# Analytical Solution for Free Vibrations of a Moderately Thick Rectangular Plate

Authors: Ivo Senjanović; Marko Tomić; Nikola Vladimir; Dae Seung Cho
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

In the present thick plate vibration theory, governing equations of force-displacement relations and equilibrium of forces are reduced to the system of three partial differential equations of motion with total deflection, which consists of bending and shear contribution, and angles of rotation as the basic unknown functions. The system is starting one for the application of any analytical or numerical method. Most of the analytical methods deal with those three equations, some of them with two (total and bending deflection), and recently a solution based on one equation related to total deflection has been proposed. In this paper, a system of three equations is reduced to one equation with bending deflection acting as a potential function. Method of separation of variables is applied and analytical solution of differential equation is obtained in closed form. Any combination of boundary conditions can be considered. However, the exact solution of boundary value problem is achieved for a plate with two opposite simply supported edges, while for mixed boundary conditions, an approximate solution is derived. Numerical results of illustrative examples are compared with those known in the literature, and very good agreement is achieved.


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## Body

## 1. Introduction

Rectangular plate is an important structural element in many modern engineering structures. Spectrum of plate natural frequencies as a continuum is very dense, and its response to an excitation can easily fall in resonance. Therefore, knowledge about plate dynamic behaviour is very important to structural designers.Theory of thin plate vibrations is very well developed, and the achievements are collected in the fundamental Szilard book [1]. Dynamics of thick plate is a more complex problem, due to the effect of shear stiffness and rotary inertia, and is still a subject of investigation. The first works on this subject are those of Reissner [2] and Mindlin [3]. They assumed that plate cross-section remains in a straight line but not normal to the plate middle surface due to shear release. Accordingly, constant shear stress distribution through the plate thickness is considered, and concept of shear correction factor is introduced in order to achieve an effective value of shear stiffness. As a result of the introduced assumptions, application of the Reissner-Mindlin theory is limited to the thick plate of moderate thickness and is known as the first-order shear deformation theory.Natural vibrations of thick plate have been a challenging problem to many researchers during the past decades. A large number of articles have been published in the meantime and a comprehensive survey of the literature up to 1994 can be found in [4]. Generally speaking, there are two main approaches for the solution of thick plate natural vibrations, that is, analytical methods for solution of differential equations of motion and numerical procedures based on the Rayleigh-Ritz energy method and FEM. In the former approach system of the governing equations of internal forces, inertia forces, displacements, and deformations is reduced to three equilibrium equations expressed with plate deflection and angles of rotations. Application of analytical methods is limited to simply supported plate at two opposite edges. Actually, total plate deflection consists of bending deflection and shear relaxation, and angles of rotation are dependent only on the bending. Analytical methods are different depending on which functions are kept as fundamental ones in further reduction of differential equations of motion. Hence, it is possible to operate with three, two, or even one function, that is, total deflection and two angles of rotation, total deflection and bending deflection, and only with total deflection, respectively.Among a large number of articles, it is worthy to mention the Wang paper [5] in which an explicit formula for natural frequencies of simply supported thick plate in the form of that for thin plate is derived. Brunelle offers a procedure for determination of natural frequencies for a simply supported plate on two opposite edges [6]. In the Xiang papers, the well-known Levy approach in thin plate theory for a simply supported plate on two opposite edges is employed to the thick plate vibration analysis [7, 8]. The semianalytical solution for natural vibration is proposed by Malekzadeh et al. [9]. The same problem is considered in the Endo and Kimura article with total deflection and bending deflection as two fundamental functions [10]. A refined plate theory with two functions, that is, bending deflection and shear deflection, is proposed by Shimpi and Patel, and it is applied only for simply supported plate at all edges [11]. The rigorous analytical solution for case of simply supported edges on two opposite edges is given in the paper of Hashemi and Arsanjani [12]. An interesting analytical solution for any combination of simply supported and clamped edges is presented by Xing and Liu [13]. It is developed under assumption that angles of rotation in vertical longitudinal plane vary in transverse direction proportionally to the total deflection, instead of actual bending deflection, and vice versa. In spite of that approximation, very good results are achieved. Xing and Liu proposed also a new variant of their plate theory in which three classical differential equations are reformulated to arrive at two equations with total deflection and generalized curvature as the basic functions [14].Since application of the analytical methods can be applied only to some combinations of boundary conditions, numerical methods are used for more complex problems of mixed boundary conditions, as well as for elastically supported edges. Mostly, the Rayleigh-Ritz method is applied, and accuracy of results depends on the chosen set of orthogonal functions for assumed natural modes. In the paper of Liew et al., two-dimensional polynomials are used for natural modes [15]. The static Timoshenko beam functions are very often applied for plate longitudinal and transverse directions, for instance, Dawe and Roufaeil [16] and Cheung and Zhou [17]. Recently, vibration problems of elastically restrained edges are in the focus [18–20]. As it is well known, the finite element method is a very powerful tool for structural analysis, and several finite elements for the Mindlin plate have been developed and incorporated in commercial FEM software. A recent formulation of a new finite element for a thick plate is proposed by Falsone and Settineri [21].In order to extend applicability of analytical methods for the solution of natural vibrations of a thick plate for any combination of boundary conditions, in the present paper, an advanced moderately thick plate theory is presented. The single differential equation of motion is derived with bending deflection as the basic function. The method of separation of variables is used, and a rigorous solution is obtained. Application of the method is illustrated in the case of different boundary conditions, and results are compared with those known in the relevant literature.


