Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies

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Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies N. Kolarevic, M. Nefovska-Danilovic n, M. Petronijevic Department for Engineering Mechanics and Theory of Structures, Faculty of Civil Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia

a r t i c l e i n f o

abstract

Article history: Received 17 March 2015 Received in revised form 16 June 2015 Accepted 18 June 2015 Handling Editor: S. Ilanko

The dynamic stiffness matrix of completely free rectangular Mindlin plate element is presented in this paper. The system of three coupled equations of motion is transformed into two uncoupled equations introducing a boundary layer function. The dynamic stiffness matrix is derived by use of the superposition method and the projection method. Using the proposed method natural frequencies of individual plates and plate assemblies with arbitrary boundary conditions are computed and validated against the results available in the literature and the finite element analysis. High efficiency and accuracy of the results are demonstrated. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction Vibration of plates is of great importance in engineering design, since plates are widely used as components of many engineering structures, such as aircraft wings, floor slabs, building walls, amongst others. The conventional Finite element method [1] (FEM) is nowadays most frequently used to calculate free vibration characteristics and dynamic response of such structures. The FEM is approximate numerical method based on the assumed frequency independent polynomial shape functions. Using the FEM the structure is meshed in order to represent the geometry, materials and boundary conditions. In addition, the mesh size is influenced by the highest frequency of interest in the analysis. In parallel with the expansion of the FEM, analytical methods were also developed for free transverse vibration analysis of rectangular plates [2]. These methods are based on the exact solutions of the governing equations of motion, and cover only special set of boundary conditions. Leissa [3] carried out a comprehensive and accurate free vibration analysis of rectangular plates based on classical (Kirchhoff’s) theory. He presented the analytical results for six types of boundary conditions having two opposite edges simply supported (Levy-type plates), while for other 15 possible combinations of boundary conditions the natural frequencies were solved using the Ritz method. Gorman [4] and Gorman and Ding [5] developed an analytical solution using the superposition method, based on the forced vibration problem solutions to obtain free transverse vibration characteristics of a completely free rectangular Kirchhoff’s and Mindlin plate, respectively. Based on the Raylegh–Ritz method Liew et al. [6] computed natural frequencies of rectangular plate using first order shear deformation theory applicable to all possible boundary conditions. Xing and Liu [7] used the direct separation of variables to obtain closed-form solutions for the free vibrations of rectangular Mindlin plate with clamped and simply supported edges. All of these analytical-type methods were applied in the free vibration analysis of an individual plate and cannot be easily extended to analyze complex plate-

n

Corresponding author. E-mail address: [email protected] (M. Nefovska-Danilovic).

http://dx.doi.org/10.1016/j.jsv.2015.06.031 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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like structures having non-uniform geometrical and material properties. In such cases the FEM is undoubtedly a powerful method to compute the free vibration characteristics. However, for structures subjected to high frequency excitations, the number of finite elements in the analysis significantly increases, which takes greater computational time and effort to solve the problem. To obtain more accurate and reliable results in such cases, the dynamic stiffness method (DSM) can be used. The dynamic stiffness method is based on the exact analytical solutions of the governing equations of motion, which results in the exact frequency dependent shape functions of a dynamic stiffness element. The corresponding dynamic stiffness matrices are also frequency dependent and can be developed explicitly. Consequently, the discretization is minimized and influenced only by the change in the geometry and material properties of structural elements. The DSM is well developed for one-dimensional elements (beams and bars) [8–11] and Levy-type plates based on Kirchhoff’s theory [12,13]. The effect of shear deformation was accounted for in the development of the dynamic stiffness matrix of two edge infinite plate element by Anderson and Kennedy [14]. Boscolo and Banerjee [15–17] made a step forward and derived the elements of the dynamic stiffness matrices for Levy-type isotropic and composite Mindlin plates in the explicit form, which significantly increased the efficiency and speed of the developed computer program. Moreover, the dynamic stiffness matrices based on higher order shear deformation theory [18] and layer-wise theory [19] for composite plate assemblies with two opposite edges simply supported were also derived in the explicit form. Exact solutions of the governing equations of motion for both transverse and in-plane free vibration of plates cannot be obtained without the limitation of the boundary conditions to the simply supported opposite edges. To overcome this, the displacement field of plate is presented in the infinite Fourier series form. In practice, the series are truncated, which induces some error. Consequently, the obtained solutions are approximate, but their accuracy can be easily controlled by varying the number of terms in the series representation of the displacement field. In addition, general boundary conditions which represent the forces and displacements along the plate boundary are continuous functions of spatial variables x and y. In order to relate the boundary forces to the boundary displacements, the general boundary conditions have to be replaced by discrete boundary conditions. This can be accomplished by using the Projection method developed by Kevorkian and Casimir [20,21] who derived the dynamic stiffness matrix of rectangular plate element based on Kirchhoff’s plate theory. Banerjee et al. [22] derived the dynamic stiffness matrix of isotropic rectangular Kirchhoff’s plate for the most general case solving the bi-harmonic equation of motion. They computed the natural frequencies and mode shapes for an individual square or rectangular plate with three types of boundary conditions. The dynamic stiffness matrix for completely free rectangular isotropic plate element undergoing in-plane vibration was developed by Nefovska-Danilovic and Petronijevic [23] and applied in the free vibration and dynamic response analysis of plate assemblies with any possible type of boundary conditions. This paper presents a method for development of the dynamic stiffness matrix for completely free rectangular plate element based on Mindlin plate theory. Three coupled equations of motion were transformed into two uncoupled equations of motion introducing a boundary layer function [24]. In order to avoid operating with large-size matrices, the transverse displacement of the plate element was expressed as a superposition of four symmetry contributions. The displacement field was given in the trigonometric series form for each symmetry contribution. The dynamic stiffness matrices for each symmetry contribution were developed using the Projection method [20,21]. Afterwards, the dynamic stiffness matrix of completely free dynamic stiffness element was derived superposing the dynamic stiffness matrices of the four symmetry contributions. In order to obtain the global dynamic stiffness matrix of plate assemblies, similar assembly procedure as in the FEM was developed [23]. The proposed method enables transverse free vibration analysis of rectangular isotropic plate assemblies having arbitrary boundary conditions and nonuniform geometrical and material properties. The Wittrick–Williams’s algorithm [25] was exploited here to determine the natural frequencies of plate assemblies. The natural frequencies and mode shapes were validated against the exact solutions from the literature [15], the results obtained by Liew et al. [6], Xing and Liu [7] and the results obtained by using the finite element software Abaqus [26]. Moreover, the influence of some plate parameters and boundary conditions on the free vibration characteristics was presented and discussed.

Fig. 1. Displacement field and corresponding forces of plate element.

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2. Dynamic stiffness elements for free transverse vibration 2.1. Basics of Mindlin plate theory Assuming that there is no deformation in the mid plane, Fig. 1, the displacement field of a plate is given as uðx; y; z; t Þ ¼ zϕy ðx; y; t Þ vðx; y; z; t Þ ¼  zϕx ðx; y; t Þ wðx; y; z; t Þ ¼ wo ðx; y; t Þ

(1)

where wo ðx; y; t Þ is the transverse displacement in the mid plane, while ϕx ðx; y; t Þ and ϕy ðx; y; t Þ are the rotations about the x and y axes, respectively. Plate forces expressed in terms of plate displacement and rotations are given by the following expressions: ! ! ∂ϕy ∂ϕy ∂ϕ ∂ϕ Mx ¼ D  ν x ; My ¼ D  x þ ν ∂x ∂y ∂y ∂x  o   o  ∂w ∂w þ ϕy ; T y ¼ kGh  ϕx T x ¼ kGh ∂x ∂y ! ∂ ϕ 1ν ∂ϕ y  x : (2) M xy ¼ M yx ¼ D 2 ∂y ∂x The equations of motion obtained from the equilibrium of forces, using the constitutive and kinematic relations can be written as ! ∂2 wo ∂ϕy ∂ϕx ∂2 wo ∂2 wo  þ kGh þ  ρh 2 ¼ 0 2 2 ∂x ∂y ∂x ∂y ∂t !  o  2 2 2 ∂ ϕy 1  ν ∂ ϕy 1 þ ν ∂ ϕx ∂w ρh3 ∂2 ϕy þ D þ  ϕ ¼0 kGh y  2 2 2 2 ∂x∂y ∂x 12 ∂t 2 ∂x ∂y !  o  2 ∂2 ϕx 1  ν ∂2 ϕx 1 þ ν ∂ ϕy ∂w ρh3 ∂2 ϕx D  þ  ϕ ¼0 (3) þ kGh x  2 2 2 ∂x 2 ∂x∂y ∂y 12 ∂t 2 ∂y density, E is Young’s modulus, G is the shear modulus, ν is the Poisson’s ratio of the where h is plate thickness, ρ is mass  3 plate material, D ¼ Eh =12 1  ν2 is plate flexural stiffness and k is the shear correction factor. 2.2. General solution Eq. (3) represents the system of three coupled partial differential equations. In order to find a solution for the system, a boundary layer function [24] was introduced as

ψ¼

∂ϕx ∂ϕy þ : ∂x ∂y

(4)

Using the boundary layer function, system of Eq. (3) is transformed into two uncoupled equations with respect to transverse displacement and boundary layer function: ! ρh ρh3 2 € o ρh3 ρh ⃜ o 2 2 o € o ¼0 D∇ ∇ w  D þ w þ ρhw U ∇ w þ kGh 12 12 kGh 1ν 2 ρh € ∇ ψ  kGhψ  ψ ¼ 0; D 2 12 3

(5)

where ∇2 ¼

∂2 ∂2 þ ; ∂x2 ∂y2

€o¼ w

∂2 wo ; ∂t 2

⃜ o

w ¼

∂4 wo : ∂t 4

(6)

It can be easily shown that the rotations ϕx ðx; y; t Þ and ϕy ðx; y; t Þ can be expressed in terms of the transverse displacement and the boundary layer function as   ρh3 € ∂ ρh € o 1  ν ∂ψ kGhϕx þ D∇2 wo D w þ kGhwo þ D ϕx ¼ ∂y kGh 12 2 ∂x   3 ρh € ∂ ρh € o 1  ν ∂ψ D∇2 wo  D w þkGhwo þ D : (7) kGhϕy þ ϕ ¼ ∂x kGh 12 y 2 ∂y Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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Introducing harmonic representation of the transverse displacement wo and boundary layer function ψ as ^ wo ðx; y; tÞ ¼ wðx; y; ωÞeiωt

ψ ðx; y; tÞ ¼ ψ^ ðx; y; ωÞeiωt ;

(8)

^ where ω is angular frequency, wðx; y; ωÞ and ψ^ ðx; y; ωÞ are the amplitudes of displacement and boundary layer function in the frequency domain, the Fourier transform of Eq. (5) can be expressed as ^ c1 ω2 ∇2 w ^ þ c2 w ^ ¼0 ∇2 ∇2 wþ ∇2 ψ^ þ c3 ψ^ ¼ 0: Coefficients ci , i¼1, 2, 3, in Eq. (9) are equal: c1 ¼

ρh kGh

þ

ρh3 12D

;

c2 ¼

ρhω2 ρh3 ω2 D

12kGh

(9) !

