1) In the introduction it should be mentioned clearly what is new in the paper relative to the previous work of the same Authors.
2) Grammatical, spelling, and syntactic check is required for the manuscript. Some minor mistakes have been detected.

3) The comprehensive frequency parameter should be explained better.
4) The “Discussion” and “Conclusions” parts of the manuscript should be more detailed and analyzed.
5) The process of equation solving, should be more detailed. How the boundary conditions are applied?
6) There is only a comparison with the Finite element Method. A comparison of the results should be made with another analytical or experimental investigation in order to improve the quality of the paper.

7) In the finite element analysis there is no information about the type of elements used, the mesh, and the convergence for the results.

8) In the results section there is nothing about Figures 2-5. An interpretation of the results should be given.
9) Emphasis should be given, in the results section, on the benefits of the obtained results with reference to engineering applications.

Author Response

Reviewer: 1

Before the Editor makes a decision, I suggest that the authors must take into account the following corrections:

Our response: We thank the reviewer for this accurate summary of our work and the important comments detailed below.

The title should be rewritten to be friendly.

Our response: We thank the reviewer for the important comment. According to the suggestion, we have rewritten the title of the present paper. The new title is presented here for easy reading of the reviewer. “New analytical free vibration solutions of thin plates using the Fourier series method”.

The author should explain the novelty clearly in the abstract and conclusion.

Our response: We thank the reviewer for the insightful comment. To better explain the novelty of the paper, abstract section and conclusion section have been modified and added in a revised manuscript highlighted in yellow. The revised ones are also presented here:

Abstract: This article aims at analytically solving free vibration problem of rectangular thin plates with one corner free and its opposite two adjacent edges rotationally-restrained, which is difficult to handle by conventional semi-inverse approaches such as the Levy solution and Naiver solution, etc. Based on the classical Fourier series theory, this work presents a first endeavor to treat the two-dimensional half-sinusoidal Fourier series, which is quite similar to the Navier’s form solution, as the solution form of plate deflection. Via utilizing the orthogonality of the present trial function and the Stoke’s transformation technique, the present solution procedure converts the complicated plate problem into solving sets of linear algebra equations, which heavily decreases the difficulties. Therefore, the present approach enables one to solve the title problem in a unified, simple and straightforward way, which is very easily implemented by researchers. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through utilizing different types of Fourier series and it can be extended for further dynamic/static analysis of plates under different shear deformation theories. Moreover, without any extra derivation process, new precise analytical free vibration solutions for plates under three non-Levy-type boundary conditions are also obtained by choosing different rotating fixed coefficients. Consequently, we present more than 400 comprehensive free vibration results for plates with classical/non-classical boundaries, all the present results are confirmed by FEM/analytical solutions and can be used as benchmark data for further research.

Conclusion

A new two-dimensional improved Fourier series approach is developed to seek exact analytical free vibration solution of thin plates under classical/non-classical edge conditions. The significant merits of the present method differing from other analytical methods are: (1) it provides a more simple and straightforward solution procedure for precise plate free vibration analysis since it avoids some complicated mathematical manipulations such as the transformation procedure in finite integral transform method or superposition procedure in symplectic superposition approach; (2) within the framework of Fourier series theory, it reduces the mathematical difficulty of plate problem by converting the boundary value problems of higher-order partial differential equation into solving linear algebra equations; (3) it provides more precise analytical solutions for mechanical problems of moderately-thick/thick plates under more complicated boundary conditions via using different types of Fourier series. The present trial function exactly satisfies both the governing vibration formula and the non-classical boundaries after determining the unknown constants with obvious physics meaning. In addition, new analytical solutions for the CCFF, CSFF and SSFF plates are also acquired by choosing rotating fixed coefficients introduced. Consequently, we provided new exact results, including 50 mode shapes and more than 400 natural frequencies, for intractable title problem. In the present parametric analysis, it is found that both the aspect ratio and boundary restraint significantly influence the free vibration behaviors of plates. All the analytical results are precise enough since they agree well with the results obtained via using the FEM and other analytical methods, which are promising to validate solutions of other methods.

The paper should be carefully revised for punctuation, grammar, spelling mistakes and sentences structuring. For example, in abstract:

“it’s opposite” should be “its opposite”

“such as Levy solution” should be “such as the Levy solution”

“form of plates deflection” should be “form of plate deflection”

“to solve title “ should be “to solve the title”

“different type Fourier series” should be “different types of Fourier series”

“under different shear deformation theory” should be “under different shear deformation theories”

Our response: We would like to thank the reviewer for careful and thorough reading of this manuscript and for the thoughtful comments and constructive suggestions, which help to improve the quality of this manuscript. According to the suggestion, we have carefully revised the paper and made modifications, for example:

All the other modifications are highlighted in yellow, please check the revised manuscript.