## 2. Basic Equations

A thick rectangular plate is considered in the Cartesian coordinate system with corresponding displacements, that is, deflectionw and angles of rotation in longitudinal and transverse vertical plane, like in the Timoshenko beam theory [22], Figure 1. Deflection of thick plate can be decomposed into bending deflection and shear deflection, according to the beam theory [23] and plate theory [24]
(1)w(x,y,t)=wb(x,y,t)+ws(x,y,t).
Angles of rotation of plate cross-sections in the lower frequency domain are predominantly caused by bending deflection:
(2)ψx=-∂wb∂x,ψy=-∂wb∂y,
while shear deflection makes contribution only to rotation of plate generatrix. Hence, bending moments and twist moments are functions of plate bending curvature:(3a)Mx=-D(∂2wb∂x2+ν∂2wb∂y2),(3b)My=-D(∂2wb∂y2+ν∂2wb∂x2),(3c)Mxy=Myx=-(1-ν)D∂2wb∂x∂y,where(4)D=Eh312(1-ν2)
is plate flexural rigidity, and h, E, and ν are plate thickness, Young’s modulus of elasticity, and Poisson’s ratio, respectively. The shear forces are results of shear deformations:
(5)Qx=S∂ws∂x,Qy=S∂ws∂y,
where S=kGh is shear rigidity and k is shear coefficient.Figure 1
Displacements of rectangular plate.The vibrating plate is loaded with normal inertia load and distributed bending moments:(6)q=m-∂2w∂t2,mx=-J∂3wb∂x∂t2,my=-J∂3wb∂y∂t2,
where m-=ρh is the plate mass per unit area and J=ρI=ρ(h3/12) is the mass moment of inertia of cross-section per unit breadth.


## 3. Differential Equations of Motion

In order to derive the differential equation of motion, let us consider the equilibrium of moments and forces:(7)∂Mx∂x+∂Mxy∂y=Qx+mx,∂My∂y+∂Myx∂y=Qy+my,∂Qx∂x+∂Qy∂y=-q.
Substituting expressions for forces and loads, (3a), (3b), (3c), (5), and (6), respectively, into (7) and taking (1) into account, after some rearrangement, yield
(8)∂ws∂x=-DS(∂3wb∂x3+∂3wb∂x∂y2)+JS∂3wb∂x∂t2,(9)∂ws∂y=-DS(∂3wb∂y3+∂3wb∂x2∂y)+JS∂3wb∂y∂t2,(10)∂2ws∂x2+∂2ws∂y2-m-S∂2wb∂t2-m-S∂2ws∂t2=0.
Equations (8), (9), and (10) represent the system of three partial differential equations with two unknowns, that is, wb and ws. However, (8) and (9) can be reduced to one equation by integrating the former per x and the latter per y:
(11)ws=-DS(∂2wb∂x2+∂2wb∂y2)+JS∂2wb∂t2+fy(y,t),(12)ws=-DS(∂2wb∂y2+∂2wb∂x2)+JS∂2wb∂t2+fx(x,t).
It is obvious that the undetermined integration functions have to be the same, that is, fy(y,t)=fx(x,t)=f(t), where f(t) represents rigid body translation. Furthermore, by substituting (8), (9), and (11) or (12) with f(t) into (10), one differential equation of motion is obtained with the unknown bending deflection:
(13)DΔΔwb-J(1+m-DJS)∂2∂t2Δwb+m-∂2∂t2(wb+JS∂2wb∂t2)=-m-d2f(t)dt2,
where Δ(·)=∂2(·)/∂x2+∂2(·)/∂y2 is the Laplace differential operator. Once wb is determined, the total deflection according to (1) and (11) yields
(14)w=wb+JS∂2wb∂t2-DS(∂2wb∂x2+∂2wb∂y2)+f(t).
Referring to (13), bending deflection consists of the homogenous solution and a particular integral due to the disturbing function f(t). Putting wp=wb(t) one finds from (13),
(15)wb(t)+JSd2wb(t)dt2=-f(t).
Furthermore, by substituting (15) into (14), the disturbing function f(t) is cancelled.Since it does not appear in the expression for the total deflection (14) and derivatives of the bending deflection, by which angles of rotation and sectional forces are determined, one can get f(t)=0. Hence, differential equation (13) becomes that homogenous one.


