Solution of thin rectangular plate vibrations for all combinations of boundary conditions

Accepted Manuscript Solution of thin rectangular plate vibrations for all combinations of boundary conditions M. Eisenberger, A. Deutsch PII:

S0022-460X(19)30196-8

DOI:

https://doi.org/10.1016/j.jsv.2019.03.024

Reference:

YJSVI 14696

To appear in:

Journal of Sound and Vibration

Received Date: 14 November 2018 Revised Date:

17 March 2019

Accepted Date: 29 March 2019

Please cite this article as: M. Eisenberger, A. Deutsch, Solution of thin rectangular plate vibrations for all combinations of boundary conditions, Journal of Sound and Vibration (2019), doi: https:// doi.org/10.1016/j.jsv.2019.03.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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M. Eisenberger1 and A. Deutsch1,2

Faculty of Civil and Environmental Engineering, Technion - Israel Inst. of Technology, Haifa, Israel 2 CUBUS Engineering Software Israel, Raanana, Israel

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Solution of Thin Rectangular Plate Vibrations for all Combinations of Boundary Conditions

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Abstract

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1. Introduction

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The equation of motion for thin rectangular plate was formulated in its correct form about 200 years ago. Exact solution, for some cases, and high accuracy solution for few more cases, have been obtained along the years since. For several other cases, high accuracy solutions were composed of separate solutions for symmetric and anti-symmetric cases. In this work a new high accuracy solution, that covers all the possible combinations is presented. It is obtained by using carefully chosen series that solve the partial differential equations of motion, for all possible combinations of edge conditions. Examples of the new solutions are given and compared with approximate solutions for these cases. Keywords: Thin Plates; Vibrations; Natural Frequencies; Analytical Solution

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At the end of the 18th century Chladni [1] experimentally demonstrated the vibrations modes of plates. It took around 20 more years for the French mathematician, Sophie Germain (published few years later [2]), to derive the equation of motion for plates. For rectangular plates one of the following four classical edge conditions are possible: C - clamped, where the deflection and slope are zero, S - Simply supported, where the deflection and bending moment are zero, F - free where the bending moment and shear are zero, and G - guided, where the slope and the shear are zero. For square plates the number of combinations for the first 3 (i.e. C, S, and F) is 21, and including also the G cases of edge condition it adds up to 55 cases. The first solution was obtained for a plate which is simply supported along all 4 edges Preprint submitted to Elsevier

March 29, 2019

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by Navier, around 1820 [3], and is known by this name. The next step, was the solution method by Levy in 1899 [4], where 5 more cases were solved analytically. These are the cases with two opposite edges simply supported, and the other two have: SS conditions (back to Navier’s solution), CC, CS, CF, SF, and FF. All these are known as Levy’s solutions. Around 50 years ago, the cases with G type condition were studied. Bert and Malik [5] listed and ordered the 55 cases for a square plate, and the same order was preserved by other researchers and also in the current work. Out of the 55 cases, analytical and exact solutions (in terms of infinite series) are available only for 27 cases as follows: cases 1-6 (Navier and Levy), 22-25 (two opposite edges S, and G condition on one or two of the remaining edges), 26-33 ( S edge condition on one edge and G condition on the opposite edge, and the remaining 2 can be any of the 4 possibilities), and 34-42 ( two opposite edges with G conditions). Thus, the remaining 28 cases (cases 7-21 and 43-55) were not solved analytically until the last few years. In Table 1 we list all the cases and group them as above. The cases for which there are exact solutions are marked (E), and those with approximate or numerical solutions only are marked (A/N). The vibrations of plates were the subject of hundreds of research publications. In 1969 Leissa [6] published a very comprehensive account of the research on the topic up the that time. Since then, many more were added and only very few of these will be listed below, especially those that presented exact solutions. Analytical solutions were given fully by Leissa [6,7] for cases 1-6, and by Bert and Malik [5] for all the other 21 cases with exact solutions cases 26-42. The next significant step was the method of superposition that was developed by Gorman [8,9]. In his method Gorman divided the solution into 4 parts and dealt with each separately: plates that deform in a symmetric mode with respect to the two local axes through the center of the plate, plates with antisymmetric modes for these two local axis, and the two remaining combinations of symmetric-antisymmetric and antisymmetric-symmetric modes of deformation. For each of these four sub-problems Gorman carefully chose a set of ”building blocks”, that were superimposed to form the needed solution, which satisfies the plate domain and all the boundary conditions. Later on it was generalized and termed the superposition method [10]. In all, the solutions for many of the cases number 7 to 21 were presented by Gorman. Recently, Lim et. al. [11] used simplectic elasticity approach to solve 2

