Analytical solutions of free vibration for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation based on the symplectic superposition method

Based on the symplectic superposition method, the free vibration models of rectangular and right-angle triangle plates on the Winkler elastic foundation are established in the present paper, and the modes and frequencies are studied. In addition, the theoretical calculation model and finite element analysis model of rectangular thin plate and right-angle triangle plate on elastic foundation are established by using Mathematica software and ABAQUS software. It proves that the symplectic superposition method converges very fast and has a good consistenc y w ith the finite element simulation results. Analytical results show that foundation stiffness, aspect ratio, and boundary condition have great influences on vibration frequency and mode shape for structures. This paper solved the free vibration problem of rectangular plate and right-angle triangle plate on elastic foundation by using symplectic superposition method. Compared with the inverse or semi-inverse method, this method avoids the process of assuming the form about the solution, hence the result of this method is completely rational.


INTRODUCTION
The free vibration of the plate structure is an important issue in practical engineering applications.However, the analytical solution can be obtained only for a few simple boundary conditions and simple rectangular plate currently due to the complexity of free vibration and the difficulty of mathematical derivation.Hence, numerical methods are used frequently to solve complex boundary conditions and other different shapes.Present work mainly discusses the free vibration problem of a representative clamped rectangular plate and right-angle triangle plate on the Winkler elastic foundation with complex boundary conditions, the problem is solved under the Hamilton system and the analytical solution is obtained.This method can be able to solve the problems such as free vibration, bending, and buckling of plates and shells under different boundary conditions, which are difficult to be solved by traditional methods.And this method can be used in any fields using plates and shells, such as aerospace, bridge structure, construction engineering, and foundation panel.
For free vibration problem, Gorman et al. [ 1 -6 ] proposed a superposition method to solve the free vibration problem of thin plates under different boundary conditions, which lays a foundation for the establishment of symplectic superposition method.Fan et al. [ 7 ] used the Reissner-Mindlin theory and Timoshenko beam theory to simulate the stiffener plate based on the non-uniform rational Bspline isogeometric analysis methods, and they studied dimensionless frequency and mode shape of the stiffener plate.Wang et al. [ 8 ] proposed a new high order continuous Chebyshev spectral element method to analyze the free vibration of rectangular thin plates with angles and internal cuts, and they studied the effects of cut size, position, and eccentricity on the free vibration characteristics of plates.Li et al. [ 9 ] used a lossless vibration correlation technique based on the energy method to predict the free vibration behavior of carbon fiber reinforced composite anisogrid stiffened cylinder.Ahmadi et al. [ 10 ] studied the free vibration of multi-nanometer systems under different boundary conditions, and discussed the effects of coupling stiffness, non-local parameters, and the number of multinanometer beams on the multi-nanometer systems.Xu et al. [ 11 ] studied the nonlinear free vibration response of magnetoelectroelastic composite plates based on the high-order shear deformation theory, and analyzed the effects of width-thickness ratio, lengththickness ratio, magnetic potential, and electric potential on the free vibration by using parameterization.Jafari-Talookolaei et al. [ 12 ] studied the modal and chord-free vibration of a rotating laminar beam, and discussed the effects of hub radius, angular velocity, layering structure, and material anisotropy on the vibration characteristics by using the finite element method.Mahmoud [ 13 ] studied the free vibration of non-uniform and stepped circumferential functionally gradient beams under different boundary conditions by using lumped mass transfer matrix method.Guo et.al. [ 14 ] proposed a method based on Walsh series discretization to study the free vibration of composite spherical shells with general boundary conditions.By using the non-local first-order shear deformation theory, Pham et al. [ 15 ] studied the effects of non-local parameters, power index, thickness-width ratio, length-thickness ratio, and other parameters on the free vibration of functionally graded double-curved nanoshells.By using the numer ical spectrum-Chebyshev method, Guo et al. [ 16 ] studied the influence of material properties and geometric dimensions of laminated structures on the free vibration frequency of the whole structure.
