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On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space
Accepted Manuscript
On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space Rui Li , Pengcheng Wang , Zekun Yang , Jiaqi Yang , Linghui Tong PII: DOI: Reference:
S0307-904X(17)30562-0 10.1016/j.apm.2017.09.011 APM 11956
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
7 January 2017 2 August 2017 6 September 2017
Please cite this article as: Rui Li , Pengcheng Wang , Zekun Yang , Jiaqi Yang , Linghui Tong , On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.09.011
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Highlights Free vibration of cantilever plates is solved in the symplectic space. Analytic free vibration solutions of rectangular cantilever plates are obtained. More similar problems are expected to be solved by the present
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approach.
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On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space Rui Lia,b,*, Pengcheng Wanga, Zekun Yanga,c, Jiaqi Yanga,c, and Linghui Tonga,c a
State Key Laboratory of Structural Analysis for Industrial Equipment, Department of
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Engineering Mechanics, International Research Center for Computational Mechanics, Dalian University of Technology, Dalian 116024, PR China b
Department of Mechanical Engineering, Northwestern University, Evanston, IL
60208, USA c
School of Naval Architecture & Ocean Engineering, Dalian University of Technology,
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Dalian 116024, PR China *
Corresponding author.
Tel.: +86 15840912019.
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E-mail address: [email protected], [email protected] (R. Li). Abstract: In this paper, we obtain accurate analytic free vibration solutions of
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rectangular thin cantilever plates by using an up-to-date rational superposition method in the symplectic space. To the authors’ knowledge, these solutions were not available
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in the literature due to the difficulty in handling the complex mathematical model. The Hamiltonian system-based governing equation is first constructed. The
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eigenvalue problems of two fundamental vibration problems are formed for a cantilever plate. By symplectic expansion, the fundamental solutions are obtained. Superposition of these solutions are equal to that of the cantilever plate, which yields
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the analytic frequency equation. The mode shapes are then readily obtained. The developed method yields the benchmark analytic solutions with fast convergence and satisfactory accuracy by rigorous derivation, without assuming any trial solutions; thus, it is regarded as rational, and its applicability to more boundary value problems of partial differential equations represented by plates’ vibration, bending and buckling may be expected. Keywords: analytic solution; free vibration; cantilever plate; symplectic space. 2
ACCEPTED MANUSCRIPT Nomenclature In-plane dimensions of plate Flexural rigidity of plate Coefficients of series expansion Thickness of plate Hamiltonian matrix Unit matrix Symplectic matrix Bending moments per unit distance in plate Equivalent shear forces per unit distance in plate Modal displacement Rectangular coordinates Eigenvectors Coefficient functions of eigenvectors State vectors Eigenvalues Poisson’s ratio Derivative of w with respect to y Mass density of plate Circular frequency
ρ
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a, b D E, F, G, H h H I J Mx, My Vx, Vy w x, y X(x), Y'(y) Y(y), X'(x) Z, Z' μ, μ'
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1. Introduction
In both mechanical and civil engineering, thin plates play an important role. They
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are generally used as support structures such as bridge decks, floor slabs, foundation beds, and aircraft panels. In elasticity, thin plates are well described by the Kirchhoff
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plate theory, which has been an indispensable part of solid mechanics. In applied mathematics, the above mechanics model is affiliated with a class of biharmonic
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equations, which are the fourth-order partial differential equations (PDEs). With proper boundary constraints, the associated boundary value problems arise. It is well
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known that such a high-order PDE is normally hard to solve analytically, especially for that with specific boundary conditions. Taking rectangular thin plates as examples, their free vibration problems have long been the key issues in elasticity and there is a fairly large literature on the topic. However, except the plates with two opposite edges simply supported, few of the others have been analytically solved by the classical approaches. Some newly proposed analytic methods are found in recent publications, which focus on specific plates without two opposite edges simply supported, such as those with point supports or with all edges free. However, the rectangular thin cantilever plates, characterized by involving free corners and both clamped and free 3
ACCEPTED MANUSCRIPT edges, have not been well treated, and there have been no accurate analytic solutions found in the literature, although they are of much importance in engineering applications, such as the overhanging roofs, airplane wings, cantilever bridges and balconies, and even in micro/nano technology, such as the atomic force microscope cantilevers. This paper aims to explore the analytic free vibration solutions of such a classical but extremely difficult class of problems. There are various solution methods for rectangular thin cantilever plates’ free
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vibration, most of which are, of course, approximate or numerical, as introduced in Leissa’s famous technical report [1]. The Ritz method was used by Barton [2] to determine approximate solutions for the frequencies and modes of vibration of rectangular and skew cantilever plates, where the deflection functions correspond to those defining the normal modes of vibration of a beam. Claassen [3] applied the
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Fourier sine-series method to obtain approximate solutions for the transverse vibrations of a thin rectangular cantilever plate. Rajalingham et al. [4] obtained closed form approximation of vibration modes of rectangular cantilever plates by the variational reduction method, where the plate PDE was reduced to two simultaneous ordinary differential equations and their boundary conditions. Liang et al. [5]
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presented a simple procedure using an empirical added mass formulation and Rayleigh-Ritz method to analyze the vibration frequencies and mode shapes of
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submerged cantilever plates. Ergin and Uğurlu [6] then investigated the dynamic characteristics, such as natural frequencies and mode shapes, of cantilever plates
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partially submerged in a fluid by use of a boundary-integral equation method together with the method of images. Seok et al. [7] performed an analysis of the free transverse
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vibrations of a cantilevered plate by means of a variational approximation procedure. Looker and Sader [8] proposed a simple uniformly valid expression for the
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fundamental flexural vibration frequency of a thin rectangular cantilever plate using an accurate variational approach based on an energy minimization principle and a singular perturbation solution. Rostami et al. [9] recently provided accurate approximate solution for in-plane vibrations of rotating orthotropic cantilever plates using the Extended Kantorovich Method. There are some other popular numerical solution methods for plate problems. Lindsay et al. [10] studied the vibration of a thin plate with a collection of clamped patches, and obtained detailed information on the limiting behavior in the form of an asymptotic expansion in the limit of small patch radius. Civalek et al. [11-13] developed the discrete singular convolution method for 4
ACCEPTED MANUSCRIPT the free vibration analysis of both thin and thick plates. The plate buckling problems were analyzed by Civalek [14] and Malekzadeh et al. [15,16] by employing the differential quadrature method. Zenkour et al. [17,18] investigated the buckling and free vibration of plates using approximate displacement solutions, which depend on the boundary conditions on the plate edges. Except for the approximate or numerical methods, there are few analytic methods handling the plate problems. Gorman [19] employed the semi-inverse superposition
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method to analyze the first five symmetric and anti-symmetric free vibration modes of a cantilever plate for a wide range of aspect ratios. Zhong et al. [20] used the double finite integral transform method to derive the eigenfrequencies and vibration modes of rectangular thin cantilever plates. The method was also applied by Tian et al. [21] to bending of the rectangular thick plates. Xing and Liu [22,23] proposed a method of
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direct separation of variables to obtain both the in-plane and out-of-plane free vibration solutions of rectangular thin or thick plates. Lim et al. [24] developed the symplectic elasticity approach for free vibration of rectangular Levy-type thin plates employing the Hamiltonian variational principle with Legendre’s transformation. The approach was proposed by Yao et al. [25] and has been extended to many other fields,
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as described in a comprehensive review by Lim and Xu [26]. It should be noted that the conventional symplectic method has not exhibited the ability to analytically solve opposite edges.
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the vibration of the plates with free edges while without two simply supported
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In recent years, we have proposed a novel analytic symplectic superposition method, which has shown applicability to general mechanics problems of the plates
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by obtaining the analytic solutions of some thin/thick plates’ bending, vibration and buckling problems [27-29]. In this paper, we develop the symplectic superposition
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method further to explore accurate analytic free vibration solutions of isotropic rectangular thin cantilever plates. The procedure starts from solving the Hamiltonian system-based governing equation of two fundamental problems, where the mathematical techniques in the symplectic space are applied, including the separation of variables, which is sometimes invalid in the Euclid space, and symplectic eigen expansion. Based on the obtained analytic solutions, skillful superposition is imposed to yield the analytic solution of the original problem. There are some boundary conditions to be satisfied in equating the superposition of the fundamental solutions to the final solution of a cantilever plate, which leads to the frequency equation. One will 5
ACCEPTED MANUSCRIPT see in the following that the developed method derives the analytic solutions in an elegant rational way, because there are no predetermined solutions but rigorous derivations from the governing equation. This constitutes the main advantage of the method, enabling one to obtain the analytic solutions which were not available in the past, including those developed in this paper. Many numerical results are presented to show the convergence and accuracy of the new analytic solutions by excellent comparison with those from the literature, if any, and the finite element method
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(FEM). They are expected to serve as the benchmarks for validation of the other methods.
