrix Kαβ of each contribution (see Eq. (28)) for this load case allows associating each peak of the response to a symmetry property. These properties are given on the last line of the table. They are related to the property of the U displacement of the corresponding natural mode of vibration.
5. Conclusion The main contribution of this paper was to describe an analytical procedure to build a dynamic stiffness matrix for the in-plane vibrations of a free orthotropic plate. This method is an extension of the procedure previously described for out of plane flexural vibrations but, here, the strong formulation of the harmonic problem implies two coupled partial derivative equations. The procedure is based on analytical solutions obtained with mathematical software for free edge boundary conditions. These solutions are described with Levy's series for each symmetry contribution and Gorman's decomposition of the free boundaries. This DSM allowed determining the in-plane harmonic response of rectangular orthotropic plates with high precision and without meshing. The method was validated with the FEM whose results converged very well with the DSM results for several load cases and with results of the literature concerning natural frequencies. The main advantages of the formulation are its precision, rapidity and efficiency.
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[10] D.J. Gorman, Free in-plane vibration analysis of rectangular plate by the method of superposition, Journal of Sound and Vibration 272 (2004) 831–851. [11] D.J. Gorman, Free vibration analysis of the completely free rectangular plate by the method of superposition, Journal of Sound and Vibration 57 (3) (1978) 437–447. [12] D.J. Gorman, Accurate analytical type solutions for the free in-plane vibration of clamped and simply supported rectangular plates, Journal of Sound and Vibration 276 (2004) 331–333. [13] Y.F. Xing, B. Liu, Exact solutions for the free in-plane vibrations of rectangular plates, International Journal of Mechanical Sciences 51 (2009) 246–255. [14] D. Jingtao, L.Li. Wen, J. Guoyong, Y. Tiejun, L. Zhigang, An analytical method for the in-plane vibration analysis of rectangular plate with elastically restrained edges, Journal of Sound and Vibration 306 (2007) 908–927. [15] J.W. Seok, H.F. Tiersten, H.A. Scarton, Free vibration of rectangular cantilever plate—part 2: in-plane motion, Journal of Sound and Vibration 271 (2004) 147–158. [16] R.L. Woodcock, R.B. Bhat, I.G. Stiharu, Effect of ply orientation on the in-plane vibration of single-layer composite plates, Journal of Sound and Vibration 312 (2008) 94–108. [17] II Park Chan, Frequency equation for the in-plane vibration of a clamped circular plate, Journal of Sound and Vibration 313 (2008) 325–333. [18] D.J. Gorman, Exact solution for free in-plane vibration of rectangular plate with two opposite edges simply supported, Journal of Sound and Vibration 294 (2006) 131–161. [19] P.H. Kulla, Continuous elements, some practical examples, ESTEC Workshop Proceedings: Modal Representation of Flexible Structures by Continuum Methods, 1989. [20] A.Y.T. Leung, Dynamic Stiffness and Substructures, Springer, London, 1993. [21] U. Lee, Spectral Element Method in Structural Dynamics, Wiley, Singapore, 2009. [22] J.B. Casimir, C. Duforet, T. Vinh, Dynamic behaviour of structures in large frequency range by continuous element methods, Journal of Sound and Vibration 267 (2003) 1085–1106. [23] J.R. Banerjee, H. Su, D.R. Jackson, Free vibration of rotating tapered beams using the dynamic stiffness method, Journal of Sound and Vibration 298 (2006) 1034–1054. [24] J.R. Banerjee, W.D. Gunawardana, Dynamic stiffness matrix development and free vibration analysis of a moving beam, Journal of Sound and Vibration 303 (2007) 135–143. [25] D. Tounsi, J.B. Casimir, M. Haddar, Dynamic stiffness formulation for circular rings, Computers and Structures 112–113 (2012) 258–265. [26] M. Boscolo, J.R. Banerjee, Dynamic stiffness method for exact inplane free vibration analysis of plates and plate assemblies, Journal of Sound and Vibration 330 (2011) 2928–2936. [27] J.B. Casimir, M.C. Nguyen, I. Tawfiq, Thick shells of revolution: derivation of the dynamic stiffness matrix of continuous elements and application to a tested cylinder, Computers and Structures 85 (2007) 1845–1857. [28] D. Tounsi, J.B. Casimir, S. Abid, I. Tawfiq, M. Haddar, Dynamic stiffness formulation and response analysis of stiffened shells, Computers and Structures 132 (2014) 75–83. [29] F. Fazzolari, A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells, Composites Part B: Engineering 62 (2014) 143–158. [30] O. Ghorbel, J.B. Casimir, L. Hammami, I. Tawfiq, M. Haddar, Dynamic stiffness formulation for free orthotropic plates, Journal of Sound and Vibration 346 (2015) 361–375. [31] M. Nefovska-Danilovic, M. Petronijevic, In-plane free vibration and response analysis of isotropic rectangular plates using the dynamic stiffness method, Computers and Structures 152 (2015) 82–95. [33] F.W. Williams, W.H. Wittrick, An automatic computational procedure for calculating natural frequencies of skeletal structures, International Journal of Mechanical Sciences 12 (9) (1970) 781–791.
