Analysis of Free Oscillations of Circular Plates With Variable Thickness Based on the Symmetry Method

Based on symmetry and factorization methods, a general analytical solution to the fourth-order differential equation has been derived for a problem on the free axisymmetric oscillations of a circular plate of variable thickness. The law of the thickness change is the concave parabola h=H0(1–μρ)2, where µ is a constant coefficient that determines the degree of plate concaveness. The solution has been given by the Bessel functions of zero and the first order of the actual and imaginary argument. A circular ring plate has been considered whose inner contour is rigidly fixed and whose outer edge is free, for three values of the µ coefficient. We have determined the first three natural values for the problem (frequency numbers) and their natural functions (oscillation shapes). It has been shown that the natural frequencies of the first three shapes of oscillations decrease, with the increase in concaveness (increase in µ), to varying degrees, determined by the number of the frequency number λi (i=1, 2, 3). At µ=1.21417 and µ=1.39127, the frequencies decrease, compared to the case of µ=0.5985, by (1;1.3) %, (17.6;24) %, (22.85;30.35) %, respectively, for λ1, λ2, λ3. One can see a significant drop in frequency on the higher shapes of oscillations (λ2, λ3) and a slight drop in the basic shape (λ1). We have established the values and coordinates of extreme deflections (antinodes of oscillations) and the indicative coordinates of the nodal cross-sections. The reported numerical parameters, along with the frequency indicators, are a means of identifying the oscillational properties of a plate when it is studied in practice. We have built the graphic dependences for radial σr and tangential σθ cyclical stresses at the basic shape for each of the three variants of the concaveness of a parabolic plate. It has been established that the increase in the ratio of edge thickness, that is, concaveness, leads to an increase in σr in the cross-sections outside the end constraint. These stresses, which operate far from the free edge, for example at the end constraint or the area of the maximum σθ, are greater than σθ in varying degrees. Because of this, these stresses pose a major threat in terms of the cyclical strength of the plate when σr reaches destructive values. We have pointed to the possibility to provide, by increasing the concaveness of the parabolic plate, the optimal ratio between the value of σr at the end constraint and σr operating away from the fastening. This ratio, approximately equal to 1, is ensured at µ=1.39127 considered in the current work.


Introduction
This paper reports the implementation of the symmetry method in studying the free oscillations of the continuous ring or circular plates of variable thickness. Circular plates are widely used in many technical applications in engineering and construction [1,2]. Plate-type elements of the membrane type have been used for medical purposes [3]. The plates are employed in high-speed rotational engineering systems as one of the main structural elements [4]. Typically, these elements are subject to destructive stresses caused by resonant oscillations during operation. Free oscillation analysis is necessary to enhance the technical resource of a structure as a whole by adjusting the resonance, which is possible if the natural frequencies are known. At the same time, there is an issue of finding a closed solution to the problems on the free oscillations of plates, addressed by a large body of research [5][6][7]. The above problems remain unresolved in many cases, despite their practical and theoretical significance. The relevance of the current work stems from the practice-dictated requests to provide for the desired opera-tional resource of plate elements or nodes, which is impossible without a clear understanding of their cyclical stressed and strained state. This requirement could be primarily met on the basis of a theoretical analysis of oscillations.