2. Governing equation and fundamental solutions by symplectic
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eigen expansion
The governing equation for free vibration of a thin isotropic plate in the rectangular coordinate system x, y can be expressed in the Hamiltonian system as [27,30]
Z y HZ ,
0 F 2 2 x
1 , 0
T
F
, H , Q T Q F 0
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w,
equivalent shear force Vy :
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moment parallel to the
x
the circular frequency,
2
0
0
derivative of
T
Z
2
include
with respect to y : , opposite of the
w
, where Vy D 3 w y3 2 3 w x 2y , and bending
axis: M y . is the mass density, D
equation (2.1), the matrix
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, 2 D 1 x 0
and G . The components of the state vector 0 1 D
the modal displacement
h 2 D 1 2 4 x 4
G
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Z w, , T , M y
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where
(2.1)
h
the plate thickness,
the flexural rigidity, and the Poisson’s ratio. In H
0
satisfies HT JHJ , where J I 2
symplectic matrix in which I 2 is the
2 2
I2 0
unit matrix. This means that
is the H
is a
Hamiltonian matrix with differential operators. By equation (2.1), the free vibration problem of a thin plate is actually described in the symplectic space [25] such that the separation of variables and symplectic eigen expansion are valid and the rational analytic solutions can be obtained without predetermining the solution forms, which cannot be realized with the classical 6
ACCEPTED MANUSCRIPT semi-inverse method in the Euclidean space. As shown in figure 1(a), what we consider is a rectangular thin cantilever plate. The origin, O, is at a corner of the plate. The plate edges with lengths
a
and
b,
x
and y axes are directed along the
respectively. The solution approach here starts
from dividing the problem into two fundamental problems, as shown in figure 1(b, c), whose solutions will be obtained by symplectic eigen expansion in the following. Superposition of the fundamental solutions yields the solution of the cantilever plate.
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It must be pointed out that each fundamental solution has the constants to be determined by equating the superposition of the fundamental problems’ supports at each edge with the support at the associated edge of the cantilever plate. The generated equations are used to determine the natural frequencies, which will be
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elaborated in section 3. [Figure 1]
In figure 1(b), the plate is slidingly clamped at x 0 ,
My
y b
y 0
and simply
and bending moment denoted by
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supported at y b ; the slope denoted by
xa, y 0,
are imposed along y 0 and y b , respectively. In figure 1(c), the same plate w x x 0
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is subject to the slopes
and
w x x a
distributed along x 0 and
xa,
respectively. The above two fundamental problems are solved below.
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For the first fundamental problem as shown in figure 1(b), solving equation (2.1) is first reduced to solving by applying
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HX x X x
dY y dy Y y
Z X x Y y
and the eigenvalue problem
via variable separation, where
function of y only, the eigenvalue, and
Y y
X x w x , x , T x , M y x
T
is a the
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eigenvector. The eigenvalue problem is solved under the boundary conditions w x x x 0, a Vx x x 0, a 0 ,
Vx x, y
where
Vx x
is the factor of the equivalent shear force
which is equal to D 3 w x, y x3 2 3 w x, y xy 2 . It is not difficult
to obtain the following eigenvalues: n1 n2 R , n 2 n2 R , n3 n2 R , n 4 n2 R
for
n 0,1, 2, 3,
(2.2)
, where n n a , and R h D . It is noted that there are four
groups of eigenvalues given in equation (2.2), which depend on the non-negative 7
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n.
Corresponding to equation (2.2), we have the eigenvectors T
X n1 x 1,
n2 R , D R n2 1 n2 R , D R n2 1 cos n x ,
T
X n2 x 1, n2 R , D R n2 1 n2 R , D R n2 1 cos n x , T
X n3 x 1,
n2 R , D R n2 1 n2 R , D R n2 1 cos n x ,
(2.3)
T
X n4 x 1, n2 R , D R n2 1 n2 R , D R n2 1 cos n x .
yields the corresponding
exp
Yn1 y cn1 exp
2 n
Yn3 y cn3
2 n
R y , Y
Y y
such that
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dY y dy Y y
For each ,
exp
R y,
R y , Yn 2 y cn 2 exp n2 R y , n 4 y cn 4
n2
(2.4)
where cn1 to cn 4 are the constants to be determined. The solution of equation (2.1) is thus written as
(2.5)
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Z X x Y y ,
where X x
, Xn1 x , Xn2 x , Xn3 x , X n4 x ,
Y y
, Yn1 y , Yn 2 y , Yn3 y , Yn 4 y ,
,
(2.6)
T
.
and the bending moment M y
y 0
the cosine series, i.e. 0,
y 0
y 0
En cos n x , w y b 0, M y
n 0,1,2,
y b
n 0,1,2,
Fn cos n x ,
(2.7)
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Vy
are respectively expanded to
y b
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imposed slope
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At the slidingly clamped edge y 0 and the simply supported edge y b , the
where En and Fn are the coefficients of expansion. Substituting the solution (2.5)
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into equation (2.7) to determine the constants cn1 to cn 4 , the dimensionless modal displacement for the first fundamental problem, w1 x, y , is a
F sec cos y sech ch y
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w1 x , y
1 E0 sh y tanh ch y sec sin 1 y 2 0
n 1,2,3,
cos nx 2
E sech sh n
n
n
1 y n 2 n 2 2 n
sech n sh n 1 y n 2 n 2 2 n
(2.8)
Fn sech n ch n y sech n ch n y ,
where x x a ,
yy b ,
b a ,
F0 aF0 D ,
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Fn aFn D ,
a2 h D ,
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For the second fundamental problem as shown in figure 1(c), the coordinates exchange should be applied such that
x
( y ) and
a
( b ) are respectively replaced by
y ( x ) and b ( a ). Therefore, solving the modified governing equation Z x HZ
is first reduced to solving dX x dx X x and the eigenvalue problem HY y Y y by applying Z Y y X x via variable separation, where X x
a
function
of
only,
x
Y y w y , w x y , Vx y , M x y h 2 D 1 2 4 y 4 Q 0
T
the
eigenvalue,
G F T Q F
the eigenvector; H
0 and F 2 2 2 D 1 y y 0
2
and
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is
2
1 . 0
, in which
The eigenvalue
w y y b M y y
y b
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problem is solved under the boundary conditions w y y y 0 Vy y y 0 0 and 0 . The obtained eigenvalues are
n1 n2 R , n2 n2 R , n3 n2 R , n4 n2 R
(2.9)
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for n 1,3,5, , where n n 2b . Corresponding to equation (2.9), we have the eigenvectors
T
2 n2 R , D R n 21 n R , D R n 12 cos n y ,
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Yn1 y 1,
T
Yn2 y 1, R , D R 1 R , D R 1 cos n y , 2 n
2 n
2 n
T
R , D R 1 R , D R 1 cos n y , 2 n
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Yn3 y 1,
2 n
2 n
2 n
2 n
(2.10)
T
Yn4 y 1, n2 R , D R n2 1 n2 R , D R n2 1 cos n y .
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that
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For each , dX x dx X x yields the corresponding X x such
X x c exp X n1 x cn1 exp n3
n3
Rx , X x c exp
Rx ,
n2 Rx , X n2 x cn 2 exp n2 Rx , n2
n4
n4
n2
(2.11)
where cn1 to cn 4 are the constants to be determined. The solution of the second fundamental problem is thus written as Z Y y X x ,
where
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(2.12)
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,
x , X n2 x , X n3 x , X n4 x, X x , X n1
.
At the slidingly clamped edges x 0 and
xa,
(2.13)
T
the imposed slopes w x x 0 and
w x x a are respectively expanded to the half cosine series, i.e. Vx
w Gn cos n y , Vx x x 0 n 1,3,5,
0,
x 0
xa
0,
w H n cos n y . x x a n 1,3,5,
(2.14)
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where Gn and H n are the coefficients of expansion. Substituting the solution (2.12) into equation (2.14) to determine the constants cn1 to cn 4 , the dimensionless modal displacement for the second fundamental problem, w2 x, y , is w2 x , y b
n 1,3,5,
1 n y cos 4 2 2
H n csch n ch n 2 2
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2 n 2 2 2 n 2 2 Gn csch n ch n 1 x n csch n ch n 1 x n (2.15) n n 2 2 2 2 2 n 2 2 x n csch n ch n n 2 2
2 n 2 2 x n , n
where n n2 2 4 2 and n n2 2 4 2 .