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In-plane dynamic stiffness matrix for a free orthotropic plate O. Ghorbel a, J.B. Casimir b,n, L. Hammami a, I. Tawfiq b, M. Haddar a a Ecole Nationale d'Ingénieurs de Sfax, Université de Sfax, Laboratoire de Mécanique, Modélisation et Production, Route Soukra Km 3.5, BP 1173, 3038 Sfax, Tunisia b Institut Supérieur de Mécanique de Paris, LISMMA-Quartz, 3 rue Fernand Hainaut, 93407 Saint-Ouen, France
a r t i c l e i n f o
abstract
Article history: Received 3 June 2015 Received in revised form 16 November 2015 Accepted 18 November 2015 Handling Editor: S. Ilanko
The aim of this paper is to describe a procedure for computing the dynamic stiffness matrix relative to the in-plane effect for an orthotropic rectangular plate. The dynamic stiffness matrix is calculated for free edge boundary conditions. The formulation is based on strong solutions for the equations of motion for an orthotropic plate obtained with the Levy series and a Gorman decomposition of the free boundary conditions. The results obtained for the in-plane harmonic response are validated by the Finite Element Method. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Dynamic Stiffness Method Orthotropic plate In-plane Vibrations Harmonic response
1. Introduction Structural vibrations are one of the main problems investigated by mechanical engineering in the domain of transport. Much research has focused on the vibration analysis of plates [1,2]. In-plane vibration is very important for understanding the energy transmission between two coupled plates [3–5] and modeling sandwich composite plates [6]. Bardell et al. [7] presented a significant contribution to understanding in-plane vibrations by using the Rayleigh–Ritz method to calculate the in-plane vibrational frequencies for plates with simply supported edges. Farag and Pan [8] expanded a solution for the in-plane vibration of a rectangular plate when the opposing edges are clamped. Yufei Zhang et al. [9] used a double Fourier cosine series to present a series solution for the in-plane vibration analysis of an orthotropic rectangular plate with elastically restrained edges and compared the analytic results with those obtained using the finite element method. Gorman extended the problem of lateral vibrations when analytical types of solutions are used for the in-plane vibration of rectangular plates [10], making it possible to apply the superposition method [11]. Gorman also presented the analytical method of in-plane solutions for the natural frequencies and for the mode shapes of simply supported and clamped rectangular plates [12]. In addition, Xing et al. [13] presented the exact solution for the in-plane natural frequencies of a rectangular plate when its opposing edges are simply supported. Du et al. [14] applied the Fourier series method to analyze the in-plane vibration of a rectangular plate with elastically restrained edges while Seok et al. [15] analyzed the free in-plane vibration for a rectangular cantilever plate. They used a variational method and an equation of motion for analyzing in-plane vibrations of thin orthotropic plates. Woodcock et al. [16] expanded the Hamilton principle and the Rayleigh–Ritz method to study the effects of the ply orientation on in-plane vibrations. Park [17] used the separation of the variables to derive the n
Corresponding author. E-mail address: [email protected] (J.B. Casimir).
http://dx.doi.org/10.1016/j.jsv.2015.11.028 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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equations for the clamped circular plate. Moreover, Gorman presented the exact solution for a rectangular plate when its two opposing edges are simply supported and the others are clamped or free [18]. The Dynamic Stiffness Method has proved efficiency for analyzing the harmonic response for complex structures composed of simple elements [19–21]. This meshless method characterized by the absence of structural discretization is based on exact solutions of the harmonic motion equations for free boundary conditions. The method uses the dynamic stiffness matrix KðωÞ that links the boundary displacements denoted U and the external forces applied on the boundaries denoted F, according to KðωÞ U ¼ F
(1)
Dynamic stiffness matrices have been developed for many elements. Casimir et al. [22] and Banerjee et al. [23,24] developed the dynamic stiffness matrix of various beams, while Tounsi et al. studied circular rings [25]. Boscolo and Banerjee [26] focused on in-plane vibration of isotropic plates according to the assumption that two opposite sides are simply supported. Some researchers have also investigated the dynamic stiffness matrix of shells [27–29]. The main objective of this paper is to develop the dynamic stiffness matrix of an orthotropic plate for in-plane vibrations according to the assumption that all the edges of the plate are free. Natural boundary conditions for the four edges are required to deal with future assemblies. In order to achieve this formulation, we used Gorman decompositions of four symmetry contributions and Levy type solutions, as explained in a previous paper concerning flexural vibrations [30]. A such approach has been recently used by NefovskaDanilovic and Petronijevic for the problem of in-plane vibrations of isotropic plates [31]. The dynamic stiffness matrix KðωÞ is built by superposing four symmetry contributions. Following this, the validation of the formulation is achieved by comparisons of harmonic responses obtained by the Finite Element Method.