In the introduction, all authors should be given by family name only for example: “D. J. Gorman” must be “Gorman” …….

Our response: We thank the reviewer for the important comment. We have modified our manuscript according to the reviewer comments, please check the revised manuscript.

In the article, all refs. should be given by family name and distance then number for example: “Yilong Wang[1]” must be “Wang [1]” …….

Our response: We appreciate the reviewer for the important comment. We fully agree with the reviewer, and modify all references in introduction part according to the reviewer’s suggestion. Please check the revised manuscript.

Why, the equations (8), (9) and (10) have 4 equations? Should be give more explanations

Our response: We appreciate the reviewer for the important comment. Why equations (8), (9) and (10) have 4 equations is for the following reasons:(1) formulas for the fourth order derivatives of the deflection (such asand ) are used to substituted into Eq. (3) to obtain Fourier coefficient ; (2) expressions for ，， and are used to derive relationships of some unknown constants, such as ; (3)expressions for are used to satisfy the bending moment requirements along the rotationally-restrained edges, and expressions for the ， are used to compose bending moments and then satisfy the bending moment requirements along the free edges.

The literature survey must be improved on adding some relevant references as: Analytical solution for a free vibration of a thermoelastic hollow sphere. Mech. Based Des. Struct. Mach. 2015, 43, 265-276, doi:10.1080/15397734.2014.956244.

Wave propagation in a generalized thermoelastic plate using eigenvalue approach. Journal of Thermal Stresses 2016, 39, 1367-1377, doi:10.1080/01495739.2016.1218229.

Vibrational behavior of thermoelastic rotating nanobeams with variable thermal properties based on memory-dependent derivative of heat conduction model. Arch Appl Mech 2022, doi:10.1007/s00419-022-02110-8.

Our response: We thank the reviewer for the important comment. All the references suggested by the reviewer are added in the text and reference section and highlighted in yellow in revised manuscript. The added sentences and references are also addressed here:

“Based on a nonlocal elasticity theory and the generalized heat conduction model with phase delays, Ahmed [17] provides Sinusoidal form solutions for the thermomechanical problem of rotating size-dependent nanobeams with clamped–clamped boundary conditions. ………… Based on the generalized thermoelasticity theory, Abbas [22] obtains the analytical solution for the temperature, displacement components, and stresses of thermoelastic material plates by utilizing an eigenvalue approach. Via using the same method, the author [23] also provides new accurate analytical solutions for the free vibration problem of thermoelastic hollow sphere, in which the dispersion relations for various types of possible vibration modes of the hollow sphere are well derived.”

Abouelregal AE, Atta D, Sedighi HM. Vibrational behavior of thermoelastic rotating nanobeams with variable thermal properties based on memory-dependent derivative of heat conduction model. Archive of Applied Mechanics 2022:1–24.
Abbas, Ibrahim A, Abdalla, Abo-El-Nour N, Alzahrani, Faris S, et al. Wave propagation in a generalized thermoelastic plate using eigenvalue approach. Journal of Thermal Stresses 2016;39:1367–77.
Abbas, Ibrahim, A. Analytical Solution for a Free Vibration of a Thermoelastic Hollow Sphere. Mechanics Based Design of Structures & Machines 2015;43:265–76.

Author Response File: Author Response.pdf

Reviewer 2 Report

Before the Editor makes a decision, I suggest that the authors must take into account the following corrections:

1. The title should be rewritten to be friendly.

2. The author should explain the novelty clearly in the abstract and conclusion.

3. The paper should be carefully revised for punctuation, grammar, spelling mistakes and sentences structuring. For example in abstract:

- “it’s opposite” should be “its opposite”

- “such as Levy solution” should be “such as the Levy solution”

- “form of plates deflection” should be “form of plate deflection”

- “to solve title “ should be “to solve the title”

- “different type Fourier series” should be “different types of Fourier series”

- “under different shear deformation theory” should be “under different shear deformation theories”

4. In the introduction, all authors should be given by family name only for example: “D. J. Gorman” must be “Gorman” …….

- 5. In the article, all refs. should be given by family name and distance then number for example: “Yilong Wang[1]” must be “Wang [1]” …….