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3. Determination of natural frequencies and mode shapes As depicted in section 2, superposition of the two fundamental solutions, (2.8) and
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(2.15), yields the solution of a cantilever plate, which contains the undetermined coefficients of expansion. To determine the natural frequencies, a set of equations are
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formed by making the above superposition satisfy the real boundary conditions at each edge of the cantilever plate.
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At the free edge y 0 of the cantilever plate, the superposition of the fundamental solutions has satisfied the boundary condition of zero equivalent shear
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force, Vy
y 0
0 . To satisfy another boundary condition that the bending moment M y
be zero, we equate the superposition of the bending moments of the fundamental problems at y 0 to zero, which yields, after simplification,
tan tanh E0 sec sech F0
n 1,3,5,
32 2 4 Gn H n n 4 4 16 2 4
0,
and
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(3.1)
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i i sech i i 2 i 2 2 sech i i 2 i 2 2 Fi
2 16 i i 2 4 2 2 i 2 n 2 4 1 G cos i H 0, n n 2 4 4 2 2 2 2 16 n 4i
(3.2)
n 1,3,5,
for i 1, 2,3, , where Fi aFi D , i i 2 2 , and i i 2 2 . At the clamped edge y b , the zero modal displacement boundary condition has
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been satisfied. To satisfy another boundary condition that the slope be zero, we equate the superposition of the slopes of the fundamental problems at y b to zero, which yields
sec sech E0 tan tanh F0
n 1,3,5,
3 3 n 4 n Gn H n sin 0, n 4 4 16 2 4 2
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and
(3.3)
sech i i 2 i 2 2 sech i i 2 i 2 2 Ei i tanh i i tanh i Fi
n 1,3, 5,
3 2 2 2 8n n 4i G cos i H n 0 2 n 16 2 4 4 n 2 4i 2 2
(3.4)
M
n sin 2
for i 1, 2,3, .
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At the free edge x 0 , the boundary condition that the bending moment M x be zero requires
16 3i 2i 2
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i 512 3 6 E0 64 i 3 3 3 sin F0 2 2 2 ii csch i csch i i i 2 i 2 2 sh i ch i i i 2 i 2 2 ch i sh i Gi 2 2 2 2 2 2 2 2 ii i i 2 i 2 2 csch i i i 2 i 2 2 csch i H i 2 2 n 1, 2,3,
16 2
4
4
i
2
4n 2
2 2
2 i 2 2 2 2 4 3 2 2 2 8 4 i n 1 En 4i i 4n sin 2
(3.5)
for i 1,3,5, , where i i 2 2 4 2 and i i 2 2 4 2 . Similarly, at
Fn 0
xa,
the boundary condition that M x be zero requires
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ACCEPTED MANUSCRIPT i 512 3 6 E0 64 i 3 3 3 sin F0 2 2 2 ii i i 2 i 2 2 csch i i i 2 i 2 2 csch i Gi 2 2 2 2 ii csch i csch i i i 2 i 2 2 sh i ch i i i 2 i 2 2 ch i sh i H i 2 2 2 2 2 2
16 3i 2i 2 cos n
n 1,2,3,
16 2
4
4
i
2
4n 2
2 2
2 i 2 2 2 2 4 3 2 2 2 8 4 i n 1 En 4i i 4n sin Fn 0 2
for i 1,3,5, .
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(3.6) Equations (3.1)-(3.6) constitute a set of infinite linear equations with respect to Em , Fm , Gn , and H n ( m 0,1, 2,
; n 1,3,5,
). The existence of nonzero solutions
requires that the determinant of the coefficient matrix be zero, which leads to an
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equation for determination of the natural frequencies. Noting that a finite number of equations are necessary for practical calculation, any accuracy can be achieved by taking adequate series terms. For convenience, the same number of series terms, denoted by N , are adopted throughout, i.e., E0 , E1 , E2 , , EN , F0 , F1 , F2 , , FN , G0 , G1 , G2 ,
, GN , and
H0 , H1 , H 2 ,
, HN
are taken into account. The mode shape
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corresponding to a natural frequency is obtained by first substituting the frequency solution back into equations (3.1)-(3.6), solving for a set of non-trivial solutions for
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their summation.
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Em , Fm , Gn , and H n , substituting them into equations (2.8) and (2.15), and then
4. Benchmark numerical results and discussion
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It is necessary to validate the above analytic solutions by giving some numerical results such that our solutions can be used as benchmarks for future comparison.
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Actually, there is insufficient literature on the analytic free vibration solutions of rectangular thin cantilever plates. The monograph by Leissa [1] has provided some useful results obtained by different approximate/numerical methods. Among those we choose a series method to compare with ours because it has the best accuracy in the monograph, as verified by our FEM, by which the convergent results are also tabulated to compare with all our solutions to offer more solid validation. Table 1 lists the first ten frequency parameter solutions for the rectangular thin cantilever plates with the Poisson’s ratio 0.3 under ten different side ratios, with b a 12
ACCEPTED MANUSCRIPT ranging from 0.5 to 5. That is, a hundred numerical frequency results are given to furnish a comprehensive view of the accuracy of our analytic method. It is seen that the present solutions agree well with [1]. However, the solutions by our analytic method are more accurate, which is confirmed by perfect agreement with FEM. The series terms are taken to be 100 in our derivation. This is to guarantee the convergence of the five significant digits, as shown in Table 1. As the extreme cases, we study the convergence of the plate with b a 0.5 and 5, respectively. As tabulated
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in Table 2, with increasing N , the solutions converge fast, and tens of terms can achieve accurate enough results. For FEM, ABAQUS software [31] has been used, where the thickness-to-width ratio of the plates h a is 106 (i.e., taking a 1 and h 106 ), with the plates’ mass density 106 , Young’s modulus 1.092 1019 , and
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Poisson’s ratio 0.3. S4R thin shell element is chosen, and the mesh size is uniformly set to be 1/100 a . Figure 2 plots the first ten mode shape solutions, where the coordinate system is in accordance with that in figure 1. Perfect agreement with FEM is observed. The accurate enough solutions presented here are expected to serve as the benchmarks for validation of various approximate/numerical methods.
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It should be noted that the present method is suitable for the conservative problems. If the damping exists, the method is not applicable, and certain weak
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artificial dissipative numerical methods could be employed, such as the generalized multi-symplectic method [32,33], which has been proved to be a robust
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[34-38].
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structure-preserving method for the damping infinite dimensional dynamic problems
[Table 1] [Table 2] [Figure 2]
5. Conclusion Analytic free vibration solutions of rectangular thin cantilever plates are obtained using the symplectic superposition method. The solution procedure involves leading the governing equations to the Hamiltonian system, solving the fundamental problems in the symplectic space via separation of variables and symplectic eigen expansion, and superposing the fundamental solutions to yield the final analytic solutions. The 13
ACCEPTED MANUSCRIPT advantage of the developed method is its rationality without predetermining the solution forms. This enables more analytic solutions of some difficult problems to be explored, which cannot be obtained by the classical methods. The comprehensive numerical results are tabulated and plotted to provide the benchmarks which are useful for validation of other methods. Ongoing work is on some more complex problems such as those for anisotropic monolayer plates and laminated composites.
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Acknowledgments
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (grant no. 11302038) and National Basic Research Program of China (grant no. 2014CB046506). R.L. gratefully acknowledges the supports from the Young Elite Scientist Sponsorship Program by CAST (No.
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2015QNRC003) and Young Science and Technology Star Program of Dalian.
References
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7.
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3.
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2.