## 4. Boundary Conditions

The following boundary conditions are considered at the transverse edge atx=a.Simply supported edge:(16)(w)x=a=0,(Mx)x=a=-D(∂2wb∂x2+ν∂2wb∂y2)x=a=0.Fixed edge:(17)(w)x=a=0,(ψx)x=a=-(∂wb∂x)x=a=0.Free edge:(18)(Mx)x=a=-D(∂2wb∂x2+ν∂2wb∂y2)x=a=0,(Q-x)x=a=(Qx+∂Mxy∂y)x=a=0,
where Q-x is the shearing force comprising ordinary transverse shear force and the effect of torsional moment as in the thin plate theory [1]. According to (3c), (5), and (8), the shearing force expressed with bending deflection reads
(19)Q-x=-D[∂3wb∂x3+(2-ν)∂3wb∂x∂y2]+J∂3wb∂x∂t2.
Boundary conditions for longitudinal edge at y=b can be specified in an analogous way.


## 5. Differential Equation of Natural Vibrations

Natural vibrations are assumed to be harmonic, and therefore,(20)wb(x,y,t)=Wb(x,y)sinωt,
where ω is the natural frequency and Wb is the mode amplitude. In this case, differential equation of motion (13) is reduced to the frequency domain. At the same time, the second term in (13) can be expressed in a more appropriate way:
(21)1+m-DJS=1+2(1-ν)k,
so that the final form of (13) reads
(22)DΔΔWb+ω2J(1+2(1-ν)k)ΔWb+ω2m-(ω2JS-1)Wb=0.
Amplitude of the total deflection, according to (14), yields
(23)W=(1-ω2JS)Wb-DS(∂2Wb∂x2+∂2Wb∂y2)
and amplitude of rotation angles according to (2)
(24)Ψx=-∂Wb∂x,Ψy=-∂Wb∂y.
Equation (22) can be reduced to the modified Timoshenko beam equation expressed with bending deflection [23].


## 6. Analytical Solution of Natural Vibrations

Differential equation of natural vibrations (22) can be solved by the method of separation of variables [25]. In that case, the bending deflection is presented as a product of two functions,
(25)Wb(x,y)=X(x)Y(y),
each depending only on one variable. By substituting (25) into (22), one obtains
(26)X′′′′Y+2X′′Y′′+XY′′′+a~(X′′Y+XY′′)+b~XY=0,
where
(27)a~=ω2JD[1+2(1-ν)k],b~=ω2m-D(ω2JS-1).
Furthermore, according to the procedure presented in [13], each of the single variable functions can be assumed in exponential form
(28)X=eμx,Y=eλx.
By inserting (28) into (26), a characteristic equation is obtained
(29)(μ2+λ2)2+a~(μ2+λ2)+b~=0.
It can be reduced to one unknown by substitution of μ2+λ2=r2. Hence, characteristic equation (29) takes the following forms
(30)r4+a~r2+b~=0.
Two roots of (30) read
(31)r12=R12,r22=-R22,
where
(32)R12=(a~2)2-b~-a~2,R22=(a~2)2-b~+a~2.
Hence, there are two solutions for μ2+λ2, that is,(33a)μ2+λ2=R12,(33b)μ2+λ2=-R22,and each of the unknown parametersμ and λ has four solutions, two real and two imaginary:(34a)μ1,2=±α,(34b)μ3,4=±iβ,(35a)
λ
1,2
=
±
η
,
(35b)
λ
3,4
=
±
i
ϑ
.Parametersα, β, η, and ϑ are real and positive quantities. Their mutual dependence is obtained by inserting μi and λi in (33a) and (33b) in all combinations as follows.μ
1,2 and λ1,2 in (33a),   μ3,4 and λ3,4 in (33b),(36a)α2+η2=R12,(36b)β2+ϑ2=R22,μ
1,2 and λ3,4 in (33b),   μ3,4 and λ1,2 in (33a),(37a)η2-β2=R12,(37b)ϑ2-α2=R22,μ
1,2 and λ3,4 in (33a),   μ3,4 and λ1,2 in (33b),(38a)α2-ϑ2=R12,(38b)β2-η2=R22.Substitution ofμ1,2 and λ1,2 into (33b) and μ3,4 and λ3,4 into (33a) does not give real results, and therefore, that combination is not acceptable.By the determined parametersα, β, η, and ϑ, the single variable functions (25) can be presented in the following form:
(39)X=A1sinhαx+A2coshαx+A3sinβx+A4cosβx,(40)Y=B1sinhηy+B2coshηy+B3sinϑy+B4cosϑy.


