Ghadirian, H., Ghazavi, M., Khorshidi, K. (2014). Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method. Mechanics of Advanced Composite Structures, 1(1), 6170. doi: 10.22075/macs.2014.280
Hossein Ghadirian; Mohammad Reza Ghazavi; Korosh Khorshidi. "Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method". Mechanics of Advanced Composite Structures, 1, 1, 2014, 6170. doi: 10.22075/macs.2014.280
Ghadirian, H., Ghazavi, M., Khorshidi, K. (2014). 'Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method', Mechanics of Advanced Composite Structures, 1(1), pp. 6170. doi: 10.22075/macs.2014.280
Ghadirian, H., Ghazavi, M., Khorshidi, K. Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method. Mechanics of Advanced Composite Structures, 2014; 1(1): 6170. doi: 10.22075/macs.2014.280
Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method
^{1}Faculty of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran
^{2}Department of Mechanical Engineering, Faculty of Engineering, Arak University, 3815688349, Arak, Iran
Abstract
In this paper free vibration analysis of two rectangular isotropic plates, which are connected to each other by two translational and rotational springs along the edges, are investigated. The equation of motion and associated boundary and continuity conditions are derived using the extended Hamilton principle. To solve the eigenvalue problem, the Ritz method is utilized. Numerical investigations are presented to show some applications of this method. In this research two types of problems are investigated: first, vibration of a continuous plate and second, free vibration of two hinged plates. This approach is usually referred to as the artificial spring method, which can be regarded as a variant of the classical penalty method. In order to validate the results, the achieved results are compared to results which are presented in literatures.
Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method
H. Ghadirian^{a*}, M.R. Ghazavi^{a}, K. Khorshidi^{b}
^{a}Faculty of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran
^{b}Department of Mechanical Engineering, Faculty of Engineering, Arak University,3815688349, Arak, Iran
paper INFO
ABSTRACT
Paper history:
Received 7 August 2014
Received in revised form 5 October 2014
Accepted 7 October 2014
In this paper free vibration analysis of two rectangular isotropic plates, which are connected to each other by two translational and rotational springs along the edges, are investigated. The equation of motion and associated boundary and continuity conditions are derived using the extended Hamilton principle. To solve the eigenvalue problem, the Ritz method is utilized. Numerical investigations are presented to show some applications of this method. In this research two types of problems are investigated: first, vibration of a continuous plate and second, free vibration of two hinged plates. This approach is usually referred to as the artificial spring method, which can be regarded as a variant of the classical penalty method. In order to validate the results, the achieved results are compared to results which are presented in literatures.
The plate is one of the most common structural elements that are encountered by either scientific or technological interest. It’s widely utilized in aerospace, marine, mechanical, electrical, nuclear and civil engineering structures. The vibration analysis of plates is one amongst the most important issues in coming up with this sort of structure. Vibration characteristics of plates were extensively studied by other researchers. Plates with completely different shapes, boundary conditions and complicating effects were thought of and also the frequency parameters were investigated in some monographs [1, 2], normal texts [35] and review papers [6, 7].
In engineering applications, plates with various complications were investigated. These effects include elastically restrained boundaries, presence of elastically or rigidly connected masses, point supports, variable thickness, anisotropic material, stiffeners[8, 9], interior openings[10] and line hinged [1113], that are used to serve different purposes required in a structure. In Refs. [1315] general studies on the vibration of plates with point supports have been deliberated and the vibration of plates with line supports have been studied in Refs. [1319]. For instance, a line hinge in a plate can be used to expedite folding of gates, or the opening of doors and hatches [20]. The hinge can also be used to simulate a through crack prior to the edge misalignment.