- 6. Why, the equations (8), (9) and (10) have 4 equations? Should be give more explanations

7. The literature survey must be improved on adding some relevant references as:

-Analytical solution for a free vibration of a thermoelastic hollow sphere. Mech. Based Des. Struct. Mach. 2015, 43, 265-276, doi:10.1080/15397734.2014.956244.

- Wave propagation in a generalized thermoelastic plate using eigenvalue approach. Journal of Thermal Stresses 2016, 39, 1367-1377, doi:10.1080/01495739.2016.1218229.

- Vibrational behavior of thermoelastic rotating nanobeams with variable thermal properties based on memory-dependent derivative of heat conduction model. Arch Appl Mech 2022, doi:10.1007/s00419-022-02110-8.

Author Response

Response to Reviewer’s 2 Comments

Reviewer: 2

This manuscript presents an analytical solution for free vibration of rectangular thin plates under classical/non-classical edge conditions. The following concerns, however, must be addressed:

Our response: We thank the reviewer for this accurate summary of our work and the important comments detailed below.

A careful reading of the text should be done to suppress language errors.

Our response: We greatly appreciate the reviewer’s careful reading. The language of the manuscript is improved and highlighted in yellow in the revised manuscript. Please refer to the revised manuscript.

In “1. Introduction”, the application background is lacking. This may leave the readers confused about the significance of the study.

Our response: We thank the reviewer for the important comment. According to the reviewer’s suggestion, we have made some modifications of introduction part to point out the significance of the study. The modifications are also presented here for easy reading of the reviewer.

“Rectangular thin plate is considered as the basic structural element in practical applications because of its relevance in various engineering fields such as civil and structural engineering, mechanical engineering, naval and aerospace, etc. The extensive application of such structures needs to investigate the dynamic characteristics of thin plates to establish an accurate and reliable design. It is confirmed that free vibration of plate causes reduction in structure stiffness which reduces its load carrying capacity, as a result may cause premature structure failure. Simultaneously, a clear understanding of natural frequencies and associated mode shapes is the basis for reliable design toward avoiding plate’s resonance problem. However, it is difficult to deal with free vibration problem of plates under various boundary conditions, especially for the ones with complex non-classical boundaries.”

The authors have performed a series of numerical simulations. Please detail the establishment of the numerical models. It is not appropriate to show only the results without describing the model establishment.

Our response: We thank the reviewer for pointing this out. We have added the following sentence in Comprehensive frequency parameter and mode shape results section to provide the information for finite element modeling. It is also presented here for easy reading of the reviewer.

“It is worth noting that the reliable FEM results for comparison are acquired by using ABAQUS 6.13 (2013), where 4-node thin shell element S4R is employed. To guarantee the accuracy of the FEM results, we testify the convergence for the FEM results of CCFF plate with aspect ratio b/a=1 as is shown in Table 1. It is found that most of the obtained FEM results for converge when the mesh size is 1/400 of the plate length. Thus, the uniform mesh size with short edge length of 1/400 is adopted throughout the present study.”

Table 1 Convergence of the dimensionless frequency parameter obtained by FEM for the CCFF square plates with different mesh sizes.

	Mode
	first	second	third	fourth	fifth	sixth	seventh	eighth	ninth	tenth
1/100	6.9190	23.905	26.590	47.654	62.711	65.572	85.726	88.377	121.50	124.21
1/200	6.9187	23.901	26.585	47.646	62.711	65.539	85.694	88.341	121.38	124.08
1/300	6.9185	23.900	26.584	47.642	62.705	65.532	85.686	88.332	121.35	124.06
1/400	6.9184	23.900	26.583	47.640	62.702	65.529	85.682	88.328	121.34	124.05
1/500	6.9183	23.900	26.583	47.639	62.702	65.529	85.682	88.327	121.34	124.05

The color legend is lacking in Figs. 2–5. Without color legend, the color changes in these figures are meaningless. Please add.

Our response: We thank the reviewer for the important comment. Actually, the present study provided mode shape figures for the non-dimensional frequency parameter of CCFF, CSFF, and SSFF square plates. To eliminate the confusion of the reviewer, we presented new mode shape figures without color, which were modified in the revise manuscript. The modification is also shown below for easy reading of the reviewer. We greatly appreciate the understanding of the reviewer.