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Journal of Structural Stability and Dynamics 9, 267-284. Civalek Ö, Ersoy H. 2009 Free vibration and bending analysis of circular Mindlin plates using singular convolution method. Communications in Numerical Methods in Engineering 25, 907-922. Civalek Ö, Ozturk B. 2010 Vibration analysis of plates with curvilinear quadrilateral domains by discrete singular convolution method. Structural Engineering and Mechanics 36, 279-299. Civalek Ö. 2004 Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Engineering Structures 26, 171-186. Malekzadeh P, Haghighi MRG, Beni AA. 2012 Buckling analysis of functionally graded arbitrary straight-sided quadrilateral plates on elastic foundations. Meccanica 47, 321-333. Malekzadeh P, Shojaee M. 2013 Buckling analysis of quadrilateral laminated plates with carbon nanotubes reinforced composite layers. Thin-Walled Structures 71, 108-118. Zenkour AM. 2000 Natural vibration analysis of symmetrical cross-ply laminated plates using a mixed variational formulation. European Journal of Mechanics A-Solids 19, 469-485. Zenkour AM. 2001 Buckling and free vibration of elastic plates using simple and mixed shear deformation theories. Acta Mechanica 146, 183-197. Gorman DJ. 1976 Free vibration analysis of cantilever plates by method of superposition. Journal of Sound and Vibration 49, 453-467. Zhong Y, Zhao X-F, Li R. 2013 Free vibration analysis of rectangular cantilever plates by finite integral transform method. International Journal for Computational Methods in Engineering Science and Mechanics 14, 221-226. Tian B, Li R, Zhong Y. 2015 Integral transform solutions to the bending problems of moderately thick rectangular plates with all edges free resting on elastic foundations. Applied Mathematical Modelling 39, 128-136. Xing Y, Liu B. 2009 Exact solutions for the free in-plane vibrations of rectangular plates. International Journal of Mechanical Sciences 51, 246-255. Liu B, Xing Y. 2011 Exact solutions for free vibrations of orthotropic rectangular Mindlin plates. Composite Structures 93, 1664-1672. Lim CW, Lu CF, Xiang Y, Yao W. 2009 On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. International Journal of Engineering Science 47, 131-140. Yao W, Zhong W, Lim CW. 2009 Symplectic elasticity. Singapore: World Scientific. Lim CW, Xu XS. 2010 Symplectic elasticity: theory and applications. Applied Mechanics Reviews 63, 050802. Li R, Wang P, Tian Y, Wang B, Li G. 2015 A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates. Scientific Reports 5, 17054. Li R, Tian Y, Wang P, Shi Y, Wang B. 2016 New analytic free vibration solutions of rectangular thin plates resting on multiple point supports. International Journal of Mechanical Sciences 110, 53-61. Li R, Ni X, Cheng G. 2015 Symplectic superposition method for benchmark flexure solutions for rectangular thick plates. Journal of Engineering Mechanics 141, 04014119. Li R, Wang B, Li G, Tian B. 2016 Hamiltonian system-based analytic
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27. 28. 29. 30.
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33. 34. 35. 36. 37.
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38.
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32.
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modeling of the free rectangular thin plates' free vibration. Applied Mathematical Modelling 40, 984-992. ABAQUS. 2013 Analysis User's Guide V6.13. Pawtucket, RI: Dassault Systèmes. Hu W, Deng Z, Han S, Zhang W. 2013 Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs. Journal of Computational Physics 235, 394-406. Hu W, Deng Z, Yin T. 2017 Almost structure-preserving analysis for weakly linear damping nonlinear Schrödinger equation with periodic perturbation. Communications in Nonlinear Science and Numerical Simulation 42, 298-312. Hu W, Deng Z, Wang B, Ouyang H. 2013 Chaos in an embedded single-walled carbon nanotube. Nonlinear Dynamics 72, 389-398. Hu W, Deng Z. 2015 Chaos in embedded fluid-conveying single-walled carbon nanotube under transverse harmonic load series. Nonlinear Dynamics 79, 325-333. Hu W, Song M, Deng Z, Zou H, Wei B. 2017 Chaotic region of elastically restrained single-walled carbon nanotube. Chaos 27, 023118. Hu W, Li Q, Jiang X, Deng Z. 2017 Coupling dynamic behaviors of spatial flexible beam with weak damping. International Journal for Numerical Methods in Engineering, DOI: 10.1002/nme.5477. Hu W, Song M, Deng Z, Yin T, Wei B. 2017 Axial dynamic buckling analysis of embedded single-walled carbon nanotube by complex structure-preserving method. Applied Mathematical Modelling, DOI: 10.1016/j.apm.2017.06.040.
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Tables Table 1
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Frequency parameters, a 2 h D , of rectangular thin cantilever plates with the edge y b clamped and the other edges free.
2
2.5
3 3.5 4
4.5 5
4th 76.298 76.307 76 27.199 27.203 27.2 17.477 17.479 12.044 12.044 12.1 9.1544 9.1546 9.21 6.6686 6.6691 4.8909 4.8912 3.7388 3.7390 3.73 2.9501 2.9502 2.3867 2.3871 2.39
5th 87.351 87.373 87.6 30.954 30.959 31.1 23.796 23.800 15.037 15.040 15.03 9.6194 9.6205 9.61 7.3746 7.3747 6.1729 6.1729 5.3085 5.3086 5.34 4.6574 4.6574 4.1493 4.1494 4.16
6th 98.679 98.700 99.2 54.183 54.193 54.3 27.383 27.391 23.129 23.131 23.2 16.948 16.949 17.0 13.114 13.116 9.6221 9.6232 7.3550 7.3557 7.36 5.8023 5.8027 4.6932 4.6935 4.70
7th 125.70 125.72 126 61.253 61.297 61.3 35.650 35.657 23.275 23.279 23.3 18.854 18.858 18.9 13.331 13.332 10.975 10.975 9.3250 9.3252 9.36 8.1073 8.1074 7.1726 7.1728 7.19
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1.5
3rd 40.722 40.723 40.88 21.284 21.289 21.29 9.5401 9.5411 5.3583 5.3586 5.37 3.4233 3.4235 3.42 2.3736 2.3736 1.9854 1.9854 1.7196 1.7196 1.72 1.5166 1.5166 1.3565 1.3570 1.35
M
1
2nd 21.403 21.404 21.52 8.5062 8.5064 8.55 5.1804 5.1804 3.7003 3.7004 3.71 2.8738 2.8738 2.87 2.3484 2.3484 1.7414 1.7415 1.3317 1.3318 1.33 1.0511 1.0511 0.85061 0.85113 0.85
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Present FEM Leissa [1] Present FEM Leissa [1] Present FEM Present FEM Leissa [1] Present FEM Leissa [1] Present FEM Present FEM Present FEM Leissa [1] Present FEM Present FEM Leissa [1]
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0.5
Modes 1st 13.971 13.972 13.972 3.4710 3.4711 3.472 1.5348 1.5348 0.85981 0.85992 0.861 0.54848 0.54836 0.549 0.37990 0.37982 0.27852 0.27860 0.21288 0.21357 0.213 0.16796 0.16721 0.13589 0.13980 0.135
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References
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b a
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8th 136.11 136.15 136.4 64.142 64.174 64.2 37.920 37.925 29.612 29.621 29.6 22.993 22.997 23.0 20.699 20.700 15.942 15.945 12.197 12.199 12.20 9.6241 9.6252 7.7845 7.7853 7.79
9th 172.34 172.37 172.8 70.961 71.003 71.1 53.148 53.180 31.731 31.736 — 27.009 27.012 27.0 21.485 21.490 16.717 16.718 14.001 14.001 14.05 12.038 12.038 10.557 10.558 10.60
10th 210.83 210.93 — 92.927 92.963 — 59.493 59.505 38.193 38.203 38.3 28.211 28.214 — 23.004 23.008 22.450 22.454 18.238 18.242 — 14.411 14.414 11.661 11.663 —
Table 2
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Convergence of the analytic frequency solutions, a 2 h D , for rectangular thin cantilever plates having b/a=0.5 and 5, with the edge y b clamped and the other edges free. b/a
Number of series terms
Modes 1st
2nd
3rd
4th
5th
6th
0.5
10
13.968
21.402
40.724
76.298
87.351
98.679
20
13.970
21.402
40.722
76.299
87.349
98.675
30
13.971
21.403
40.721
76.298
87.349
50
13.971
21.403
40.721
76.298
100
13.971
21.404
40.722
76.298
10
0.13590
0.85073
1.3578
2.3873
20
0.13589
0.85062
1.3568
2.3868
30
0.13589
0.85061
1.3566
2.3868
50
0.13589
0.85061
1.3565
2.3867
100
0.13589
0.85061
1.3565
2.3867
9th
10th
136.11
172.34
210.82
125.70
136.11
172.35
210.83
98.676
125.70
136.11
172.34
210.83
87.350
98.677
125.70
136.11
172.34
210.83
87.351
98.679
125.70
136.11
172.34
210.83
4.1535
4.6949
7.1801
7.7880
10.570
11.668
4.1502
4.6934
7.1742
7.7851
10.560
11.662
4.1496
4.6932
7.1732
7.7847
10.558
4.1493
4.6932
7.1726
7.7845
10.557
11.661 11.661
4.1493
4.6932
7.1726
7.7845
10.557
11.661
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8th
125.71
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5
7th
18
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Figures and figure captions
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AC
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Figure 1. Symplectic superposition for free vibration of a rectangular thin cantilever plate.
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Present
FEM
Mode 1
Mode 2
Mode 4
Mode 5
Mode 8
Mode 9
Mode 10
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Present
Mode 3
Mode 6
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FEM
Mode 7
AC
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Figure 2. First ten mode shapes of a square thin cantilever plate with the edge y b ( b a ) clamped and the other edges free.