Literature review and problem statement
The results of precise studies into the plates of variable thickness are very limited primarily due to the known mathematical difficulties arising from the analytical investigation of their oscillations.
Paper [8] shows that it is possible, under a free oscillation mode, based on the approximate Relay-Ritz method and Chebyshev's polynomial functions, to solve the problem on natural values. According to the proposed approach, the displacement coefficients are determined as a simplification tool for solving the problem. The paper reports the results of studying the oscillations of a plate with a linear-variable thickness on the elastic base. The specified methods, however, do not make it possible to derive an analytical solution to the problem on the oscillations of a plate with the parabolic law of thickness change. It could be argued that the reason for this is the mathematical difficulties that arise when trying to use the ratios given in the work for this purpose.
Another approach to solving a problem on natural values is considered in paper [9]. In this case, an independent conjugal coordinate method is recommended to analyze the free oscillations of a plate. However, in fact, it is unclear how the specified approach could be extended to the problem involving a plate with a solid profile, different from linear.
The free oscillations of a plate of constant thickness are studied in [10,11]. These works employed a differential method of transforming equations to derive the solutions to the problem of natural values. The authors do not consider the algorithm for finding the natural frequencies of a plate for the highest shapes of oscillations. In addition, work [10] graphically presents only the diagrams of deflections while the diagrams of cyclical stresses are lacking, which may raise doubts about the suitability of the proposed approach for practice. A circular plate was examined in paper [11] employing the Runge-Kutta method. The resulting ratios and the approach itself are difficult to apply to solve the problem on free oscillations of a variable-thickness plate.
Although article [12] provides a solution to the problem about a plate under a mode of axisymmetric free oscillations (the frequency numbers were defined), it does not give the frequency equation in an analytical form. To study the plates, the authors suggest using additional variable coefficients, which they believe are designed to improve the classical Kirchhoff plate theory. As a result, this leads to a significant complication of the problem.
The issue of fixing a plate under conditions of repeatedlyvariable operational loads deserves special attention. This is explained by the fact that the classical statement of the problem on natural values requires mandatory accounting of boundary conditions. Investigating such a situation is addressed, for example, in work [13], where a finite element method was used to analyze free oscillations of a rigidly-clamped plate.
The issues relating to the construction of analytical solutions to the fourth-order oscillation equation with variable coefficients remain unresolved. Part of the solution to this problem is given in article [14], where a similar equation was studied on the basis of approximate approachesthe classical method by Galerkin and the energy one. The statement of the problem set out in the cited article does not make it possible to use the results to consider plates with a parabolic profile.
Prospective practical application of variable-thickness plates is addressed in works [15,16] that show the importance of solving a problem on natural values when using plates under the conditions of cyclical air loads in aircraft. There remained, however, the issues related to the search for the natural frequencies of a plate under conditions of real loads; no regions of the location of nodes and antinodes of oscillations were specified. The absence of such information could lead to uncontrollable deformations of the plate.
All this suggests that it is appropriate to conduct a study on solving the problem on the oscillations of a plate of variable thickness with varying degrees of concaveness. The above literary sources employ a mathematical apparatus based only on the approximate numerical methods and approaches. The accuracy and reliability of the results, in this case, could be highly questionable. Analytical approaches, as opposed to approximate or numerical, make it possible to expand the existing estimated base of accurate solutions for the plates subject to oscillations, thereby supplementing it with new results obtained in the final form.

The aim and objectives of the study
The aim of this study is to examine the oscillations and to analyze the stressed-strained state of a circular ring plate whose thickness changes in line with the law of concave parabola h=H 0 (1-μρ) 2 with varying degrees of concaveness, determined by the values of constant μ. In practice, this would make it possible to map out the optimal design of plate-type structural elements of variable thickness according to the criteria of mass-size, resonance frequencies, movements (shapes of oscillations) and cyclical stresses by choosing appropriate values for μ.
To accomplish the aim, the following tasks have been set: -to build a general analytical solution to the fourth-order differential equation for the problem about a cyclical symmetrical bend of the circular plate of a parabolic profile with varying degrees of concaveness; -to derive, based on a general solution, the formulae for calculating cyclical stresses in a plate; -to determine, for a circular plate with a rigidly trimmed internal contour and a free one on the outside, by using the resulting solution, the first three natural values for the problem for each of the three cases of concaveness and to build, based on them, the natural functions (shapes of the plate's natural oscillations); -to construct, as an example, the diagrams of radial and tangential stresses at the basic shape of oscillations.