Due to its conceptual simplicity, wide flexibility, high reliability and computational efficiency, the Ritz technique has been widely utilized to resolve the vibration problem of rectangular plates. The Ritz procedure consists in approximating the normal displacement variable through a linear combination of generally assumed functions, commonly known as admissible functions, trial functions or basis functions, each satisfying at least the geometrical boundary conditions of the plate. The unknown constant factors of the combination can be obtained by the minimization of the energy functions of the system. Convergence to the exact solution is assured if the admissible functions are linearly independent and form a mathematically complete set. The chosen functions are not required to satisfy the natural boundary conditions, although, if they do, better convergence and accuracy might be achieved. Moreover, the properties of trial functions have a significant effect on computational efficiency and numerical stability of the solution [21]. The rather weak conditions imposed by the Ritz method and the great sensitivity of the related solution to the choice of admissible functions have prompted many researchers in evaluating and developing suitable ways of constructing the trial set for Kirchhoff plates with arbitrary boundary conditions and complicating effects. There is a so huge amount of literature on the topic which prevents to provide a comprehensive review of all the available approaches here. Restricting the analysis to rectangular plates, notable solutions include the use of characteristic beam functions, simple and orthogonal polynomials, static beam functions, Fourier sine and cosine series or their appropriate combinations [21, 22]. Since the initial works of Young [23], Warburton [24]and Leissa [25], admissible functions as products of vibrating beam eigenfunctions have been presented by many researchers. They work well in most conditions. However, because of the occurrence of overrestraint at free edges, the results obtained for plates involving one or more free edges are much less accurate [26]. Characteristic beam functions are also clearly dependent on the boundary conditions. There exist 21 different cases for rectangular plates by considering all possible combinations of classical edge conditions [25]. Therefore, their use involves a boring solution process since a specific set is required for each type of boundary. Finally, numerical instability happens when the common expressions for the beam mode shape functions are evaluated at high orders [27].
In this paper, a more general case of free vibration analysis of assembled plates is considered for preparing a model that can approximate vibration behaviour of a hinged rectangular plate or a uniformed rectangular plate, only with determination of various stiffness’s of translational and rotational springs that connect two plates to each other. In the process of solving the eigenvalue problem which is obtained from the variation method on the total system, the quasi analytical Ritz method is used. The trigonometric sets are used as admissible functions due to the fact that they are very effective from a computational point of view and their reliability and versatility for flexural vibration analysis of rectangular plates which may be subjected to various complicating factors. Finally some illustrating results are presented in tabular and graphical forms to validate the method.
2. The Determination of the Boundary Value Problem
Figure 1 represents an isotropic rectangular thin plate in the xy plane. In this figure h is thickness, aand b are length and width of the plate, respectively. A slit that is parallel to the yaxis is located at x=c, as shown in Figure 1. The plate is considered to have two spans that are separated by the slit. The total domain of the plate A is divided into two subdomains and as is shown in Figure 1. It is seen that these two subdomains are separated by the line at x=c. and are connected to each other by using linear translational and rotational springs. and are constants of translational and rotational springs, respectively. It can be assumed that the thickness and deflection of each subdomain are small compared with the wavelength of flexural vibration; consequently, thin plate theory is applicable. Throughout the remainder of the paper, the counterclockwise fourletter symbolic notation introduced by Leissa [25] is used for describing classical boundary conditions. For instance, an SFSC plate has a simply supported (zero deflection and free rotation) left edge, a free (free deflection and rotation) bottom edge, a simply supported right edge and a clamped (zero deflection and rotation) top edge, respectively.
Figure 1. A schematic diagram of hinged plates in xy plane
In order to analyse the transverse displacements of the system, it can be assumed that the vertical position of the kth (k=1, 2) plate at any time t, is defined by the function . The total kinetic energy of the system is [28]:
(1)
Where is the mass density of the kth plate and such that is constituted of both subdomains , and each respective boundary .
The total potential energy due to the elastic deformation of the plate is [28]:
(2)
where is the rigidity of the kth plate, ν is the Poisson’s ratio, is the external load, is the difference in the lateral displacement of the two plates along the slit, is the difference between the normal slopes and denotes the directional derivative of w with respect to the outward normal unit vector to the curve and s is the coordinate along the line of slit.
To derive the equations of motion, the extended Hamilton principle is used as:
(3)
where:
(4)
and δ is variational operator.