1 ;

c3 ¼

ρh3 12

ω2  kGh D1 2 ν

:

(10)

According to the superposition method, the transverse displacement and boundary layer function, as well as the rotations of rectangular plate element, are split into four contributions: symmetric–symmetric (SS), symmetric–anti-symmetric (SA), anti-symmetric– ymmetric (AS) and anti-symmetric–anti-symmetric (AA): ^ ^ SS ðx; yÞ þ w ^ SA ðx; yÞ þ w ^ AS ðx; yÞ þ w ^ AA ðx; yÞ wðx; yÞ ¼ w

ψ^ ðx; yÞ ¼ ψ^ SS ðx; yÞ þ ψ^ SA ðx; yÞ þ ψ^ AS ðx; yÞ þ ψ^ AA ðx; yÞ SS

SA

AS

AA

SS

SA

AS

AA

ϕ^ x ðx; yÞ ¼ ϕ^ x ðx; yÞ þ ϕ^ x ðx; yÞ þ ϕ^ x ðx; yÞ þ ϕ^ x ðx; yÞ ϕ^ y ðx; yÞ ¼ ϕ^ y ðx; yÞ þ ϕ^ y ðx; yÞ þ ϕ^ y ðx; yÞ þ ϕ^ y ðx; yÞ:

(11)

Using the superposition method it is possible to obtain the general solution and the corresponding dynamic stiffness matrix using only one quarter of the rectangular plate for each symmetry contribution. In the following section, the general solution of Eq. (9) will be derived for the SS case. General solutions for other three symmetry contributions can be obtained in similar way and they are given in Appendix A. 2.2.1. Symmetric–symmetric contribution (SS) In order to satisfy the defined double symmetry deformation presented in Fig. 2, as well as Eq. (9), the transverse displacement and rotations of plate element are represented in the Fourier series as ^ SS ðx; yÞ ¼ w

M X 1

W SS m ðyÞ cos

m¼0 SS

ϕ^ x ðx; yÞ ¼

M X 1 m¼0

SS

ϕ^ y ðx; yÞ ¼

M X 1 m¼1

ϕSS xm ðyÞ cos

ϕSS ym ðyÞ sin

M mπ x X mπ y 2 þ W SS m ðxÞ cos a b m¼0

ðaÞ

M mπ x X mπ y 2 SS þ ϕxm ðxÞ sin a b m¼1

ðbÞ

M mπ x X mπ y 2 SS þ ; ϕym ðxÞ cos a b m¼0

ðcÞ

(12)

where M is the number of terms in the general solution. According to Eq. (4), the boundary layer function in this case is given as

ψ^ SS ðx; yÞ ¼

M X m¼1

1

ψ SS m ðyÞ sin

M mπ x X mπ y 2 SS þ : ψ m ðxÞ sin a b m¼1

(13)

1 2 SS Considering the double symmetry deformation of plate element (Fig. 2) it is obvious that 1 W SS m ðyÞ, W m ðxÞ, ϕym ðyÞ and 2 SS 2 SS 1 SS 2 SS ϕxm ðxÞ are even functions, while 1 ψ SS ð y Þ, ψ ð x Þ, ϕ ð x Þ and ϕ ð y Þ are odd functions. ym xm m m SS

Fig. 2. Double symmetry deformation of plate element.

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By substituting the first part of Eqs. (12) and (13) into Eq. (9), the following equations are obtained for m 40: IV  II     1 W SS þ c1 2α2m 1 W SS þ α4m  c1 α2m þ c2 1 W SS m m m ¼0 

1

ψ SS m

II

   α2m c3 1 ψ SS m ¼0

(14) (15)

where αm ¼ mπ =a. Excluding the odd functions from the general solution of Eq. (14) and the even function from the solution 1 SS of Eq. (15), the solutions for 1 W SS m ðyÞ and ψ m ðyÞ can be written as     1 1 1 W SS r 1m y þ 1 A2m cosh 1 r 2m y m ðyÞ ¼ A1m cosh   1 SS ψ m ðyÞ ¼ 1 A3m sinh 1 r3m y ; (16) where 1 Aim (i¼1, 2, 3) are integration constants and 1 r 1m ;  1 r 1m ; 1 r 2m ;  1 r 2m ; 1 r 3m ;  1 r 3m are the roots of the SS corresponding characteristic equations of Eqs. (14) and (15). Using Eqs. (7), (8) and (16), the rotation functions 1 ϕxm ðyÞ 1 SS and ϕym ðyÞ are obtained in the following form:       1 SS ϕym ðyÞ ¼ 1 δ 1m 1 A 1m cosh 1 r 1m y þ 1 δ 2m 1 A 2m cosh 1 r 2m y þ 1 δ 3m 1 A 3m cosh 1 r 3m y       1 SS ϕxm ðyÞ ¼ 1 γ 1m 1 A 1m sinh 1 r 1m y þ 1 γ 2m 1 A 2m sinh 1 r 2m y þ 1 γ 3m 1 A 3m sinh 1 r 3m y ; (17) where 1

     2 12αm Gk kGh þ D 1 r im  α2m þDρω2   ; i ¼ 1; 2 δim ¼ 2 kGh 12Gk  ρh ω2      2 121 r im Gk kGh þ D 1 r im  α2m þ Dρω2 1   ; i ¼ 1; 2 γ im ¼ 2 kGh 12Gk  ρh ω2 1

SS

δ3m ¼

6D1 r 3m ð1  νÞ

1

12kGh  ρh ω 3

2

γ 3m ¼

6Dαm ð1  νÞ

(18)

12kGh  ρh ω2 3

For m ¼0, 1 ψ 0 ðyÞ ¼ 0 and 1 ϕy0 ðyÞ ¼ 0. Consequently, Eq. (15) vanishes and Eq. (14) becomes:     SS IV SS II SS 1 W0 þ c1 1 W 0 þc2 1 W 0 ¼ 0: SS

(19)

SS

The solution for 1 W 0 ðyÞ is obtained as 1

    SS W 0 ðyÞ ¼ 1 A10 cosh 1 r 10 y þ 1 A20 cosh 1 r 20 y ;

(20)

where 1 r 10 and 1 r 20 are the roots of the corresponding characteristic equation. The rotation function ϕ from Eqs. (7), (8) and (20) in the following form:     1 SS ϕx0 ðyÞ ¼ 1 γ 10 1 A10 sinh 1 r 10 y þ 1 γ 20 1 A20 sinh 1 r20 y ; 1

where

    2  r i0 kG kGh þ D 1 r i0 þ Dρω2   ; ¼ 3 2 Gk kGh  ρh12ω

SS x0 ðyÞ

is obtained (21)

1

1

SS

γ i0

i ¼ 1; 2:

(22)

The solutions for 2 W m ðxÞ, 2 ϕxm ðxÞ and 2 ϕym ðxÞ can be obtained likewise. Now, the transverse displacement and rotations for the double symmetry contribution can be written in the following form: SS

SS

^ SS ðx; yÞ ¼ w

2 h X 1 i¼1

   i Ai0 cosh 1 r i0 y þ 2 Ai0 cosh 2 r i0 x

! 3 2 X   1 Aim cosh 1 r im y cos ðαm xÞ 7 6 M 6 7 X 6 i¼1 7 ! þ 6 7 2 X 6 7     2 m¼14 2 Aim cosh r im x cos β y 5 þ 2

m

i¼1

SS

ϕ^ x ðx; yÞ ¼

2 X i¼1

1



2 

γ i0 1 Ai0 sinh 1 r i0 y þ

6 M 6 X 6 6 6 m¼14

3 X

1



i¼1

þ

! 

γ im 1 Aim sinh 1 rim y

3 X

2

γ im 2 Aim

3 cos ðαm xÞ

7 7 7 ! 7 2   7 cosh r im x sin β m y 5

i¼1

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Fig. 3. Boundary displacements and forces of a quarter segment of rectangular plate.

2 SS

ϕ^ y ðx; yÞ ¼

2 X

2





δi0 2 Ai0 sinh 2 ri0 x þ

i¼1

3 X

1



! 

δim 1 Aim cosh 1 rim y

3 sin ðαm xÞ

6 7 M 6 7 X 6 i¼1 7 ! 6 7 3 X 6 2   7 2 m¼14 2 δim Aim sinh rim x cos βm y 5 þ

(23)

i¼1

where β m ¼ mπ =b.

2.3. Dynamic stiffness matrix for the double symmetry contribution General boundary conditions which represent the displacements and forces on the boundary of a quarter segment of plate element are presented in Fig. 3. The corresponding displacement vector q^ SS and force vector Q^ SS are: 2 SS 3 2 3 ^ SS ða; yÞ T^ x ða; yÞ 2 SS 3 w 2 SS 3 6 7 Q1 6 SS 7 q1 6 ^ SS 7 6 ^ ða; yÞ 7 6 6 7 7 ð a; y Þ M SS 6 SS 7 6 ϕ 7 x 6 Q2 7 6 7 6 q2 7 6 y 7 6 7 7 6 SS 6 SS 7 6 ^ SS 7 ^ 6 6 7 SS  M xy ða; yÞ 7 6 q 7 6 ϕ ða; yÞ 7 Q 6 6 7 7 3 x 3 7 6 6 7 ^ 7: q^ SS ¼ 6 SS 7 ¼ 6 SS (24) ; Q ¼ 6 SS 7 ¼6 SS 6 7 7 ^ ðx; bÞ 7 ^ 6 q4 7 6 w 7 SS 6 6 Q 4 7 6 T y ðx; bÞ 7 6 7 6 7 6 6 7 7 6 qSS 7 6 ^ SS 7 6 Q SS 7 6 7 4 5 5 6 ϕx ðx; bÞ 7 ^ SS ðx; bÞ 7 4 5 5 6 M 6 7 y 6 7 4 5 qSS SS SS 4 5 6 Q6 ϕ^ y ðx; bÞ ^ SS ðx; bÞ M xy

These vectors are functions of spatial variables x and y and consequently the relation between them cannot be defined as in the case of one – dimensional elements. Discretization of the general boundary conditions can be accomplished by using the projection method [20,21], which is based on the projection of the general boundary conditions onto a set of projection functions hn ðsÞ: qSS i ðsÞ 

M X

〈qSS i ; hn 〉 hn ðsÞ ¼

n¼0

Q SS i ðsÞ 

M X

M X

q~ SS i hn ðsÞ;

i ¼ 1; :::; 6

n¼0

〈Q SS i ; hn 〉 hn ðsÞ ¼

n¼0

M X

SS Q~ i hn ðsÞ;

i ¼ 1; :::; 6

(25)

n¼0

where

~ 〈qSS i ; hn 〉 ¼ q i ¼ SS

2 L

Z

qSS i ðsÞhn ðsÞds; Z ~ SS 2 Q SS ðsÞhn ðsÞds 〈Q SS i ; hn 〉 ¼ Q i ¼ L s i s

(26)

are the projections of boundary displacements and forces, s is x or y and L is 2a or 2b. For the double symmetry case, the following projection functions were used: 1

hssn ðyÞ ¼ cos

nπ y ; b

2

hssn ðyÞ ¼ cos

nπ y ; b

3

hssn ðyÞ ¼ sin

nπ y b

for

x ¼ a;