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34-42 (E) 43-55 (A/N) 34 - CGSG 43 - CCCG 35 - CGCG 44 - CCSG 36 - SGGG 45 - CCGF 37 - CGGG 46 - CCGG 38 - SGFG 47 - CGCF 39 - CGFG 48 - CGSF 40 - GGFG 49 - CSGF 41 - FGFG 50 - CGGF 42 - GGGG 51 - CFGF 52 - SGFF 53 - CGFF 54 - GGFF 55 - GFFF

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26-33 (E) 26 - CSCG 27 - CSSG 28 - SSGG 29 - CSGG 30 - SSGF 31 - SCGF 32 - SGGF 33 - SFGF

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(E) 7-21 (A/N) 22-25 (E) 1 - SSSS 7 - CCCC 22 - SSSG 2 - SSSC 8 - CCCS 23 - SCSG 3 - SSSF 9 - CCCF 24 - SGSF 4 - SCSC 10 - CSCF 25 - SGSG 5 - SCSF 11 - CFCF 6 - SFSF 12 - CFSF 13 - CCSS 14 - CCSF 15 - CSSF 16 - SSFF 17 - CSFF 18 - CCFF 19 - SFFF 20 - CFFF 21 - FFFF

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Table 1: 55 Possible Cases of Edge Conditions

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exactly for the Navier and Levy type boundary cases (cases 1-6), and Li et. al. [12] used the same method to solve for the FFFF plate (case 21), and in [13] for CFFF plate (case 18). The same approach can be extended to the other cases, but has not been done yet. Three years ago, Eisenberger and Deutsch [14] presented the exact solution for cases 7, 8, and 10. In it, the solution for the vibration frequencies was taken as the static solution for a plate with ”negative stiffness” elastic foundation, and the values of the natural frequencies were found as the negative stiffness constant that will cause the plate to loose its stiffness, and yield infinite deflections due to zero loading. Banerjee and co-workers [15-17] derived the spectral dynamic stiffness matrix that enables to obtain the exact vibration frequencies for plates. This solution is also broken into the four sub-cases of symmetry and anti-symmetry as in Gorman’s solutions, and is achieved through lengthy symbolic computations. They presented the solution for cases 4, 7, 8, and 21. In this paper, we present a solution that satisfies the equation of motion 3

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in the domain of the plate, for all the 55 possible edge conditions, exactly. In the next section the equation of motion and the internal moments and shear forces are given. In section 3 the solution is derived as a combination of trigonometric functions multiplied by hyperbolic functions. Then, the boundary conditions are applied to find the complete solution. Numerical results are given from the new solution and compared with available known results in section 4. The paper ends with discussion of the novelty of the solution and its extensions. 2. Equation of Motion

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The equation of motion for thin plates is   4 ∂ 4 w0 ∂ 4 w0 ∂ 2 w0 ∂ w0 + 2 + + ρh = 0, D ∂x4 ∂x2 ∂y 2 ∂y 4 ∂t2

(1)

where w0 (x, y, t) are the out of plane deflections of the plate, D = Eh3 /12(1− ν 2 ), is the flexural stiffness of the plate, ρ is the mass density of the material, h is the plate thickness. Assuming free harmonic vibrations, we take

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w0 (x, y, t) = w(x, y)sin(ωt) ,

where ω is the natural frequency of vibrations, and we obtain,   4 ∂ 4w ∂ 4w ∂ w +2 2 2 + − ρhω 2 w = 0 . D 4 4 ∂x ∂x ∂y ∂y

(2)

(3)