Consequently, by using the Hamilton principle, the equations of motion for free vibration analysis of the coupled plates are obtained as follows:
(5)
The above equations represent the dynamical behaviour of the vibrating plates.
2.1. Classical Boundary Conditions
In this study, plates may take any classical boundary conditions, including free, simply supported and clamped. The boundary conditions along the edges and are satisfied by the following relations [17]:
(a) For a free edge:
,
(6)
(b) For a simply supported edge:
,
(7)
(c) For a clamped edge:
,
(8)
Where is the bending moment on the edges and , and is the Kirchhoff equivalent force, one can write:
At :
(9)
(10)
And similarly it could be obtained for the other edges. Applying the states before boundary conditions and continuity conditions, a set of the homogeneous equations would be obtained.
2.2. Continuity conditions for connection of two regions
At the slit location (x=c), continuity conditions along the slit line can be written as:
(11)
(12)
According to the variational principle, the coefficients of and should be zero. Consequently, the following equations are obtained:
(13)
(14)
(15)
(16)
where all of the above equations would be evaluated at x=c. Eqs. (11) and (12) express an equal moment with the opposite signs, which are applied to the edges x=c and indicate the Kirchhoff equivalent force on the common edge between two regions.
3. Eigenvalue Problem in Ritz Method
For free vibrations of the plate, the displacements can be written as:
(17)
where ω is the circular frequency of the plate. Substituting Eq. (17) into Eqs. (1) and (2), the maximum kinetic energy and the maximum potential energy are obtained. For the sake of simplicity the following dimensionless parameters are used:
,
(18)
Therefore, it can be written:
(19)
(20)
Where , and are assumed to be constant for two areas i.e.
,
(21)
Due to classical boundary conditions, the functional energy of the system is expressed as:
(22)
The Ritz approximation is:
(23)
Where the superscript k denotes the kth subdomain, are unknown coefficients and and are appropriate admissible functions satisfying at least the geometrical boundary conditions of the problem. After substituting Eqs. (19) and (20) into Eq. (22), the coefficients can be obtained by finding extremum of the functional energy as follows:
(24)
Consequently the following eigenvalue equation is obtained:
(25)
where is the dimensionless frequency.
The stiffness and mass matrices K and M are presented respectively as:
(26)
(27)
where,
(28)
(29)
(30)
In the above equations the integral statements are defined as:
(31)
(32)
(33)
(34)
(35)
(36)
where α and β denote the order of derivatives.
It is noted that , and are dimensionless coefficients. The spring stiffnesses and are selected either as the actual connecting stiffnesses, if flexible joints are represented, or as very high values compared with the adjoining plates, therefore approximating rigid connections. (For example, at a free edge, and are taken as zero, while, to approximate a hinged boundary or connection between two adjacent plates, is given some very high value and is taken as zero.)[21]
In the present study the admissible functions and are defined by means of the trigonometric set. The following trial functions which are used by Beslin and Nicolas [29] for flexural vibration of Kirchhoff plates are presented:
(37)
where the coefficients , , and are listed in Table 1.
The function is defined according to Eq. (30), where and m are replaced by and n, respectively. A subset of is plotted in Figure 2 where functions of increasing order are arranged in a matrix form [21].
It is seen that the first and third functions and have a nonzero displacement at and , respectively. The second and fourth trigonometric functions and have a free slope at the same edges at , respectively. The functions and are arranged in a similar style for .
Table 1. Coefficients of the trigonometric set
i
1
2
3
4
>4
Figure 2. The first 15 functions of the trigonometric set
As it is, the first four functions to enable one to easily satisfy any classical boundary condition by selecting a suitable combination among them. For example, the analysis of a completely free plate (FFFF) will keep all these four functions in the final sequence. If a simple support condition is imposed on the edge (FFFS plate), the function will be eliminated from the set. For a fully clamped plate (CCCC) all the couples of these functions both in and in direction will be removed. The nine combinations of classical boundary conditions are reported in Table 2 where a bullet denotes that the corresponding function must be kept in the final set. The first letter in the table refers to the edge at or , whereas the second letter refers to that at or [22].