	Mode shapes
	first	second	third	fourth	fifth
Present
FEM
	sixth	seventh	eighth	ninth	tenth
Present
FEM
Fig. 2 First ten mode shapes of the square CCFF plate with ,

	Mode shapes
	first	second	third	fourth	fifth
Present
FEM
	sixth	seventh	eighth	ninth	tenth
Present
FEM
Fig. 3 First ten mode shapes of the square CSFF thin plate with ,

	Mode shapes
	first	second	third	fourth	fifth
Present
FEM
	sixth	seventh	eighth	ninth	tenth
Present
FEM
Fig. 4 First ten mode shapes of the square SSFF plates with ,

	Mode shapes
	first	second	third	fourth	fifth




Fig. 5 First five mode shapes of the RRFF square plates with, and equaling 0.1, 0.3, 0.5 and 0.8.

In Page 9, it is mentioned that “…in Tables 1-3, it is clearly found that the acquired of CCFF plates is larger than that of CSFF and SSFF plates; values of for CSFF plates are larger than values of for SSFF plates, which reveals…” There are no relevant results for SSFF plates in Tables 1–3. Please check.

Our response: We thank the reviewer for pointing this out. Actually, we have made a spelling mistake, results in Table 4 are for SSFF plates. We have modified the title of Table 4. The title modified is also presented here for easy reading of the reviewer.

Table 4 Frequency parameter of SSFF rectangular plates with defined rotational fixity factors and aspect ratio changing from 0.5 to 5

Author Response File: Author Response.pdf

Reviewer 3 Report

This manuscript presents an analytical solution for free vibration of rectangular thin plates under classical/non-classical edge conditions. The following concerns, however, must be addressed:

1. A careful reading of the text should be done to suppress language errors.

2. In “1. Introduction”, the application background is lacking. This may leave the readers confused about the significance of the study.

3. The authors have performed a series of numerical simulations. Please detail the establishment of the numerical models. It is not appropriate to show only the results without describing the model establishment.

4. The color legend is lacking in Figs. 2–5. Without color legend, the color changes in these figures are meaningless. Please add.

5. In Page 9, it is mentioned that “…in Tables 1-3, it is clearly found that the acquired of CCFF plates is larger than that of CSFF and SSFF plates; values of for CSFF plates are larger than values of for SSFF plates, which reveals…” There are no relevant results for SSFF plates in Tables 1–3. Please check.

Author Response

Response to Reviewer’s 3 Comments

Reviewer: 3

The paper should be improved according to the following:

Our response: We thank the reviewer for this accurate summary of our work and the important comments detailed below.

1 In the introduction it should be mentioned clearly what is new in the paper relative to the previous work of the same Authors.

Our response: We thank the reviewer for the important comment. We have added and modified some sentences of the introduction part to illustrate the main novelties of the present study. The modifications are highlighted in yellow in the revised manuscript and also are presented here for easy reading of the reviewer.

“Considerable literatures confirm that the Navier’s form solutions are still accurate analytical solutions of the ideal form because of their simplicity and excellent orthogonality. It is well acknowledgement that Navier’s solution is the most commonly used in analyzing static and dynamic problems of plates since its simple solution procedure. However, Navier’s solution is only suitable for plates with all edges simply-supported. Consequently, it is of great significance to extent such effective classical approach to solve problem of plates under more complex boundaries. Motivated by the above situation, a two-dimensional improved Fourier series approach is developed to provide exact vibration analysis of plates under the complicated non-classical boundary restraints. For the first time, we treat the two-dimensional half-sinusoidal Fourier series as the trial function for the deflection of plates. Interestingly, the adopted trial function is quite similar to the Navier’s form solution and automatically satisfies the zero deflection at rotationally restrained edges, zero effective shear force at free edges and zero twisting moment at the free corner. We also make a first endeavor to introduce the Stoke’s transformation technique on deriving detailed expressions for the first four order derivatives of the deflection, in which some associated unknown constants with obvious physical meaning can also be acquired. The obtained constants can be solved by dealing with sets of linear simultaneous equations after putting the obtained derivatives of the deflection into the remain boundaries. The present solution procedure and new Fourier series expansions, which can be directly adopted for solving mechanical problems of thick plates, are the main contribution of the present study. Another advantage of the present study is that new precise free vibration results for rectangular thin plates under classical boundaries can be obtained via selecting different coefficients of the rotational springs introduced. Therefore, four different types of boundaries including RRFF, CCFF, CSFF and SSFF are considered for the investigated plate, where the free, rotationally-restrained, simply supported and clamped edge are respectively designated as F, R, S and C. The present work chooses a clockwise representation for the boundaries studied, starting from the edges of x=0. Finally, we list new precise results for the dimensionless natural frequency parameters and plot the corresponding mode shapes, all of which are testified to be qualified as new exact benchmarks after comparing with the ones solved by the conventional finite element method and other analytical methods.”