For structures on the elastic foundation, Babaei [ 17 ] studied the effects of carbon nanotubes distribution mode, elastic foundation stiffness, and geometric parameters on the nonlinear free vibration and stability of carbon nanotubes reinforced nanocomposites by using two-step perturbation technique and Galerkin method.Based on the first-order shear deformation theory, Kumar et al. [ 18 ] studied the vibration performance of a rectangular gradient functional material plate with variable thickness, and they analyzed the influence of elastic foundation on its free v ibration frequenc y.Doeva et al. [ 19 ] studied the bending performance of beams on elastic foundation based on Euler-Bernoulli beam theor y.R ahmani et al. [ 20 ] established governing equations and studied the effects of Winker-Pstolnak coefficient, damping coefficient, elastic ratio, different anti-symmetric laminate layout, and length-width ratio on elastic foundation based on Hamilton principle and Reddy plate theory.Javani et al. [ 21 ] combined the theory of first-order shear-deformed plates with the nonlinear strain-displacement relation to study the nonlinear free vibration of circular plates reinforced by graphene sheets in nanocomposites.By using the transfer matrix technique, Abdoos et al. [ 22 ] transformed the motion differential Eq. into a new linear ordinary differential Eq. set, and studied the dynamic characteristics of the action for the horizontal curved beam on the moving mass, which regarded as the elastic foundation.Kalbaran et al. [ 23 ] used generalized differential orthogonal method to study the influence of thickness function, thickness variation parameters, boundary conditions, elastic foundation parameters, and composite laminates on the nonlinear transient dynamic performance of parabolic panels for revolution structure.Keleshteri et al. [ 24 ] studied the nonlinear bending behavior of functionally gradient carbon nanocomposite ring plates with varying thickness on elastic foundation based on the third-order shear deformation theory and nonlinear strain field.Lu et al. [ 25 ] established a new in-plane vibration model of rotating rings and studied the dynamic characteristics of thin rings on elastic foundation with high rotating speed.Benferhat et al. [ 26 ] adopted the fine shear deformation theory to establish the calculation model of free vibration characteristics of simply supported functionally gradient plates on elastic foundation considering the influence of transverse shear deformation, and studied the influence of foundation stiffness coefficient on free vibration characteristics.For the free vibration problem of thin plate on elastic foundation, Shen et al. [ 27 ] studied the free vibration and forced vibration problem of Reissner-Mindlin plates on a Pasternak-type elastic foundation.Hsu et al. [ 28 ] established a differential quadrature method to obtain the vibration characteristics of rectangular plates resting on elastic foundations and carrying any number of sprung masses.Motaghian et al. [ 29 ] studied the free vibration problem of thin rectangular plates on Winkler and Pasternak elastic foundation and adopted a novel mathematical to find the exact analytical solution of free vibration of plates with mixed or fully clamped boundary conditions.Ike [ 30 ] adopted Ritz variational method to solve the bending problem of rectangular Kirchhoff plate resting on elastic foundation for the case of simply supported edges and transverse distributed load.
For symplectic superposition method, Li et al. [ 31 -37 ] studied bending, buckling, and free vibration of cylindrical/plate structures constrained by the mixed boundary conditions of clamped, simply supported, and free boundary conditions.Ma et al. [ 38 ] established a computational model for the free vibration of orthotropic rectangular thin plates and calculated the free vibration characteristics of orthotropic rectangular thin plates by combining three different boundary conditions.Jia et al. [ 39 ] established a locally defective cylindrical shell model with stepped variations of local wall thickness and thickness and investigated the influence of local non-uniform geometric parameters of wall thickness on free vibration characteristics.Qiao et al. [ 40 ] introduced partial differential equations of elastic thin plates into separable Hamilton systems and solved bending, buckling, and free vibration problems of fully supported rectangular thin plates.
In this work, the free vibration models of rectangular and right-angle triangle plates on elastic foundation are established to study the modes and frequencies by the symplectic superposition method under Hamilton system.The theoretical calculation model and finite element analysis model of rectangular thin plate and right-angle triangle plate on elastic foundation are established by using Mathematic and commercial ABAQUS software, and the calculation results are compared to verify the correctness of this work.This paper solved the free vibration problem of rectangular plate and right-angle triangle plate on elastic foundation by using symplectic superposition method.Compared with the inverse or semi-inverse method, this method avoids the process of assuming the form about the solution, hence the result of this method is completely rational.