Due to the zero determinant of the coefficient matrix in Eq. (21), the problem has a nontrivial solution. The stated determinant gives the natural frequencies of the system. It is important to note that the nontrivial solution of the system gives the mode shapes of the plate.
Table 2. Combination of the first four functions in the trigonometric set to satisfy the related boundary conditions.
Boundary
condition
FF
●
●
●
●
FS
●
●

●
FC
●
●


SF

●
●
●
SS

●

●
SC

●


CF


●
●
CS



●
CC




4. Numerical Results and Discussions
In order to examine the accuracy and applicability of the approach developed and discussed in the previous sections, numerical results were calculated for a number of different problems for which comparison values were available in the literature and also convergence studies have been carried out in the graphical and tabular form. All calculations have been performed with Poisson’s ratio .
4.1. Rectangular Plate with Different Boundary Conditions
The discussed method is applied to rectangular isotropic plates with four different combinations of classical boundary conditions. To obtain the frequency parameters corresponding to a continuous rectangular plate without slit, it is necessary that the translational and rotational stiffnesses approach infinity. For this purpose in this section dimensionless translational stiffness per unit length is assumed to be equal to the dimensionless rotational stiffness and it can be assumed to be equal to a high value ( ). The problems are solved for two cases which have wellknown closed form solutions [25], i.e., SSSS and SFSF plates; and, last, a cantilever plate (CFFF) that there is no exact solution for this case [30]. Table 3 shows the convergence of the dimensionless frequency parameter for the first eight modes.
Results are obtained by using a square selection method, i.e., the similar number of terms M=N is adopted in the series expansion, with no regard to symmetry. It is seen that the solutions monotonically decrease as the number of terms in the set increases. Table 3 also shows that the higher frequency convergent values to four digits are obtained with a different number of terms for each mode and case.
Table 3. Convergence study of the first eight frequency parameters for square isotropic plates with different classical boundary conditions.
BCs
N
Mode sequence
1
2
3
4
SSSS
6
19.7438
49.3666
49.3772
78.9872
8
19.7408
49.3544
49.3596
78.9682
10
19.7398
49.3515
49.3537
78.9627
12
19.7392
49.3501
49.3510
78.9602
Ref.[30]
19.7392
49.3480
49.3480
78.9569
SFSF
6
9.6506
16.2247
37.1184
39.0561
8
9.6362
16.1626
36.8367
38.9784
10
9.6332
16.1466
36.7684
38.9594
12
9.6321
16.1407
36.7454
38.9523
Ref.[30]
9.6314
16.1348
36.7256
38.9451
CFFF
6
3.4893
8.5887
21.3913
27.4601
8
3.4778
8.5354
21.3250
27.2712
10
3.4745
8.5209
21.3058
27.2279
12
3.4723
8.5108
21.2917
27.2038
Ref.[21]
3.47108
8.5066
21.2848
27.1990
Table 4. convergence study of the first eight values of the for a SSSS plate with an internal line hinge.
N
Mode sequence
1
2
3
4
0.1
6
16.8401
39.1790
47.4875
72.1332
8
16.7937
39.0915
47.4346
72.0232
10
16.7904
39.0859
47.4268
72.0129
12
16.7890
39.0851
47.4248
72.0106
Ref.[28]
16.7891
39.0862
47.4207
72.0098
0
6
16.1427
46.7697
49.3666
75.3799
8
16.1372
46.7504
49.3545
75.3063
10
16.1360
46.7443
49.3517
75.2949
12
16.1353
46.7416
49.3499
75.2836
Ref.[28]
16.1347
46.7381
49.3480
75.2833
Moreover, this number does not necessarily increase as the mode number increases. It is illustrated from Table 3 that the frequency parameters which are obtained by the presented method in this paper in comparison to reference [21] are higher about 0.2% at worst state. This difference is due to the discontinuity in the plate and the values of and which are assumed to be a finite value.