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On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space Rui Li , Pengcheng Wang , Zekun Yang , Jiaqi Yang , Linghui Tong PII: DOI: Reference:
S0307-904X(17)30562-0 10.1016/j.apm.2017.09.011 APM 11956
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
7 January 2017 2 August 2017 6 September 2017
Please cite this article as: Rui Li , Pengcheng Wang , Zekun Yang , Jiaqi Yang , Linghui Tong , On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.09.011
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Highlights Free vibration of cantilever plates is solved in the symplectic space. Analytic free vibration solutions of rectangular cantilever plates are obtained. More similar problems are expected to be solved by the present
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CE
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approach.
1
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On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space Rui Lia,b,*, Pengcheng Wanga, Zekun Yanga,c, Jiaqi Yanga,c, and Linghui Tonga,c a
State Key Laboratory of Structural Analysis for Industrial Equipment, Department of
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Engineering Mechanics, International Research Center for Computational Mechanics, Dalian University of Technology, Dalian 116024, PR China b
Department of Mechanical Engineering, Northwestern University, Evanston, IL
60208, USA c
School of Naval Architecture & Ocean Engineering, Dalian University of Technology,
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Dalian 116024, PR China *
Corresponding author.
Tel.: +86 15840912019.
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E-mail address: [email protected], [email protected] (R. Li). Abstract: In this paper, we obtain accurate analytic free vibration solutions of
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rectangular thin cantilever plates by using an up-to-date rational superposition method in the symplectic space. To the authors’ knowledge, these solutions were not available
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in the literature due to the difficulty in handling the complex mathematical model. The Hamiltonian system-based governing equation is first constructed. The
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eigenvalue problems of two fundamental vibration problems are formed for a cantilever plate. By symplectic expansion, the fundamental solutions are obtained. Superposition of these solutions are equal to that of the cantilever plate, which yields
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the analytic frequency equation. The mode shapes are then readily obtained. The developed method yields the benchmark analytic solutions with fast convergence and satisfactory accuracy by rigorous derivation, without assuming any trial solutions; thus, it is regarded as rational, and its applicability to more boundary value problems of partial differential equations represented by plates’ vibration, bending and buckling may be expected. Keywords: analytic solution; free vibration; cantilever plate; symplectic space. 2
ACCEPTED MANUSCRIPT Nomenclature In-plane dimensions of plate Flexural rigidity of plate Coefficients of series expansion Thickness of plate Hamiltonian matrix Unit matrix Symplectic matrix Bending moments per unit distance in plate Equivalent shear forces per unit distance in plate Modal displacement Rectangular coordinates Eigenvectors Coefficient functions of eigenvectors State vectors Eigenvalues Poisson’s ratio Derivative of w with respect to y Mass density of plate Circular frequency
ρ
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a, b D E, F, G, H h H I J Mx, My Vx, Vy w x, y X(x), Y'(y) Y(y), X'(x) Z, Z' μ, μ'
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1. Introduction
In both mechanical and civil engineering, thin plates play an important role. They
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are generally used as support structures such as bridge decks, floor slabs, foundation beds, and aircraft panels. In elasticity, thin plates are well described by the Kirchhoff
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plate theory, which has been an indispensable part of solid mechanics. In applied mathematics, the above mechanics model is affiliated with a class of biharmonic
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equations, which are the fourth-order partial differential equations (PDEs). With proper boundary constraints, the associated boundary value problems arise. It is well
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known that such a high-order PDE is normally hard to solve analytically, especially for that with specific boundary conditions. Taking rectangular thin plates as examples, their free vibration problems have long been the key issues in elasticity and there is a fairly large literature on the topic. However, except the plates with two opposite edges simply supported, few of the others have been analytically solved by the classical approaches. Some newly proposed analytic methods are found in recent publications, which focus on specific plates without two opposite edges simply supported, such as those with point supports or with all edges free. However, the rectangular thin cantilever plates, characterized by involving free corners and both clamped and free 3
ACCEPTED MANUSCRIPT edges, have not been well treated, and there have been no accurate analytic solutions found in the literature, although they are of much importance in engineering applications, such as the overhanging roofs, airplane wings, cantilever bridges and balconies, and even in micro/nano technology, such as the atomic force microscope cantilevers. This paper aims to explore the analytic free vibration solutions of such a classical but extremely difficult class of problems. There are various solution methods for rectangular thin cantilever plates’ free
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vibration, most of which are, of course, approximate or numerical, as introduced in Leissa’s famous technical report [1]. The Ritz method was used by Barton [2] to determine approximate solutions for the frequencies and modes of vibration of rectangular and skew cantilever plates, where the deflection functions correspond to those defining the normal modes of vibration of a beam. Claassen [3] applied the
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Fourier sine-series method to obtain approximate solutions for the transverse vibrations of a thin rectangular cantilever plate. Rajalingham et al. [4] obtained closed form approximation of vibration modes of rectangular cantilever plates by the variational reduction method, where the plate PDE was reduced to two simultaneous ordinary differential equations and their boundary conditions. Liang et al. [5]
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presented a simple procedure using an empirical added mass formulation and Rayleigh-Ritz method to analyze the vibration frequencies and mode shapes of
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submerged cantilever plates. Ergin and Uğurlu [6] then investigated the dynamic characteristics, such as natural frequencies and mode shapes, of cantilever plates
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partially submerged in a fluid by use of a boundary-integral equation method together with the method of images. Seok et al. [7] performed an analysis of the free transverse
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vibrations of a cantilevered plate by means of a variational approximation procedure. Looker and Sader [8] proposed a simple uniformly valid expression for the
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fundamental flexural vibration frequency of a thin rectangular cantilever plate using an accurate variational approach based on an energy minimization principle and a singular perturbation solution. Rostami et al. [9] recently provided accurate approximate solution for in-plane vibrations of rotating orthotropic cantilever plates using the Extended Kantorovich Method. There are some other popular numerical solution methods for plate problems. Lindsay et al. [10] studied the vibration of a thin plate with a collection of clamped patches, and obtained detailed information on the limiting behavior in the form of an asymptotic expansion in the limit of small patch radius. Civalek et al. [11-13] developed the discrete singular convolution method for 4
ACCEPTED MANUSCRIPT the free vibration analysis of both thin and thick plates. The plate buckling problems were analyzed by Civalek [14] and Malekzadeh et al. [15,16] by employing the differential quadrature method. Zenkour et al. [17,18] investigated the buckling and free vibration of plates using approximate displacement solutions, which depend on the boundary conditions on the plate edges. Except for the approximate or numerical methods, there are few analytic methods handling the plate problems. Gorman [19] employed the semi-inverse superposition
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method to analyze the first five symmetric and anti-symmetric free vibration modes of a cantilever plate for a wide range of aspect ratios. Zhong et al. [20] used the double finite integral transform method to derive the eigenfrequencies and vibration modes of rectangular thin cantilever plates. The method was also applied by Tian et al. [21] to bending of the rectangular thick plates. Xing and Liu [22,23] proposed a method of
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direct separation of variables to obtain both the in-plane and out-of-plane free vibration solutions of rectangular thin or thick plates. Lim et al. [24] developed the symplectic elasticity approach for free vibration of rectangular Levy-type thin plates employing the Hamiltonian variational principle with Legendre’s transformation. The approach was proposed by Yao et al. [25] and has been extended to many other fields,
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as described in a comprehensive review by Lim and Xu [26]. It should be noted that the conventional symplectic method has not exhibited the ability to analytically solve opposite edges.
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the vibration of the plates with free edges while without two simply supported
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In recent years, we have proposed a novel analytic symplectic superposition method, which has shown applicability to general mechanics problems of the plates
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by obtaining the analytic solutions of some thin/thick plates’ bending, vibration and buckling problems [27-29]. In this paper, we develop the symplectic superposition
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method further to explore accurate analytic free vibration solutions of isotropic rectangular thin cantilever plates. The procedure starts from solving the Hamiltonian system-based governing equation of two fundamental problems, where the mathematical techniques in the symplectic space are applied, including the separation of variables, which is sometimes invalid in the Euclid space, and symplectic eigen expansion. Based on the obtained analytic solutions, skillful superposition is imposed to yield the analytic solution of the original problem. There are some boundary conditions to be satisfied in equating the superposition of the fundamental solutions to the final solution of a cantilever plate, which leads to the frequency equation. One will 5
ACCEPTED MANUSCRIPT see in the following that the developed method derives the analytic solutions in an elegant rational way, because there are no predetermined solutions but rigorous derivations from the governing equation. This constitutes the main advantage of the method, enabling one to obtain the analytic solutions which were not available in the past, including those developed in this paper. Many numerical results are presented to show the convergence and accuracy of the new analytic solutions by excellent comparison with those from the literature, if any, and the finite element method
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(FEM). They are expected to serve as the benchmarks for validation of the other methods.