THE GOVERNING EQUATIONS OF FREE VIBR ATION IN HA MILTON SYSTE M 2.1. Hamilton system
The analytical free vibration solutions of clamped rectangular thin plate on elastic foundations are a classic problem in solid mechanics and has been studied extensively in the past, but so far it is very little for introducing Hamilton system into solution.And the symplectic superposition method is applied to solve free vibration problem in the Hamilton system on present work.Figure 1 shows the coordinates and dimensions of the rectangular thin plate on the Winkler elastic foundation, with where K , P , and L denote foundation stiffness, unit pressure, and settlement value, respectively.According to the bending theory of small deflection for Kirchhoff thin plate, the differential Eq. of free vibration is.
where M x , M y , and M xy are the bending moments along x and y directions and torque, respectively, and Q x , Q y are the transverse shear forces; w is the deflection function of the thin plate, ρ and ω are mass density and the natural frequency of the plate, respectively.The internal forces ( moments M x , M y , torque M xy , and shear forces Q x , Q y ) of free vibration for thin plate are in which D is the bending stiffness and expressed as D = Eh 3 12( 1 −v 2 ) , v and E are the Poisson's ratio and Young's modulus of thin plate, respectively, h is the thickness of plate; in addition, V x , V y are the equivalent shear forces.

Basic problem of free vibration for rectangular thin plate on elastic foundation
According to the content of the previous 2.1 section, yields: where T , and substituting Eq. ( 13) into Eq.( 12) , yields dY (y ) dy = μY (y ) ( 14) in which μ is the eigenvalue, and X ( x ) is the corresponding eigenvector corresponding to μ. Substituting the value of H into Eq.( 15) , the characteristic equation can be obtained, yields −μ 1 0 0 Expanding above Eq.( 16) , gets: , the solution of λ can be obtained according to Eq. ( 17) , with where Based on Eq. ( 18) , the eigenvalue solution of Eq. ( 15) is obtained as where A , B , C , and F is the undetermined constant.
Considering the value of ∂ 2 w ∂y 2 at the two opposite edges x = 0 and x = a are zero, the boundary conditions of the simply supported plate at those edges can be described as: Substituting Eq. ( 19) into Eq.( 20) , yields The determinant of coefficient matrix for Eq. ( 22) should be zero so as to ensure Eq. ( 22) have the non-zero solution, with Then α 1 = ± mπ a and α 2 = ± mπ a i are obtained, where m = 1, 2, 3, .And the eigenvalues μcan be obtained by combining Eq. ( 18) , yields: And based on symplectic eigen expansion method, the Eq. ( 13) can be written as follows: where f k m and f k −m ( k = 1, 2 ) are undetermined constants, and the solutions of constants f k m and f k −m could be determined by using the simply supported boundary conditions on the y-direction.

ANALY TICAL FREE VIBR ATION SOLU TIONS OF CL A MPED RECTANGUL AR PL ATE ON EL A STIC FOUNDATION 3.1. Determination of deflection function
where E m and F m are expansion coefficients, and the constant solution can be solved when substituting Eq. ( 26) into Eq.( 27) .Finally, the symplectic analytical solution of displacement for sub-problem 1 can be expressed as where For the sub-problem 2 depicted in Fig. 2 ( c ) , they are simply supported on four edges, and non-zero modal bending moments represented by trigonometric series are applied on the x = 0 and x = a simply supported edges, then the boundary conditions are where G n and H n are expansion coefficients, the symplectic analytical solution of displacement for sub-problem 2 can also be expressed when substituting Eq. ( 26) into Eq.( 29) , yields where And the modal displacements containing the natural frequencies and expansion coefficients of the rectangular thin plate are obtained by superimposing the solutions of Eqs. ( 28) and ( 30 ) .In order to obtain the expansion coefficients in sub-problems 1 and 2, the above superposition needs to satisfy the clamped boundary conditions on the elastic foundation firstly, which obtaining the coupled equations about the expansion coefficients.Secondly, the determinant of the coefficient matrix for the expansion coefficients must equal to zero so as to ensure the coupled equations have non-zero solutions.Finally, the analytical modal displacement solution can be obtained by the above superposition.
The rotate angles should be zero for clamped plate on four edges ( simplified as CCCC ) , hence the following equations should be satisfied, with Substituting Eq. ( 31) into equations ( 28) and ( 30 ) , yields Eqs. ( 32) -( 35) contains the expansion coefficients E m , F m , G n , and H n ( m , n = 1, 2, 3, ) .And the determinant of the coefficient matrix must equal to zero so as to ensure the expansion coefficient exists non-zero solution, then the analytical solution of free vibration for rectangular clamped thin plate on the elastic foundation are obtained.