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nπ x ; a

hssn ðxÞ ¼ cos

2

hssn ðxÞ ¼ cos

nπ x ; a

3

hssn ðxÞ ¼ sin

nπ x a

7

y ¼ b:

for

(27)

According to Eqs. (25) and (26) the projection of displacement vector q~ SS of a quarter segment of plate can be defined as h i T q~ SS ¼ q~ SS0 q~ SS1 … q~ SSm … q~ SSM ; (28) 1xð6M þ 4Þ

where

2 R 3 3 2 b ^ SS ða; yÞdy w ^ SS ða; yÞ; 1〉 〈w 7 6 2b  b 6 ^ SS 7 6 x SS 7 6 R ^ SS ða; yÞdy 7 7 16 2 b ϕ 6 ϕ 7 1 6 〈ϕ 7 ð a; y Þ; 1〉 y 6 7 6 y0 7 6 2b  b y 7 ¼ 6 y SS 7 ¼ 6 SS ¼ 6 R 7; SS ^ ðx; bÞ; 1〉 7 7 26 2 a w 6 w0 7 2 6 〈w ^ ðx; bÞdx 7  a 2a 4 5 4 5 6 7 4 R 5 SS y SS ^ SS ðx; bÞ; 1〉 2 a 〈ϕ ϕ x0 x ϕ^ ðx; bÞdx 2x

q~ SS0

2

q~ SSm

x

w SS m

w SS 0

2

3

2

3

^ SS ða; yÞ; cos 〈w

6 x SS 7 6 ^ SS ða; yÞ; 6 ϕ 7 6 〈ϕ y 6 ym 7 6 6 6 SS 7 6 SS 6 x ϕ 7 6 〈ϕ ^ 6 xm 7 6 x ða; yÞ; 7 ¼6 6 y w SS 7 ¼ 6 ^ SS ðx; bÞ; 6 〈w 6 m 7 6 SS 7 6 6 6 y ϕ 7 6 ^ SS 6 xm 7 6 〈ϕx ðx; bÞ; 4 5 4 y SS ϕym ^ SS ðx; bÞ; 〈ϕ y

cos sin cos cos sin

2

2a

a

x

3 2 ^ SS ða; yÞ cos mbπ ydy w 2b  b 6 7 7 6 R mπ y 7 ^ SS ða; yÞ cos mπ ydy 7 2 b 7 〉7 6 ϕ y b  b b 2b 7 7 6 6 R 7 7 SS 6 2 b ^ 7 mπ y 7 mπ y 〉 ϕ ð a; y Þ sin dy 6 7 b 7 6 2b  b x b 7: ¼ 6 2 Ra mπ x 7 7 SS mπ x 7 〉 ^ ð x; b Þ cos dx w 7 a 2a  a a 7 6 7 7 6 6 2 R a ^ SS 7 mπ x 7 mπ x 6 7 〉 ϕ ð x; b Þ cos dx 7 6 2a  a x a a 7 5 4 5 R SS a mπ x ^ 2 mπ x 〉 a 2a  a ϕy ðx; bÞ sin a dx mπ y 〉 b

3

Rb

Similarly, the projection of force vector Q~ SS of a quarter segment of plate is h i T Q~ SS ¼ Q~ SS0 Q~ SS1 … Q~ SSm … Q~ SSM

1xð6M þ 4Þ

(29)

;

(30)

where 2 3 3 R SS SS 2 b ^ T ða; yÞdy 〈T^ x ða; yÞ; 1〉 6 2bR  b x 7 6 7 6 2 b ^ SS 7 7 6 ^ SS 7 6 7 7 1 6 〈M 7 ð a; y Þ; 1〉 ð a; y Þdy M x 7 7 6 7 16 2b  b x Q~ SS0 ¼ 7; 7¼ 6 7¼ 6 SS R 7 26 〈T^ ðx; bÞ; 1〉 7 26 2 a T^ SS ðx; bÞdx 7 y 6 2a  a y 7 5 6 7 4 R 5 4 5 y SS SS M y0 ^ SS ðx; bÞ; 1〉 2 a ^ 〈M  M ð x; b Þdx y y 2a  a 2 3 2 3 R SS SS 2 b ^ 2 x SS 3 T ða; yÞ cos mbπ ydy 〈T^ x ða; yÞ; cos mbπ y〉 T xm 6 7 6 2bR  b x 7 6 2 b ^ SS 7 mπ y 6 x SS 7 6 ^ SS ða; yÞ; cos mπ y〉 7 7 6 2b 7 ð a; y Þ cos dy 〈 M M 6 M xm 7 6 x x  b b b 7 6 7 6 7 6 6 7 7 R SS SS 6 x SS 7 6 ^ ða; yÞ sin mπ ydy 7 ^ ða; yÞ; sin mπ y〉 7 6 2 b  M 6 M xym 7 6 〈M 6 6 7 7 xy xy  b b b 2b 6 7 6 6 7 7: ¼ 6 y SS 7 ¼ 6 ¼6 R a SS 7 2 m π x 6 T ym 7 6 〈T^ SS ðx; bÞ; cos mπ x〉 7 ^ 7 6 2a  a T y ðx; bÞos a dx 7 6 7 6 y a 7 6 7 6 y SS 7 6 6 7 7 R SS SS 6 M ym 7 6 ^ ðx; bÞ cos mπ xdx 7 ^ ðx; bÞ; cos mπ x〉 7 6 2 a  M 4 5 6 〈M y y 6 7 7 a 2a  a a y SS 4 5 4 5 M xym R SS SS mπ x 2 a mπ x ^ ^ 〈Mxy ðx; bÞ; sin a 〉 2a  a M xy ðx; bÞ sin a dx 2

x SS T x0 6 x SS 6 M x0 6 6 y SS 6 T y0 4

Q~ SSm

2

3

(31)

The superscripts x and y in Eqs. (29) and (31) refer to the plate boundaries x¼a and y¼ b, respectively. By substituting Eqs. (2) and (23) into Eqs. (29) and (31) the following relations are obtained: ~ SS CSS q~ SS ¼ D SS Q~ SS ¼ F~ CSS ;

where CSS is vector of integration constants: h 1 1 A20 2 A10 CTSS ¼ A10 …

1

2

A20

A1M

1

… A2M

1

A1m

1

A3M

1

A2m

2

A1M

(32)

1

A3m

2

A2M

2

A1m

2

A3M

2

i :

A2m

2

A3m

… (33)

~ and F~ depends on the number of terms M in the general solution. In order to make them The size of the matrices D ~ SS and F~ SS square matrices, the number of terms in Eqs. (25) and (12) should be the same. The elements of the matrices D were derived here explicitly using the Symbolic toolbox in MATLAB [27] and are given in Appendix A. By eliminating the vector CSS from Eq. (32) the relation between the projection vectors Q~ SS and q~ SS is obtained as SS

SS

~ SS q~ ; Q~ SS ¼ K D SS

(34)

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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8

~ SS is the dynamic stiffness matrix for the SS contribution, defined for a quarter segment of plate: where K D  SS   1 ~ SS ¼ F~ SS D ~ : K D

(35)

~ SS K D

is 6Mþ4. The elements of the dynamic stiffness matrix were evaluated The size of the dynamic stiffness matrix numerically from Eq. (35). The projections of displacement and force vectors q~ SA , q~ AS , q~ AA , Q~ SA , Q~ AS and Q~ AA , as well as the corresponding dynamic ~ SA , K ~ AS and K ~ AA for other symmetry contributions can be expressed likewise. The size of these matrices is stiffness matrices K D D D respectively 6Mþ 2, 6M þ2 and 6M. 2.4. Dynamic stiffness matrix of completely free rectangular plate The dynamic stiffness matrices derived for the four symmetry cases will be used here to obtain the full dynamic stiffness matrix of the completely free dynamic stiffness element. Introducing vectors Q~ o and q~ o as h i T q~ o ¼ q~ SS q~ SA q~ AS q~ AA ; h i T (36) Q~ o ¼ Q~ SS Q~ SA Q~ AS Q~ AA ; the relation between them can be written as 2

~ SS K D

6 6 6 0 ~ ~ ~ Q o ¼ Koqo ¼ 6 6 6 0 4 0

0

0

~ SA K D

0

0

~ AS K D

0

0

0

3

7 7 0 7 7q~ o ; 7 0 7 5 ~ AA K

(37)

D

~ o is the dynamic stiffness matrix obtained collecting the dynamic stiffness matrices of the four symmetry where K contributions. General boundary conditions for completely free plate are collected into displacement and force vectors: 2 3 2 3 ^ ða; yÞ w T^ x ða; yÞ 6 ^ 7 6 6 ϕy ða; yÞ 7 ^ x ða; yÞ 7 6 M 7 6 7 6 7 6 ^ 7 6 ^ xy ða; yÞ 7 6 ϕx ða; yÞ 7  M 6 7 6 7 6 7 6 ^ 7 6 7 ^ 6 wðx; bÞ 7 T y ðx; bÞ 6 7 6 7 6 7 6 ϕ ^ ðx; bÞ 7 6 ^ 6 x 7  My ðx; bÞ 7 6 7 6 7 6 7 6 ϕ 7 ^ ^ xy ðx; bÞ 7 6 M 6 y ðx; bÞ 7 7 7; Q^ ¼ 6 q^ ¼ 6 (38) 6 7: 6w 7 6  T^ x ð  a; yÞ 7 6 ^ ð  a; yÞ 7 6 7 6^ 7 6 M ^ x ð a; yÞ 7 6 ϕy ð  a; yÞ 7 6 7 6 7 6 ^ 7 6^ 7 6 Mxy ð a; yÞ 7 6 ϕ ð a; yÞ 7 6 7 6 x 7 6 7 6 ^ 7 6  T^ y ðx; bÞ 7 6 wðx; bÞ 7 6 7 6 7 6 ^ y ðx; bÞ 7 6ϕ 7 6 M 7 6 ^ x ðx;  bÞ 7 4 5 4 5 ^ ^  M xy ðx;  bÞ ϕ ðx;  bÞ y

The corresponding projections of the displacement and force vectors, as well as their sub vectors can be represented in the following form: h i T q~ ¼ q~ 0 q~ 1 ⋯ q~ n ⋯ q~ M ; h i 1 T 1 ϕy0 2 w0 2 ϕx0 3 w0 3 ϕy0 4 w0 4 ϕx0 ; q~ 0 ¼ w0 h i S A S A S A S A T q~ n ¼ 1 wSn 1 wAn 1 ϕyn 1 ϕyn 1 ϕxn 1 ϕxn ⋯ 4 wSn 4 wAn 4 ϕxn 4 ϕxn 4 ϕyn 4 ϕyn ;

Q~ n ¼

h

Q~ 0 ¼ 1 S T xn

1 A T xn

1

h

M Sxn

h T Q~ ¼ Q~ 0 1

T x0 1

1

MAxn

M x0 1

2

Q~ 1

T y0

M Sxyn

1

2

⋯

Q~ n 3

M y0

M Axyn

…

⋯

T x0 4 S T yn

3

i Q~ M ; M x0 4 A T yn

4

T y0 4

4

i M y0 ;

M Syn

4

MSyn

4

MSxyn

4

i M Axyn :

(39)

The relation between vectors q~ o and q~ is given through the transformation matrix T a ~ q~ o ¼ 12 Tq:

(40)

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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Fig. 4. (a) Plate structure, (b) displacement and rotation projections for m ¼0, and (c) displacement and rotation projections for m¼ 1.