2 Grammatical, spelling, and syntactic check is required for the manuscript. Some minor mistakes have been detected.

Our response: We greatly appreciate the reviewer’s careful reading. Grammatical, spelling, and syntactic of the paper have been carefully checked. We revised our manuscript according to the reviewer comments and highlighted revision in yellow. Please refer to the revised manuscript.

3 The comprehensive frequency parameter should be explained better.

Our response: We thank the reviewer for the important comment and suggestion. We have made modifications in “Comprehensive frequency parameter and mode shape results” part of the revised mauscript and highlighted revision in yellow. The modifications are also presented here for easy reading of the reviewer.

“As is shown in Table 2-4, the non-dimensional frequency parameter decreases as the aspect ratio increases and vice versa. Therefore, the minimum values of corresponds to the aspect ratio (5.0). The non-dimensional frequency parameter decreases rapidly when the aspect ratio is 1 or less than one. Additionally, it can be observed that the values of changes very few when changes from 2.5 to 5. It can be concluded that the aspect ratio from 0.5 to 2.0 has more influence on the natural frequencies as compared to the aspect ratio from 2.5 to 5. Through Tables 2-4, it also found that boundary conditions have great influence on the non-dimensional frequency parameter. The acquired for all boundary conditions is higher as the aspect ratio gets smaller than one. The magnitude of is always greater for CCFF than those under CSFF and SSFF when aspect ratio is same. Also, the non-dimensional frequency parameter for CSFF is higher than the SSFF for all aspect ratio. It can be concluded that the plate with more clamped edges requires more energy to vibrate. Through comparisons in Tables 2-4, it is clearly seen that all the results obtained are in good agreement with ones solved numerically or analytically, especially with those in Ref[43]. Similarly, through Tables 5-6, it is obvious the obtained increases with the increase of rotating fixed coefficient r, which indicates that the RRFF plate with stricter boundary constraints needs more energy to vibrate. The aforesaid parametric analysis indicates that both the aspect ratios and the degree of boundary restraint effect free vibration behaviors of plates significantly.”

4 The “Discussion” and “Conclusions” parts of the manuscript should be more detailed and analyzed.

Our response: We thank the reviewer for the important comment and suggestion. According to the suggestion, we have modified some sentences to give more detailed frequency parameter analysis. We did not present it here since it was already shown in our response to Comment 3. We then added some sentences to better explain the mode shape results of the present study. The sentences added are also present below for easy reading of the reviewer. We finally made modifications for the conclusion section according to the suggestion of the reviewer. The revised conclusion is also presented here:

As is shown in Figures 2-4, it is easily found that all the obtained free vibration mod shapes for the CCFF, CSFF and SSFF plates match well with the ones provided by the Finite element method. It is also obviously found the present mode shapes strictly satisfy the boundary conditions, which confirm the accuracy of the present method. Through Figure 5, it is found that the mode shapes for the RRFF plate under different degree boundary restraints change gradually, which further confirm the affections of boundary conditions on the free vibration characteristics of plates.

Conclusion

5 The process of equation solving, should be more detailed. How the boundary conditions are applied?

Our response: We appreciate the reviewer for the important comment. Actually, the given Eq. (16) in this article has been simplified already. In order to eliminate the confusion on how to apply the boundary condition, we have added and modified some equations. The modifications for the equations and the associated sentences are presented here for easy reading of the reviewer.

Expressions for the RRFF plate, in which rotationally-restrained at edges and , and its opposite corner free, are described as:

(4)

Substituting the assumed and its fourth high-order derivatives appeared in Eqs. (8)–(10) into Eq.(3) yields the following expression for the free vibration governing equation :

(12)

It is obvious that is zero since the zero twisting moment at free corner. Similarly, due to the zero deflection at rotationally-restrained edges (, ) and effective shear force at free edges (, ), it is easy to derive that

(13)

(14)

(15)

It is clearly that are the Fourier coefficients of the slope along the free edges. Obviously, and are the Fourier coefficients of bending moments along the rotationally-restrained edges, the corresponding bending moments can be directly acquired by the following formula:

(16)

One can obtain the following simplified expression for the free vibration governing equations by substituting Eqs. (13)-(15) into the Eq. (12), which is shown below:

(17)

After obtain the expression for , using the solution procedure expressed by Eqs. (20)-(28) to determine the unknown constants , , and , then one can obtain analytical solutions satisfy both the governing vibration equation and the boundaries investigated.