2. In-plane orthotropic plate equations 2.1. Internal forces–displacements relationship Let us consider an orthotropic rectangular plate characterized by its lateral dimensions 2a and 2b, and its thickness h. It is assumed that the principal material axes are parallel with the edges of the plate and with the axes of the Cartesian coordinate system denoted Ox and Oy. The in-plane displacements of any point on the middle surface of the plate along the xaxis and the y-axis are denoted u and v, respectively (see Fig. 1). The relationship of the stress/displacements is given by the following equations: 8 ∂u ∂v > > σ x ¼ Dx þ D1 > > ∂x ∂y > > > < ∂u ∂v σ y ¼ D1 þ Dy (2) ∂x ∂y > > > > > ∂u ∂v > > : σ xy ¼ Dxy ∂x þ ∂y where σx, σy and σxy are the stress tensor components. Dx, Dy, D1 and Dxy are the material constants defined by the following equation: 8 Ex > > Dx ¼ > > 1 ν > xy νyx > > > > ν Ey νyx Ex xy > < D1 ¼ ¼ 1 νxy νyx 1 νxy νyx (3) > > Ey > > > Dy ¼ > > 1 νxy νyx > > > :D ¼G xy xy
z
y
2b O
x
h 2a Fig. 1. Orthotropic plate.
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where Ex and Ey are Young's moduli along directions x and y, respectively, νxy, νyx are Poisson's ratios and Gxy is the shear modulus. The internal forces are obtained by a simple integration of the stresses over the thickness according to the following equation: Z h=2 Z h=2 Z h=2 σ x dz; N y ¼ σ y dz; Nxy ¼ σ xy dz (4) Nx ¼ h=2
h=2
h=2
2.2. In-plane equation of motion The in-plane equations of motion can be obtained by applying Hamilton's principle. They are given by Eq. (5) in the case where no distributed loads are applied inside the plate: 8 ∂Nx ∂Nxy ∂2 u > > > < ∂x þ ∂y ¼ ρh ∂t 2 (5) > ∂Nxy ∂Ny ∂2 v > > þ ¼ ρh 2 : ∂x ∂y ∂t where ρ is the mass density of the material. Eqs. (3)–(5) allow obtaining the system of Eq. (6) satisfied by the displacement components alone: 8 ∂2 u ∂2 v ∂2 u ∂2 v ∂2 u > > þ Dxy 2 þ Dxy ¼ρ 2 > Dx 2 þD1 < ∂x∂y ∂x∂y ∂x ∂y ∂t (6) > ∂2 v ∂2 u ∂2 v ∂2 u ∂2 v > > þ Dxy 2 þ Dxy ¼ρ 2 : Dy 2 þD1 ∂x∂y ∂x∂y ∂x ∂y ∂t Time t can be eliminated for harmonic responses. Therefore the displacements U and V are written as follows: ( uðx; y; tÞ ¼ Uðx; yÞejωt vðx; y; tÞ ¼ Vðx; yÞejωt
(7)
where Uðx; yÞ and Vðx; yÞ are the amplitudes of the in-plane harmonic displacement. Therefore the system of Eq. (6) reduces to 8 ∂2 U ∂2 V ∂2 U ∂2 V > 2 > > < Dx ∂x2 þ D1 ∂x∂y þ Dxy ∂y2 þ Dxy ∂x∂y ¼ ρω U (8) > ∂2 V ∂2 U ∂2 V ∂2 U > > þ Dxy 2 þDxy ¼ ρω2 V : Dy 2 þD1 ∂x∂y ∂x∂y ∂x ∂y After successive derivations, the equations of the system (8) can be decoupled into two equations, resulting in a set of uncoupled equations of motion as follows: 8 4 4 4 2 2 > > > A1 ∂ U þ A2 ∂ U þ A3 ∂U þ A4 ∂ U þ A5 ∂ U þ A6 U ¼ 0 > < ∂x4 ∂y4 ∂x2 ∂y2 ∂x2 ∂y2 (9) > ∂4 V ∂4 V ∂4 V ∂2 V > ∂2 V > B þB þB þ B þ B þ B V ¼ 0 > 5 1 2 3 4 6 2 : ∂x4 ∂y ∂y4 ∂x2 ∂y2 ∂x2 where Ai and Bi for i¼ 1,2,6 are constants dependent on material properties, mass density and circular frequency ω: Ex 1 νxy νyx Ey A2 ¼ 1 νxy νyx
A1 ¼
A3 ¼ Gxy þ
E x Ey 2
ðGxy ð1 νxy νyx ð þ νxy Ey Þ2
ð1 νxy νyx Þ ð1 νxy νyx Þ2 Gxy Ex ρω2 A4 ¼ ρω2 ð1 νxy νyx ÞGxy Ey ρω2 A5 ¼ ρω2 ð1 νxy νyx ÞGxy A6 ¼
ρ2 ω4 Gxy
B1 ¼ Gxy Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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B2 ¼ Gxy
Ey Ex
Ey B3 ¼ þ 1 νxy νyx
G2xy ð1 νxy νyx Þ Ex
B4 ¼ B5 ¼
νxy Ey þ Gxy 1 νxy νyx Ex
Gxy ρω2 ð1 νxy νyx Þ ρω2 EL
1 νxy νyx
Ey ρω2 Gxy ρω2 ð1 νxy νyx Þ Ex Ex B6 ¼
2
ρ2 ω4 ð1 νxy νyx Þ Ex
3. The in-plane dynamic stiffness matrix of an orthotropic plate The Dynamic Stiffness Method is based on the calculation of the dynamic stiffness matrix. This matrix is built with a strong solution of the harmonic problem. Free edge boundary conditions are necessary to allow assemblies between plates along their four edges. In order to obtain this strong solution, a decomposition approach is used, as was done recently for the flexural problem [30]. This approach is based on a decomposition of the problem into four symmetry contributions denoted SS (symmetric–symmetric), SA (symmetric–antisymmetric), AS (antisymmetric–symmetric) and AA (antisymmetric–antisymmetric). For each symmetry contribution, the displacement solutions U and V are obtained with two series. The main difference with Kirchhoff's flexural problem is that two components of the displacement solution have to be obtained simultaneously from the solution of the differential system of Eq. (8).