Fig. 5. Assembly procedure.

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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10

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Similarly, the relation between vectors Q~ and Q~ o can be written as Q~ o ¼ TT Q~ :

(41)

The development of the transformation matrix T is described in detail in [21] and [23] for the Kirchhoff’s plate element and for the plate element undergoing in-plane vibration, respectively. Similar procedure was carried out here to obtain the transformation matrix for the Mindlin plate element. The elements of the transformation matrix are given in Appendix A. Now it is possible to relate the vectors Q~ and q~ using Eqs. (37), (40) and (41) as ~ D q; ~ o Tq~ ¼ K ~ Q~ ¼ 12 TT K

(42)

~ D is the full dynamic stiffness matrix of the completely free rectangular plate. The size of the dynamic stiffness where K matrix depends on the number of terms M in the general solution and is equal (24  Mþ8). 2.5. Assembly procedure In order to validate the presented method based on the DSM, a computer code in MATLAB [27] was developed for the free vibration analysis of rectangular plate assemblies with arbitrary boundary conditions. Since plates are connecting along the boundary lines instead at nodes, basic unknowns are the projections of the boundary displacements and rotations. The assembly procedure is described by an example of plate assembly consisting of two plates connected along boundary line 7, Fig. 4a. To simplify the procedure, it is assumed that the number of terms in the general solution M is equal to 1. The displacement projections for each boundary line and for each term m in the general solution are given in Fig. 4b, c. They are collected into the following vectors according to Eq. (39): 2 2 3 3 1 7 wS1 wS1 61 67 7 7 6 wA1 7 6 wA1 7 6 6 7 7 6 1ϕ 7 67 7 2 2 3 3 6 yS1 7 6 ϕyS1 7 1 7 6 6 7 7 w0 w0 6 6 7 7 1 7 61 67 7 7 6 ϕyA1 7 6 ϕyA1 7 6 ϕy0 7 6 ϕy0 7 6 6 7 7 6 6 7 7 6 1ϕ 7 6 7ϕ 7 62 63 7 7 6 xS1 7 6 xS1 7 6 w0 7 6 w0 7 6 6 7 7 6 6 7 7 2 3 2 3 6 1 ϕxA1 7 6 7 ϕxA1 7 62 63 7 7 1 2 6 6 7 7 ~ ~ q q ϕ ϕ 6 6 7 7 0 0 x0 7 x0 7 6 6 7 7 1 2 1 2 1 2 q~ ¼ 4 1 5; q~ ¼ 4 2 5; q~ 0 ¼ 6 ; q~ 0 ¼ 6 ; q~ 1 ¼ 6 ⋮ 7; q~ 1 ¼ 6 ⋮ 7; (43) 7 4 6 6 7 7 66 65 7 7 q~ 1 q~ 1 6 w0 7 6 w0 7 6 6 7 wS1 7 wS1 7 67 64 7 7 6 6 7 6 ϕy0 7 6 ϕy0 7 66 65 7 7 6 6 7 7 6 wA1 7 6 wA1 7 66 65 7 7 6 6 7 7 6 w0 7 6 w0 7 6 6ϕ 7 6 5ϕ 7 4 4 5 5 6 xS1 7 6 xS1 7 5 6 6 6 7 7 ϕx0 ϕx0 6 6 ϕxA1 7 6 5 ϕxA1 7 6 6 7 7 6 6ϕ 7 6 5ϕ 7 6 yS1 7 6 yS1 7 4 4 5 5 6

ϕyA1

5

ϕyA1

where the right superscripts 1 and 2 denote plate element numbers, while the left superscripts denote the number of the boundary line. The size of the global dynamic stiffness matrix of the structure is given as N ¼ nb U ð6 UM þ2Þ;

(44)

where nb is the number of boundary lines of plate assembly and ð6 UM þ 2Þ is the number of displacement projections per boundary line. The assembly procedure for the computation of the global dynamic stiffness matrix is ~ 1 and K ~ 2 had been computed, the global dynamic schematically presented in Fig. 5. After the dynamic stiffness matrices K D D G ~ was obtained by adding the elements of the dynamic stiffness matrices K ~ 1 and K ~ 2 with the same code stiffness matrix K D D D numbers. 2.6. Boundary conditions and computation of natural frequencies and mode shapes Once the global dynamic stiffness matrix of a structure has been obtained, the natural frequencies can be determined from the following equation:   ~  (45) detK Dnn  ¼ 0; ~ D is the dynamic stiffness matrix of the structure relating the unknown displacement projections with the where K nn corresponding force projections. It is obtained by removing the rows and columns of the global dynamic stiffness matrix that correspond to the components of the constrained displacement projections. The following boundary conditions were assigned to the plate boundary:

 S – simply supported: w¼0 and ϕx ¼0 for the edge parallel to the y axis, i.e. w¼0 and ϕy ¼0 for the edge parallel to the x axis; Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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11

Table 1 Dimensionless natural frequencies of rectangular plate calculated using different number of terms in the general solution (a/b ¼0.4, ν ¼ 0.3, k ¼ 5/6, h/ b ¼0.2). Boundary conditions

Freq. number

Present solution

Error (%)

M¼3

M¼5

M¼7

M¼ 9

M¼ 11

M ¼13

M ¼15

ΩM

¼ 15

ΩM

 ΩM

¼ 3

¼ 15

CCCC

1 2 3 4 5 6 7 8

68.21 78.90 96.40 118.5 129.6 138.2 143.8 151.7

68.21 78.91 96.44 118.7 129.6 138.3 143.9 151.9

68.21 78.91 96.44 118.7 129.6 138.3 143.9 151.9

68.21 78.91 96.44 118.7 129.6 138.3 143.9 151.9

68.21 78.91 96.44 118.7 129.6 138.3 143.9 151.8

68.21 78.93 96.44 118.7 129.6 138.3 143.9 151.8

68.21 78.93 96.44 118.7 129.6 138.3 143.9 151.8

0.00 0.04 0.04 0.17 0.00 0.07 0.07 0.07

SSSS

1 2 3 4 5 6 7 8

51.08 66.08 87.70 112.8 125.3 134.0 140.4 147.9

51.13 66.24 87.88 113.2 125.4 134.3 140.7 148.3

51.14 66.29 87.93 113.3 125.4 134.3 140.8 148.3

51.14 66.30 87.94 113.3 125.4 134.3 140.8 148.4

51.15 66.32 87.95 113.3 125.4 134.4 140.8 148.4

51.15 66.33 87.96 113.3 125.4 134.4 140.8 148.4

51.15 66.33 87.96 113.4 125.4 134.4 140.8 148.4

0.14 0.38 0.30 0.53 0.08 0.30 0.28 0.34

SCSC

1 2 3 4 5 6 7 8

52.18 68.70 90.60 115.4 125.5 134.6 142.1 148.8

52.17 68.69 90.63 115.6 125.5 134.8 142.2 149.0

52.17 68.69 90.61 115.6 125.5 134.8 142.2 149.0

52.17 68.68 90.61 115.5 125.5 134.8 142.2 149.0

52.17 68.68 90.61 115.5 125.5 134.8 142.2 149.0

52.17 68.68 90.60 115.5 125.5 134.8 142.2 149.0

52.17 68.68 90.60 115.5 125.5 134.8 142.2 149.0

 0.02  0.03 0.00 0.09 0.00 0.15 0.07 0.13

CSSF

1 2 3 4 5 6 7 8

55.18 62.16 76.23 96.76 120.6 125.0 130.5 140.5

55.20 62.20 76.46 96.96 121.2 125.1 130.6 141.0

55.20 62.22 76.48 96.98 121.3 125.1 130.6 141.0

55.20 62.23 76.49 96.99 121.3 125.1 130.6 141.0

55.21 62.23 76.49 96.99 121.3 125.1 130.6 141.0

55.21 62.23 76.50 96.99 121.3 125.1 130.6 141.0

55.21 62.23 76.50 96.99 121.3 125.1 130.6 141.0

0.05 0.11 0.35 0.24 0.58 0.08 0.08 0.35

CFCF

1 2 3 4 5 6 7 8

64.60 65.87 71.64 84.40 104.4 126.0 127.3 128.0

64.61 65.87 71.69 84.69 104.6 126.0 128.2 128.3

64.62 65.88 71.69 84.72 104.6 126.0 128.2 128.4

64.62 65.88 71.69 84.73 104.6 126.0 128.2 128.4

64.62 65.88 71.69 84.73 104.6 126.0 128.2 128.4

64.62 65.88 71.69 84.73 104.7 126.0 128.2 128.4

64.62 65.88 71.69 84.73 104.7 126.0 128.2 128.4

0.03 0.02 0.07 0.39 0.29 0.00 0.70 0.31

CSFF

1 2 3 4 5 6 7 8

19.49 28.82 47.69 69.06 73.65 79.39 94.95 102.8

19.50 28.82 47.70 69.06 73.68 79.35 95.04 102.7

19.52 28.82 47.70 69.05 73.68 79.33 95.04 102.7

19.52 28.82 47.70 69.05 73.68 79.33 95.04 102.7

19.52 28.82 47.70 69.05 73.69 79.33 95.04 102.7

19.52 28.82 47.70 69.05 73.69 79.33 95.04 102.7

19.52 28.82 47.70 69.05 73.69 79.32 95.04 102.7

0.15 0.00 0.02  0.01 0.05  0.09 0.09  0.10

 C – clamped: w¼0, ϕx ¼0 and ϕy ¼0; and  F – free: w a0, ϕx a0 and ϕy a0. ~ D contain transcendental functions, it is difficult to calculate the natural Since the elements of the dynamic stiffness matrix K nn frequencies using Eq. (45). To avoid numerical difficulties and missing the potentially coincident natural frequencies, the Wittrick– Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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12

Table 2 Dimensionless natural frequencies of rectangular plate compared to the results obtained using FEM and the results from the literature (a/b¼ 0.4, ν ¼ 0.3, k ¼ 5/6, h/b ¼ 0.2) Boundary Freq. Present FEM (Abaqus) conditions number solution M ¼15 8x20 elements

Liew et al. [6] 20x50 elements

40x100 elements

80x200 elements

Xing and Liu [7]

Ω

Error (%) Ω

Error (%)

Ω

Error (%)

Ω

Error (%)

Ω

Error (%) Ω

Error (%)

