Seeking analytic free vibration solutions of rectangular thick plates without two parallel simply supported edges is of significance for an insight into the performances of related engineering devices and structures as well as their rapid design. A challenging set of problems concern the vibrating plates with a free corner, i.e., those with two adjacent edges free and the other two edges clamped or simply supported or one of them clamped and the other one simply supported. The main difficulty in solving one of such problems is to find a solution meeting both the boundary conditions at each edge and the condition at the free corner, which is unattainable using a conventional analytic method. In this paper, for the first time, we extend a novel symplectic superposition method to free vibration of rectangular thick plates with a free corner. The analytic frequency and mode shape solutions are both obtained and presented via comprehensive numerical and graphic results. The rigorousness in mathematical derivation and rationality of the method (without any predetermination for the solutions) guarantee the validity of our analytic solutions, which themselves are also validated by the reported results and refined finite element analysis.

References

References
1.
Leissa
,
A. W.
,
1969
,
Vibration of Plates, Office of Technology Utilization
,
NASA
,
Washington DC
.
2.
Reissner
,
E.
,
1945
, “
The Effect of Transverse Shear Deformation on the Bending of Elastic Plates
,”
ASME J. Appl. Mech.
,
12
(
2
), pp.
A69
A77
.
3.
Mindlin
,
R. D.
,
1951
, “
Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates
,”
ASME J. Appl. Mech.
,
18
(
1
), pp.
31
38
.
4.
Reddy
,
J. N.
,
1984
, “
A Simple Higher-Order Theory for Laminated Composite Plates
,”
ASME J. Appl. Mech.
,
51
(
4
), pp.
745
752
.
5.
Reddy
,
J. N.
,
1984
, “
A Refined Nonlinear-Theory of Plates With Transverse-Shear Deformation
,”
Int. J. Solids Struct.
,
20
(
9–10
), pp.
881
896
.
6.
Wu
,
K.
, and
Zhu
,
W.
,
2018
, “
A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate
,”
ASME J. Vib. Acoust.
,
140
(▪), p.
011002
.
7.
Wang
,
C. Y.
,
2015
, “
Vibrations of Completely Free Rounded Rectangular Plates
,”
ASME J. Vib. Acoust.
,
137
(
2
), p.
024502
.
8.
Leamy
,
M. J.
,
2016
, “
Semi-Exact Natural Frequencies for Kirchhoff-Love Plates Using Wave-Based Phase Closure
,”
ASME J. Vib. Acoust.
,
138
(
2
), p.
021008
.
9.
Waksmanski
,
N.
,
Pan
,
E.
,
Yang
,
L.-Z.
, and
Gao
,
Y.
,
2014
, “
Free Vibration of a Multilayered One-Dimensional Quasi-Crystal Plate
,”
ASME J. Vib. Acoust.
,
136
(
4
), p.
041019
.
10.
Lai
,
S. K.
, and
Xiang
,
Y.
,
2012
, “
Buckling and Vibration of Elastically Restrained Standing Vertical Plates
,”
ASME J. Vib. Acoust.
,
134
(
1
), p.
014502
.
11.
Malekzadeh
,
P.
, and
Karami
,
G.
,
2004
, “
Vibration of Non-Uniform Thick Plates on Elastic Foundation by Differential Quadrature Method
,”
Eng. Struct.
,
26
(
10
), pp.
1473
1482
.
12.
Cho
,
D. S.
,
Kim
,
B. H.
,
Kim
,
J. H.
,
Vladimir
,
N.
, and
Choi
,
T. M.
,
2015
, “
Forced Vibration Analysis of Arbitrarily Constrained Rectangular Plates and Stiffened Panels Using the Assumed Mode Method
,”
Thin-Walled Struct.
,
90
(▪), pp.
182
190
.
13.
Zhou
,
D.
,
Cheung
,
Y. K.
,
Au
,
F. T. K.
, and
Lo
,
S. H.
,
2002
, “
Three-Dimensional Vibration Analysis of Thick Rectangular Plates Using Chebyshev Polynomial and Ritz Method
,”
Int. J. Solids Struct.
,
39
(▪), pp.
6339
6353
.
14.
Pradhan
,
K. K.
, and
Chakraverty
,
S.
,
2015
, “
Transverse Vibration of Isotropic Thick Rectangular Plates Based on New Inverse Trigonometric Shear Deformation Theories
,”
Int. J. Mech. Sci.
,
94–95
(▪), pp.
211
231
.
15.
Ye
,
T.
,
Jin
,
G.
,
Su
,
Z.
, and
Chen
,
Y.