3.1. Symmetry contributions Any displacement solution U can be decomposed into four contributions with respect to symmetry considerations. It can be written as follows: Uðξ; ηÞ ¼ U SS ðξ; ηÞ þ U AA ðξ; ηÞ þ U SA ðξ; ηÞ þ U AS ðξ; ηÞ
(10)
where ξ and η are the dimensionless coordinates defined by the following equation: x ξ¼ ; a
η¼
y b
(11)
The first subscript, S or A, indicates if the contribution is symmetric or antisymmetric, respectively, around the coordinate ξ. The second subscript indicates the symmetry property about the coordinate η. For example, UAS is the contribution of the displacement U which is antisymmetric around ξ and symmetric around η, that is to say: U AS ðξ; ηÞ ¼ U AS ð ξ; ηÞ ¼ U AS ðξ; ηÞ ¼ U AS ð ξ; ηÞ
(12)
Similar considerations can be made for the V component of the displacement but U and V are coupled by Eq. (8). These equations show that if U is symmetric around a spatial coordinate, V should be antisymmetric around this coordinate, and if U is antisymmetric around a spatial coordinate, then V should be symmetric around this coordinate. Therefore the component displacement V is split into four contributions as is U with Eq. (10), but the notation is quite different. For example, the symmetric–symmetric contribution of V is denoted VAA. The subscripts always indicate the symmetry property of the corresponding U component of the displacement.
3.2. Strong solutions Two series are used for each symmetry contribution to obtain a strong solution of the harmonic problem. These series are defined as the Levy series for the flexural problem but the parameters in the trigonometric functions were chosen to obtain Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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a non-vanishing solution along the four edges of the plate. These series are given in the following equation: 8 1 1 X X > 4n 3 4n 3 > > πη þ πξ 1SS U n ðξÞ cos 2SS U n ðηÞ cos U SS ðξ; ηÞ ¼ > > > 4 4 > n¼1 n¼1 > > > 1 1 > > > V ðξ; ηÞ ¼ X 1 V ðξÞ sin 4n 3 πη þ X 2 V ðηÞ sin 4n 3 πξ > > SS SS n SS n > > 4 4 > n¼1 n¼1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ 1AA U n ðξÞ sin 2AA U n ðηÞ sin > > U AA ðξ; ηÞ ¼ > 4 4 > n ¼ 1 n ¼ 1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ V AA ðξ; ηÞ ¼ 1AA V n ðξÞ cos 2AA V n ðηÞ cos > > < 4 4 n¼1 n¼1 1 1 X X > 4n 3 4n 3 > > ð Þ ð Þ πη þ πξ U ξ; η ¼ 1 U ξ sin 2SA U n ðηÞ cos > n SA SA > 4 4 > > n¼1 n¼1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ V SA ðξ; ηÞ ¼ 1SA V n ðξÞ cos 2SA V n ðηÞ sin > > 4 4 > > n ¼ 1 n ¼ 1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ U AS ðξ; ηÞ ¼ 1AS U n ðξÞ cos 2AS U n ðηÞ sin > > 4 4 > > n¼1 n¼1 > > > 1 1 > X X > 4n 3 4n 3 > > πη þ πξ V AS ðξ; ηÞ ¼ 1AS V n ðξÞ sin 2AS V n ðηÞ cos > > 4 4 : n¼1 n¼1
5
(13)
The left-superscript 1 or 2 indicates whether the function depends on the coordinate ξ or the coordinate η. Introducing the series (13) into Eq. (9) gives ordinary differential equations satisfied by the functions kαβ U n and kαβ V n , where k is 1 or 2, αβ is SS, AA, SA or AS, see the following equation: 8 2k 4k > d U > A A d αβ U n 4 A2 2 2 A5 4 A6 k > > ϕ4 αβ n þ ϕ2 k21 3 þ ϕ2 b2 4 þ k k b þ b U ¼0 > 1 1 > A1 A1 A1 A1 A1 αβ n > dξ4 dξ2 > > > > 4k 2k > > d αβ U n A1 A4 2 A5 d αβ U n 4 2 2 4 A6 k > 2 2 A3 > þ k1 ϕ4 k1 b ϕ2 þ b U ¼0 > < dη4 þ ϕ k1 A2 þb A2 A4 A2 A2 αβ n dη2 (14) 2k 4k > d αβ V n d αβ V n > 4 B2 2 2 B5 4 B6 k > 4 2 2 B3 2 2 B4 > ϕ þ ϕ k1 þ ϕ b þ k1 k1 b þb V ¼0 > > B1 B1 B1 B1 B1 αβ n > dξ4 dξ2 > > > 4k 2k > > d αβ V n > B1 B4 2 B5 d αβ V n 4 2 2 4 B6 k > 2 2 B3 > þ k1 ϕ4 k1 b ϕ2 þ b V ¼0 > : dη4 þ ϕ k1 B2 þ b B2 B4 B1 B1 αβ n dη2 3Þπ with ϕ ¼ ba and k1 ¼ ð4n . The closed form solutions are given by the following equations: 4 8 1 1 1γ n ξ > 7e γ n ξ þ Bn e1κn ξ 7 e 1κn ξ > αβ U n αβ ðξÞ ¼ An e > > > > > 2 U ðηÞ ¼ C n e2γ n η 7 e 2γn η þDn e2κn η 7e 2κn η < αβ n αβ 1 > V ðξÞ ¼ An e1γ n ξ 7e 1γ n ξ þ Bn e1κn ξ 7 e 1κn ξ > > > αβ n αβ > > > : 2αβ V n ðηÞ ¼ C n e2γ n η 7e 2γ n η þ Dn e2κn η 7 e 2κn η αβ
(15)
The sign 7 changes with the contribution and the 1 γ n , 2 γ n , 1 κ n and 2 κ n are the complex roots of the characteristic equations. They are obtained by the following equations: 8 2 > A1 A2 A3 ðA1 b ρω2 Þ > 2 2 > k B þ þ b ρω2 > 1 1 > pffiffiffiffiffiffi > B B B 1 1 1 > 1 > þ Δ1 =ϕ4 > γn ¼ 2 > > ϕ > > > > > 2 > > A1 A2 A3 ðA1 b ρω2 Þ 2 2 > > k1 B1 þ þ b ρω2 pffiffiffiffiffiffi > > B B B > 1 1 1 2 < γ ¼ Δ1 =ϕ4 n 2 ϕ (16) > > A1 A2 A3 ðA1 a2 ρω2 Þ > 2 2 2 > > k B þ þ a ρω 1 > 1 pffiffiffiffiffiffi >1 B1 B1 B1 > > κn ¼ þ Δ2 =2 > > 2 > > > ϕ > > A1 A2 A3 ðA1 a2 ρω2 Þ : > 2 > k B þ þ a2 ρω2 pffiffiffiffiffiffi > 1 1 > >2 B B B 1 1 1 > > Δ2 =2 : κn ¼ ϕ2 Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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where Δ1 and Δ2 are defined by Eqs. (17) and (19), respectively: 2
Δ1 ¼ d 4ce where
(17)
8 c ¼ A1 ϕ4 > > < 2 2 d ¼ ϕ2 k1 A3 þ ϕ2 b A4 > > : e ¼ k4 A b2 k4 A þA b4 1 5
1 2
(18)
6
Δ2 ¼ o2 4pq
(19)
where 8 o ¼ A2 > < 2 2 p ¼ ϕ2 k1 A3 þ ϕ2 b A5 > : q ¼ k4 A b2 k4 A þ A b4 1 1
1 4
(20)
6
3.3. Edge in-plane displacements The procedure for calculating the dynamic stiffness matrix (DSM) consists of a sequence of symbolic computations of equations that link coefficients An ; Bn ; C n and Dn to the projections of the edge displacement and external force on a functional basis. First, the edge displacements are obtained from the U and V solutions inside the plate given by Eqs. (13) and (15). These displacements are linked to the coefficients according to a symbolic matrix equation. For example, the edge displacements for the symmetric–symmetric contribution are such that 8 pffiffiffi 1 1 X X > 4n 3 2 > 1 2 > πη þ ð 1Þn þ 1 U ð 1; η Þ ¼ U ð 1 Þ cos U ð η Þ > SS SS n SS n > 4 2 > > n¼1 n¼1 > p ffiffiffi > 1 1 > X X > 2 4n 3 1 2 > > ð 1Þn þ 1 πη þ > V SS ð1; ηÞ ¼ SS V n ð1Þ sin SS V n ðηÞ < 2 4 n¼1 n ¼ 1 pffiffiffi (21) 1 1 X X > 4n 3 2 1 2 > nþ1 > πξ ð 1Þ U ð ξ; 1 Þ ¼ U ð ξ Þ þ U ð 1 Þ cos > SS SS n SS n > 4 2 > > n¼1 n¼1 > pffiffiffi : > 1 1 > X X > 4n 3 2 > 1 2 nþ1 > ð Þ ð Þ ð Þ πξ ð 1Þ V ξ; 1 ¼ V ξ þ V 1 sin > SS SS n SS n : 4 2 n¼1
n¼1
and after limiting the Levy series to N terms, we obtain 0
A1
1
B B1 C B C B C B C1 C B C 0 1 B D1 C U SS ð1; ηÞ B C B C B C B V SS ð1; ηÞ C B ⋮ C B C ¼ ∂HSS B C B U SS ðξ; 1Þ C BA C @ A B NC B C V SS ðξ; 1Þ B BN C B C BC C B NC B C @ DN A where
0
∂H 11 ðηÞ
B ∂H ðηÞ B 21 ∂HSS ¼ B B ∂H 31 ðξÞ @ ∂H 41 ðξÞ
⋯ ⋯ ⋯ ⋯
∂H 1;4ðN 1Þ þ 4 ðηÞ
(22)
1
∂H 2;4ðN 1Þ þ 4 ðηÞ C C C ∂H 3;4ðN 1Þ þ 4 ðξÞ C A
(23)
∂H 4;4ðN 1Þ þ 4 ðξÞ