Numerical results and discussions for rectangular thin plate
The ABAQUS software is selected to simulate the free vibration of clamped rectangular resting on elastic foundation to verify the feasibility and accuracy of the present analytical method.In the finite element calculation, S4R shell element is adopted, the size of the mesh is set as 0.01a uniformly, the thickness to width ratio h / a , density ρ, Young's modulus E, Poisson's ratio ν, and foundation stiffness k are 10 −6 , 10 4 kg/m 3 ,1.092× 10 10 Pa, 0. Table 2 shows the effect of foundation stiffness on first ten-order natural frequencies of the CCCC rectangular thin plate with aspect ratio b / a = 1 .The greater the foundation stiffness is, the greater is the free vibration frequency of rectangular thin plate, which is in accordance with classic vibration theory.Table 3 shows the convergence of the first ten-order normalized frequencies ( ωa 2 √ ρh / D ) for CCCC rectangular thin plate on the elastic foundation with aspect ratio b / a = 1 and b / a = 5 .It can be found that five significant digits can be converged when setting the number of series terms as 100, hence number of series terms are 100 used in examples.Figure 3 shows the first ten-order mode shapes for a CCCC rectangular thin plate on the elastic foundation with aspect ratio b / a = 1 .Compar ed with the finite element simulation results, the maximum relative error between symplectic superposition method and FEM is less than 1%, hence present analytical symplectic superposition method can be used as a benchmark for other methods.

ANALY TICAL FREE VIBR ATION SOLU TIONS OF RIGHT-ANGLE TRIANGLE PL ATE ON EL A STIC FOUNDATION 4.1. Determination of deflection function
The vibration problem of right-angle triangle plate on elastic foundation with size of right-angle sides a and b are shown in Fig. 4 .Based on the symplectic superposition method under Hamilton system, the original problem can be divided into three sub-problems.From Fig. 4 , sub-problems 1 and 2 are simply supported rectangular plates on four edges in xoy coordinate system, sub-problem 3 is located in the local coordinate system x 1 o 1 y 1 , and the hypotenuse of the triangle just overlaps with the axis of the local coordinate system.In a word, the superposition sum of modal displacements and modal bending moments for three sub-problems is equivalent to the original problem for right-angle triangle plate.
From Fig. 4 , they are simply supported on x = 0, x = a , and y = 0 edges for sub-problem 1, and non-zero modal displacement and modal bending moment expressed by triangular series are applied on y = b edge, then the boundary conditions at the y direction are   where E m and F m are expansion coefficient.The symplectic analytical solution w 1 ( x , y ) can be obtained by using Eq. ( 36) as  where For sub-problem 2, they are simply supported on y = 0, y = b , and x = a edges.And non-zero modal displacements and modal bending moment expressed by triangular series are applied on x = 0 edge, then the boundary conditions in the x direction are where G n and H n are expansion coefficient.The symplectic analytical solution w 2 ( x , y ) can be obtained by using Eq. ( 38) as where For sub-problem 3, they are simply supported on x 1 = 0, x 1 = a 1 , and y 1 = b 1 edges.And modal displacement and modal bending moment expressed by triangular series are applied on y 1 = 0 edge, the boundary conditions on the y 1 direction are where K p and L p are expansion coefficient.The symplectic analytical solution w 3 ( x , y ) can be obtained by using Eq. ( 40) as where And the relation between the xoy and x 1 o 1 y 1 coordinate systems are Based on Eq. ( 42) , Eq. ( 41) transformed into the xoy coordinate system is where Thus, symplectic analytical solutions of the three sub-problems with expansion coefficients E m , F m , G n , H n , K p , and L p are obtained.And the coupled equations of expansion coefficients can be obtained by superimposing the analytical solutions of the three subproblems to satisfy the actual boundary conditions.