4.2. Rectangular Plate with an Internal Line Hinge
The problem is the transverse vibration of two rectangular plates which hinged together along they coordinate at x=c. There are some literatures about this problem. As it was referred to previously, to approximate a hinged connection between two adjacent plates, is given some very high value and is taken as zero. In this case some results of a convergence study of the values of the frequency are presented in Table 4 and Table 5. The first eight values of are presented for a square SSSS plate with an internal line hinge located at two different positions, namely and . It can be observed that with an increase in iteration number the frequency parameters converge monotonically. From Table 4 it can be observed that the results are in a good agreement with exact frequency parameters for a SSSS plate with an internal line hinge presented in Ref. [28].
Similar results for plates with different boundary conditions, aspect ratios and position of line hinge are presented in Table 5. A convergence study is shown in Figure 3 and Figure 4 for fundamental frequency in terms of translational and rotational frequencies that vary from 0 to .
These figures are presented in two cases in terms of boundary conditions; Case 1: symmetric boundary conditions (SSSS, SFSC) and Case 2: asymmetric boundary conditions (CCFF, CCSS).
(a1)
(a2)
(b1)
(b2)
Figure 1 Fundamental frequency parameter for symmetric boundary conditions as (1) translational and (2) both stiffnesses equally vary from 0 to for (a)SSSS plates and (b)SFSC plates
It should be noted that the first natural frequency of vibration is named as fundamental frequency and its magnitude is less than all of the other frequencies. In Figure 3 and Figure 4, the “a” and “b” letters introduce the types of boundary conditions and their indices are used to show the variation of the translational and rotational stiffnesses which are specified in each figure. As it can be seen, the fundamental frequency oscillates in a range which is very small when only translational frequency varies, but when both the translational and rotational stiffnesses vary, the fundamental frequency converges monotonically to a specific value. This result is due to the symmetry in boundary conditions which yields this fact that in the low values of a good accuracy of can be obtained.
Table 5. The first ten values of the for a rectangular plate with different boundary conditions and aspect ratios
BSc
b/a
Mode sequence
1
2
3
4
SSSS
1/2
1/3
3.7002
4.2606
11.0746
11.4277
0
2.4345
3.6877
11.3615
15.0393
1/3
1.5930
3.6902
10.4245
11.9999
1/3
1/3
2.3530
4.9276
7.2380
12.9054
0
1.0761
2.3474
6.6699
7.2363
1/3
0.7038
2.3549
4.6153
7.3915
SFSF
1/2
1/3
3.5258
7.6768
8.2681
16.5416
0
2.2370
7.4994
11.3763
18.0312
1/3
1.5060
7.5078
10.0619
17.0707
1/3
1/3
1.5587
3.6584
4.7732
10.0594
0
0.9886
4.7513
5.0260
8.5270
1/3
0.6654
4.4529
4.7517
10.1410
CFFF
1/2
1/3
6.6122
10.4308
14.1414
21.8162
0
6.6445
13.4687
14.8999
19.5932
1/3
1/3
4.3687
4.6372
9.1419
12.8157
0
4.3766
6.6159
8.9322
9.5142
(a1)
(a2)
(b1)
(b2)
Figure 2 Fundamental frequency parameter for asymmetric boundary conditions as (1) translational and (2) both stiffnesses equally vary from 0 to for (a)CCFF plates and (b)CCSS plates
On the other hand, for asymmetric boundary conditions such as CCFF and CCSS, for the cases in which only varies or both and vary, the fundamental frequency parameter converges monotonically to a specific value as is shown in Figure 2.