The functions ∂Hij are obtained by a symbolic computation. 3.4. External forces A similar symbolic computation is achieved for the external forces applied on each edge of the plate. First, the force/ displacement relationships given by Eqs. (2) and (4) are used to evaluate internal forces from the displacement solutions U and V. Then, free boundary conditions are used to obtain the relations between the external forces and the coefficients An , Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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Table 1 Functional basis. SS Uð1; ηÞ V ð1; ηÞ Uðξ; 1Þ V ðξ; 1Þ
AA
cos ðmπηÞ
ð2m 1Þπη 2 cos ðmπξÞ
ð2m 1Þπη 2 cos ðmπηÞ
sin
ð2m 1Þπξ 2
ð2m 1Þπξ 2 cos ðmπξÞ
ð2m 1Þπη 2 cos ðmπηÞ
sin
sin
ð2m 1Þπξ 2
cos ðmπηÞ
ð2m 1Þπη 2 ð2m 1Þπξ sin 2 cos ðmπξÞ
sin
cos ðmπξÞ
sin
AS
sin
sin
SA
Bn , C n , Dn . These relations are defined by 0
1 A1 BB C B 1 C B C B C1 C 0 1 B C Nx ð1; ηÞ BD C B N ð1; ηÞ C B 1C B xy C B C B C B C B Ny ðξ; 1Þ C ¼ ∂GSS B ⋮ C @ A BA C B NC Nxy ðξ; 1Þ B C B BN C B C BC C @ NA DN
(24)
where 0
∂G11 ðηÞ B ∂G ðηÞ B 21 ∂GSS ¼ B B ∂G31 ðξÞ @ ∂G41 ðξÞ
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
∂G1;4ðN 1Þ þ 4 ðηÞ
1
∂G2;4ðN 1Þ þ 4 ðηÞ C C C ∂G3;4ðN 1Þ þ 4 ðξÞ C A ∂G4;4ðN 1Þ þ 4 ðξÞ
(25)
Functions ∂Gij are obtained by a symbolic computation and stored in a matrix. 3.5. Functional basis The external forces and edge displacement functions are projected on a functional basis. The DSM gives the exact relations between these projections. To do this, a functional basis is defined on each edge of the plate and for each contribution. For example, in the case of the symmetric–symmetric contribution, the functional basis on the edge ξ ¼ 1 is given by the functions cos mπη for m A f1 ,.., Ng. Table 1 gives all the functional bases used. The edge displacement projections on these bases are computed by symbolic computational software for each symmetry contribution. These projections still depend on the unknown parameters Ai ; Bi ; C i and Di . They are stored in four matrices denoted HSS ; HAA ; HSA and HAS . For example, the symmetric–symmetric matrix is such that 0
1 1 SS U 0 B 2 C B SS U 0 C B C B 1 C 0 1 B SS U 1 C A1 B C B 1V C BB C B SS 1 C B 1C B 2 C B C B C U B C1 C B SS 1 C B C B 2 C BD C B SS V 1 C B 1C B C B C B C C ⋮ B C ¼ HSS B B ⋮ C B1 C BA C B SS U N 1 C B NC B C B C B1 C B BN C B SS V N 1 C B C B C B C B2 C @ CN A B SS U N 1 C B C DN B2 V C B SS N 1 C B 1 C B C @ SS V N A 2 SS V N
(26)
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and is given by the following equation: 0 R 1 1
1 H 11 ðηÞ R 1 1 2 1 H 21 ðξÞ
2
dη
B B dξ B B R1 B H ðηÞ cos ðπηÞ dη B 1 31 πη B R1 B dη B 1 H 41 ðηÞ sin 2 B B ⋮ B B R1 B 1 H 11 ðηÞ cos ððN 1ÞπηÞ dη B B R ð2N 3Þ B 1 πη dη HSS ¼ B 1 H 21 ðηÞ sin B 2 B R1 B H ðξÞ cos ððN 1ÞπξÞ dξ B 1 11 B B R1 ð2N 3Þ B πξ dξ B 1 H 21 ðξÞ sin 2 B B B R1 ð2N 1Þ B πη dη H ð η Þ sin B 1 31 2 B BR ð2N 1Þ @ 1 πη dη 1 H 41 ðξÞðηÞ sin 2
⋯ ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
1 2
R1
1 R 1 1 2 1
H 1;4ðN 1Þ þ 4 ðηÞ dη
1
C C C C R1 C H ðηÞ cos ðπηÞ dη C 1 3;4ðN 1Þ þ 4 πη C R1 C dη H ð η Þ sin C 4;4ðN 1Þ þ 4 1 2 C C ⋮ C R1 C C 1 H 1;4ðN 1Þ þ 4 ðηÞ cos ððN 1ÞπηÞ dη C C R1 ð2N 3Þ C πη dη C 1 H 2;4ðN 1Þ þ 4 ðηÞ sin C 2 C R1 C H ðξÞ cos ððN 1ÞπξÞ dξ C 1 1;4ðN 1Þ þ 4 C C R1 ð2N 3Þ πξ dξ C C 1 H 2;4ðN 1Þ þ 4 ðξÞ sin 2 C C R1 C ð2N 1Þ πη dη C 1 H 3;4ðN 1Þ þ 4 ðηÞ sin C 2 C C R1 ð2N 1Þ A πη dη 1 H 4;4ðN 1Þ þ 4 ðηÞ sin 2 H 2;4ðN 1Þ þ 4 ðξÞ dξ
(27)
The projections of the forces applied are computed in the same way and stored in four matrices GSS ; GAA ; GSA and GAS . 3.6. Dynamic stiffness matrix The last step consists in calculating the DSM for each contribution by eliminating the parameters Ai ; Bi ; C i and Di using a numerical inversion of the matrices Gαβ . Therefore the DSM of the αβ-symmetry contribution is given by 1 Kαβ ¼ Hαβ Gαβ