2. Governing equation and fundamental solutions by symplectic
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eigen expansion
The governing equation for free vibration of a thin isotropic plate in the rectangular coordinate system x, y can be expressed in the Hamiltonian system as [27,30]
Z y HZ ,
0 F 2 2 x
1 , 0
T
F
, H , Q T Q F 0
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w,
equivalent shear force Vy :
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moment parallel to the
x
the circular frequency,
2
0
0
derivative of
T
Z
2
include
with respect to y : , opposite of the
w
, where Vy D 3 w y3 2 3 w x 2y , and bending
axis: M y . is the mass density, D
equation (2.1), the matrix
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, 2 D 1 x 0
and G . The components of the state vector 0 1 D
the modal displacement
h 2 D 1 2 4 x 4
G
M
Z w, , T , M y
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where
(2.1)
h
the plate thickness,
the flexural rigidity, and the Poisson’s ratio. In H
0
satisfies HT JHJ , where J I 2
symplectic matrix in which I 2 is the
2 2
I2 0
unit matrix. This means that
is the H
is a
Hamiltonian matrix with differential operators. By equation (2.1), the free vibration problem of a thin plate is actually described in the symplectic space [25] such that the separation of variables and symplectic eigen expansion are valid and the rational analytic solutions can be obtained without predetermining the solution forms, which cannot be realized with the classical 6
ACCEPTED MANUSCRIPT semi-inverse method in the Euclidean space. As shown in figure 1(a), what we consider is a rectangular thin cantilever plate. The origin, O, is at a corner of the plate. The plate edges with lengths
a
and
b,
x
and y axes are directed along the
respectively. The solution approach here starts
from dividing the problem into two fundamental problems, as shown in figure 1(b, c), whose solutions will be obtained by symplectic eigen expansion in the following. Superposition of the fundamental solutions yields the solution of the cantilever plate.
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It must be pointed out that each fundamental solution has the constants to be determined by equating the superposition of the fundamental problems’ supports at each edge with the support at the associated edge of the cantilever plate. The generated equations are used to determine the natural frequencies, which will be
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elaborated in section 3. [Figure 1]
In figure 1(b), the plate is slidingly clamped at x 0 ,
My
y b
y 0
and simply
and bending moment denoted by
M
supported at y b ; the slope denoted by
xa, y 0,
are imposed along y 0 and y b , respectively. In figure 1(c), the same plate w x x 0
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is subject to the slopes
and
w x x a
distributed along x 0 and
xa,
respectively. The above two fundamental problems are solved below.
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For the first fundamental problem as shown in figure 1(b), solving equation (2.1) is first reduced to solving by applying
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HX x X x
dY y dy Y y
Z X x Y y
and the eigenvalue problem
via variable separation, where
function of y only, the eigenvalue, and
Y y
X x w x , x , T x , M y x
T
is a the
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eigenvector. The eigenvalue problem is solved under the boundary conditions w x x x 0, a Vx x x 0, a 0 ,
Vx x, y
where
Vx x
is the factor of the equivalent shear force
which is equal to D 3 w x, y x3 2 3 w x, y xy 2 . It is not difficult
to obtain the following eigenvalues: n1 n2 R , n 2 n2 R , n3 n2 R , n 4 n2 R
for
n 0,1, 2, 3,
(2.2)
, where n n a , and R h D . It is noted that there are four
groups of eigenvalues given in equation (2.2), which depend on the non-negative 7
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n.
Corresponding to equation (2.2), we have the eigenvectors T
X n1 x 1,
n2 R , D R n2 1 n2 R , D R n2 1 cos n x ,
T
X n2 x 1, n2 R , D R n2 1 n2 R , D R n2 1 cos n x , T
X n3 x 1,
n2 R , D R n2 1 n2 R , D R n2 1 cos n x ,
(2.3)
T
X n4 x 1, n2 R , D R n2 1 n2 R , D R n2 1 cos n x .
yields the corresponding
exp
Yn1 y cn1 exp
2 n
Yn3 y cn3
2 n
R y , Y
Y y
such that
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dY y dy Y y
For each ,
exp
R y,
R y , Yn 2 y cn 2 exp n2 R y , n 4 y cn 4
n2
(2.4)
where cn1 to cn 4 are the constants to be determined. The solution of equation (2.1) is thus written as
(2.5)
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Z X x Y y ,
where X x
, Xn1 x , Xn2 x , Xn3 x , X n4 x ,
Y y
, Yn1 y , Yn 2 y , Yn3 y , Yn 4 y ,
,
(2.6)
T
.
and the bending moment M y
y 0
the cosine series, i.e. 0,
y 0
y 0
En cos n x , w y b 0, M y
n 0,1,2,
y b
n 0,1,2,
Fn cos n x ,
(2.7)
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Vy
are respectively expanded to
y b
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imposed slope
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At the slidingly clamped edge y 0 and the simply supported edge y b , the
where En and Fn are the coefficients of expansion. Substituting the solution (2.5)
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into equation (2.7) to determine the constants cn1 to cn 4 , the dimensionless modal displacement for the first fundamental problem, w1 x, y , is a
F sec cos y sech ch y
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w1 x , y
1 E0 sh y tanh ch y sec sin 1 y 2 0
n 1,2,3,
cos nx 2
E sech sh n
n
n
1 y n 2 n 2 2 n
sech n sh n 1 y n 2 n 2 2 n
(2.8)
Fn sech n ch n y sech n ch n y ,
where x x a ,
yy b ,
b a ,
F0 aF0 D ,
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Fn aFn D ,
a2 h D ,
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For the second fundamental problem as shown in figure 1(c), the coordinates exchange should be applied such that
x
( y ) and
a
( b ) are respectively replaced by
y ( x ) and b ( a ). Therefore, solving the modified governing equation Z x HZ
is first reduced to solving dX x dx X x and the eigenvalue problem HY y Y y by applying Z Y y X x via variable separation, where X x
a
function
of
only,
x
Y y w y , w x y , Vx y , M x y h 2 D 1 2 4 y 4 Q 0
T
the
eigenvalue,
G F T Q F
the eigenvector; H
0 and F 2 2 2 D 1 y y 0
2
and
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is
2
1 . 0
, in which
The eigenvalue
w y y b M y y
y b
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problem is solved under the boundary conditions w y y y 0 Vy y y 0 0 and 0 . The obtained eigenvalues are
n1 n2 R , n2 n2 R , n3 n2 R , n4 n2 R
(2.9)
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for n 1,3,5, , where n n 2b . Corresponding to equation (2.9), we have the eigenvectors
T
2 n2 R , D R n 21 n R , D R n 12 cos n y ,
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Yn1 y 1,
T
Yn2 y 1, R , D R 1 R , D R 1 cos n y , 2 n
2 n
2 n
T
R , D R 1 R , D R 1 cos n y , 2 n
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Yn3 y 1,
2 n
2 n
2 n
2 n
(2.10)
T
Yn4 y 1, n2 R , D R n2 1 n2 R , D R n2 1 cos n y .
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that
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For each , dX x dx X x yields the corresponding X x such
X x c exp X n1 x cn1 exp n3
n3
Rx , X x c exp
Rx ,
n2 Rx , X n2 x cn 2 exp n2 Rx , n2
n4
n4
n2
(2.11)
where cn1 to cn 4 are the constants to be determined. The solution of the second fundamental problem is thus written as Z Y y X x ,
where
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(2.12)
ACCEPTED MANUSCRIPT Y y , Yn1 y , Yn2 y , Yn3 y , Yn4 y ,
,
x , X n2 x , X n3 x , X n4 x, X x , X n1
.
At the slidingly clamped edges x 0 and
xa,
(2.13)
T
the imposed slopes w x x 0 and
w x x a are respectively expanded to the half cosine series, i.e. Vx
w Gn cos n y , Vx x x 0 n 1,3,5,
0,
x 0
xa
0,
w H n cos n y . x x a n 1,3,5,
(2.14)
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where Gn and H n are the coefficients of expansion. Substituting the solution (2.12) into equation (2.14) to determine the constants cn1 to cn 4 , the dimensionless modal displacement for the second fundamental problem, w2 x, y , is w2 x , y b
n 1,3,5,
1 n y cos 4 2 2
H n csch n ch n 2 2
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2 n 2 2 2 n 2 2 Gn csch n ch n 1 x n csch n ch n 1 x n (2.15) n n 2 2 2 2 2 n 2 2 x n csch n ch n n 2 2
2 n 2 2 x n , n
where n n2 2 4 2 and n n2 2 4 2 .