Analytical solution of CCC right-angle triangle plate on elastic foundation
The modal displacement on the y = b , x = 0, and y 1 = 0 edges should be zero for CCC right-angle triangle plate, namely w And the rotate angle of the y = b , x = 0, and y 1 = 0 edges should be zero for CCC r ight-angle tr iangle plate, namely     For the simply supported right-angle triangle plate on all edges ( SSS ) , the zero modal displacement condition is the same as Eq. ( 43) on y = b , x = 0, and y 1 = 0 edges.Besides, the bending moments also are zero on the three simply supported edges, i.e.
Other type of right-angle triangle plates on elastic foundation with clamped and simply supported boundary conditions are CSS, CCS, CSC, SSC, SCC, and SCS in Fig. 5 ( a ) -( f ) , where the alphabet "C" represents clamped condition and "S" denotes simply supported condition.And the modal displacements for above right-angle triangle plates with combined boundaries can be obtained according to 4.1 and 4.2 charters.In addition, this basic Hamilton system can also be used to solve the free vibration problem of right-angle triangle plate on elastic foundation with one side free and the other two sides clamped in the Fig. 6 .The difference is the boundary condition equation Figure 6 ( a ) denotes the FCC right-angle triangle plate on elastic foundation, and the equivalent shear force on the free side ( x = 0 ) equals to zero, i.e.
and the following equation is obtained Equations ( 44b) , ( 44c ) , ( 45b ) , ( 45c ) , and ( 46b ) are considered to solve the unknown numbers and obtain the vibration shapes of FCC right-angle triangle plate on elastic foundation at corresponding natural frequencies.
Equations ( 44b) , ( 44c ) , ( 45a ) , ( 45b ) , and ( 46c ) are considered to solve the unknowns and obtain the mode shapes of CCF right-angle triangle plate on elastic foundation with corresponding natural frequencies.Table 4 , Table 5 , Table 6 , and Table 7 are the analytical solutions of the first ten-order frequencies of CCC, SSS, FCC, and CCF rightangle triangle plates on elastic foundation at different aspect ratio, respectively.It can be found that present symplectic superposition method is in good agreement with FEM method.In addition, each-order vibration frequency decreases with increasing of aspect ratio, which can be attributed to fact that the increase amount of stiffness is greater than that of mass for right-angle triangle plate.
Table 8 and Table 9 denote the results of FCC and CCF right-triangle thin plates on elastic foundation with aspect ratio b / a = 1 and b / a = 5 when the series term converges to five significant digits, respectively.From Table 8 and Table 9 , the vibration frequency is converged when 30 is adopted as Fourier series terms.And the number of Four ier ser ies terms selecting as 100 is enough convergence for examples.
Table 10 , Table 11 , Table 12 , and Table 13 show effect of foundation stiffness K on first ten-order frequencies of CCC, SSS, FCC, and CCF right-angle triangle plates, respectively.From the Table 10 to Table 13 , the free v ibration frequenc y of above different rightangle triangle plate increases with the increasing of foundation stiffness, which can be attributed to fact that the plate's total stiffness increases when increasing the foundation stiffness, then the vibration frequency increases subsequently according to vibration theory.
Figure 8 , Fig. 9 , Fig. 10 , and Fig. 11 show the comparison of mode shape between the present symplectic superposition method and FEM solution for SSS, CCC, FCC, and CCF plate on the elastic foundation, respectively.It can be found that there is a good consistency for two different methods from Fig. 8 to Fig. 11 .And it can be seen from the above analytical and simulation results that the natural frequency of CCC triangle is greater than that of SSS right-angle triangle plate with the same aspect ratio, which is just in accordance with actual engineering case.In addition, the vibration frequency of FCC right-angle triangle plate is greater than that of CCF right-angle triangle plate, which can be attributed to fact that the FCC plate with clamped hypotenuse is of greater stiffness and mass comparing with that of FCC plate, and the stiffness takes up more significant effect.