(28)
4. Results and discussion 4.1. Symmetric–symmetric contribution We consider a carbon-epoxy plate with Young's moduli Ex ¼ 18:1 GPa and Ey ¼ 50:9 GPa, shear modulus Gxy ¼ 11 GPa, Poisson's ratio νxy ¼ 0:8 and mass density ρ ¼ 1526 kg=m3 . Its dimensions are a ¼ 0:5 m; b ¼ 0:25 m and h ¼ 0:002 m. Free– free–free–free (FFFF) boundary conditions are considered. The sole symmetric–symmetric contribution is isolated by using symmetric–symmetric in-plane loads. Uniform and identical harmonic loads are defined on the edges located on ξ ¼ 1 and ξ ¼ 1, that is to say: F x ð1; ηÞ ¼ F x ð 1; ηÞ ¼ 1
(29)
Fig. 2 shows this loading. The harmonic response along the x-direction is evaluated at point A located on coordinates (1,0). Fig. 3 shows this response over the frequency range [0, 10 000 Hz] and gives a comparison with the responses obtained by finite element models. Four-node Discrete Kirchhoff Quadrilateral (DKQ) elements are used in the FEM for two meshing refinements (40 40 and 80 80 elements).
Fig. 2. Symmetric–symmetric loads.
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−20 FEM 40x40 FEM 80x80 1 DSM
−40 −60
20 log10|U|
−80 −100 −120 −140 −160 −180 −200 −220
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
Frequency (Hz) Fig. 3. Harmonic response to a symmetric–symmetric load.
−120 FEM 40x40 FEM 80x80 FEM 100x100 1 DSM
−140
20 log10|U|
−160
−180
−200
−220
−240 8000
8100
8200
8300
8400
8500
8600
8700
8800
8900
9000
Frequency (Hz) Fig. 4. Harmonic response on [8000 Hz, 9000 Hz].
Very good convergence can be seen between the two approaches in the [0, 6000 Hz] frequency range, but beyond 6000 Hz, the precision of the FEM responses decreases. Fig. 4 shows more detailed responses in the [8000 Hz, 9000 Hz] frequency ranges. A third FEM using 100 100 DKQ elements is used to prove the convergence. Table 2 gives the computation times for calculating the response for one frequency and shows the performances of the present formulation compared to FEM.
4.2. Distributed force The next case studied is a plate subjected to a uniform distributed force on its right edge only, see Fig. 5. This force is defined by F x ð1; ηÞ ¼ 1
(30)
The material properties and the dimensions of the plate are the same as the previous one. Fig. 6 shows the harmonic response in the direction x of point A located at (1,0) . Very good convergence between the FEM results and DSM results can be observed. For frequencies higher than 5000 Hz, the FEM mesh must be fine enough to reach the precision of the DSM. Fig. 7 gives a more detailed representation of the curves. Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i
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Table 2 Computation times. Model
1 frequency calculation time
1 CEM 20 20 FEM 40 40 FEM 80 80 FEM 100 100 FEM
0.014 0.015 s 0.069 s 0.45 s 1.2 s
F y
z
x
Fig. 5. Distributed force.
0 FEM 20x20 FEM 40x40 1 DSM
20 log10|U|
−50
−100
−150
−200
−250
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Frequency (Hz)
Fig. 6. Harmonic response for a distributed load.
−80 1 DSM FEM 50 x 50 FEM 100 x 100 FEM 200 x 200
−100 −120
20log10|U|
−140 −160 −180 −200 −220 −240 8000
8100
8200
8300
8400
8500
8600
8700
8800
8900
9000
Frequency (Hz) Fig. 7. Harmonic response in the frequency range [8000 Hz, 9000 Hz].