All the other modifications are highlighted in yellow, please check the revised manuscript. We greatly appreciate the understanding of the reviewer.

6 There is only a comparison with the Finite element Method. A comparison of the results should be made with another analytical or experimental investigation in order to improve the quality of the paper.

Our response: We appreciate the reviewer for the important comment. We agree that it is necessary to compare the present results with those obtained by other analytical methods or experimental investigation. We thus made further literature review, and found some comparable results obtained by other analytical methods. Unfortunately, we did not find suitable experimental results. We greatly appreciate the understanding of the reviewer. The comprehensive comparison has been incorporated in Tables 1-3 which are shown below. The reference added and corresponding sentence modified are highlighted in yellow in revised manuscript and presented here for easy reading of the reviewer.

“Consequently, we present more than 400 comprehensive free vibration results for plates with classical/non-classical boundaries, all the present results are confirmed by FEM/analytical solutions and can be used as benchmark data for further research.”

“Aim at testifying the capability of the proposed approach in predicting free vibration response for rectangular thin plates with classic/non-classic type edge conditions, new precise comprehensive results including frequency parameters and the associated mode shapes are in comparison with FEM results and the existing analytical solutions[16,43].”

“Through the comparisons in Tables 2-4, it is clearly seen that all the results obtained are in good agreement with ones solved numerically or analytically, especially with those in Ref[43].”

“Excellent agreement is found between the present numerical/graphical results with FEM solutions and other analytical solutions, which confirms the qualification of the present method in solving complex plate problems.”

“All the analytical results are precise enough since they agree well with the results obtained via using the FEM and other analytical methods, which are promising to validate solutions of other methods.”

[16] Li WL, Zhang X, Du J, Liu Z. An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. Journal of Sound and Vibration 2009;321:254–69.

[43] Zhang J, Lu J, Ullah S, Gao Y, Zhao D, Jamal A, et al. Free vibration analysis of thin rectangular plates with two adjacent edges rotationally-restrained and the others free using finite Fourier integral transform method. Structural Engineering and Mechanics 2021;80:455–62.

Table 2 Frequency parameterof CCFF rectangular plates with defined rotational fixity factors and aspect ratiochanging from 0.5 to 5.

		Mode
		first	second	third	fourth	fifth	sixth	seventh	eighth	ninth	tenth
0.5	Present	17.118	36.372	73.402	90.902	115.20	131.62	158.98	208.74	219.42	248.45
	Ref[43]	17.118	36.372	73.402	90.902	115.20	－	－	－	－	－
	FEM	17.116	36.363	73.391	90.894	115.17	131.60	158.93	208.73	219.36	248.46
1	Present	6.9191	23.903	26.587	47.651	62.706	65.531	85.697	88.342	121.35	124.05
	Ref[43]	6.9191	23.903	26.587	47.651	62.706	－	－	－	－	－
	FEM	6.9184	23.900	26.583	47.640	62.702	65.529	85.682	88.328	121.34	124.05
1.5	Present	4.9677	13.235	23.278	30.112	34.138	52.236	56.033	62.667	72.788	78.511
1.5	FEM	4.9673	13.233	23.275	30.110	34.132	52.227	56.029	62.666	72.781	78.498
2	Present	4.2795	9.0929	18.351	22.726	28.800	32.904	39.745	52.184	54.855	62.112
	Ref[43]	4.2795	9.0929	18.351	22.726	28.800	－	－	－	－	－
	FEM	4.2793	9.0918	18.349	22.724	28.795	32.901	39.738	52.181	54.846	62.111
2.5	Present	3.9726	7.1296	13.038	21.728	22.798	26.402	33.479	34.675	43.607	50.053
2.5	FEM	3.9725	7.1287	13.037	21.726	22.796	26.399	33.474	34.675	43.600	50.049
3	Present	3.8149	6.0344	10.143	16.364	22.227	24.570	25.416	30.169	35.642	37.386
3	FEM	3.8148	6.0338	10.142	16.362	22.225	24.567	25.414	30.164	35.641	37.380
3.5	Present	3.7249	5.3602	8.3946	12.940	19.077	22.202	24.211	26.928	28.171	33.423
3.5	FEM	3.7248	5.3598	8.3937	12.939	19.076	22.200	24.208	26.925	28.167	33.417
4	Present	3.6692	4.9175	7.2540	10.721	15.406	21.228	22.251	23.714	26.642	28.644
4	FEM	3.6692	4.9171	7.2533	10.720	15.405	21.226	22.249	23.712	26.639	28.642
4.5	Present	3.6327	4.6127	6.4673	9.2024	12.885	17.543	22.045	23.047	23.585	25.704
4.5	FEM	3.6328	4.6125	6.4668	9.2015	12.884	17.541	22.043	23.046	23.585	25.701
5	Present	3.6075	4.3952	5.9014	8.1166	11.086	14.844	19.383	22.081	23.066	24.665
5	FEM	3.6075	4.3949	5.9009	8.1158	11.085	14.843	19.382	22.080	23.064	24.663