## 7. Boundary Conditions in terms of Separated Variables

Based on known functionsX(x) and Y(y), the total deflection (23) takes the following form;
(41)W=(1-ω2JS)XY-DS(X′′Y+XY′′).
For specification of boundary conditions at the transverse and longitudinal edges, where x=a and y=b, respectively, the total deflection can be presented in the following ways:
(42)W(a,y)=[(1-ω2JS-DSY′′Y)X(a)-DSX′′(a)]Y,W(x,b)=[(1-ω2JS-DSX′′X)Y(b)-DSY′′(b)]X.
Angles of rotation, (2), have simple expressions:(43a)Ψx(a,y)=-X′(a)Y,(43b)Ψy(y,b)=-Y′(b)X.In a similar way, the bending moments, (3a), (3b), and (3c), take the following form:
(44)Mx(a,y)=-D[X′′(a)+νX(a)Y′′Y]Y,My(x,b)=-D[Y′′(b)+νY(b)X′′X]X.The shearing forces, according to (19), yield
(45)Q-x(a,y)=-D[X′′′(a)+ω2JDX′(a)+(2-ν)X′(a)Y′′Y]Y,Q-y(x,b)=-D[Y′′′(b)+ω2JDY′(b)+(2-ν)Y′(b)X′′X]X.In the previous formulae for thex and y directions, ratios Y′′/Y and X′′/X are functions which make specifying boundary conditions as algebraic equations impossible. Their scalar values are obtained only in the special case of simply supported edges [26]. Since
(46)X=sinmπxa,Y=sinnπyb,
one obtains
(47)X′′X=-(mπa)2,Y′′Y=-(nπb)2,
where m and n are the total number of mode half waves in x and y direction, respectively. Equation (47) can be used for arbitrary boundary conditions as the first rough approximation. Furthermore, according to (39) and (40), functions X and Y consist of the hyperbolic and trigonometric part, that is,
(48)X=Xh+Xt,X′′=α2Xh-β2Xt,Y=Yh+Yt,Y′′=η2Yh-ϑ2Yt.
By taking into account (47) and (48), the expressions for the geometric boundary conditions read as follows.Deflection:(49)W(a,y)=[p1Xh(a)+p2Xt(a)]Y,(50)W(x,b)=[q1Yh(b)+q2Yt(b)]X,
where
(51)p1=1-ω2JS+DS[(nπb)2-α2],p2=1-ω2JS+DS[(nπb)2+β2],q1=1-ω2JS+DS[(mπa)2-η2],q2=1-ω2JS+DS[(mπa)2+ϑ2].Angles of rotation:(52)Ψx(a,y)=-X′(a)Y,Ψy(x,b)=-XY′(b).In a similar way, the bending moments (44) can be written in the following form:
(53)Mx(a,y)=-D[r1Xh(a)-r2Xt(a)]Y,(54)My(x,b)=-D[s1Yh(b)-s2Yt(b)]X,
where
(55)r1=α2-ν(nπb)2,r2=β2+ν(nπb)2,s1=η2-ν(mπa)2,s2=ϑ2+ν(mπa)2.
Finally, for the shearing forces (45), one can write
(56)Q-x(a,y)=-D[u1dXhdx|x=a-u2dXtdx|x=a]Y,Q-y(x,b)=-D[v1dYhdy|y=b-v2dYtdy|y=b]X,
where
(57)u1=α2+ω2JD-(2-ν)(nπb)2,u2=β2-ω2JD+(2-ν)(nπb)2,v1=η2+ω2JD-(2-ν)(mπa)2,v2=ϑ2-ω2JD+(2-ν)(mπa)2.