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For the solution we shall need also the boundary condition along the edges of the plate. In Kirchoff’s thin plate theory we have for the slope, bending moment, and shear force (with the contribution of torsional moments [18]), along the edges parallel to the y and x axes, respectively, the following expressions [6]: ∂w(x, y) , (4) Φx = ∂x ∂w(x, y) Φy = , (5) ∂y  2  ∂ w(x, y) ∂ 2 w(x, y) Mx = −D +ν , (6) ∂x2 ∂y 2 4

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My = −D  Qx = −D  Qy = −D

∂ 2 w(x, y) ∂ 2 w(x, y) + ν ∂y 2 ∂x2

 ,

(7)

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∂ 3 w(x, y) ∂ 3 w(x, y) + (2 − ν) ∂x3 ∂x∂y 2



∂ 3 w(x, y) ∂ 3 w(x, y) + (2 − ν) ∂y 3 ∂x2 ∂y



,

(8)

.

(9)

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These values are calculated along the edges where either x or y coordinate are constant.

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3. Solution

The solution for this partial differential equation is assumed as given below, and it has to satisfy the equilibrium equation all over the plate, w(x, y) =

∞ X

¯ m · Ym + X

∞ X

Xn · Y¯n .

(10)

n=0

m=0

(11) (12) (13) (14)

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¯ m and Y¯n are trigonometric functions The two functions X     ¯ m = sin (λa x) · sin mπ + cos(λa x) · cos mπ , X 2 2  nπ   nπ  Y¯n = sin (λb y) · sin + cos(λb y) · cos , 2 2 with mπ , λa = a nπ λb = . b The two functions Ym and Xn are hyperbolic functions

Ym = Am cosh (α1 y) + Bm cosh (α2 y) + Cm sinh (α1 y) + Dm sinh (α2 y) (15) Xn = En cosh (α3 x) + Fn cosh (α4 x) + Gn sinh (α3 x) + Hn sinh (α4 x) (16)

with

s α1 =

r

λ2a −

5

ρhω 2 , D

(17)

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α2 =

r

λ2a + s

α3 =

r

λ2b − s λ2b +

ρhω 2 , D ρhω 2 . D

(18)

(19)

(20)

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α4 =

r

ρhω 2 , D

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s

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3.1. The Boundary Conditions The solution that we have so far is dependent on 8 unknown constants for any value of m and n in the summation in Eqs. (15-16). These are Am , Bm , Cm , Dm , En , Fn , Gn , and Hn . The plate has 4 edges and on each of them we can prescribe 2 quantities, making for the 8 unknowns. It is convenient to number the edges in the counterclockwise direction, as given below in Table 2 for all four edges, and from every row, i.e. edge, only two relations can be utilized, depending on the particular edge restraints.

Deflection w(− a2 , y) w(x, − 2b ) w( a2 , y) w(x, 2b )

Rotation Φx (− a2 , y) Φy (x, − 2b ) Φx ( a2 , y) Φy (x, 2b )

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Edge 1 2 3 4

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Table 2: Possible Edge Conditions

Moment Mx (− a2 , y) My (x, − 2b ) Mx ( a2 , y) My (x, 2b )

Reaction Qx (− a2 , y) Qy (x, − 2b ) Qx ( a2 , y) Qy (x, 2b )

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The expanded expressions for the deflection, slope, bending moment, and shear that are obtained by substitution of the solution Eq. (10) into Eqs. (4 - 9), and are given in Appendix A. As an example, we derive the equation that will be obtained when we apply zero deflection along edge 2. After substitution of y = −b/2 in Eq. (10) for the deflection along the edge, we have

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b w x, − 2

 =

∞ X

¯m × X

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m=0

2

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h α2 b α1 b ) + Bm cosh( ) Am cosh( 2 2 α1 b α2 b i −Cm sinh( ) − Dm sinh( ) 2 2 ∞ X + Y¯n | y=− b ×

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n=0 En cosh(α3 x) + Fn cosh(α4 x)  +Gn sinh(α3 x) + Hn sinh(α4 x) .