5. Conclusions
This paper presents the formulation of an analytical model for the dynamic behaviour of rectangular isotropic plates, with an arbitrarily located slit and classical boundaries. The equations of motion and associated boundary and compatibility conditions are derived by using the extended Hamilton principle. An approach has been presented to solve the free vibration of the previously mentioned plates in a direct variational and numerical way. A not complicated, computationally efficient and accurate method has been developed for the determination of natural frequencies and modal shapes. The approach is the trigonometric Ritz method which is based on a simple, stable and computationally efficient set of admissible trigonometric functions and has been presented for free vibration analysis of rectangular Kirchhoff plates. The versatility and reliability of the present approach have been shown in various states of the slotted plate with an arbitrarily selected subset of complicating factors. Very accurate and stable solutions have been obtained for all cases with lesser computational effort in comparison with the other similar methods. Consequently, the present analysis shows that the trigonometric Ritz method is a valuable way for solving transverse free vibrations of thin rectangular plates and is easily applicable to a wide class of problems with complicating effects. To investigate the effect of the line slit and its location on the vibration behaviour, parametric studies have been performed. It is valuable to note that by using a modified version of this method, the static deflection problems and buckling can be analysed. On the other hand, this method can be easily generalized for analysing problems that include anisotropic plates.
References
[1] Leissa AW, Vibration of plates ,DTIC Document, 1969.
[2] B. RD, 1984 Formulas for Natural Frequency and Mode Shape, Malabar, FL: Robert E. Krieger Publishing Company.
[3] Gorman DJ, Vibration analysis of plates by the superposition method, World Scientific, 1999.
[4] Reddy JN, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, 2003.
[5] Timoshenko S, WoinowskyKrieger S, Theory of plates and shells, McGrawhill New York, 1959.
[6] Leissa A, Recent research in plate vibrations, 19731976: complicating effects, Shock and Vibration Inform. Shock Vib Digest 1978; 10: 2135.
[7] Qatu MS, Sullivan RW, Wang W, Recent research advances on the dynamic analysis of composite shells: 2000–2009, Compos Struct 2010; 93: 1431.
[8] Lee H, Ng T, Vibration of symmetrically laminated rectangular composite plates reinforced by intermediate stiffeners, Compos Struct 1994; 29: 405413.
[9] Lee H, Ng T, Effects of Torsional and Bending Restraints of Intermediate Stiffeners on the Free Vibration of Rectangular Plates, Struct Mech 1995; 23: 309320.
[10] Huang C, Leissa A, Chan C, Vibrations of rectangular plates with internal cracks or slits, Int J Mech Sci 2011; 53: 436445.
[11] Grossi RO, Raffo J, Natural vibrations of anisotropic plates with several internal line hinges, Acta Mech 2013; 224: 26772697.
[12] Quintana MV, Grossi RO, Free Vibrations of a Trapezoidal Plate with an Internal Line Hinge, The Scientific World Journal 2014.
[13] Wang C, Xiang Y, Wang C, Buckling and vibration of plates with an internal line hinge via the Ritz method, Proceedings of the First AsianPacific Congress on Computational Mechanics 2001, 16631672.
[14] Du J, Liu Z, Li WL, Zhang X, Li W, Free inplane vibration analysis of rectangular plates with elastically pointsupported edges, Vib Acoust 2010; 132: 031002.
[15] Liew K, Lam K, Effects of arbitrarily distributed elastic point constraints on vibrational behaviour of rectangular plates, sound vibration 1994; 174: 2336.
[16] Kim C, Dickinson S, The flexural vibration of line supported rectangular plate systems, sound and vibration 1987; 114:129142.
[17] Xiang Y, Zhao Y, Wei G, Levy solutions for vibration of multispan rectangular plates, Int J Mech Sci 2002; 44: 11951218.
[18] Zhou D, Vibrations of pointsupported rectangular plates with variable thickness using a set of static tapered beam functions, Int J Mech Sci 2002; 44: 149164.
[19] Zhou D, Cheung Y, Free vibration of line supported rectangular plates using a set of static beam functions, sound vibration 1999; 223: 231245.
[20] Quintana M, Grossi RO, Free vibrations of a generally restrained rectangular plate with an internal line hinge, Appl Acoust 2012; 73: 356365.
[21] Dozio L, On the use of the trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates, ThinWalled Struct 2011; 49: 129144.
[22] Dozio L, Inplane free vibrations of singlelayer and symmetrically laminated rectangular composite plates, Compos Struct 2011; 93:17871800.