The original differential equation and its general solution for a plate with a thickness of h=H 0 (1-μρ) 2
For a plate that abides the law of changing the thickness h=H 0 (1-μρ) 2 , the equation of the shapes of natural oscillations takes the form [17] ( ) f is the frequency of natural oscillations; H 0 is the thickness in the center of a plate; ν, γ, E are the Poisson coefficient, the specific weight, the elasticity module of the plate materi-al, respectively; g is the acceleration of gravity. Hereafter, the v coefficient is accepted equal to 1/3, which is true for most structural metallic materials.
The order IV equation in notation (1) makes it possible, based on a factorization method, to replace it with two order II equations and then the solution to equation (1) could be defined as the sum of the general solutions to these two equations W=W 1 +W 2 , where W 1 is the solution to equation (3) at the plus sign before the root, and W 2 -at the minus sign. Using a variable replacement (3) could be rewritten as The equation (4) is similar in structure to the equation of the shapes of longitudinal oscillations of a rod with a variable cross-section, with a diameter of D(x), so the solution to it could be derived by using the symmetry method. Since D x /D is generally independent of the D 0 coefficient from ratios (5), this coefficient could be thought to be arbitrary, not affecting the outcome of the general solution. Note that the natural boundaries of the variable ρ=0÷1 for equations (3) are matched with the boundaries of the variable x≥0 for equations (4) at any ρ≥0 that do not exceed the marked limits of ρ. To derive a final solution to the problem, it is required that the function W=W 1 +W 2 should meet the boundary conditions at x 1 and x 2 .
The equation (4) at D set by law (5) is not solvable in elementary or known special functions. However, its solution is easy to find by approximating D(x) with some function D 1 (x), in which the solution could be defined in a closed form. As a result, the issue of the accuracy of solving the problem is transferred to the successful choice of the law D 1 (x) in the sense of its satisfactory approximation to D(x) over the required interval (x 1 ÷x 2 ). The symmetry method allows this choice to be implemented because, according to this method, the expression for D 1 (x) may contain a series of uncertain coefficients, the selection of which enables the required approximation.
The author of article [17] derived an approximation function D 1 (x) for a given problem, based on the symmetry method, in the following form 1 2 0.21 . 0.2483 It follows from the analysis of graphic dependences given in the work that solving a problem based on (6) yields acceptable results for technical applications. Given this, we shall for further analysis use the calculations that were earlier obtained by the author in article [17] to compute natural frequencies and to build the shapes of oscillations of a thin plate. In this case, to construct the first three shapes of oscillations, we shall employ the function of deflections, expressed through the variable x according to the dependence x=-ln (1-μρ), recorded in a slightly modified form: where ( ) ( ) Here, at known coefficients A/B 1 , B/B 1 , A 1 /B 1 , the B 1 multiplier is freely selected and is used in this case to normalize W i , so that W i (ρ=ρ 2 )=W i (x=x 2 )=1.

Formulae for calculating cyclical stresses in a plate
To analyze the stressed-strained state of a plate, we shall use the expressions, known from a theory of plates, to determine the maximally thick radial σ r and tangential (circular) σ θ normal stresses [18]. ; where W r , W rr are the derivatives from radius r; is the cylindrical rigidity. Following the transition to the relative variable ρ=r/R and onwards, to the variable x=-ln (1-μρ), and after the introduction of ν=1/3 to (8), we obtain The minus sign is omitted at σ 0 due to the cyclicality of the stresses σ r and σ θ . Given that W=W 1 +W 2 ; 2 Substitution of the obtained expressions W 1 ʹ, W 2 ʹ from article [17], ratios (10) in formulae (9) leads to the desired dependences for cyclical stresses: The Bessel's functions [ J(αx), Y(αx), I(βx), K(βx)] are denoted [ J, Y, I, K] for convenience.