(21)

In the second summation we have the hyperbolic functions which are dependent on x. Using the expansions in Appendix B, we can write 

b w x, − 2

 =

∞ X

¯m × X

m=0

α1 b α2 b ) + Bm cosh( ) 2 2 α1 b α2 b i −Cm sinh( ) − Dm sinh( ) 2 2 ∞ X + cos(nπ) × Am cosh(

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h

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n=0

h

En

∞ X

¯ m + Fn c3 X

m=0 ∞ X

+Gn

m=0

7

∞ X

¯m c4 X

m=0 ∞ X

¯ m + Hn s3 X

m=0

i ¯ s4 Xm ,

(22)

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or,

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  ∞ X b ¯m × = w x, − X 2 m=0 h α2 b α1 b ) + Bm cosh( ) Am cosh( 2 2 α1 b α2 b i −Cm sinh( ) − Dm sinh( ) 2 2 ∞ X ∞ X ¯m × + cos(nπ) X n=0 m=0

(23)

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[En c3 + Fn c4 + Gn s3 + Hn s4 ] ,

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and finally,   ∞ X b ¯m × X w x, − = 2 m=0 h α1 b α2 b Am cosh( ) + Bm cosh( ) 2 2 α1 b α2 b −Cm sinh( ) − Dm sinh( ) 2 2 ∞ i X + cos(nπ) (En c3 + Fn c4 + Gn s3 + Hn s4 ) .

(24)

n=0

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¯ m is dependent on x, the edge deflection can be zero only if we Then, as X have h α2 b α1 b 0 = Am cosh( ) + Bm cosh( ) 2 2 α1 b α2 b ) − Dm sinh( ) − Cm sinh( 2 2 ∞ X
i + cos(nπ) En c3 + Fn c4 + Gn s3 + Hn s4 . (25) n=0

In this equation we have all the 8 unknowns for a given value of m and n. In a similar way we can derive the equations for the remaining 7 edge conditions. We obtain that the number of equations is (4 ∗ m + 4 ∗ n) for all combinations of boundary conditions. The vibration frequencies of the plate 8

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are found when the determinant of the coefficient matrix of the system of equation is equal 0. This was done by a simple search to identify sign change of the determinant, and then by halving the interval until the difference is less than a preset criteria (in this case the number of digits as shown in the results). For repeated eigenvalues the determinant is very close to 0, and one can identify these by checking that it is almost zero, but still does not change sign. The vibration mode shape is then calculated as the deflection of the plate w(x, y) for the particular frequency. 4. Numerical Results

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The exact vibration frequencies and mode shapes for three cases are presented here. These are cases 44 - CCSG, 48 - CGSF, and 49 - CSGF. All these three are cases with no known exact solutions. The example plates are square. For each case the first ten normalized frequencies, ω ˆ = ω (ρ h/D)1/2 , are given in Tables 3-5. The values are compared with the highly accurate approximate Rayleigh-Ritz results from [19], and the results from FE analysis from the commercial code ANSYS, using 100X100 element grid. For the first six frequencies, the vibration mode shapes are presented in Figures 1 and 2. In Figure 3 the convergence as a function of the number of terms in the solution for m=n is shown for all the 3 cases (normalized by the values for 100 terms). It is seen that using more then 30 terms will yield errors of less then 0.01% for all cases. The results in Tables 3-5, for cases 44, 48, and 49, respectively, were calculated using 100 terms and they are exact for the number of digits that are given in the Tables. In Table 6 we present the results for case 21 - a completely free plate (FFFF). The first 3 zero frequencies representing rigid body motion are not shown. This case was solved by Gorman [9] and these results are given together with the values from finite element analysis. It can be seen that Gorman’s results are the same (rounded to 4 digits), and all the approximate FE values are higher then the exact values. 5. Concluding remarks The exact analytical solution to the problem of thin plate vibrations with all the possible combinations of boundary conditions is given. In this solution there is no need to separate symmetric and anti-symmetric cases and all are obtained using just one expression (Eq. (10). The choice of separation of variables in the solution enables to represent the edge conditions 9

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Table 3: Case 44 - CCSG - The First 10 Normalized Vibration Frequencies

18.348461 41.250101 52.630459 74.083998 85.145116 106.838155 116.814183 127.658174 149.089356 169.426667

18.349 41.251 52.632 74.086 85.147 106.843 -

FEM - ANSYS 9,395 D.O.F. 18.351299 41.261050 52.661952 74.115197 85.211931 106.976256 116.886725 127.785538 149.326810 169.574375