CONCLUSIONS
The symplectic superposition method based on Hamilton system is used to solve the free vibration problem of clamped rectangular thin plate and right-angle triangle plate on elastic foundation.The analytical solution of the original problem is obtained by applying modal displacements and modal bending moments on the boundary of the sub-problems, which is equivalent to the original problem.
The innovation of present method is that it is not need to assume the form of the solution in advance and is universal.Besides, some important conclusions are obtained as follows: ( 1 ) The greater the foundation stiffness is, the greater is the free vibration frequency of rectangular thin plate.
( 2 ) Vibration frequency decreases with increasing of aspect ratio as the increase amount of stiffness is greater than that of mass for right-angle triangle plate.( 3 ) Vibration frequency of FCC right triangle is greater than that of CCF right triangle as the FCC plate with clamped hypotenuse is of greater stiffness and mass comparing with that of FCC plate, and the stiffness takes up more significant effect.

Figure 1
Figure 1 Configuration of a rectangular thin plate on the elastic foundation.

Figure 2
Figure 2 Symplectic superposition of free vibration problem for a clamped rectangular thin plate on the elastic foundation.

From Fig. 2 ,
the original clamped plate is divided into two sub-problems [Fig. 2 ( b ) and Fig. 2 ( c ) ], which are simply supported on opposite edges based on symplectic superposition method.For the sub-problem 1 as depicted in Fig. 2 ( b ) , they are simply supported on the four edges, and the non-zero modal bending moments represented by trigonometric series are applied on the y = 0 and y = b simply supported edges, then the boundary conditions are Downloaded from https://academic.oup.com/jom/article/doi/10.1093/jom/ufad032/7395018 by guest on 29 May 2024 400 • Journal of Mechanics , 2023, Vol.39 3 and 200N/m, respectively.As shown in Table 1 , the analytical solutions of the first ten-order frequencies with aspect ratio b / a = 1, b / a = 1.5, b / a = 2, b / a = 2.5, b / a = 3, b / a = 3.5, b / a = 4, b / a = 4.5, and b / a = 5 for CCCC rectangular thin plates on elastic foundation are presented, which have a good agreement with that applying finite element method ( FEM ) .

Figure 3
Figure 3 First ten-order of mode shapes for a CCCC rectangular thin plate on the elastic foundation with aspect ratio b/a b/ a = 1 a = 1 .

Figure 4
Figure 4 Symplectic superposition of free vibration problem for a CCC right-angled triangle plate on the elastic foundation.

Figure 5
Figure 5 Mixed boundary conditions of right-angled triangle plate on the elastic foundation.

Figure 6
Figure 6 Clamped right-angled triangle plate on the elastic foundation with one free edge.

Figure 7
Figure 7 Convergence of the first ten-order normalized frequencies ( ωa 2 √ ρh /D ) for the SSS and CCC right-angled triangle plates on the elastic foundation with aspect ratio b/a = 1 . cos

Figure 8
Figure 8 First ten-order of mode shapes for a CCC right-angled triangle plate on the elastic foundation with aspect ratio b / ba a = 1 .

Figure 9 Figure 10 Figure 11
Figure 9 First ten-order of mode shapes for a SSS right-angled triangle plate on the elastic foundation with aspect ratio b / ba a = 1 .