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4.3. Harmonic response for a concentrated force An important step of validation consists in evaluating the results when more than one term is necessary in the series development of the loads. This can be achieved using concentrated forces. The next load case implies such a load. A harmonic concentrated force located at ξ ¼ 1 and η ¼ 1 is used and the response is computed in the x-direction at point A located at coordinates (1,0). The projections of the concentrated force on the functional basis given in Table 1 are defined by the following equation: 8 1 > > > F0 ¼ > > 2b > > < ð 1Þn F Sm ¼ (31) > b > > n > > ð 1Þ A > > : Fm ¼ b These projections are such that F ð1; ηÞ ¼ F 0 þ
M X
F Sm cos mπη þ
m¼1
M X m¼1
π F Am sin ð2m 1Þ η 2
(32)
The response is evaluated at point Pðξ ¼ 1; η ¼ 1Þ. Fig. 8 shows this response and a comparison with finite element responses over 7000 Hz–8000 Hz range. −80 FEM 50 x 50 FEM 100 x 100 FEM 200 x 200 1 DSM
−100
20 log10|U|
−120 −140 −160 −180 −200 −220 −240 7000
7100
7200
7300
7400
7500
7600
7700
7800
7900
8000
Frequency (Hz) Fig. 8. Harmonic response for a concentrated load.
20 1 DSM
0
20log10|U|
-20
-40
-60
-80
-100
-120
0
50
100
150
200
250
300
Frequency (Hz) Fig. 9. Harmonic response of the Woodcock plate.
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Table 3 Eigenfrequencies. Model
F1
F2
F3
F4
F5
F6
F7
F8
Woodcock 1 DSM 20 20 4-node FEM 40 40 4-node FEM 10 10 8-node FEM 20 20 8-node FEM Symmetry properties
109.86 109.97 110.03 109.88 109.83 109.82 SS
126.23 126.15 126.26 126.17 126.14 126.13 AS
138.31 138.30 138.73 138.43 138.34 138.33 SA
205.00 204.99 206.43 205.39 205.08 205.04 AA
223.85 223.64 224.80 223.89 223.61 223.58 AS
225.19 225.20 226.39 225.44 225.18 225.12 SS
236.27 236.17 237.73 236.61 236.35 236.24 SA
252.38 252.31 253.13 252.35 252.14 252.09 AA
Very good convergence can be observed between the FE results and the DSM response obtained with 9 terms in the series development. 4.4. Natural frequencies This section gives a comparison with the available literature concerning the in-plane vibrations of orthotropic plates. No reference was found regarding the in-plane harmonic response for free boundary conditions, therefore the comparison is restricted to the calculation of natural frequencies. The in-plane natural frequencies of an orthotropic plate were given in the article of Woodcock et al. [16] cited in the Introduction of our paper. A modal analysis could be performed using the William–Wittrick algorithm [33]. Instead, the harmonic response curve is used because there is no damping in the model and the modal density is poor. A fine frequency sweep is used and the positions of the peaks are detected. The Woodcock plate is a 0:376 0:376 m2 square plate with a thickness of 1:077 mm. The constitutive material is called Material I in the paper. It is a material with Young's moduli Ex ¼ 13:8 GPa and Ey ¼ 0:9 GPa, shear modulus Gxy ¼ 0:71 GPa, Poisson's ratio νxy ¼ 0:2 and mass density ρ ¼ 100 kg=m3 . Free–free–free–free (FFFF) boundary conditions are considered. Two concentrated forces are applied in the corner located on coordinates (1,1). One in x direction and one in y direction are defined. The harmonic response is processed along direction x at the point located on coordinates (1,0). It is shown in Fig. 9. Eight natural frequencies are obtained on [0,300 Hz]. These frequencies are given on line 3 of Table 3. The other lines give Woodcock's results (Table 8 of his paper for a 0 angle ply) and the results of modal analysis performed with finite element models. Very good agreement between all the results is shown. The use of the Dynamic Stiffness Matrix Kαβ of each contribution (see Eq. (28)) for this load case allows associating each peak of the response to a symmetry property. These properties are given on the last line of the table. They are related to the property of the U displacement of the corresponding natural mode of vibration.
5. Conclusion The main contribution of this paper was to describe an analytical procedure to build a dynamic stiffness matrix for the in-plane vibrations of a free orthotropic plate. This method is an extension of the procedure previously described for out of plane flexural vibrations but, here, the strong formulation of the harmonic problem implies two coupled partial derivative equations. The procedure is based on analytical solutions obtained with mathematical software for free edge boundary conditions. These solutions are described with Levy's series for each symmetry contribution and Gorman's decomposition of the free boundaries. This DSM allowed determining the in-plane harmonic response of rectangular orthotropic plates with high precision and without meshing. The method was validated with the FEM whose results converged very well with the DSM results for several load cases and with results of the literature concerning natural frequencies. The main advantages of the formulation are its precision, rapidity and efficiency.
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Please cite this article as: O. Ghorbel, et al., In-plane dynamic stiffness matrix for a free orthotropic plate, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.11.028i