CCCC

1 2 3 4 5 6 7 8

68.21 78.93 96.44 118.7 129.6 138.3 143.9 151.8

69.07 79.61 97.36 120.6 134.1 142.2 147.6 155.2

1.26 0.86 0.95 1.60 3.47 2.82 2.57 2.24

68.35 79.03 96.59 119.0 130.3 138.9 144.5 152.4

0.21 0.13 0.16 0.25 0.54 0.43 0.42 0.40

68.24 78.95 96.48 118.8 129.7 138.5 144.1 152.0

0.04 0.02 0.04 0.08 0.08 0.14 0.14 0.13

68.22 78.93 96.45 118.7 129.6 138.3 144.0 151.9

0.02 0.00 0.01 0.00 0.00 0.00 0.07 0.07

68.21 78.92 96.44 118.7 129.6 138.3 144.0 151.8

0.00  0.01 0.00 0.00 0.00 0.00 0.07 0.00

68.69 80.28 98.27 120.6 130.4 140.5 145.7 154.8

0.70 1.71 1.90 1.60 0.62 1.59 1.25 1.98

SSSS

1 2 3 4 5 6 7 8

51.15 66.33 87.96 13.35 125.4 134.4 140.8 148.4

51.69 66.79 88.75 115.2 130.1 138.4 144.5 151.8

1.06 0.69 0.90 1.63 3.75 2.98 2.63 2.29

51.24 66.41 88.10 113.7 126.1 135.0 141.4 148.9

0.18 0.12 0.16 0.31 0.56 0.45 0.43 0.34

51.18 66.35 88.01 113.4 125.6 134.5 141.0 148.5

0.06 0.03 0.06 0.04 0.16 0.07 0.14 0.07

51.16 66.34 87.98 113.4 125.4 134.4 140.8 148.4

0.02 0.02 0.02 0.04 0.00 0.00 0.00 0.00

51.64 67.49 89.30 114.5 125.4 134.6 141.5 148.7

0.96 1.75 1.52 1.02 0.00 0.15 0.50 0.20

51.82 67.94 89.87 115.0 125.5 134.7 142.1 149.0

1.31 2.43 2.17 1.46 0.08 0.22 0.92 0.40

SCSC

1 2 3 4 5 6 7 8

52.17 68.68 90.60 115.5 125.5 134.8 142.2 149.0

52.68 69.13 91.44 117.4 130.2 138.8 145.9 152.3

0.98 0.66 0.93 1.65 3.75 2.97 2.60 2.21

52.25 68.75 90.74 115.8 126.2 135.4 142.8 149.5

0.15 0.10 0.16 0.26 0.56 0.44 0.42 0.34

52.19 68.69 90.64 115.6 125.7 134.9 142.4 149.1

0.04 0.02 0.04 0.09 0.16 0.07 0.14 0.07

52.17 68.68 90.61 115.6 125.6 134.8 142.3 149.0

0.00 0.00 0.01 0.09 0.08 0.00 0.07 0.00

51.68 68.67 90.60 115.5 125.5 134.8 142.2 149.0

 0.94  0.02 0.00 0.00 0.00 0.00 0.00 0.00

52.60 69.60 91.73 116.6 125.6 135.1 143.2 149.6

0.82 1.34 1.25 0.95 0.08 0.22 0.70 0.40

CSSF

1 2 3 4 5 6 7 8

55.21 62.23 76.50 96.99 121.3 125.1 130.6 141.0

55.42 62.38 76.63 97.20 121.8 126.3 131.7 141.9

0.38 0.24 0.17 0.22 0.41 0.96 0.84 0.64

55.34 62.33 76.58 97.13 121.6 125.9 131.3 141.6

0.24 0.16 0.11 0.14 0.25 0.64 0.54 0.43

55.24 62.25 76.52 97.03 121.4 125.3 130.8 141.2

0.05 0.03 0.03 0.04 0.08 0.16 0.15 0.14

55.22 62.24 76.51 97.00 121.4 125.2 130.7 141.1

0.02 0.02 0.01 0.01 0.08 0.08 0.08 0.07

55.21 62.23 76.51 97.00 121.3 125.1 130.6 141.0

0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00

–

CFCF

1 2 3 4 5 6 7 8

64.62 65.88 71.69 84.73 104.7 126.0 128.2 128.4

65.66 66.80 72.26 85.12 105.5 130.4 130.8 132.8

1.61 1.40 0.80 0.46 0.76 3.49 2.03 3.43

64.80 66.04 71.78 84.79 104.8 126.8 128.7 128.9

0.28 0.24 0.13 0.07 0.10 0.64 0.39 0.39

64.67 65.93 71.71 84.74 104.7 126.2 128.4 128.5

0.08 0.08 0.03 0.01 0.00 0.16 0.16 0.08

64.64 65.90 71.69 84.73 104.7 126.1 128.2 128.4

0.03 0.03 0.00 0.00 0.00 0.08 0.00 0.00

64.64 65.90 71.69 84.74 104.7 126.0 128.2 128.4

0.03 0.03 0.00 0.01 0.00 0.00 0.00 0.00

–

CSFF

1 2 3 4 5 6 7 8

19.52 28.82 47.70 69.05 73.69 79.32 95.04 102.7

19.54 28.84 47.96 70.13 74.73 80.30 95.85 105.2

0.10 0.07 0.55 1.56 1.41 1.24 0.85 2.43

19.52 28.82 47.74 69.22 73.85 79.48 95.17 103.1

0.00 0.00 0.08 0.25 0.22 0.20 0.14 0.39

19.52 28.82 47.71 69.09 73.73 79.36 95.07 102.8

0.00 0.00 0.02 0.06 0.05 0.05 0.03 0.10

19.52 28.82 47.70 69.06 73.69 79.33 95.05 102.7

0.00 0.00 0.00 0.01 0.00 0.01 0.01 0.00

19.52 28.82 47.70 69.05 73.68 79.32 95.04 102.7

0.00 0.00 0.00 0.00  0.01 0.00 0.00 0.00

–

Williams algorithm [25] was used to detect the number of natural frequencies which are lower than a trial frequency. When the natural frequencies were determined, the mode shapes were obtained using the well-known procedure.

3. Numerical validation Free vibration characteristics of individual plates and plate assemblies computed using the proposed method are validated against the results available in the literature, for different aspect and thickness ratios b/a and h/b, respectively, as well as considering different types of boundary conditions. Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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13

Fig. 6. First four mode shapes of CFCF rectangular plate.

Fig. 7. First four mode shapes of CSSF rectangular plate.

3.1. Rectangular plate In order toffi check the convergence of the proposed method, the first eight dimensionless natural frequencies pffiffiffiffiffiffiffiffiffiffiffi

Ω ¼ ωb2 ρh=D of rectangular plate (a/b¼0.4) have been calculated for six types of boundary conditions, using different number of terms M in the general solution. The results are presented in Table 1. The higher the frequency required in the analysis, the larger number of terms should be used for the solution. Nevertheless, the differences between the first eight Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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N. Kolarevic et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Fig. 8. First four mode shapes of CSFF rectangular plate.

natural frequencies computed by using 3 and 15 terms in the general solution are practically negligible – far less than 1 percent. The dimensionless natural frequencies are compared with the results from the literature [6,7], given in Table 2. As can be seen from Table 2, the results obtained using the proposed method are in excellent agreement with the results from the literature [6,7]. In addition, the finite element analysis was performed using Abaqus S4R type plate element and four different mesh sizes: 8  20, 20  50, 40  100 and 80  200 elements. The results are also presented in Table 2. As the number of finite elements increases, the natural frequencies computed using the FEM converge to the present solutions. The first four mode shapes of rectangular plate for CFCF, CSSF and CSFF boundary conditions calculated using the proposed method are presented in Figs. 6–8, respectively. 3.2. Square plate For further validation, free vibration characteristics of Levy-type square plate (a/b ¼1, h/b ¼0.1, ν ¼0.3, k ¼5/6) are computed using the proposed method for SSSS, SCSC and SFSF boundary conditions. The results are presented in Table 3 and are compared with the exact solutions reported in [15]. The differences between the natural frequencies obtained by these two solutions are negligible. The dimensionless natural frequencies of the square plate (a/b¼1, h/b¼0.05, 0.1, 0.2) have also been computed using the proposed method based on Mindlin and Kirchhoff’s plate theory for six types of boundary conditions (FFFF, CFFF, SFFF, SSCF, CCCF) that could not be computed by using the Boscolo and Banerjee’s exact solution [15]. The results are summarized in Table 4. The natural frequencies are in very good agreement with the results from the literature [7]. Decreasing the thickness ratio h/b, the difference between the natural frequencies computed using the Kirchhoff’s and Mindlin plate theory decreases. For h/b ¼0.05 the error is less than 5 percent for the first five natural frequencies and for all boundary conditions considered in Table 4. For higher modes, the error increases and varies from 5 to 10 percent, depending on the type of boundary conditions. In the case when the thickness ratio is equal 0.2, the error is significant even for the first free vibration mode, and substantially increases for higher modes, especially for clamped boundary conditions. Influence of the thickness ratio and boundary conditions on the difference between the natural frequencies computed using Kirchhoff’s and Mindlin plate theory is presented in Fig. 9 for the 1st, 5th and 10th mode shape.

3.3. Stepped plate Free vibration analysis of stepped plate shown in Fig. 10 has been carried out to demonstrate the efficiency and accuracy of the proposed method. Edges parallel to the y axis are either simply supported or clamped, while edges parallel to the x Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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15

Table 3 Dimensionless natural frequencies of square Levy-type plate (a/b¼ 1, h/b ¼ 0.1, ν ¼0.3, k ¼5/6). Boundary conditions

Freq. number

Present solution (M¼ 9)

Boscolo and Banerjee [15]

SSSS

1 2, 3 4 5, 6 7, 8 9, 10 11 12

19.0602 45.4568 69.7676 85.0397 106.641 133.613 139.991 152.530

19.0650 45.4827 69.7944 85.0380 106.683 133.621 140.057 152.608

SCSC

1 2 3 4 5 6 7 8 9 10 11 12

26.6603 49.1010 59.2185 78.8062 86.8378 101.367 112.060 118.917 134.596 148.310 149.988 155.766

26.6683 49.1129 59.2102 78.8130 86.8440 101.371 112.058 118.922 134.595 148.316 149.990 155.788

SFSF

1 2 3 4 5 6 7 8 9 10 11 12

9.4462 15.3920 33.8529 36.3463 42.7956 62.1435 66.1953 76.6244 82.4024 92.6158 101.031 110.022

9.4406 15.3893 33.8599 36.3570 42.7927 62.1467 66.1965 76.6330 82.4088 92.6128 101.029 110.024

axis are simply supported, clamped or free. The results computed using the proposed method, are summarized in Table 5 together with the results from the literature [15] and the results obtained by using the Abaqus [26]. Again, the results computed using the presented theory, show excellent agreement with the exact results computed by Boscolo and Banerjee [15]. The first 16 natural frequencies computed using mesh size of 100  100 finite elements are also in excellent agreement with the natural frequencies computed using the proposed method. In addition, the number of dynamic stiffness elements, which is influenced only by the plate geometry, has been minimized. In this case the number of dynamic stiffness elements is equal to 5, while the number of boundary lines is equal to 16. Consequently, according to Eq. (44) the size of the global dynamic stiffness matrix of the steeped plate structure is 16  (6  Mþ 2). For M¼5 and M¼15 the size of the global dynamic stiffness matrix is 512 and 1472, respectively, while the number of degrees of freedom in the finite element models is equal 30603. Obviously, applying the presented method the number of unknowns and the computational time have been significantly decreased in comparison with the FEM, without losing the accuracy and reliability of the results.