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1 2 3 4 5 6 7 8 9 10

Current Value R-R Method [19]

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Mode

Mode

Current Value R-R Method [19] 15.292358 21.896030 45.056231 49.681081 57.395415 82.025000 88.084160 103.789272 111.946517 124.964766

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1 2 3 4 5 6 7 8 9 10

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Table 4: Case 48 - CGSF - The First 10 Normalized Vibration Frequencies

15.293 21.897 45.058 49.684 57.400 82.031 -

10

FEM - ANSYS 9,595 D.O.F. 15.295723 21.899100 45.066524 49.714196 57.426177 82.057772 88.143037 103.930797 112.081973 125.037901

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Table 5: Case 49 - CSGF - The First 10 Normalized Vibration Frequencies

6.600454 19.952949 31.674627 47.031067 53.631309 75.997293 80.077205 92.710118 107.538142 126.887296

6.601 19.954 31.677 47.034 53.632 76.003 -

FEM - ANSYS 9,595 D.O.F. 6.600863 19.955145 31.686669 47.042460 53.657334 76.069268 80.107471 92.775629 107.659239 126.962440

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Current Value R-R Method [19]

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Mode

Mode

Current Value Superposition [9] 3.292143 4.805964 6.105709 8.558151 15.233045 15.689330 17.032250 19.242576 26.050581 29.105547

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1 2 3 4 5 6 7 8 9 10

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Table 6: Case 21 - FFFF - The First 10 Normalized Vibration Frequencies

3.292 4.806 6.106 8.558 15.23 15.69 17.03 19.24 26.05 29.11

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FEM - ANSYS 10,198 D.O.F. 3.292268 4.806526 6.106885 8.559230 15.241903 15.692413 17.038837 19.253333 26.059511 29.140262

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12 3

2

1

Mode Number

52.630459

41.250101

45.056231

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31.674627

19.952949

6.600454

Case 49 - CSGF

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21.896030

15.292358

18.348461

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Case 48 - CGSF

Case 44 - CCSG

Figure 1: Normalized Vibration Modes 1-3 for the Example Cases

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13 6

5

4

Mode Number

106.838155

74.083998

82.025000

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75.997293

53.631309

47.031067

Case 49 - CSGF

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57.395415

49.681081

74.083998

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Case 48 - CGSF

Case 44 - CCSG

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Figure 2: Normalized Vibration Modes 4-6 for the Example Cases - continued

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Figure 3: Convergence of the First 6 Normalized Vibration Modes for the Example Cases

44 - CCSG 0.0001

RELATIVE ERROR [%]

0

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-0.0001 -0.0002 -0.0003

-0.0005 0

20

ω1

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-0.0004

ω2

40 60 NUMBER OF TERMS

ω2

ω4

80

ω5

100

ω6

48 - CGSF

-0.01 -0.03 -0.05 -0.07

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RELATIVE ERROR [%]

0.01

-0.09 -0.11

0

20

ω2

AC C

RELATIVE ERROR [%]

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ω1

40 60 NUMBER OF TERMS ω2

ω4

80

ω5

100

ω6

49 - CSGF

0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07

0

20

ω1

ω2

40 60 NUMBER OF TERMS ω2

14

ω4

80

ω5

ω6

100

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for deflection, slope, shear forces, and bending moments in terms of common functions (as in Eq. (24)). Then when we apply the 8 particular boundary condition on the edges, we obtain the set of equations that we need in order to determine all the terms in the complete solution. This solution for all types of boundary conditions was possible as we were able to expand the hyperbolic functions into trigonometric series. The same idea can be used also in many other problems.