Table 3 Frequency parameterof CSFF rectangular plates with defined rotational fixity factorsand aspect ratio changing from 0.5 to 5.

		Mode
		first	second	third	fourth	fifth	sixth	seventh	eighth	ninth	tenth
0.5	Present	8.5061	30.954	64.140	70.962	92.922	128.79	139.47	199.83	202.51	208.60
	Ref[43]	8.5062	30.955	64.142	70.962	92.927	－	－	－	－	－
	FEM	8.5030	30.943	64.137	70.950	92.899	128.79	139.45	199.81	202.50	208.59
1	Present	5.3508	19.073	24.673	43.087	52.706	63.754	77.488	83.660	106.29	120.34
	Ref[43]	5.3509	19.075	24.671	43.087	52.707	－	－	－	－	－
	FEM	5.3499	19.071	24.675	43.076	52.706	63.754	77.476	83.652	106.28	120.33
1.5	Present	4.4122	10.870	22.852	25.572	32.347	48.278	49.368	62.439	71.359	72.932
1.5	FEM	4.4117	10.869	22.850	25.570	32.340	48.270	49.364	62.437	71.351	72.920
2	Present	4.0292	7.8658	15.849	22.573	27.800	29.251	37.927	47.071	51.891	61.980
	Ref[43]	4.0294	7.8661	15.850	22.576	27.801	－	－	－	－	－
	FEM	4.0290	7.8649	15.847	22.572	27.798	29.250	37.924	47.071	51.887	61.979
2.5	Present	3.8434	6.4013	11.522	19.657	22.430	25.953	31.191	32.724	41.880	46.016
2.5	FEM	3.8433	6.4005	11.520	19.656	22.428	25.949	31.189	32.718	41.872	46.015
3	Present	3.7409	5.5654	9.1527	14.784	21.990	22.884	24.843	29.525	32.902	36.284
3	FEM	3.7409	5.5650	9.1523	14.785	21.990	22.883	24.841	29.523	32.901	36.281
3.5	Present	3.6792	5.0420	7.7087	11.830	17.544	22.149	23.969	25.093	27.675	32.667
3.5	FEM	3.6791	5.0416	7.7079	11.829	17.543	22.147	23.966	25.092	27.671	32.662
4	Present	3.6393	4.6931	6.7580	9.9093	14.261	19.828	22.154	23.584	26.245	26.958
4	FEM	3.6392	4.6928	6.7574	9.9084	14.260	19.826	22.152	23.583	26.242	26.956
4.5	Present	3.6121	4.4497	6.0968	8.5895	12.010	16.405	21.614	22.259	23.298	25.486
4.5	FEM	3.6121	4.4494	6.0962	8.5887	12.009	16.404	21.613	22.257	23.297	25.483
5	Present	3.5927	4.2735	5.6176	7.6419	10.402	13.945	18.278	22.041	22.938	23.570
5	FEM	3.5928	4.2733	5.6171	7.6412	10.401	13.944	18.278	22.040	22.936	23.569

Table 4 Frequency parameter of SSFF rectangular plates with defined rotational fixity factors and aspect ratio changing from 0.5 to 5