4. 4 .Figure 7
Figure7shows the convergence of the first order normalized frequencies ( ωa2 √ ρh / ρh D D ) for the SSS and CCC right-angle triangle plates on the elastic foundation with aspect ratio b / a = 1 .From Fig.7, both two curves are converged when Fourier series terms is greater than 50, which indicates the feasibility of symplectic superposition method.Table4, Table5, Table6, and Table7are the analytical solutions of the first ten-order frequencies of CCC, SSS, FCC, and CCF rightangle triangle plates on elastic foundation at different aspect ratio, respectively.It can be found that present symplectic superposition method is in good agreement with FEM method.In addition, each-order vibration frequency decreases with increasing of aspect ratio, which can be attributed to fact that the increase amount of stiffness is greater than that of mass for right-angle triangle plate.Table8and Table9denote the results of FCC and CCF right-triangle thin plates on elastic foundation with aspect ratio b / a = 1 and b / a = 5 when the series term converges to five significant digits, respectively.From Table8and Table9, the vibration frequency is converged when 30 is adopted as Fourier series terms.And the number of Four ier ser ies terms selecting as 100 is enough convergence for examples.Table10, Table11, Table12, and Table13show effect of foundation stiffness K on first ten-order frequencies of CCC, SSS, FCC, and CCF right-angle triangle plates, respectively.From the Table10to Table13, the free v ibration frequenc y of above different rightangle triangle plate increases with the increasing of foundation stiffness, which can be attributed to fact that the plate's total stiffness increases when increasing the foundation stiffness, then the vibration frequency increases subsequently according to vibration theory.Figure8, Fig.9, Fig.10, and Fig.11show the comparison of mode shape between the present symplectic superposition method and FEM solution for SSS, CCC, FCC, and CCF plate on the elastic foundation, respectively.It can be found that there is a good consistency for two different methods from Fig.8to Fig.11.And it can be seen from the above analytical and simulation results that the natural frequency of CCC triangle is greater than that of SSS right-angle triangle plate with the same aspect ratio, which is just in accordance with actual engineering case.In addition, the vibration frequency of FCC right-angle triangle plate is greater than that of CCF right-angle triangle plate, which can be attributed to fact that the FCC plate with clamped hypotenuse is of greater stiffness and mass comparing with that of FCC plate, and the stiffness takes up more significant effect.

Table 1
Downloaded from https://academic.oup.com/jom/article/doi/10.1093/jom/ufad032/7395018 by guest on 29 May 2024 Analytical solutions for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation • 401 Normalized frequency solution ( ωa 2 √ ρh /DD ) of the CCCC rectangular thin plate on the elastic foundation.

Table 2
Effect of foundation stiffness on first ten-order natural frequency of the CCCC rectangular thin plate with aspect ratio b/a = 1 .

Table 3
Convergence of the normalized frequencies ( ωa 2 √ ρh /D ) for CCCC rectangular thin plate on the elastic foundation with aspect ratio b/a = 1 and b = 5 .

Table 4
Downloaded from https://academic.oup.com/jom/article/doi/10.1093/jom/ufad032/7395018 by guest on 29 May 2024 Analytical solutions for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation • 407 Normalized frequency solutions ( ωa 2 √ ρh /D ) of the CCC right-angled triangle plate on the elastic foundation.

Table 5
Normalized frequency solutions ( ωa 2 √ ρh /D ) of the SSS right-angled triangle plate on the elastic foundation.

Table 6
Normalized frequency solutions ( ωa 2 √ ρh /D ) of the FCC right-angled triangle plate on the elastic foundation.

Table 7
Normalized frequency solutions ( ωa 2 √ ρh /D ) of the CCF right-angled triangle plate on the elastic foundation.

Table 8
Convergence of the normalized frequencies ( ωa 2 √ ρh /D ) for the FCC right-angled triangle plate on the elastic foundation with aspect ratio b/a = 1 and b/a = 5 .

Table 9
Convergence of the normalized frequencies ( ωa 2 √ ρh /D ) for the CCF right-angled triangle plate on the elastic foundation with aspect ratio b/a = 1 and b/a = 5 .

Table 10
Frequency solutions of the CCC right-angled triangle plate with different foundation stiffness ( b/a = 1 ) .

Table 11
Frequency solutions of the SSS right-angled triangle plate with different foundation stiffness ( b/a = 1 ) .

Table 12
Frequency solutions of the FCC right-angled triangle plate with different foundation stiffness ( b/a = 1 ) .

Table 13
Downloaded from https://academic.oup.com/jom/article/doi/10.1093/jom/ufad032/7395018 by guest on 29 May 2024 Analytical solutions for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation • 411 Frequency solutions of the CCF right-angled triangle plate with different foundation stiffness ( b/a = 1 ) .