## 8. Natural Vibrations of Clamped Plate, CCCC

Each of the four rectangular plate edges can be simply supported, clamped, or free. That results in a quite large number of possible combinations of boundary conditions. However, in order to illustrate applicability of the developed theory, it is sufficient to analyse a few typical boundary value problems.Hence, let us first consider vibrations of a clamped plate. Boundary conditions at transverse edges read(58)W(a,y)=0,Ψx(a,y)=0,W(0,y)=0,Ψx(0,y)=0.
According to (39), (49), and (52), (58) lead to the following expressions:
(59)p1(A1sinhαa+A2coshαa)+p2(A3sinβa+A4cosβa)=0,A1αcoshαa+A2αsinhαa+A3βcosβa-A4βsinβa=0,(60)p1A2+p2A4=0,αA1+βA3=0.
From (60) it yields that
(61)A4=-p1p2A2,A3=-αβA1.
Substitution of (61) into (59) gives a system of two algebraic equations which can be written in the following matrix form
(62)[p1βsinhαa-p2αsinβap1β(coshαa-cosβa)p2α(coshαa-cosβa)p2αsinhαa+p1βsinβa]×{A1A2}={00}.
The nontrivial solution is obtained if determinant of (62) is zero, after some manipulations, which leads to the following frequency equation:
(63)2p1p2αβ(1-coshαacosβa)+[(p2α)2-(p1β)2]sinhαasinβa=0.
In the homogenous system of algebraic equations (62), one of the integration constants A1 and A2 is arbitrary, while the other is determined by the first or the second equation. From the first equation one can write
(64)A1=p1β(coshαa-cosβa),A2=-p1βsinhαa+p2αsinβa.
Constants A3 and A4 are obtained by employing (61). In that, way function X, (39), for mode shapes in longitudinal direction is defined.Boundary conditions for the clamped longitudinal edges read(65)W(x,b)=0,Ψy(x,b)=0,W(x,0)=0,Ψy(x,0)=0.
Following the previous procedure, the final frequency equation can be written in an analogous way:
(66)2q1q2ηϑ(1-coshηbcosϑb)+[(q2η)2-(q1ϑ)2]sinhηbsinϑb=0.
For integration constants, one can write
(67)B1=q1ϑ(coshηb-cosϑb),B1=-q1ϑsinηb+q2ηsinϑb,B3=-ηϑB1,B4=-q1q2B2.
In that way, function Y, (40), is defined. Now, the bending deflection mode can be determined Wb=XY and then the complete deflection mode by employing (41).In frequency equations (63) and (66), five quantities are unknown, that is, ω, α, β, η, and ϑ, and therefore, it is necessary to specify three additional conditions. For that purpose, six relations are on disposal, that is, (36a), (36b), (37a), (37b), (38a), and (38b). Frequency ω is the basic variable, and it is convenient to choose parameter β of the trigonometric functions of the longitudinal plate generatrix as the second basic variable, since it remains in the case of simply supported edges. The other parameters α, η, and ϑ can be expressed with β. Summing up (36a) and (38b) or (36b) and (38a) yields
(68)α2+β2=R12+R22=R2.
Furthermore, one finds from (37a) and (36b) that(69a)η=R12+β2,(69b)ϑ=R22-β2.It is interesting, that summation of square of (69a) and (69b) gives an expression similar to (68), that is,
(70)η2+ϑ2=R12+R22=R2.
The previous relations of vibration parameters are geometrically dependent based on Pythagoras’ theorem and therefore can be constructed as shown in Figure 2. Assuming the value of ω, R1 and R2 are determined and presented as catheti of the right triangle ABC with hypotenuse R. The circumscribed circle of rectangle ABCD is drawn. Furthermore, value of parameter β is assumed, and α is defined as cathetus of triangle AEC, according to (68). Value of β=CE¯ is rotated so that η is obtained as the hypotenuse of triangle CDG according to (69a). Parameter ϑ is determined as the cathetus of triangle BCF, according to (69b). Finally, η=DG¯ and ϑ=BF¯ are rotated and their intersection, point H, lies at the circle according to (70). In that way, the plan of the vibration parameters is closed. It changes depending on boundary conditions and vibration mode.Figure 2
Plan of vibration parameters.Now,α, η, and ϑ from (68) and (69a) and (69b) can be substituted into frequency equations (63) and (66). In the previous way, system of five nonlinear algebraic equations is reduced to two equations with unknown parameters ω and β. It can be solved by a Newton-Raphson iteration procedure. For that purpose, it is necessary to specify the numbers of mode half waves m and n in x and y directions, respectively. Hence, m, n is actually mode identification number. In case of clamped plate, m and n take values 2, 3, 4,…. Furthermore, an expected value of ω has to be prescribed as the initial value in the iteration procedure. Also, a realistic value for β has to be assumed. A set of few mathematical solutions for ω and β is usually obtained, but only that solution which manifests the same numbers of the mode half waves m and n as the prescribed ones is physically correct.Accuracy of the natural frequencies can be increased if calculation is repeated with ratios (47) of the condensed functions. For that purpose, the average values based on the 1-norm can be applied:
(71)X′′X⟶-∫0a|X′′|dx∫0a|X|dx=-(m~πa)2,Y′′Y⟶-∫0b|Y′′|dy∫0b|Y|dy=-(n~πb)2,
where m~ and n~ are numbers analogous to m and n. The previous integrals can be solved numerically.Calculation of natural frequencies is performed for rectangular plate of aspect ratioa/b=0.4, thickness ratio h/b=0.001 and 0.1, and shear coefficient k=5/6. The obtained values of frequency parameter Λ=(ωb2/π2)ρh/D are listed in Tables 1 and 2 for thin and thick plates, respectively. PS0 is the present solution obtained with given m and n, while PS1 and PS2 are the first and the second iterative solution determined with average values of the functions of separated variables, (71), respectively. In that procedure, m and n change values to floating point values m~ and n~.Table 1
Frequency parameterΛ for case CCCC, a/b=0.4, h/b=0.001.
Mode:m,n
2,2
2,3
2,4
2,5
2,6
3,2
3,3
3,4
PS0
14.910
17.445
22.228
29.374
38.795
39.923
42.601
47.225
Xing and Liu [13]
14.910
17.445
22.228
29.374
38.795
39.927
42.601
47.225
Liew et al. [15]
14.972
17.608
22.427
29.553
38.951
39.943
42.671
47.349Table 2
Frequency parameterΛ for case CCCC, a/b=0.4, h/b=0.1.
Mode:m,n
2,2
2,3
2,4
2,5
3,2
2,6
3,3
3,4
PS0
10.696
12.025
14.929
19.199
23.353
24.037
24.171
25.830
m
~
1
,
n
~
1
1.1154, 1.3764
1.0898, 2.3560
1.0548, 3.3528
1.0153, 4.3556
1.7244, 1.3198
1.6138, 4.8451
1.7051, 2.2904
1.6745, 3.2868
PS1
10.578
11.949
14.714
18.658
22.602
23.376
23.720
25.682
m
~
2
,
n
~
2
1.1000, 1.3381
1.1033, 2.2034
1.1097, 3.0362
1.1208, 3.8440
1.6454, 1.3064
1.1103, 4.6693
1.6674, 2.1632
1.6978, 2.9921
PS2
10.572
11.918
14.652
18.585
22.597
23.317
23.658
25.520
Xing and Liu [13]
10.544
11.921
14.702
18.657
22.530
23.393
23.646
25.643
Liew et al. [15]
10.702
12.352
15.257
19.195
22.627
23.861
23.972
26.198Results for the thin plate, Table1, are obtained directly without iteration. They are compared with Xing’s solution [13] and Liew’s solution [15]. The former is determined analytically in closed form, and for the latter the Rayleigh-Ritz method is employed. PS0 and Xing’s values are the same, while Liew’s values are somewhat higher.For thick plate, Table2, stable values of vibration parameter Λ are obtained in two iteration steps. They are very close to the Xing’s values, while Liew’s values are somewhat higher. Values of m and n are moderately reduced in the first step of iteration and very little in the second one. Even the first approximate values of natural frequencies, PS0, are very close to the Liew values. Generally speaking, if natural frequency does not converge to the expected value, it is necessary to adjust values of m and n.For illustration, the first natural mode of the thick plate is presented by deflection of longitudinal generatrix in the middle of the plate, Figure3. Similar diagrams can be drawn for transverse generatrix. Total deflection W(x,b/2) consists of bending deflection Wb(x,b/2) and shear deflection Ws(x,b/2). Their boundary values exist as a result of ignoring integration function f(t), Section 3. However, the boundary values are of the close magnitudes but opposite signs, and therefore, they are almost cancelled. As a result, the boundary conditions are approximately satisfied, that is, W(0,y)=W(a,y)=ε(y), where the shape of discrepancy function ε(y) is similar to that of shear deflection Ws(xi,y). The same consideration is valid for the longitudinal edges. Actually, the boundary conditions for deflection are satisfied in the mean according to the introduced 1-norms (71). Furthermore, it is obvious from the boundary slope of the bending deflection that the angles of rotation are zero as specified by the boundary conditions. The shape of the shear deflection is similar to the bending moment of clamped beam, since it depends on curvatures. Indeed, the main shear part in the formula for total deflection, (23), can be expressed with the bending moments, (3a), (3b), and (3c):
(72)W=(1-ω2JS)Wb+1(1+ν)S(M1+M2).
Shear contribution to total deflection is increased for higher modes due to shortening of wave length.Figure 3
Deflection of plate longitudinal generatrix, case CCCC, the 1st mode,a/b=0.4, h/b=0.1.The shape of the first natural mode is shown in Figure4, where negligible small hold-down of the plate corners can be noticed due to the previous explained reasons.Figure 4
The first natural mode, case CCCC,a/b=0.4, h/b=0.1; m,n=2,2.