4. Conclusions The dynamic stiffness matrix for a completely free rectangular plate element based on Mindlin theory has been developed in the paper. The general solution of the governing equations of motion, presented in the Fourier series form, has been obtained using the boundary layer function and the superposition method. General boundary conditions have been discretized using the projection method. The dynamic stiffness matrix has been derived by eliminating the integration constants from the projections of the general boundary conditions. The global dynamic stiffness matrix of the plate assembly has been obtained using an assembly procedure that is similar to the one used in the FEM. The Wittrick–Williams algorithm has been implemented to compute the natural frequencies. The accuracy of the proposed method is independent of the number of dynamic stiffness elements used in the analysis. It is influenced only by the number of terms in the general solution. Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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16

Table 4 Dimensionless natural frequencies of square plate (a/b ¼1, ν ¼ 0.3, k ¼ 5/6). Boundary conditions

Freq. number

Kirchhof’s theory

h/b ¼0.05

h/b ¼0.1

Present solution (M¼ 9)

Present solution (M¼ 9)

h/b ¼ 0.2 Liew et.al. [6]

Present solution (M ¼9)

Liew et al. [6]

FFFF

1 2 3 4,5 6,7 8 9 10

13.46 19.59 24.27 34.08 61.11 63.68 69.28 77.17

13.1384 19.4198 24.0231 33.7090 59.4103 60.7050 66.2192 74.0351

12.7308 18.9403 23.3278 31.9109 55.3585 55.7181 60.633 67.6817

12.7190 18.9437 23.3248 31.9222 55.3527 55.7139 60.6329 67.6749

11.6999 17.4059 21.1940 27.5714 45.1092 45.2290 48.2139 53.7162

11.7014 17.4001 21.1940 27.5737 45.1051 45.2324 48.2170 53.7133

CFFF

1 2 3 4 5 6 7 8 9 10

3.47 8.50 21.28 27.20 30.94 54.18 61.24 64.14 70.95 92.93

3.4524 8.3433 20.9542 26.6124 30.0168 51.7862 59.1226 61.9037 67.8496 87.7009

3.4284 8.0556 20.0911 25.5095 28.2427 47.5187 54.1118 57.0128 61.5426 78.1348

3.4307 8.0615 20.0906 25.4991 28.2468 47.5340 54.1190 57.0157 61.5676 78.1485

3.3445 7.3484 17.5738 22.4647 23.8912 38.3482 42.6038 45.3969 48.3218 59.5442

3.3399 7.3479 17.5738 22.4682 23.8894 38.3444 42.6051 45.3962 48.3275 59.5463

SFFF

1 2 3 4 5 6 7 8 9 10

6.63 14.90 25.37 25.99 48.43 50.58 58.73 65.17 87.93 89.06

6.5212 14.7687 24.7423 25.5095 46.9433 49.1010 56.6772 63.1025 83.3854 84.5842

6.3774 14.4810 23.7114 24.5985 43.9224 46.1041 52.5534 58.4753 75.1140 76.1929

6.3698 14.4787 23.7186 24.5960 43.9099 46.0980 52.5616 58.4705 75.1235 76.2062

6.0058 13.4860 21.0621 21.9972 36.6699 38.5520 42.8915 47.3868 58.2955 58.9700

6.0007 13.4848 21.0558 21.9954 36.6616 38.5546 42.8864 47.3820 58.2899 58.9660

SSFF

1 2 3 4 5 6 7 8 9 10

3.36 17.31 19.29 38.19 51.03 53.47 72.92 74.62 104.7 107.2

3.3086 17.0703 19.0362 37.1135 49.6764 52.0739 69.9114 71.5897 99.7844 102.1340

3.2846 16.5668 18.5088 35.1235 46.6076 48.8373 63.9416 65.5000 89.1394 91.1534

3.2856 16.5602 18.5055 35.1269 46.6013 48.8427 63.9373 65.5125 89.1393 91.1626

3.1767 15.1643 16.9144 30.1007 38.7917 40.6138 50.7793 52.0859 67.4660 68.9764

3.1800 15.1577 16.9175 30.0993 38.7974 40.6105 50.7771 52.0928 67.4617 68.9767

SSCF

1 2 3 4 5 6 7 8 9 10

16.78 31.17 51.31 64.01 67.61 101.18 105.49 117.24 122.51 153.68

16.5428 30.3525 49.7723 61.7599 64.8287 95.6127 99.6885 111.0050 114.9850 142.6040

15.9674 28.7701 45.9843 56.9169 58.9308 84.6082 87.3653 98.0342 99.5207 121.8963

15.9838 28.7847 46.0042 56.9229 58.9324 84.6259 87.3825 98.0328 99.5300 121.9113

14.2652 24.6704 36.9097 45.7445 46.1521 63.4741 64.0615 71.9000 72.9800 86.4700

14.2626 24.6701 36.9182 45.7506 46.1592 63.4843 64.0577 71.9099 72.9877 86.4814

CCCF

1 2 3 4 5 6 7 8 9 10

23.91 40.08 63.19 76.70 80.55 116.6 122.2 134.4 140.2 172.8

23.3518 38.5999 60.4173 72.9802 76.1449 108.3670 113.5460 125.1020 129.0340 157.2290

22.0811 35.6270 54.2077 65.2602 67.1063 92.9275 96.3560 106.9290 108.1760 130.2021

22.0941 35.6243 54.2197 65.2628 67.1153 92.9322 96.3767 106.9322 108.1965 130.2432

18.6287 28.7581 40.9495 49.6285 49.7364 66.5309 67.2022 74.7184 75.7733 88.8902

18.6328 28.7640 40.9539 49.6293 49.7349 66.5300 67.2031 74.7169 75.7729 88.8896

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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17

Fig. 9. The difference between the Kirchhoff’s and Mindlin theory for different boundary conditions.

Fig. 10. Geometry of stepped plate.

The presented method extends the application of the DSM to rectangular Mindlin plates with arbitrary boundary conditions, which was previously limited to to Levy type plates. Based on the proposed method a computer code has been developed using MATLAB. The natural frequencies of individual plates and plate assemblies have been computed for different types of boundary conditions. The results have been validated against the available results in the literature and the results obtained using Abaqus. High convergence and accuracy for various types of boundary conditions has been accomplished adopting only 3–5 terms in the general solution. It significantly decreased the size of the dynamic stiffness matrix and consequently saved the computational time in comparison with the FEM. However, for higher frequencies, a larger number of terms in the general solution is required. As expected, by increasing the thickness ratio h/b, the difference between the results obtained using the Kirchhoff’s and Mindlin plate theory significantly increases, which is more pronounced for clamped boundaries. Besides its high accuracy and low computational time, the main advantage of the proposed method is the capability to compute free vibration characteristics of rectangular plate assemblies with non-uniform geometrical and material properties, having any possible boundary conditions. In addition, the use of the boundary layer function, which resulted in the uncoupling the governing equations of motion, enables the proposed method to be more efficiently applied to higher order plate theories, as well as to free vibration analysis of composite plates and plate assemblies with transversely isotropic layers.

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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18

Table 5 Natural frequencies (in Hz) of stepped square plate (E ¼71200 MPa,ν ¼0.3, ρ ¼ 2.7 t/m3, a/b ¼1, b1 ¼0.3b, b2 ¼ 0.1b, b3 ¼ 0.2b, b/h1 ¼10, h2/h1 ¼1.5, k ¼ 5/6). Boundary conditions

Present solution (5dynamic stiffness elements)

Boscolo and Banerjee [15]

FEM (Abaqus) 100  100 elements

M¼ 5

M¼7

M¼9

M ¼11

M ¼13

M ¼15

SSSS

501.70 1167.3 1249.4 1817.7 2144.4 2338.3 2765.9 2803.5 3431.7 3626.3 3650.7 3877.1 4002.2 4611.5 4719.7 4738.7

501.80 1167.6 1249.7 1818.5 2144.9 2338.8 2767.5 2804.5 3432.0 3627.2 3653.1 3878.4 4003.8 4613.1 4723.7 4739.0

501.90 1167.8 1249.8 1818.9 2145.1 2339.0 2768.3 2804.9 3432.2 3627.6 3654.4 3878.9 4004.5 4613.8 4725.5 4739.2

502.00 1167.9 1249.9 1819.1 2145.3 2339.1 2768.8 2805.2 3432.3 3627.8 3655.1 3879.2 4004.9 4614.2 4726.4 4739.3

502.00 1168.0 1250.0 1819.3 2145.4 2339.1 2769.2 2805.4 3432.3 3627.9 3655.5 3879.4 4005.1 4614.4 4727.0 4739.3

502.00 1168.0 1250.0 1819.4 2145.5 2339.2 2769.4 2805.5 3432.4 3628.0 3655.8 3879.5 4005.3 4614.6 4727.4 4739.4

502.07 1168.2 1250.1 1819.7 2145.7 2339.3 2770.1 2805.9 3432.5 3628.2 3656.8 3879.8 4005.7 4615.1 4728.6 4739.5

502.11 1168.5 1250.5 1820.2 2147.2 2341.2 2771.4 2807.5 3437.2 3633.0 3658.7 3883.9 4009.9 4618.8 4732.2 4748.7

SCSC

675.70 1335.6 1498.3 2032.0 2387.1 2535.8 2938.6 3067.7 3658.0 3839.8 3865.4 4092.5 4215.0 4859.9 4864.7 5049.1

675.80 1335.8 1498.4 2032.5 2387.5 2536.2 2939.4 3069.2 3658.9 3840.1 3867.8 4094.0 4216.2 48561.7 4868.9 5050.3

675.90 1336.0 1498.5 2032.8 2387.6 2536.4 2939.7 3069.9 3659.3 3840.3 3869.0 4094.6 4216.8 4862.4 4870.7 5050.8

675.90 1336.0 1498.6 2032.9 2387.7 2536.5 2940.0 3070.4 3659.4 3840.4 3869.6 4094.9 4217.1 4862.8 4871.6 5051.1

675.90 1336.1 1498.6 2033.0 2387.8 2536.6 2940.1 3070.7 3659.6 3840.4 3870.1 4095.1 4217.2 4863.0 4872.2 5051.2

675.90 1336.1 1498.7 2033.1 2387.8 2536.7 2940.2 3070.9 3659.6 3840.5 3870.3 4095.2 4217.3 4863.2 4872.6 5051.3

676.00 1336.2 1498.7 2033.3 2388.0 2536.9 2940.5 3071.5 3659.8 3840.6 3871.3 4095.6 4217.7 4863.6 4873.7 5051.6

676.10 1336.6 1499.5 2034.1 2389.8 2539.1 2942.1 3073.4 3664.4 3846.4 3873.5 4099.7 4222.6 4868.0 4877.4 5061.2

CFCF

498.50 586.40 1004.2 1331.1 1431.6 1817.8 1863.3 2329.0 2490.4 2543.2 2969.9 3034.7 3560.9 3669.5 3700.2 3775.4

498.60 586.70 1004.3 1331.4 1432.0 1818.2 1863.5 2329.5 2492.0 2544.3 2971.4 3035.2 3561.8 3673.8 3700.6 3776.0

498.70 586.90 1004.4 1331.5 1432.2 1818.3 1863.5 2329.8 2492.6 2544.5 2971.8 3035.3 3562.0 3674.7 3700.7 3776.3

498.70 586.90 1004.4 1331.6 1432.4 1818.4 1863.6 2329.9 2492.8 2544.6 2971.9 3035.3 3562.1 3675.0 3700.8 3776.6

498.80 587.00 1004.4 1331.7 1432.5 1818.4 1863.6 2330.0 2493.0 2544.6 2972.0 3035.3 3562.1 3675.1 3700.9 3776.6

498.80 587.00 1004.4 1331.8 1432.5 1818.4 1863.6 2330.0 2493.0 2544.6 2972.0 3035.4 3562.1 3675.1 3701.0 3776.6

–

499.00 587.23 1004.6 1332.8 1433.6 1819.1 1864.3 2332.6 2495.6 2545.3 2974.1 3037.9 3564.4 3676.9 3707.3 3782.9

Acknowledgments The authors thank to the Ministry of Science and Technology, Republic of Serbia, for the financial support for this research within the Project TR 36046.