[23] Young D, Vibration of rectangular plates by the Ritz method, appl mech 1950; 17:448453.
[24] Warburton G, The vibration of rectangular plates, Proceedings of the Institution of Mechanical Engineers 1954; 168: 371384.
[25] Leissa A, The free vibration of rectangular plates, sound vibration, 31 (1973) 257293.
[26] S. Bassily, S. Dickinson, On the use of beam functions for problems of plates involving free edges, appl mech 1975; 42 : 858.
[27] Goncalves P, Brennan M, Elliott S, Numerical evaluation of highorder modes of vibration in uniform Euler–Bernoulli beams, sound vibration 2007; 301: 10351039.
[28] Xiang Y, Reddy J, Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory, sound vibration 2003; 263: 285297.
[29] Beslin O, Nicolas J, A hierarchical functions set for predicting very high order plate bending modes with any boundary conditions, sound vibration 1997; 202:633655.
[30] Xing Y, Liu B, New exact solutions for free vibrations of thin orthotropic rectangular plates, Compos Struct 2009; 89: 56757.
References
Leissa AW, Vibration of plates ,DTIC Document, 1969.
[2] B. RD, 1984 Formulas for Natural Frequency and Mode Shape, Malabar, FL: Robert E. Krieger Publishing Company.
[3] Gorman DJ, Vibration analysis of plates by the superposition method, World Scientific, 1999.
[4] Reddy JN, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, 2003.
[5] Timoshenko S, WoinowskyKrieger S, Theory of plates and shells, McGrawhill New York, 1959.
[6] Leissa A, Recent research in plate vibrations, 19731976: complicating effects, Shock and Vibration Inform. Shock Vib Digest 1978; 10: 2135.
[7] Qatu MS, Sullivan RW, Wang W, Recent research advances on the dynamic analysis of composite shells: 2000–2009, Compos Struct 2010; 93: 1431.
[8] Lee H, Ng T, Vibration of symmetrically laminated rectangular composite plates reinforced by intermediate stiffeners, Compos Struct 1994; 29: 405413.
[9] Lee H, Ng T, Effects of Torsional and Bending Restraints of Intermediate Stiffeners on the Free Vibration of Rectangular Plates, Struct Mech 1995; 23: 309320.
[10] Huang C, Leissa A, Chan C, Vibrations of rectangular plates with internal cracks or slits, Int J Mech Sci 2011; 53: 436445.
[11] Grossi RO, Raffo J, Natural vibrations of anisotropic plates with several internal line hinges, Acta Mech 2013; 224: 26772697.
[12] Quintana MV, Grossi RO, Free Vibrations of a Trapezoidal Plate with an Internal Line Hinge, The Scientific World Journal 2014.
[13] Wang C, Xiang Y, Wang C, Buckling and vibration of plates with an internal line hinge via the Ritz method, Proceedings of the First AsianPacific Congress on Computational Mechanics 2001, 16631672.
[14] Du J, Liu Z, Li WL, Zhang X, Li W, Free inplane vibration analysis of rectangular plates with elastically pointsupported edges, Vib Acoust 2010; 132: 031002.
[15] Liew K, Lam K, Effects of arbitrarily distributed elastic point constraints on vibrational behaviour of rectangular plates, sound vibration 1994; 174: 2336.
[16] Kim C, Dickinson S, The flexural vibration of line supported rectangular plate systems, sound and vibration 1987; 114:129142.
[17] Xiang Y, Zhao Y, Wei G, Levy solutions for vibration of multispan rectangular plates, Int J Mech Sci 2002; 44: 11951218.
[18] Zhou D, Vibrations of pointsupported rectangular plates with variable thickness using a set of static tapered beam functions, Int J Mech Sci 2002; 44: 149164.
[19] Zhou D, Cheung Y, Free vibration of line supported rectangular plates using a set of static beam functions, sound vibration 1999; 223: 231245.
[20] Quintana M, Grossi RO, Free vibrations of a generally restrained rectangular plate with an internal line hinge, Appl Acoust 2012; 73: 356365.
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