## 9. Natural Vibrations of Plate with Two Simply Supported and Two Clamped Edges, CCSS

Boundary conditions at the transverse edges read(73)W(a,y)=0,Mx(a,y)=0,W(0,y)=0,Ψx(0,y)=0.
According to (49), (53), and (52), one can write
(74)p1(A1sinhαa+A2coshαa)+p2(A3sinβa+A4cosβa)=0,r1(A1sinhαa+A2coshαa)-r2(A3sinβa+A4cosβa)=0,p1A2+p2A4=0,αA1+βA3=0.
The previous system of algebraic equations leads to the following frequency equation:
(75)(1+p1p2r2r1)tanhαa-αβ(p2p1+r2r1)tanβa=0.
The frequency equation for transverse direction can be written in the analogous way since the boundary conditions are the same in both cases:
(76)(1+q1q2s2s1)tanhηb-ηϑ(q2q1+s2s1)tanϑb=0.
The integration constants Ai and Bi, i=1,2,3,4, are expressed by the same formulas as in the previous cases, (61), (64), and (67).Since one of two opposite edges is simply supported and another is clamped, numbers of mode half waves take valuesm=1.5,2.5,… and n=1.5,2.5,…. The numerical procedure for determining natural frequencies is the same as in the case of clamped plate.Values of vibration parameterΛ for thin and thick plates are presented in Tables 3 and 4, respectively. In the former cases results obtained by all three methods are the same. For thick plates the initial values of Λ are very close to the final values. Also, PS2 values are very close to Xing’s values. However, Liew’s values are somewhat higher as in the previous example. The second natural mode is shown in Figure 5, where only slight release of the corner of clamped two adjacent edges is registered.Table 3
Frequency parameterΛ for case CCSS, a/b=0.4, h/b=0.001.
Mode:m,n
1.5,1.5
1.5,2.5
1.5,3.5
1.5,4.5
2.5,1.5
2.5,2.5
1.5,5.5
2.5,3.5
PS0
10.651
13.484
18.462
25.603
32.573
34.858
35.416
40.247
Xing and Liu [13]
10.651
13.484
18.462
25.603
32.573
34.858
35.416
40.247
Liew et al. [15]
10.669
13.524
18.507
25.641
32.579
34.888
35.436
40.279Table 4
Frequency parameterΛ for case CCSS, a/b=0.4, h/b=0.1.
Mode:m,n
1.5,1.5
1.5,2.5
1.5,3.5
1.5,4.5
2.5,1.5
2.5,2.5
1.5,5.5
2.5,3.5
PS0
8.4956
10.382
13.643
17.997
21.176
22.393
23.096
24.542
m
~
1
,
n
~
1
1.0802, 1.1907
1.0601, 2.1725
1.0346, 3.1684
1.0071, 4.1691
1.814, 1.1603
1.8312, 2.1326
0.9780, 5.1712
1.8136, 3.1290
PS1
8.4735
10.363
13.559
17.776
20.920
22.275
22.687
24.562
m
~
2
,
n
~
2
1.0769, 1.1828
1.0708, 2.1158
1.0670, 3.0304
1.0679, 3.9286
1.8063, 1.1603
1.8199, 2.0978
1.0724, 4.8150
1.8373, 3.0150
PS2
8.4730
10.359
13.549
17.762
20.921
22.266
22.676
24.533
Xing and Liu [13]
8.4681
10.359
13.559
17.778
20.905
22.265
22.694
24.566
Liew et al. [15]
8.5213
10.504
13.752
17.978
20.950
22.412
22.883
24.818Figure 5
The second natural mode, case CCSS,a/b=0.4, h/b=0.1; m,n=1.5,2.5.