W (x, y) =

∞ X

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Deflection, Slopes, Bending Moments, and Shears

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Appendix A

¯m × X

m=0

Am cosh(α1 y) + Bm cosh(α2 y)
+Cm sinh(α1 y) + Dm sinh(α2 y) ∞ X + Y¯n × n=0

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En cosh(α3 x) + Fn cosh(α4 x)
+Gn sinh(α3 x) + Hn sinh(α4 x)

(A.26)

∂W (x, y) ∂x ∞  mπ  i  mπ  X h − sin(λa x) · cos × = λa cos (λa x) · sin 2 2 m=0

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φx =

AC C

(Am cosh(α1 y) + Bm cosh(α2 y) + Cm sinh(α1 y) + Dm sinh(α2 y)) ∞ X + Y¯n × n=0

[α3 En sinh(α3 x) + α4 Fn sinh(α4 x) + α3 Gn cosh(α3 x) +α4 Hn cosh(α4 x)]

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(A.27)

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φy

∞ ∂W (x, y) X ¯ = = Xm × (α1 Am sinh(α1 y) ∂y m=0

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+α2 Bm sinh(α2 y) + α1 Cm cosh(α1 y) + α2 Dm cosh(α2 y)) ∞  nπ  i h  nπ  X − sin(λb y) · cos × + λb cos (λb y) · sin 2 2 n=0 En cosh(α3 x) + Fn cosh(α4 x) + Gn sinh(α3 x)  +Hn sinh(α4 x) (A.28)  ∂ 2 w (x, y) ∂ 2 w (x, y) +ν ∂x2 ∂y 2 ∞ X ¯ m × Am cosh(α1 y) + Bm cosh(α2 y) = − λ2a X m=0

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Mx − = D


+Cm sinh(α1 y) + Dm sinh(α2 y) ∞ X Y¯n × α32 En cosh(α3 x) + α42 Fn cosh(α4 x) +
sinh(α3 x) + α42 Hn sinh(α4 x)

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n=0 +α32 Gn ∞ X

¯m × X

+ν

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 2 m=0 α1 Am cosh(α1 y) + α22 B m cosh(α2 y)  +α12 Cm sinh(α1 y) + α22 Dm sinh(α2 y) ∞ X  − νλ2b Y¯n × En cosh(α3 x) + Fn cosh(α4 x) n=0

AC C

 +Gn sinh(α3 x) + Hn sinh(α4 x)

16

(A.29)

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∂ 2 w (x, y) ∂ 2 w (x, y) = + ν ∂y 2 ∂x2 ∞ X ¯m × = X 



m=0

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My − D

h α12 Am cosh(α1 y) + α22 B m cosh(α2 y)

−

∞ X

sinh(α1 y) +

α22 Dm

sinh(α2 y)

i

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+α12 Cm

h λ2b Y¯n × En cosh(α3 x) + Fn cosh(α4 x)

n=0

−

∞ X

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i +Gn sinh(α3 x) + Hn sinh(α4 x) ¯m × νλ2a X

m=0 Am cosh(α1 y) + Bm cosh(α2 y)  +Cm sinh(α1 y) + Dm sinh(α2 y) ∞ X + ν Y¯n ×

AC C

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TE D

n=02 α3 En cosh(α3 x) + α42 Fn cosh(α4 x)  +α32 Gn sinh(α3 x) + α42 Hn sinh(α4 x)

17

(A.30)

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 ∂ 3 w (x, y) ∂ 3 w (x, y) + (2 − ν) ∂x3 ∂x∂y 2 ∞  mπ   mπ  i X h = λ3a − cos (λa x) · sin + sin(λa x) · cos × 2 2 m=0 

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Qx − = D

(Am cosh(α1 y) + Bm cosh(α2 y) + Cm sinh(α1 y) + Dm sinh(α2 y)) ∞ X + Y¯n × α33 En sinh(α3 x) + α43 Fn sinh(α4 x)

SC

n=0

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+α33 Gn cosh(α3 x) + α43 Hn cosh(α4 x) ∞ X  mπ   mπ  − sin(λa x) · cos ]× + (2 − ν) λa [ cos (λa x) · sin 2 2 m=0  2 α1 Am cosh(α1 y) + α22 B m cosh(α2 y)  +α12 Cm sinh(α1 y) + α22 Dm sinh(α2 y) ∞ X  − λ2b Y¯n × α3 En sinh(α3 x) + α4 Fn sinh(α4 x) n=0

AC C

EP

TE D

 +α3 Gn cosh(α3 x) + α4 Hn cosh(α4 x)

18

(A.31)