Appendix A Explicit expressions for the coefficients of the D and F matrices are given below for SS contribution. Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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19

A.1. SS contribution e SS consists of four different blocks of matrices: Matrix D 2

½DSS nm 4x4 6 h SS i 6 ½DSS  nm 6x4 ~ ¼6 D 6 ⋮ 4 ½DSS nm 6x4

½DSS nm 4x6

⋯

½DSS nm 6x6 ⋮

⋯ ⋱

½DSS nm 6x6

⋯

For m Z 1 and n Z1 blocks 2

½DSS nm 6x6

m

ð  1Þ U I SS 1m 6 b 6 60 6 6 6 ð  1Þm 1 SS SS 6 6 b γ 1m I 5m SS  SS  ½Dnm 6x6 ¼ 6 6 ISS 6 9m cosh 1 r 1m b 6 a 6 SS  SS  6 I9m 1 SS 6 a γ 1m sinh 1 r 1m b 6  SS  4 ISS SS 1 13m 1 r 1m b a δ 1m cosh

m

ð  1Þ b

½DSS nm 4x6

3

7 7 ½DSS nm 6x6 7 7 ⋮ 5 ½DSS  nm 6x6

and

½F SS nm 6x6

ISS 2m

2

; ð6M þ 4Þð6M þ 4Þ

½F SS nm 4x4 6 h SS i 6 ½F SS  nm 6x4 ¼6 F~ 6 ⋮ 4 ½F SS nm 6x4

ð  1Þ 1 SS SS 2m I 6m b

ð  1Þ b

I SS 9m a

0

γ

 SS  cosh 1 r 2m b  SS  SS I SS 1 9m 1 r 2m b a γ 2m sinh  SS  SS SS I 13m 1 1 δ cosh r 2m b 2m a

0

m

SS U 1 3m U ISS 7m

γ

SS ISS 9m 1 3m a

γ

sinh

SS ISS 13m 1 3m a

δ

0

m

SS 9m

SS 9m

SS I SS 1 M xy;1m 13m a

sinh

1 SS r 1m b

1

SS





1 SS r 3m b

 SS  cosh 1 r 3m b

m

SS

I SS

T y;2m 9m a sinh

1





1 SS r 2m b

 SS  SS I SS 1  1 M y;2m 9m r 2m b a cosh  SS  SS SS I 1 1 M xy;2m 13m sinh r 2m b a

SS

6 1 M SS ð  1Þm I SS 6 1m x;1m b

6 ½F SS nm 4x6 ¼ 6 0 4 0

m

M x;3m ð b1Þ I SS 3m

0  SS  SS I SS 1 T y;3m 9m sinh r 3m b a  SS  SS SS I 1  1 M y;3m 9m cosh r 3m b a  SS  SS I SS 1 1 M xy;3m 13m sinh r 3m b a

0

0

0

m SS 1 M x;2m ð b1Þ I SS 2m

m SS 1 M x;3m ð b1Þ I SS 3m

0 0

SS I SS 4m 2 2m b

δ

SS I SS 8m 2 2m b

γ

½F SS nm 4x6

3

7 7 ½F SS nm 6x6 7 7 ⋮ 5 ½F SS nm 6x6





 SS sinh 2 r 2m a  SS  cosh 2 r 2m a

0

ð  1Þm 2 a

SS



2 SS r 2m a

0

ð  1Þm 2 SS SS 2m I 15m a

δ 1m ISS 14m

δ

 SS  sinh 2 r 1m a  SS  SS I SS 2 M x;1m 4m cosh 2 r 1m a b  SS  SS I 2 8m  2 MSS r 1m a xy;1m b sinh

ð6M þ 4Þð6M þ 4Þ

 SS  sinh 2 r 2m a  SS  SS I SS 2 M x;2m 4m cosh 2 r 2m a b  SS  SS SS I  2 M xy;2m 8m sinh 2 r 2m a b

0

0

SS

3

0

 SS  7 7 SS ISS 4m 2 δ 3m sinh 2 r 3m a 7 7 b 7   SS 7 SS I8m 2 SS γ 3m cosh 2 r 3m a 7 b 7 7 7 7 0 7 7 7 0 7 7 5 m SS ð  1Þ 2 SS δ I 3m 16m a

SS

SS

2 SS I 4m T x;2m b

SS

m

2 SS I4m T x;3m b 2



0 SS

m

m

0

0

0

0

ð  1Þm SS a I 10m

ð  1Þm SS a I 11m

0

0

0

0

0 0

0 0

0 0

0

 2 M y;1m ð a1Þ I SS 10m

SS

0 SS

M x;20

I SS 2m b

0

3

SS

m

 2 M y;3m ð a1Þ I SS 12m

0

7 7 7 7 5

3

7 7 7 7 0 7 7 7 0 7 SS I SS 7 2  M y;20 11m a 5 0

0

0

0

0

0

0

 2 M y;10 10m a

0

0

  cosh 2 r SS 1m a   2 SS δ1m sinh 2 rSS 1m a I SS 10m a

m

 2 M y;2m ð a1Þ I SS 11m

0 1

7 7 7 7 7 5

7 07 7 7 05 0 0

m

7 7 7

3

0

SS

3

 SS  7 sinh 2 r 3m a 7

0

0



2 SS r 3m a

0

0

0



2 SS r 3m a

 2 M y;3m ð a1Þ ISS 12m

0

  cosh 1 r SS 2m b   1 SS γ 2m sinh 1 r SS 2m b

M x;3m 4m cosh b

SS ISS  2 M xy;3m 8m b

0

0

I SS



 2 M y;2m ð a1Þ I SS 11m

0

I 2m b

SS

sinh

 2 M y;1m ð a1Þ I SS 10m

0

SS For m ¼ 0 and n ¼ 0 blocks ½DSS nm 4x4 and ½F nm 4x4 are valid: 2 SS SS I 1m

cosh

ð  1Þm SS a I 11m

SS For m ¼ 0 and n Z 1 blocks ½DSS nm 6x4 and ½F nm 6x4 are valid: 2 SS 3 2 I 1m I SS 2m 0 0 0 b b 6 7 6 1 SS ISS 60 7 0 0 0 6 M x;10 1m 6 7 b 6 6 ISS 7 I SS 6m 1 SS 60 6 5m 1 γ SS 7 0 6 6 b 1m b γ 2m 0 7 SS SS ½Dnm 6x4 ¼ 6 7; ½F nm 6x4 ¼ 6 ISS I SS 60 60 7 10m 11m 0 6 6 7 a a 6 6 7 40 60 7 0 0 0 4 5 SS SS I14m 2 SS I 15m 2 SS 0 δ1m b δ2m 0 0 b

6 b 6 60 6 ½DSS nm 4x4 ¼ 6   6 cosh 1 r SS 1m b 4   1 SS γ 1m sinh 1 r SS 1m b

I SS 4m b

ð  1Þm SS a I 10m

1

0 0

⋯

SS

SS For m Z 1 and n ¼ 0 blocks ½DSS nm 4x6 and ½F nm 4x6 are valid: 2 ð  1Þm SS ð  1Þm SS I I 2m b 6 b 1m 6 0 0 6 ½DSS nm 4x6 ¼ 6 0 40

2

½F SS nm 6x6

2 SS I 4m T x;1m b

0

M x;2m ð b1Þ ISS 2m 0

1

⋯ ⋱

  cosh 2 r SS 1m a   SS SS ISS 4m 2 δ 1m sinh 2 r 1m a b  SS  SS ISS 8m 2 γ 1m cosh 2 r 1m a b

2 6 SS 1 6 M x;1m ð b1Þ ISS 1m 6 6 0 6 6  SS  ½F SS nm 6x6 ¼ 6 1 SS I 6 T y;1m a sinh 1 r 1m b   6 1 SS I 6  M y;1m a cosh 1 r SS1m b 4  

½F SS nm 6x6 ⋮

ISS 4m b

0 m

⋯

are valid: 0

0

½F SS nm 4x6

SS

I SS

3   cosh 2 r SS 2m a  7 7 2 SS δ2m sinh 2 rSS 2m a 7 7 SS 7 I 11m 7 a 5 0

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

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20

2 ½F SS nm 4x4

0

6 6 1 M SS ISS 1m 6 x;10 b ¼6 6 1 T SS sinh1 r SS b 6 y;10 1m 4 1 SS   1 M SS cosh r 1m b y;10

1

I SS

2m MSS x;20 b

  sinh 1 r SS 2m b 1 SS   1 M SS y;20 cosh r 2m b

1 SS T y;20

SS

1

1 SS M x;i0 ¼  νD1 r SS i0 γ i0

1

1 SS M x;im ¼ Dðαm 1 δim  ν1 r SS im γ im Þ

1

1 SS M y;i0 ¼  D1 r SS i0 γ i0

1

1 SS M y;im ¼ Dðναm 1 δim  1 r SS im γ im Þ

1

1 1 SS M xy;im ¼ D1 2 νð1 r SS im δim þ αm γ im Þ

SS

SS

SS SS

SS

SS

SS

  sinh 2 r SS 1m a   2 SS M x;10 cosh 2 r SS 1m a

2 SS T x;10

0

0 I SS

10m  2 MSS y;10 a

SS

  3 sinh 2 r SS 2m a  7 2 SS 7 M x;20 cosh 2 r SS 2m a 7 7 7 0 7 5 SS 2 SS I 11m  M y;20 a 2 SS T x;20

2

2 M x;i0 ¼ D2 r SS i0 δi0

2

2 2 SS M x;im ¼ Dð2 r SS im δim  νβ m γ im Þ

2

2 M y;i0 ¼ Dν2 r SS i0 δi0

2

2 2 SS M x;ym ¼ Dðν2 r SS im δim  β m γ im Þ

2

2 SS M xy;im ¼ D1 2 νð  βm 2 δim  2 γ SS im r im Þ;

SS

SS

SS

SS

SS

SS

SS

SS

2 SS T x;i0

SS

i ¼ 1; 2

δ

2 SS ¼ khGð2 r SS i0 Þ i0 þ

8 9 8 9 < 1 T SS ¼ khGð1 δSS  αm Þ = < 2 T SS ¼ khGð2 r SS þ 2 δSS Þ = im im x;im x;im im SS SS 1 SS 2 SS : ; : ; T x;3m ¼ khG1 δim T x;3m ¼ khG2 δim  1 SS 1 SS T y;i0 ¼ khG 1 r SS i0  γ i0 Þ n    1 SS n 2 SS  2 SS