## 10. Natural Vibrations of Plate with Mixed Boundary Conditions, CFSS

Boundary conditions at transverse edges are the same as in the previous case. Therefore, frequency equation (75), with corresponding integration constants, (61), (64), is valid also in the present case.Boundary conditions at longitudinal edges read(77)W(x,b)=0,My(x,b)=0,My(x,0)=0,Q-y(x,0)=0.
The expanding form of (77) is the following:
(78)q1(B1sinhηb+B2coshηb)+q2(B3sinϑb+B4cosϑb)=0,s1(B1sinhηb+B2coshηb)-s2(B3sinϑb+B4cosϑb)=0,s1B2-s2B4=0,ηv1B1-ϑv2B3=0.
The previous system of algebraic equations leads to the following frequency equation:
(79)(1+q2q1s1s2)tanhηb-ηv1ϑv2(q2q1+s2s1)tanϑb=0.
The integration constants are the following:
(80)B1=coshηb+q2s1q1s2cosϑb,B2=-sinhηb-ηq2v1ϑq1v2sinϑb,B3=ηv1ϑv2B1,B4=s1s2B2.
Based on the expected mode shape, the following values of modal parameters are taken into account: m=1.5,2.5,… and n=0.5,1.5,….The obtained results for thin and thick plates are listed in Tables5 and 6, respectively. Values of the frequency parameter Λ for thin plate are determined by iteration since in this case free edge convergence is not as fast as for the other boundary conditions. Very good agreement with Liew’s values can be noticed. Calculation of natural vibration is also performed by NASTRAN [27] with mesh density of the finite element model 8×16. That is sufficient to determine the first 8 natural frequencies accurately. The obtained results are quite close to those of PS2 and Liew’s values.Table 5
Frequency parameterΛ for case CFSS, a/b=0.4, h/b=0.001.
Mode:m,n
1.5,0.5
1.5,1.5
1.5,2.5
1.5,3.5
1.5,4.5
2.5,0.5
2.5,1.5
2.5,2.5
PS2
9.8873
11.3994
14.6341
19.7738
26.9092
31.7086
33.4171
36.0825
Liew et al. [15]
9.8742
11.3458
14.4898
19.5392
26.6235
31.6979
33.3971
35.7883
FEM
9.8589
11.2012
14.0840
18.7437
25.3267
31.6571
33.0734
33.9035Table 6
Frequency parameterΛ for case CFSS, a/b=0.4, h/b=0.1.
Mode:m,n
1.5,0.5
1.5,1.5
1.5,2.5
1.5,3.5
1.5,4.5
2.5,0.5
2.5,1.5
2.5,2.5
PS0
7.8297
8.1701
9.3195
12.722
17.589
20.561
21.422
22.422
m
~
1
,
n
~
1
1.0895, 0.4950
1.0845, 0.8354
1.0706, 1.4737
1.0411, 2.6646
1.0095, 3.9345
1.8465, 0.7878
1.8393, 1.2686
1.8309, 2.0644
PS1
7.9335
8.9499
11.148
14.537
18.882
20.497
21.299
22.970
m
~
2
,
n
~
2
1.0791, 0.6096
1.0678, 1.2959
1.0531, 2.2285
1.0499, 3.2175
1.0566, 4.2231
1.8037, 0.8579
1.8054, 1.3982
1.8120, 2.2943
PS2
7.9310
8.9585
11.183
14.602
18.976
20.484
21.305
23.010
Liew et al. [15]
7.9407
8.9700
11.135
14.462
18.761
20.459
21.357
23.112
FEM
8.0190
8.8660
10.834
14.056
18.278
20.771
21.397
22.662The fifth and sixth natural modes determined by the present theory are shown in Figures6 and 7, where the mode identification number is m,n=1.5,4.5 and 2.5,0.5, respectively. Intention is to point out the transition of natural vibrations from a dominantly large number of mode half waves in transverse direction to longitudinal direction. A slight release of the corner of clamped and free edge can be noticed. The same natural modes obtained by FEM analysis are presented in Figures 8 and 9 [27]. There is no other visual difference between the mode shapes determined in these two ways. They can be distinguished mainly by the smooth and straight lines of the deformed mesh.Figure 6
The fifth natural mode, case CFSS,a/b=0.4, h/b=0.1; m,n=1.5,4.5, PS2.Figure 7
The sixth natural mode, case CFSS,a/b=0.4, h/b=0.1; m,n=2.5,0.5, PS2.Figure 8
The fifth natural mode, case CFSS,a/b=0.4, h/b=0.1; m,n=1.5,4.5, FEM.Figure 9
The sixth natural mode, case CFSS,a/b=0.4, h/b=0.1; m,n=2.5,0.5, FEM.


## 11. Conclusion

An advanced vibration theory of moderately thick plate vibrations is presented. The three traditional partial differential equations of motion with total deflection and angles of rotation are reduced to one equation with bending deflection as the basic unknown acting as a potential function. Its general solution is determined exactly by the method of separation of variables. The frequency equations are exact in the case of simply supported plate and simply supported two opposite edges. These two special problems are elaborated in detail in [26]. An approximate but very reliable solution is obtained for arbitrary boundary conditions which are satisfied in the mean, except angles of rotations that are satisfied exactly in any case. For the first time, an analytical solution is obtained for the case of a free edge where Levy’s approach is not applicable.As pointed out, the new theory can be used for vibration analysis of moderately thick plate. Very thick plates, where coupling of bending shear and in-plane shear modes appears at higher natural frequencies, can be analysed by the method described in [28]. Actually, very thick plates are elastic bodies, and 3D analysis is more effective. By the proposed method, results obtained by numerical methods, as the Rayleigh-Ritz method, can be verified. Also, finite elements for thick plate can be tested. The differential equation of motion is derived in the time domain and can be applied for determining both harmonic and transient force responses.Difference between the proposed moderately thick plate theory and the known theories is obvious as can be seen from the outline of the latter presented in the Appendix.



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