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 3  Qy ∂ w (x, y) ∂ 3 w (x, y) − = + (2 − ν) D ∂y 3 ∂x2 ∂y 3 ¯ m × α Am sinh(α1 y) + α3 B sinh(α2 y) = X 1 2 m
3 3 +α1 Cm cosh(α1 y) + α2 Dm cosh(α2 y) ∞  nπ   nπ   X  + λ3b − cos (λb y) · sin + sin(λb y) · cos × 2 2 n=0

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(En cosh(α3 x) + Fn cosh(α4 x) + Gn sinh(α3 x) + Hn sinh(α4 x)) ∞  X ¯m × + (2 − ν) − λ2a X

Appendix B

(A.32)

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m=0  α1 Am sinh(α1 y) + α2 Bm sinh(α2 y)  +α1 Cm cosh(α1 y) + α2 Dm cosh(α2 y) ∞  nπ   nπ   X  λb cos (λb y) · sin + − sin(λb y) · cos × 2 2 n=02 α3 En cosh(α3 x) + α42 Fn cosh(α4 x)  +α32 Gn sinh(α3 x) + α42 Hn sinh(α4 x)

Hyperbolic Functions Expansions

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In this appendix the expansion of Eqs. (15) and (16) are derived. Rewriting Xn

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Xn = En cosh (α3 x) + Fn cosh (α4 x) + Gn sinh (α3 x) + Hn sinh (α4 y) (B.1) We are interested in expanding the four cosh and sinh hyperbolic functions ¯ m . This is done as follows: in terms of the trigonometric functions in X cosh(αi x) =

∞ X

Am · sin

m=1,3,5...

 mπx  a

· sin

 mπ  2

∞  mπx   mπ  X B0 + + Bm · cos · cos , 2 a 2 m=2,4,6...

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(B.2)

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with

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Z a  mπ   mπx  2 2 · sin dx = 0, (B.3) Am = cosh(αi x) · sin a −a a 2 2 Z a  mπ   mπx  4 a αi sinh( α2i a ) 2 2 Bm = ·cos dx = cosh(αi x)·cos , (B.4) a −a a 2 m2 π 2 + αi2 a2 2

∞ X

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and utilizing trigonometric identities, we finally arrive at the following simpler expression  mπ  4 a α sinh( αi a ) i 2 ¯m, · ·X cosh(αi x) = Tm · cos 2 2 + α2 a2 2 m π i m=0,1,2...

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2

with



Tm =

(B.5)

1 2

if m = 0 1 otherwise.

Similarly we obtain for sinh(αi x) ∞ X

 mπ  4 a α cosh( αi a ) i 2 ¯m, · ·X Tm · sin sinh(αi x) = 2 2 2 2 2 m π + α a i m=0,1,2...

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and

2

cosh(αi y) =

∞ X

Tn · cos2

 nπ  4 b α sinh( αi b ) i 2 · 2 2 · Y¯n , 2 n π + αi2 b2

(B.7)

Tn · sin2

 nπ  4 b α cosh( αi b ) i 2 · 2 2 · Y¯n , 2 2 2 n π + αi b

(B.8)

n=0,1,2... ∞ X

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sinh(αi y) =

n=0,1,2...

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with

(B.6)

 Tn =

1 2

if n = 0 1 otherwise.

Here we can rewrite the expressions for the hyperbolic functions sinh and cosh in a compact form as sinh(α1 y) =

∞ X n=0

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s1 Y¯n ,

(B.9)

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sinh(α4 x) = cosh(α1 y) =

m=0 ∞ X n=0 ∞ X

¯m, s4 X

(B.11) (B.12)

c1 Y¯n ,

(B.13)

c2 Y¯n ,

(B.14)

¯m, c3 X

(B.15)

¯m, c4 X

(B.16)

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cosh(α2 y) =

m=0 ∞ X

¯m, s3 X

(B.10)