Determining the natural frequencies, shapes of oscillations, and cyclical stresses
For the intended use of circular plates of variable thickness in technical devices, in accordance with the objectives of the current study, three options of thickness h=H 0 (1-μρ) 2 were selected. The ratios of the limiting thicknesses h(ρ 1 )/ h(ρ 2 ) at ρ=ρ 1 =0.1 and ρ=ρ 2 =0.5 are assigned as 1.8; 5; 8. The graphic illustrations (Fig. 1) show the plates with these parameters under numbers 1; 2; 3, which corresponds to the coefficient values μ i =0.5985; 1.21417; 1.39127 (i=1, 2, 3), respectively. Based on ratios (7) and boundary conditions for a circular plate with rigid fastening at ρ=ρ 1 =0.1 and free at ρ=ρ 2 =0.5, the following results were obtained.
We determined the frequency numbers λ i as the solutions to the corresponding frequency equation (Table 1).  Table 1 The oscillations parameters of the plate of different concaveness ( ) ( ) Based on (7), according to the obtained λ i , we constructed the shapes (deflections) of natural oscillations (Fig. 2-4). With the help of (7), (11) and λ i , the diagrams of the radial and tangential cyclical stresses were built for the main shape of oscillations (Fig. 5, 6). Table 1 gives the coordinates of the antinodes of oscillations ρ mi and the corresponding values of maximum deflections W i (ρ mi ). The estimated data, given in Table 1, as well as the graphic dependences, shown in Fig. 2-6, provide quite clear information about the oscillations of the plate considered.

Discussing the results of solving the set problem
By using the function of deflections W i , constructed on the basis of the symmetry method, and by applying the built algorithm, one could derive analytical expressions to calculate the radial σ r and circular σ θ stresses. That provides for an opportunity to study the distribution of stresses and to generally analyze the stressed-strained state of a plate. The resulting graphic dependences for the shapes of oscillation of the plate make it possible to determine the location of nodes and antinodes, to estimate the magnitude of deflections.
According to the results of our calculations, it is possible to confirm the obvious that the degree of concaveness affects the weight of a plate. One could see that, based on the profiles (Fig. 1), the increase in the ratio h(ρ 1 )/h(ρ 2 ) makes it possible to reduce the mass of the plate element when designing. Based on the data from Table 1, the natural frequencies of the first three shapes of oscillations decrease to varying degrees with a rise in concaveness, determined by the number of the frequency number λ i (i=1, 2, 3). Thus, it follows from the data in Table 1 that at h(0.1)/h(0.5)=1.8; 5; 8 (μ=0.5985; 1.21417; 1.39127) the frequencies decrease, compared to μ 1 , respectively, for cases μ 2 and μ 3 , by (1.0; 1.3) % for λ 1 , by (17.6;24) % for λ 2 , by (22.85;30.35) % for λ 3 . These indicators are a benchmark for studying the oscillations of a given type of plates. One could see a significant drop in frequency on the higher shapes (λ 2 , λ 3 ) and a slight drop at the basic shape(λ 1 ). To date, only one attempt to analytically determine a single frequency coefficient λ 1 is known, but based on a laborious and less accurate series method, and only for the plate of a linear-variable thickness [19].
Data from Table 1 could be applied to establish that the values of the normalized extreme deflections for cases μ 2 , μ 3 (h(0.1)/h(0.5)=5; 8) are much lower compared to the plate at μ 1 (h(0.1)/h(0.5)=1.8). This fact is important for judging the effect of concaveness on the cylindrical rigidity of a plate under a mode of the axisymmetric cyclical bend. In this case, plate number 3 (μ=μ 3 ) has obviously the least rigidity. Table 1 could be used to determine the coordinates of the extreme values of deflections (the antinodes of oscillations) and Fig. 3, 4 -to find the indicative coordinates of the nodes. These parameters are a means, along with frequency indicators, to identify the oscillatory properties of a plate when it is studied in practice.
The following should be noted as regards the pattern of distribution of stresses σ r (Fig. 5) and σ θ (Fig. 6) for the three variants of the circular plate. The stresses σ r that operate far from the free edge, such as at the end constraint or in the region of maximal σ θ , are greater to varying degrees than σ θ . Therefore, these stresses are the main threat in terms of cyclical strength (resistance to fatigue) of the plate when σ r reaches destructive values in the process of oscillations. The experiments set to determine cyclical strength by initiating intense axisymmetric resonance oscillations in such plates confirmed the validity of the stated assumption [20] because during such experiments the fatigue crack was always located perpendicular to the direction of the action of maximal σ r . As the ratio of limiting boundary thicknesses increases, the maximum value of the radial stress σ r is closer to the value of σ r at the end constraint. For example, for h(0.1)/h(0.5)=5, the stress σ r has an extremum only at the end constraint (ρ=0.1); so such a plate cannot be used for fatigue tests. At h(0.1)/h(0.5)=8, there is an extremum σ r ≈0.93 at ρ≈0.3. In this case, the ratio σ r (0.1)/ σ r (0.3)=1.075 is little different from 1. It is obvious that with a further increase in concaveness (increase in μ) one could obtain the types of plates required for these purposes, with a ratio of σ r (0.1)/ σ r (0.3)<1.
Our study of the problem about the natural oscillations of a plate with the assigned profile has demonstrated the flexibility of the symmetry method. The main feature of the current study is the technique whereby the equation of the shape of natural oscillations could be greatly simplified. The consequence of simplification is the possibility of obtaining an analytically closed solution to the formulated problem.
The algorithm for solving the problem in the case of the original parameters of the plate, which are different from those adopted in the estimation example, remains unchanged. It may be necessary, however, to modify the approximation function. However, according to the symmetry method, whose flexibility and multivariance has been confirmed in the current work, this circumstance does not present fundamental difficulties.
A feature of the obtained theoretical results is the real practical use of the symmetry method and the algorithm of its application to solve the problem about the oscillations of a plate with different-variant thickness h=H 0 (1-μρ) 2 . The practical significance of our results is the possibility to directly use the estimated data obtained in the current work, in particular for the rational design of resonance sound and ultrasound systems based on plates as acoustically active elements.