		Mode
		first	second	third	fourth	fifth	sixth	seventh	eighth	ninth	tenth
0.5	Present	6.6436	25.374	58.740	65.177	89.093	113.14	131.50	185.64	189.60	202.12
	Ref[16]	6.644	25.370	58.739	65.183	89.084	113.17	－	－	－	－
	Ref[43]	6.6437	25.375	58.741	65.179	89.096	－	－	－	－	－
	FEM	6.6401	25.366	58.729	65.175	89.065	113.13	131.47	185.63	189.55	202.13
1	Present	3.3668	17.314	19.292	38.208	51.034	53.485	72.964	74.626	104.71	107.22
	Ref[16]	3.370	17.321	19.293	38.203	51.032	53.497	－	－	－	－
	Ref[43]	3.3670	17.316	19.293	38.210	51.035	－	－	－	－	－
	FEM	3.3659	17.313	19.292	38.199	51.033	53.485	72.945	74.614	104.71	107.21
1.5	Present	2.2328	9.5393	16.679	24.543	26.993	44.058	48.175	51.149	61.061	68.835
1.5	FEM	2.2321	9.5374	16.678	24.540	26.987	44.047	48.173	51.148	61.052	68.821
2	Present	1.6607	6.3435	14.684	16.293	22.273	28.291	32.875	46.416	47.401	50.531
	Ref[43]	1.6609	6.3438	14.685	16.295	22.274	－	－	－	－	－
	FEM	1.6605	6.3427	14.682	16.292	22.271	28.291	32.870	46.415	47.395	50.530
2.5	Present	1.3195	4.7306	10.316	15.789	18.845	20.100	27.184	30.659	37.051	45.407
2.5	FEM	1.3192	4.7296	10.314	15.789	18.843	20.097	27.179	30.658	37.043	45.406
3	Present	1.0937	3.7679	7.8075	13.722	15.734	18.629	21.949	23.967	31.060	32.321
3	FEM	1.0935	3.7674	7.8068	13.720	15.734	18.627	21.947	23.966	31.058	32.320
3.5	Present	0.93401	3.1304	6.2428	10. 680	15.513	16.705	17.882	21.828	24.305	27.372
3.5	FEM	0.93375	3.1297	6.2418	10. 679	15.512	16.704	17.880	21.825	24.303	27.367
4	Present	0.81491	2.6776	5.1872	8.6529	13.220	15.555	17.240	19.093	20.477	24.784
4	FEM	0.81468	2.6770	5.1863	8.6518	13.219	15.555	17.239	19.092	20.475	24.780
4.5	Present	0.72280	2.3397	4.4321	7.2357	10.892	15.228	15.742	16.910	19.424	21.101
4.5	FEM	0.72260	2.3391	4.4314	7.2346	10.891	15.227	15.742	16.909	19.422	21.100
5	Present	0.64944	2.0779	3.8673	6.2000	9.1985	12.915	15.471	16.562	17.498	18.740
5	FEM	0.64926	2.0775	3.8667	6.1990	9.1975	12.914	15.471	16.562	17.497	18.738

7 In the finite element analysis there is no information about the type of elements used, the mesh, and the convergence for the results.

Table 1 Convergence of the dimensionless frequency parameter obtained by FEM for the CCFF square plates with different mesh sizes.

	Mode
	first	second	third	fourth	fifth	sixth	seventh	eighth	ninth	tenth
1/100	6.9190	23.905	26.590	47.654	62.711	65.572	85.726	88.377	121.50	124.21
1/200	6.9187	23.901	26.585	47.646	62.711	65.539	85.694	88.341	121.38	124.08
1/300	6.9185	23.900	26.584	47.642	62.705	65.532	85.686	88.332	121.35	124.06
1/400	6.9184	23.900	26.583	47.640	62.702	65.529	85.682	88.328	121.34	124.05
1/500	6.9183	23.900	26.583	47.639	62.702	65.529	85.682	88.327	121.34	124.05

8 In the results section there is nothing about Figures 2-5. An interpretation of the results should be given.

Our response: We appreciate the reviewer for the important comment. We have added some sentences to give the interpretation of the obtained mode shape results. We did not present them here since they were already shown in our response to Comment 3. Please check the response to Comment 3.

9 Emphasis should be given, in the results section, on the benefits of the obtained results with reference to engineering applications.

Our response: We thank the reviewer for the important comment. According to the suggestion of the reviewer, we have added some sentences into “Comprehensive frequency parameter and mode shape results” part and highlighted them in yellow. The sentences added are also presented here for easy reading of the reviewer.

Obviously, all the present the non-dimensional natural frequency parameters and the associated vibration mode shapes can be adopted for engineers and scientists for academic and practical applications.The laws indicated by the present parametric analysis enable designers to effectively avoid resonance problems of rectangular thin plate structures.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Reviewer 3 Report

This manuscript was revised according to the comments, which could meet the requirement of acceptance.

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New Analytical Free Vibration Solutions of Thin Plates Using the Fourier Series Method

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