,
2014
, “
A Modified Fourier Solution for Vibration Analysis of Moderately Thick Laminated Plates With General Boundary Restraints and Internal Line Supports
,”
Int. J. Mech. Sci.
,
80
(▪), pp.
29
46
.
16.
Jin
,
G.
,
Su
,
Z.
,
Shi
,
S.
,
Ye
,
T.
, and
Gao
,
S.
,
2014
, “
Three-Dimensional Exact Solution for the Free Vibration of Arbitrarily Thick Functionally Graded Rectangular Plates With General Boundary Conditions
,”
Compos. Struct.
,
108
(▪), pp.
565
577
.
17.
Zhang
,
H.
,
Shi
,
D.
, and
Wang
,
Q.
,
2017
, “
An Improved Fourier Series Solution for Free Vibration Analysis of the Moderately Thick Laminated Composite Rectangular Plate With Non-Uniform Boundary Conditions
,”
Int. J. Mech. Sci.
,
121
(▪), pp.
1
20
.
18.
Li
,
R.
,
Wang
,
P.
,
Tian
,
Y.
,
Wang
,
B.
, and
Li
,
G.
,
2015
, “
A Unified Analytic Solution Approach to Static Bending and Free Vibration Problems of Rectangular Thin Plates
,”
Sci. Rep.
,
5
(▪), p.
17054
.
19.
Li
,
R.
,
Tian
,
Y.
,
Wang
,
P.
,
Shi
,
Y.
, and
Wang
,
B.
,
2016
, “
New Analytic Free Vibration Solutions of Rectangular Thin Plates Resting on Multiple Point Supports
,”
Int. J. Mech. Sci.
,
110
(▪), pp.
53
61
.
20.
Wang
,
B.
,
Li
,
P.
, and
Li
,
R.
,
2016
, “
Symplectic Superposition Method for New Analytic Buckling Solutions of Rectangular Thin Plates
,”
Int. J. Mech. Sci.
,
119
(▪), pp.
432
441
.
21.
Xing
,
Y.
, and
Liu
,
B.
,
2009
, “
Characteristic Equations and Closed-Form Solutions for Free Vibrations of Rectangular Mindlin Plates
,”
Acta Mech. Solida Sin.
,
22
(▪), pp.
125
136
.
22.
Xing
,
Y.
, and
Liu
,
B.
,
2009
, “
Closed Form Solutions for Free Vibrations of Rectangular Mindlin Plates
,”
Acta Mech. Sin.
,
25
(▪), pp.
689
698
.
23.
Li
,
R.
,
Wang
,
P.
,
Xue
,
R.
, and
Guo
,
X.
,
2017
, “
New Analytic Solutions for Free Vibration of Rectangular Thick Plates With an Edge Free
,”
Int. J. Mech. Sci.
,
131–132
(▪), pp.
179
190
.
24.
Yao
,
W.
,
Zhong
,
W.
, and
Lim
,
C. W.
,
2009
,
Symplectic Elasticity
,
World Scientific
,
Singapore
.
25.
Lim
,
C. W.
, and
Xu
,
X. S.
,
2010
, “
Symplectic Elasticity: Theory and Applications
,”
ASME Appl. Mech. Rev.
,
63
(
5
), p.
050802
.
26.
Lim
,
C. W.
,
Lu
,
C. F.
,
Xiang
,
Y.
, and
Yao
,
W.
,
2009
, “
On New Symplectic Elasticity Approach for Exact Free Vibration Solutions of Rectangular Kirchhoff Plates
,”
Int. J. Eng. Sci.
,
47
(
1
), pp.
131
140
.
27.
Lim
,
C. W.
,
2010
, “
Symplectic Elasticity Approach for Free Vibration of Rectangular Plates
,”
Adv. Vib. Eng.
,
9
(
2
), pp.
159
163
.
28.
Li
,
R.
,
Zhong
,
Y.
, and
Li
,
M.
,
2013
, “
Analytic Bending Solutions of Free Rectangular Thin Plates Resting on Elastic Foundations by a New Symplectic Superposition Method
,”
Proc. R. Soc. A-Math. Phys. Eng. Sci.
,
469
(
2153
), p.
20120681
.
29.
ABAQUS,
2013
,
Analysis User's Guide V6.13
,
Dassault Systèmes
,
Pawtucket, RI
.
30.
Liew
,
K. M.
,
Xiang
,
Y.
, and
Kitipornchai
,
S.
,
1993
, “
Transverse Vibration of Thick Rectangular Plates-1. Comprehensive Sets of Boundary Conditions
,”
Comput. Struct.
,
49
(
1
), pp.
1
29
.
You do not currently have access to this content.