Conclusions
1. A combination of the symmetry method and the factorization technique has helped to build a general analytical solution to the order IV differential equation for the problem about the cyclical axisymmetric bend of a circular plate, whose thickness changes in line with the law of concave parabola h=H 0 (1-μρ) 2 at varying degrees of its concaveness.
2. Based on the general solution, we have derived formulae to calculate the cyclical stresses in a plate. The feature of the formulae is their compactness achieved by transforming the known expressions containing derivatives from the displacement function of second-order for natural variable ρ to the form containing only the first derivatives for the auxiliary variable x(ρ). The transformed formulae are convenient for directly using the deflection function W=W(x) in them, obtained as a general solution.
3. We have determined the first three natural values for the problem (frequency numbers) and natural functions (oscillation shapes) for a circular plate with a rigid fastening of the inner contour (ρ 1 =0.1) (1/0.1) and free on the external contour (ρ 2 =0.5). It has been shown that the natural frequencies of the first three shapes of oscillations decrease to varying degrees with the increase in concaveness, determined by the number of the frequency number λ i (i=1, 2, 3). At μ=1.214171 and μ=1.39127, the frequencies decrease, compared to the case of μ=0.598, by (1; 1.3) %, (17.6; 24) %, (22.85; 30.35) %, respectively. One could see a significant drop in frequency on the higher shapes of oscillations (λ 2 , λ 3 ) and a slight drop at the basic shape (λ 1 ). We have established the magnitudes and coordinates of extreme deflections (the antinodes of oscillations) and the indicative coordinates of the nodes. These numerical parameters, along with the frequency indicators, are a means of identifying the oscillatory properties of a plate when it is studied in practice. 4. We have constructed the diagrams (graphic dependences) of radial σ r and tangential σ θ cyclical stresses at the basic shape for each of the three variants of the parabolic plate concaveness. It has been established that the increase in the ratio of the edge thickness h(0.1)/h(0.5), that is concaveness, leads to an increase in σ r in the cross-sections outside the end constraint. At h(0.1)/h(0.5)=1.8;5, the stresses σ r have an extremum σ r =0.93 at ρ=0.3. The ratio σ r (0.1)/σ r (0.3)=1.075 in this case is little different from 1. It has been concluded that a further increase in concave-ness would lead to the values σ r (0.1)/σ r (0.3)<1, which makes it possible, in particular, to use such plates to experimentally determine the limits of materials endurance under a complex stressed state.