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sinh(α3 x) =

n=0 ∞ X

s2 Y¯n ,

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sinh(α2 y) =

∞ X

n=0

cosh(α3 x) = cosh(α4 x) =

∞ X

m=0 ∞ X m=0

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s1 =

sin2 ( nπ ) cosh( α21 b ) 2 4 b α1 Tn , n2 π 2 + α1 2 b2 sin2 ( nπ ) cosh( α22 b ) 2 4 b α2 Tn , n2 π 2 + α2 2 b2

s2 =

) cosh( α23 a ) sin2 ( mπ 2 , m2 π 2 + α3 2 a2 sin2 ( mπ ) cosh( α24 a ) 2 s4 = 4 a α4 Tm , m2 π 2 + α4 2 a2 cos2 ( nπ ) sinh( α21 b ) 2 c1 = 4 b α1 Tn , n2 π 2 + α1 2 b2 cos2 ( nπ ) sinh( α22 b ) 2 c2 = 4 b α2 Tn , n2 π 2 + α2 2 b2 cos2 ( mπ ) sinh( α23 a ) 2 c3 = 4 a α3 Tm , m2 π 2 + α3 2 a2 cos2 ( mπ ) sinh( α24 a ) 2 c4 = 4 a α4 Tm . m2 π 2 + α4 2 a2

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s3 = 4 a α3 Tm

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(B.17) (B.18) (B.19) (B.20) (B.21) (B.22) (B.23) (B.24)

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References

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[1] Chladni EFF. 1802 Die Akustik, Leipzig, Germany: Breitkopf und Hartel.

[2] Germain S. 1821 Recherches sur la theorie des surfaces elastiques. Paris.

[3] Navier LMH. 1819 Resume des Lecons de Mechanique,Ecole Polytechnique, Paris.

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[4] Levy M. 1899 Sur l’equlibre elastique d’une plaque rectangulaire. C. R. Acad. Sci. 129, 535-539.

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[5] Bert CW., Malik M. 1994 Frequency equations and modes of free vibrations of rectangular plates with various edge conditions. Proc. Inst. Mech. Eng., Part C: Jour. Mech. Engng. Sci. 208, 308-319. [6] Leissa, A. 1969 Vibration of plates. NASA.

[7] Leissa A. 1973 The free vibration of rectangular plates. J. Sound Vibr. 31, 257-293.

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[8] Gorman DJ. 1982 Free Vibration Analysis of Rectangular Plates. New York, NY: Elsevier. [9] Gorman DJ. 1978 Free vibration analysis of the completely free rectangular plate by the method of superposition. J. Sound Vibr. 57, 437-447.

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[10] Gorman DJ. 1999 Vibration Analysis of Plates by the Superposition Method. Singapore: World Scientific.

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[11] Lim CW, Lu CF. 2009 On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchoff plates. Int. Jour. Engng. Sci. 47, 131-140. [12] Li R., Wang B., Li G., Tian B. 2016, Hamiltonian system-based analytic modeling of the free rectangular thin plates free vibration. Appl. Math. Modelling, 40, 984-992.

[13] Li R., Wang P., Yang Z., Yang J., Tong L. 2018 On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space. Appl. Math. Modelling, 53, 310-318.

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[14] Eisenberger M, Deutsch A. 2015 Static Analysis for Exact Vibration Analysis of Clamped Plates. Int. Jour. Struct. Stab. Dynamics, 15, 1540030, (DOI: 10.1142/S0219455415400301).

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[15] Liu X., Banerjee JR. 2015 The spectral-dynamic stiffness method: A novel approach for exact free vibration analysis of plate-like structures. In Proc. 15th int. conf. on civil, structural and environmental engng. comp. (eds. J. Kruis, Y. Tsompanakis and BHV. Topping) Stirlingshire, Scotland, Civil-Comp Press.

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[16] Liu X., Banerjee JR. 2016 Free vibration analysis for plates with arbitrary boundary conditions using a novel spectral-dynamic stiffness method. Computers & Structures, 164, 108-126. [17] Banerjee JR., Papkov SO., Liu X., Kennedy D., 2015 Dynamic stiffness matrix of a rectangular plate for the general case. J. Sound Vibr. 342, 177-199 (doi.org/10.1016/j.jsv.2014.12.031). [18] Szilard, R. 2004 Theories and application of plate analysis. John Wiley &Sons, New York.

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[19] Ilanko S., Monterrubio LE. 2014 The Rayleigh-Ritz Method for Structural Analysis. Hoboken NJ: J. Wiley & Sons.

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