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3. Determination of natural frequencies and mode shapes As depicted in section 2, superposition of the two fundamental solutions, (2.8) and
ED
(2.15), yields the solution of a cantilever plate, which contains the undetermined coefficients of expansion. To determine the natural frequencies, a set of equations are
PT
formed by making the above superposition satisfy the real boundary conditions at each edge of the cantilever plate.
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At the free edge y 0 of the cantilever plate, the superposition of the fundamental solutions has satisfied the boundary condition of zero equivalent shear
AC
force, Vy
y 0
0 . To satisfy another boundary condition that the bending moment M y
be zero, we equate the superposition of the bending moments of the fundamental problems at y 0 to zero, which yields, after simplification,
tan tanh E0 sec sech F0
n 1,3,5,
32 2 4 Gn H n n 4 4 16 2 4
0,
and
10
(3.1)
ACCEPTED MANUSCRIPT tanh 2 i 2 2 2 tanh 2 i 2 2 2 E i i i i i i i
i i sech i i 2 i 2 2 sech i i 2 i 2 2 Fi
2 16 i i 2 4 2 2 i 2 n 2 4 1 G cos i H 0, n n 2 4 4 2 2 2 2 16 n 4i
(3.2)
n 1,3,5,
for i 1, 2,3, , where Fi aFi D , i i 2 2 , and i i 2 2 . At the clamped edge y b , the zero modal displacement boundary condition has
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been satisfied. To satisfy another boundary condition that the slope be zero, we equate the superposition of the slopes of the fundamental problems at y b to zero, which yields
sec sech E0 tan tanh F0
n 1,3,5,
3 3 n 4 n Gn H n sin 0, n 4 4 16 2 4 2
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and
(3.3)
sech i i 2 i 2 2 sech i i 2 i 2 2 Ei i tanh i i tanh i Fi
n 1,3, 5,
3 2 2 2 8n n 4i G cos i H n 0 2 n 16 2 4 4 n 2 4i 2 2
(3.4)
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n sin 2
for i 1, 2,3, .
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At the free edge x 0 , the boundary condition that the bending moment M x be zero requires
16 3i 2i 2
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i 512 3 6 E0 64 i 3 3 3 sin F0 2 2 2 ii csch i csch i i i 2 i 2 2 sh i ch i i i 2 i 2 2 ch i sh i Gi 2 2 2 2 2 2 2 2 ii i i 2 i 2 2 csch i i i 2 i 2 2 csch i H i 2 2 n 1, 2,3,
16 2
4
4
i
2
4n 2
2 2
2 i 2 2 2 2 4 3 2 2 2 8 4 i n 1 En 4i i 4n sin 2
(3.5)
for i 1,3,5, , where i i 2 2 4 2 and i i 2 2 4 2 . Similarly, at
Fn 0
xa,
the boundary condition that M x be zero requires
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ACCEPTED MANUSCRIPT i 512 3 6 E0 64 i 3 3 3 sin F0 2 2 2 ii i i 2 i 2 2 csch i i i 2 i 2 2 csch i Gi 2 2 2 2 ii csch i csch i i i 2 i 2 2 sh i ch i i i 2 i 2 2 ch i sh i H i 2 2 2 2 2 2
16 3i 2i 2 cos n
n 1,2,3,
16 2
4
4
i
2
4n 2
2 2
2 i 2 2 2 2 4 3 2 2 2 8 4 i n 1 En 4i i 4n sin Fn 0 2
for i 1,3,5, .
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(3.6) Equations (3.1)-(3.6) constitute a set of infinite linear equations with respect to Em , Fm , Gn , and H n ( m 0,1, 2,
; n 1,3,5,
). The existence of nonzero solutions
requires that the determinant of the coefficient matrix be zero, which leads to an
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equation for determination of the natural frequencies. Noting that a finite number of equations are necessary for practical calculation, any accuracy can be achieved by taking adequate series terms. For convenience, the same number of series terms, denoted by N , are adopted throughout, i.e., E0 , E1 , E2 , , EN , F0 , F1 , F2 , , FN , G0 , G1 , G2 ,
, GN , and
H0 , H1 , H 2 ,
, HN
are taken into account. The mode shape
M
corresponding to a natural frequency is obtained by first substituting the frequency solution back into equations (3.1)-(3.6), solving for a set of non-trivial solutions for
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their summation.
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Em , Fm , Gn , and H n , substituting them into equations (2.8) and (2.15), and then
4. Benchmark numerical results and discussion
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It is necessary to validate the above analytic solutions by giving some numerical results such that our solutions can be used as benchmarks for future comparison.
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Actually, there is insufficient literature on the analytic free vibration solutions of rectangular thin cantilever plates. The monograph by Leissa [1] has provided some useful results obtained by different approximate/numerical methods. Among those we choose a series method to compare with ours because it has the best accuracy in the monograph, as verified by our FEM, by which the convergent results are also tabulated to compare with all our solutions to offer more solid validation. Table 1 lists the first ten frequency parameter solutions for the rectangular thin cantilever plates with the Poisson’s ratio 0.3 under ten different side ratios, with b a 12
ACCEPTED MANUSCRIPT ranging from 0.5 to 5. That is, a hundred numerical frequency results are given to furnish a comprehensive view of the accuracy of our analytic method. It is seen that the present solutions agree well with [1]. However, the solutions by our analytic method are more accurate, which is confirmed by perfect agreement with FEM. The series terms are taken to be 100 in our derivation. This is to guarantee the convergence of the five significant digits, as shown in Table 1. As the extreme cases, we study the convergence of the plate with b a 0.5 and 5, respectively. As tabulated
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in Table 2, with increasing N , the solutions converge fast, and tens of terms can achieve accurate enough results. For FEM, ABAQUS software [31] has been used, where the thickness-to-width ratio of the plates h a is 106 (i.e., taking a 1 and h 106 ), with the plates’ mass density 106 , Young’s modulus 1.092 1019 , and
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Poisson’s ratio 0.3. S4R thin shell element is chosen, and the mesh size is uniformly set to be 1/100 a . Figure 2 plots the first ten mode shape solutions, where the coordinate system is in accordance with that in figure 1. Perfect agreement with FEM is observed. The accurate enough solutions presented here are expected to serve as the benchmarks for validation of various approximate/numerical methods.
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It should be noted that the present method is suitable for the conservative problems. If the damping exists, the method is not applicable, and certain weak
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artificial dissipative numerical methods could be employed, such as the generalized multi-symplectic method [32,33], which has been proved to be a robust
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[34-38].
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structure-preserving method for the damping infinite dimensional dynamic problems
[Table 1] [Table 2] [Figure 2]
5. Conclusion Analytic free vibration solutions of rectangular thin cantilever plates are obtained using the symplectic superposition method. The solution procedure involves leading the governing equations to the Hamiltonian system, solving the fundamental problems in the symplectic space via separation of variables and symplectic eigen expansion, and superposing the fundamental solutions to yield the final analytic solutions. The 13
ACCEPTED MANUSCRIPT advantage of the developed method is its rationality without predetermining the solution forms. This enables more analytic solutions of some difficult problems to be explored, which cannot be obtained by the classical methods. The comprehensive numerical results are tabulated and plotted to provide the benchmarks which are useful for validation of other methods. Ongoing work is on some more complex problems such as those for anisotropic monolayer plates and laminated composites.
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Acknowledgments
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (grant no. 11302038) and National Basic Research Program of China (grant no. 2014CB046506). R.L. gratefully acknowledges the supports from the Young Elite Scientist Sponsorship Program by CAST (No.
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2015QNRC003) and Young Science and Technology Star Program of Dalian.
References
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7.
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3.
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2.