1 SS 2 SS 1 SS 1 SS T y;im ¼ khG 1 r SS T y;im ¼ khG  β m  2 γ SS im  γ im Þ T y;3m ¼ khG  γ 3m Þ im Þ T y;3m ¼ khG  γ 3m Þ Integrals: ISS 1m

¼

ISS 3m ¼

Rb b

Rb

1



cosh rSS 1m y cos

nπ y dy ¼ b

  nπ y cosh 1 rSS 3m y cos b dy ¼

8 sinhðb1 r SS 1m Þ > ; n¼0 > < 1 rSS 1m

8 sinhðb1 rSS 2m Þ > ; n¼0 > < 1 rSS 1 SS  2m nπ y ¼  b cosh r 2m y cos b dy ¼ 2b2 1 rSS ð  1Þn sinhðb1 rSS Þ > 2m 2m > ; n 40 2 : n2 π 2 þ ðb1 rSS 2m Þ ( 0; ma n Rb mπ y nπ y I SS 4m ¼  b cos b cos b dy ¼ b; m ¼ n

2 2b 1 r SS ð  1Þn sinhðb1 r SS > 1m 1m Þ > ; n4 0 2 : n2 π 2 þ ðb1 r SS 1m Þ 8 1 r SS sinh b ð Þ 3m > ; n¼0 > < 1 rSS 3m

2 2b 1 r SS ð  1Þn sinhðb1 r SS > 3m 3m Þ > ; n4 0 2 : n2 π 2 þ ðb1 r SS 3m Þ 8 < 0; n ¼ 0 1 SS  Rb nπ y 2bnπ ð  1Þn sinhðb1 r SS ISS 1m Þ 5m ¼  b sinh r 1m x sin b dy ¼ :  n2 π 2 þ ðb1 rSS Þ2 ; n 4 0 1m 8 < 0; n ¼ 0 1 SS  Rb nπ y 2bnπ ð  1Þn sinhðb1 r SS ISS 3m Þ 7m ¼  b sinh r 3m x sin b dy ¼ :  n2 π 2 þ ðb1 rSS Þ2 ; n 4 0 3m ( 0; ma n Ra mπ x nπ x ISS 9m ¼  a cos a cos a dx ¼ a; m ¼ n

b

8 < 0; n ¼ 0   nπ y 2bnπ ð  1Þn sinhðb1 r SS sinh 1 r SS 2m Þ 2m x sin b dy ¼ :  ; n4 0 2 n2 π 2 þ ðb1 r SS 2m Þ 8 > < 0; ma n Rb mπ y nπ y b; m ¼ n a 0 I SS 8m ¼  b sin b sin b dy ¼ > : 0; m ¼ n ¼ 0 8 sinhða2 r SS 1m Þ > > ; n¼0 < 2 rSS   R 1m a 2 SS nπ x ¼ cosh r x cos dx ¼ I SS 2 2 SS 10m 1m 2a r 1m ð  1Þn sinhða2 r SS a a 1m Þ > ; n40 > 2 : n2 π 2 þ ða2 r SS 1m Þ 8 2 SS sinh a r ð Þ > 3m > ; n¼0 < 2 rSS 2 SS  Ra 3m nπ x I SS 2 2 r SS ð  1Þn sinh a2 r SS 12m ¼  a cosh r 3m x cos a dx ¼ 2a ð 3m Þ 3m > ; n40 > : SS 2 n2 π 2 þ ða2 r 3m Þ 8 < 0; n ¼ 0 2 SS  Ra nπ x 2anπ ð  1Þn sinhða2 r SS I SS 1m Þ 14m ¼  a sinh r 1m x sin a dx ¼ :  n2 π 2 þ ða2 rSS Þ2 ; n 40 I SS 6m ¼

8 sinhða2 r SS > 2m Þ > ; n¼0 < 2 rSS  2m SS SS nπ x I11m ¼  a cosh r2m x cos a dx ¼ 2a2 2 rSS ð  1Þn sinhða2 rSS Þ 2m 2m > ; n 40 > : SS 2 n2 π 2 þ ða2 r 2m Þ 8 > < 0; ma n Ra mπ x nπ x a; m ¼ na 0 ISS 13m ¼  a sin a sin a dx ¼ > : 0; m ¼ n ¼ 0 8 < 0; n ¼ 0 2 SS  Ra nπ x 2anπ ð  1Þn sinhða2 r SS ¼ sinh r x sin dx ¼ ISS 2m Þ 15m 2m a a :  n2 π 2 þ ða2 rSS Þ2 ; n 4 0 2m Ra

Rb

I SS 2m

2

Rb

b

1m

I SS 16m ¼

Ra a

  nπ x sinh 2 r SS 3m x sin a dx ¼

A.2. SA contribution

^ SA ðx; yÞ ¼ w

M X

1

W SA m ðyÞ cos

m¼0 M X

mπ x a

þ

ϕSA ym ðyÞ sin

mπ x a

þ

mπ x 1 SA ψ^ SA ðx; yÞ ¼ ψ m ðyÞ sin a m¼1

þ

ϕ^ x ðx; yÞ ¼

m¼0 SA

ϕ^ y ðx; yÞ ¼

M X m¼1 M X

1

1

:

2anπ ð  1Þn sinhða2 r SS 3m Þ n2 π 2 þ ða2 r SS 3m Þ

2

; n 40

M mπ x X ð2m  1Þπ y 2 þ W SA m ðxÞ sin a 2b m¼1

ϕSA xm ðyÞ cos

SA

8 < 0; n ¼ 0

M X

2

ϕSA xm ðxÞ cos

ð2m  1Þπ y 2b

2

ϕSA ym ðxÞ sin

ð2m 1Þπ y 2b

2

ψ SA m ðxÞ cos

ð2m 1Þπ y 2b

m¼1 M X m¼1 M X m¼1

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

N. Kolarevic et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 2

SA

SA

SA

21 SA

W m ðxÞ,2 ϕxm ðxÞ, 1 ϕxm ðyÞ, 1 ψ m ðyÞ are even functions, while 1 W m ðyÞ, 1 ϕym ðyÞ,2 ϕym ðxÞ, and 2 ψ m ðxÞ are odd functions: SA

SA

1

2 X 1

W SA m ðyÞ ¼

  Aim sinh 1 r im y ;

i¼1 1

1

ψ

ϕSA xm ðyÞ ¼

1

ϕSA ym ðyÞ ¼

1 SA m ðyÞ ¼ A3m 3 X 1

2



2

i¼1

δim 1 Aim

2 X 2

  Aim cosh 2 r im x

i¼1 2

γ im 1 Aim cosh 1 rim y ;

3 X 1

SA

W SA m ðxÞ ¼

  cosh 1 r 3m y ; 

SA

  sinh 1 r im y ;

2

i¼1

ψ

  sinh 2 r 3m x

2 SA m ðxÞ ¼ A3m 3 X 2

ϕSA xm ðxÞ ¼









γ im 2 Aim cosh 2 r im x

i¼1 3 X 2

ϕSA ym ðxÞ ¼

δim 2 Aim sinh 2 rim x

i¼1

Solution for the AS contribution can be obtained from the SA contribution by replacing x and y coordinates as well as ϕx and ϕy rotations in the above equations. A.3. AA contribution

^ AA ðx; yÞ ¼ w

M X

1

W AA m ðyÞ sin

m¼1 M X

AA

ϕ^ x ðx; yÞ ¼

1

ϕ xm ðyÞ sin

ð2m 1Þπ x 2a

þ

1

AA ym ðyÞ cos

ð2m  1Þπ x 2a

þ

ð2m  1Þπ x 2a

þ

m¼1 M X

AA

ϕ^ y ðx; yÞ ¼

m¼1 M X

ψ^ AA ðx; yÞ ¼

1

m¼1 1

AA

AA

AA

M ð2m  1Þπ x X ð2m  1Þπ y 2 þ W AA m ðxÞ sin 2a 2b m¼1

AA

ϕ

AA

ψ m ðyÞ cos

M X

2

m¼1

AA

M X

m¼1

AA

2

ϕ ym ðxÞ sin

2

ψ m ðxÞ cos

m¼1 M X

AA

ϕ xm ðxÞ cos

AA

AA

AA

AA

2 X 2

ð2m  1Þπ y 2b ð2m  1Þπ y 2b ð2m 1Þπ y 2b

AA

AA

ϕ xm ðyÞ,2 ϕ ym ðxÞ, 1 ψ m ðyÞ and 2 ψ m ðxÞ are even functions, while 1 W m ðyÞ,2 W m ðxÞ, 2 ϕ xm ðxÞ and 1 ϕ ym ðyÞ are odd functions: 1

AA

W m ðyÞ ¼

2 X 1

  Aim sinh 1 r im y ;

2

W m ðxÞ ¼

  cosh 1 r 3m y ;

2

ψ

i¼1 1

1

1

ψ

1 AA m ðyÞ ¼ A3m

AA

ϕ xm ðyÞ ¼ AA

ϕ ym ðyÞ ¼

3 X





1

γ im 1 Aim cosh 1 rim y ;

1

  sinh 1 r im y ;

i¼1

2

i¼1 3 X

δim 1 Aim

i¼1

2

2 AA m ðxÞ ¼ A3m AA

3 X

AA

3 X

ϕ xm ðxÞ ¼

ϕ y m ðx Þ ¼

  Aim sinh 2 r im x

  cosh 2 r 3m x 2









γ im 2 Aim sinh 2 rim x

i¼1 2

δim 2 Aim cosh 2 r im x

i¼1

A.4. Transformation matrix T 2 2

3

TSS 6 SA 7 6T 7 7 ; T¼6 6 TAS 7 4 5 TAA ð24M þ 8Þð24M þ 8Þ 2 6 6 6 6 6 SA T ¼6 6 6 6 4

6 6 6 6 6 SS T ¼6 6 6 6 4

3

tSS 0

7 7 7 7 7 7 7 7 7 5

tSS ⋱ tSS ⋱ tSS 3

tSA 0

ð6M þ 4Þð24M þ 8Þ

7 7 7 7 7 7 7 7 7 5

tSA ⋱ tSA ⋱ tSA

ð6M þ 2Þð24M þ 8Þ

Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i

N. Kolarevic et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

22

2 6 6 6 AA T ¼6 6 6 4

3

tAA

7 7 7 7 7 7 5

⋱ tAA ⋱ tAA

2

1 60 6 tSS 0 ¼6 40

ð6M Þð24M þ 8Þ

0 1

0 0

0 0

1 0

0 1

0 0

0 0

0

1

0

0

0

1

0

3 7 7 7 5

0 0 0 1 0 0 0  1 48  0 0 1 0 0 0 1 0 tSA 0 ¼ 0 0 0 1 0 0 0 1 28

2

3

1 60 6 6 60 tSS ¼ 6 60 6 6 40

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

07 7 7 07 7 07 7 7 05

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

2

1

6x24

3

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

60 6 6 60 SA t ¼6 60 6 6 40

0 0

0 0

1 0

0 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

0 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

2

0

0

0

0

0

0

0

0

0

0

0

1

0

7 7 7 7 7 7 7 7 5 624

3

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

60 6 6 60 AA t ¼6 60 6 6 40

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

07 7 7 07 7 07 7 7 05

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

624

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N. Kolarevic et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Please cite this article as: N. Kolarevic, et al., Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.06.031i