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Journal of Structural Stability and Dynamics 9, 267-284. Civalek Ö, Ersoy H. 2009 Free vibration and bending analysis of circular Mindlin plates using singular convolution method. Communications in Numerical Methods in Engineering 25, 907-922. Civalek Ö, Ozturk B. 2010 Vibration analysis of plates with curvilinear quadrilateral domains by discrete singular convolution method. Structural Engineering and Mechanics 36, 279-299. Civalek Ö. 2004 Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Engineering Structures 26, 171-186. Malekzadeh P, Haghighi MRG, Beni AA. 2012 Buckling analysis of functionally graded arbitrary straight-sided quadrilateral plates on elastic foundations. Meccanica 47, 321-333. Malekzadeh P, Shojaee M. 2013 Buckling analysis of quadrilateral laminated plates with carbon nanotubes reinforced composite layers. Thin-Walled Structures 71, 108-118. Zenkour AM. 2000 Natural vibration analysis of symmetrical cross-ply laminated plates using a mixed variational formulation. European Journal of Mechanics A-Solids 19, 469-485. Zenkour AM. 2001 Buckling and free vibration of elastic plates using simple and mixed shear deformation theories. Acta Mechanica 146, 183-197. Gorman DJ. 1976 Free vibration analysis of cantilever plates by method of superposition. Journal of Sound and Vibration 49, 453-467. Zhong Y, Zhao X-F, Li R. 2013 Free vibration analysis of rectangular cantilever plates by finite integral transform method. International Journal for Computational Methods in Engineering Science and Mechanics 14, 221-226. Tian B, Li R, Zhong Y. 2015 Integral transform solutions to the bending problems of moderately thick rectangular plates with all edges free resting on elastic foundations. Applied Mathematical Modelling 39, 128-136. Xing Y, Liu B. 2009 Exact solutions for the free in-plane vibrations of rectangular plates. International Journal of Mechanical Sciences 51, 246-255. Liu B, Xing Y. 2011 Exact solutions for free vibrations of orthotropic rectangular Mindlin plates. Composite Structures 93, 1664-1672. Lim CW, Lu CF, Xiang Y, Yao W. 2009 On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. International Journal of Engineering Science 47, 131-140. Yao W, Zhong W, Lim CW. 2009 Symplectic elasticity. Singapore: World Scientific. Lim CW, Xu XS. 2010 Symplectic elasticity: theory and applications. Applied Mechanics Reviews 63, 050802. Li R, Wang P, Tian Y, Wang B, Li G. 2015 A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates. Scientific Reports 5, 17054. Li R, Tian Y, Wang P, Shi Y, Wang B. 2016 New analytic free vibration solutions of rectangular thin plates resting on multiple point supports. International Journal of Mechanical Sciences 110, 53-61. Li R, Ni X, Cheng G. 2015 Symplectic superposition method for benchmark flexure solutions for rectangular thick plates. Journal of Engineering Mechanics 141, 04014119. Li R, Wang B, Li G, Tian B. 2016 Hamiltonian system-based analytic
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modeling of the free rectangular thin plates' free vibration. Applied Mathematical Modelling 40, 984-992. ABAQUS. 2013 Analysis User's Guide V6.13. Pawtucket, RI: Dassault Systèmes. Hu W, Deng Z, Han S, Zhang W. 2013 Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs. Journal of Computational Physics 235, 394-406. Hu W, Deng Z, Yin T. 2017 Almost structure-preserving analysis for weakly linear damping nonlinear Schrödinger equation with periodic perturbation. Communications in Nonlinear Science and Numerical Simulation 42, 298-312. Hu W, Deng Z, Wang B, Ouyang H. 2013 Chaos in an embedded single-walled carbon nanotube. Nonlinear Dynamics 72, 389-398. Hu W, Deng Z. 2015 Chaos in embedded fluid-conveying single-walled carbon nanotube under transverse harmonic load series. Nonlinear Dynamics 79, 325-333. Hu W, Song M, Deng Z, Zou H, Wei B. 2017 Chaotic region of elastically restrained single-walled carbon nanotube. Chaos 27, 023118. Hu W, Li Q, Jiang X, Deng Z. 2017 Coupling dynamic behaviors of spatial flexible beam with weak damping. International Journal for Numerical Methods in Engineering, DOI: 10.1002/nme.5477. Hu W, Song M, Deng Z, Yin T, Wei B. 2017 Axial dynamic buckling analysis of embedded single-walled carbon nanotube by complex structure-preserving method. Applied Mathematical Modelling, DOI: 10.1016/j.apm.2017.06.040.
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Tables Table 1
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Frequency parameters, a 2 h D , of rectangular thin cantilever plates with the edge y b clamped and the other edges free.
2
2.5
3 3.5 4
4.5 5
4th 76.298 76.307 76 27.199 27.203 27.2 17.477 17.479 12.044 12.044 12.1 9.1544 9.1546 9.21 6.6686 6.6691 4.8909 4.8912 3.7388 3.7390 3.73 2.9501 2.9502 2.3867 2.3871 2.39
5th 87.351 87.373 87.6 30.954 30.959 31.1 23.796 23.800 15.037 15.040 15.03 9.6194 9.6205 9.61 7.3746 7.3747 6.1729 6.1729 5.3085 5.3086 5.34 4.6574 4.6574 4.1493 4.1494 4.16
6th 98.679 98.700 99.2 54.183 54.193 54.3 27.383 27.391 23.129 23.131 23.2 16.948 16.949 17.0 13.114 13.116 9.6221 9.6232 7.3550 7.3557 7.36 5.8023 5.8027 4.6932 4.6935 4.70
7th 125.70 125.72 126 61.253 61.297 61.3 35.650 35.657 23.275 23.279 23.3 18.854 18.858 18.9 13.331 13.332 10.975 10.975 9.3250 9.3252 9.36 8.1073 8.1074 7.1726 7.1728 7.19
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1.5
3rd 40.722 40.723 40.88 21.284 21.289 21.29 9.5401 9.5411 5.3583 5.3586 5.37 3.4233 3.4235 3.42 2.3736 2.3736 1.9854 1.9854 1.7196 1.7196 1.72 1.5166 1.5166 1.3565 1.3570 1.35
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1
2nd 21.403 21.404 21.52 8.5062 8.5064 8.55 5.1804 5.1804 3.7003 3.7004 3.71 2.8738 2.8738 2.87 2.3484 2.3484 1.7414 1.7415 1.3317 1.3318 1.33 1.0511 1.0511 0.85061 0.85113 0.85
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Present FEM Leissa [1] Present FEM Leissa [1] Present FEM Present FEM Leissa [1] Present FEM Leissa [1] Present FEM Present FEM Present FEM Leissa [1] Present FEM Present FEM Leissa [1]
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0.5
Modes 1st 13.971 13.972 13.972 3.4710 3.4711 3.472 1.5348 1.5348 0.85981 0.85992 0.861 0.54848 0.54836 0.549 0.37990 0.37982 0.27852 0.27860 0.21288 0.21357 0.213 0.16796 0.16721 0.13589 0.13980 0.135
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References
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b a
17
8th 136.11 136.15 136.4 64.142 64.174 64.2 37.920 37.925 29.612 29.621 29.6 22.993 22.997 23.0 20.699 20.700 15.942 15.945 12.197 12.199 12.20 9.6241 9.6252 7.7845 7.7853 7.79
9th 172.34 172.37 172.8 70.961 71.003 71.1 53.148 53.180 31.731 31.736 — 27.009 27.012 27.0 21.485 21.490 16.717 16.718 14.001 14.001 14.05 12.038 12.038 10.557 10.558 10.60
10th 210.83 210.93 — 92.927 92.963 — 59.493 59.505 38.193 38.203 38.3 28.211 28.214 — 23.004 23.008 22.450 22.454 18.238 18.242 — 14.411 14.414 11.661 11.663 —
Table 2
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ACCEPTED MANUSCRIPT
Convergence of the analytic frequency solutions, a 2 h D , for rectangular thin cantilever plates having b/a=0.5 and 5, with the edge y b clamped and the other edges free. b/a
Number of series terms
Modes 1st
2nd
3rd
4th
5th
6th
0.5
10
13.968
21.402
40.724
76.298
87.351
98.679
20
13.970
21.402
40.722
76.299
87.349
98.675
30
13.971
21.403
40.721
76.298
87.349
50
13.971
21.403
40.721
76.298
100
13.971
21.404
40.722
76.298
10
0.13590
0.85073
1.3578
2.3873
20
0.13589
0.85062
1.3568
2.3868
30
0.13589
0.85061
1.3566
2.3868
50
0.13589
0.85061
1.3565
2.3867
100
0.13589
0.85061
1.3565
2.3867
9th
10th
136.11
172.34
210.82
125.70
136.11
172.35
210.83
98.676
125.70
136.11
172.34
210.83
87.350
98.677
125.70
136.11
172.34
210.83
87.351
98.679
125.70
136.11
172.34
210.83
4.1535
4.6949
7.1801
7.7880
10.570
11.668
4.1502
4.6934
7.1742
7.7851
10.560
11.662
4.1496
4.6932
7.1732
7.7847
10.558
4.1493
4.6932
7.1726
7.7845
10.557
11.661 11.661
4.1493
4.6932
7.1726
7.7845
10.557
11.661
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8th
125.71
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5
7th
18
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Figures and figure captions
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Figure 1. Symplectic superposition for free vibration of a rectangular thin cantilever plate.
19
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Present
FEM
Mode 1
Mode 2
Mode 4
Mode 5
Mode 8
Mode 9
Mode 10
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Present
Mode 3
Mode 6
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FEM
Mode 7
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Figure 2. First ten mode shapes of a square thin cantilever plate with the edge y b ( b a ) clamped and the other edges free.
20