Analysis of Free Oscillations of Round Thin Plates of Variable Thickness With a Point Support

This paper reports the derived general analytical solution to the IV-order differential equation with variable coefficients for the problem on free axisymmetric oscillations of a circular plate of variable thickness. The plate thickness changes along the radius ρ in line with the parabolic law h=H0(1–µρ)2. When building a solution, the synthesis of the factorization method and the symmetry method was used. The factorization method has enabled us to represent the solution to the original IV-order equation as the sum of the solutions to the two respectively constructed II-order equations. The method of symmetry has produced precise solutions to these two equations.<br>The problem on a point fixation of the plate has been considered as a boundary case of the problem on the rigid fixation of the inner contour of a circular plate whose ρ→0. To this end, the general solution has been transformed into the form that pre-meets the conditions on a rigid point support. The result of such a transformation is a simpler solution, with only two permanent integration variables instead of four. As a result, the frequency equation for a plate under any conditions on the outer contour is significantly simplified because it is derived from the second-order determinant. The frequency equation for a plate with a point support and with a free edge at µ=1.39127, which corresponds to the ratio of the limit thicknesses equal to 10.8, yielded the first five eigenvalues λi (i=1÷5). The oscillation shapes have been constructed as a graphic illustration for λi (i=1÷3). The numerical values of amplitude ratios have been given, as well as coordinates (relative radii) of the oscillation antinodes and nodal circles for each of the five oscillation shapes (i=1÷5). The derived numerical values of the oscillation parameters could in practice be used to initially identify the type of an oscillatory system and its possible characteristics for the case when a plate is fixed inside the inner contour of the small diameter. The criterial ratio of the fastening contour diameter to the plate diameter could serve the same purpose. If this ratio is equal to or less than 0.2, then it is permissible to assume that it is a point-type fastening. In this case, it is possible to calculate the circular plate oscillations with its internal contour fixed using the algorithm set out for a plate with a point support<br>


Introduction
A series of problems on the natural oscillations of circular plates of various thicknesses have been considered in studies [1][2][3]. They also outline issues that reveal the importance and essence of the problem, review the ways to solve it with appropriate literary references.
In practice, there are cases when a circular plate in engineering structures is fixed inside the inner contour whose diameter is much smaller than that of the outer one. This situation necessitates considering a problem on the oscillations of a solid plate with a point support. The transition from natural conditions inside the internal contour to the conditions of point fastening is predetermined by the desire to simplify the plate estimation model without a significant loss of accuracy. For a plate of constant thickness with a support in the center, the results of solving the problem are reported in [4][5][6][7]. There is a basic issue related to a variable-thickness platethe search for a general analytical solution based on the original differential equation of free oscillations. If it is solved, it is possible to consider a series of applied problems on the oscillations of solid or circular, free, or fixed-in-different ways plates, including those with a point support. The relevance of the issue of calculating the oscillations of a plate of variable thickness with a point support is due to the need to improve the efficiency of estimation models, which is inextricably linked to their simplification, as is noted for the case of point fastening.

Literature review and problem statement
Our analysis of publications [8][9][10][11][12][13][14][15] has revealed that the solution to the problem of free oscillations of the plate with a point support is very general in the scientific papers related to the field of applied plate theory. At the same time, the focus of research is on ways and finding analytical solutions to the problem of free oscillations. In this aspect, one should 2  note the use of both a variety of classic approaches to solving the specified problem and modern algorithms. However, the numerous solution methods reviewed and the particular results obtained are difficult to use to analyze the oscillations of a round plate of variable thickness with a point support. The reason for this seems to be the difficulty of finding an accurate analytical solution to the IV-order differential equation with variable coefficients that will describe the oscillations of a variable-thickness plate. 5 natural frequencies for a plate with two point supports were obtained in work [8] based on the theory by Uffland-Mindlin. However, no variant for a variable thickness plate has been found.

Для задачi про вiльнi вiсесиметричнi коливання кругової пластинки зi змiнною товщиною отримано загальний аналiтичний розв'язок диференцiального рiвняння IV зi змiнними коефiцiєнтами. Товщина пластинки змiнюється вздовж радiуса ρ за параболiчним законом h=H 0 (1-μρ)
The "cloud" method and the plate theory by Midlin were considered in paper [9] to solve a problem on the free oscillations of a plate with a point support. In this case, the authors considered plates of different shapes and with different fixation, except for a thin plate of variable thickness. As a result, the reported results of the study are unsuitable for use in solving a problem on the oscillations of a plate with a point support.
Study [10] considers a numerical approach to solving a problem on the oscillations of a complicated plate of rectangular configuration. Given the problem statement and the algorithm to solve it, the resulting expressions cannot be used for a plate with a point support.
Works [11][12][13] give analytical solutions to a problem on the free oscillations of rectangular plates. To find a solution, numerical approaches are useda simplex superposition method, a method of finite elements, and an integrated method of transformation. It is obvious that the proposed approaches are difficult to use for analytical research into the plates of variable thickness with a point support.
A paired coordinate method is used in [14] to calculate the natural frequencies and build the plate oscillation shapes. However, the considered case of a thin plate with cutouts is difficult to adapt to the problem involving a solid plate with a point support. The same refers to the analysis of the ratios from article [15]. There, the Relay-Ritz method was proposed to construct the first oscillation shape of a thick plate. The cited article does not include any variants for a plate with a point support.
Thus, the above review of the scientific literature testifies to the absence of publications on the analytical solution to the problem on the oscillations of a round plate of variable thickness with a point support. The reason, of course, is not the lack of demand, as evidenced by the noted interest in the plates, including those with point supports, but, apparently, the difficulties associated with finding a solution to mathematical problems. Building an analytical solution to the problem on the free oscillations of a round plate with a point support would complement the theory of thin plates with new results, which, in itself, has scientific and applied value.

The aim and objectives of the study
The aim of this study is to derive an analytical solution to the problem on the free oscillations of a round plate, which changes in line with the law of concave parabola if its center includes a rigid point support.
To accomplish the aim, the following tasks have been set: -to derive, based on the synthesis of symmetry and factorization methods, a general analytical solution to the differential equation for a problem on the free axisymmetric oscillations of a plate whose thickness h=H 0 (1-µρ) 2 ; -to transform the general solution to the form that meets the conditions on the support; -to derive a frequency equation, calculate natural frequencies, and build the first three oscillation shapes for a plate with a free contour.

Differential equation and its overall solution for a variable thickness plate
For a plate whose thickness h=H 0 (1-µρ) 2 , work [3] gives a differential equation for the shapes of natural axisymmetric bending oscillations, which can be represented in the following form Here, W=W(ρ) are the plate displacements (deflections); µ is an arbitrary constant. ρ=r/R is the relative variable; r is the variable radius; R is the plate radius.

( )
f is the natural oscillation frequency; H 0 is the thickness in the plate center; ν, γ, E are, respectively, the Poisson coefficient, specific weight, a material's elasticity module; g is the acceleration of gravity. Hereafter, the Poisson coefficient is accepted to equal 1/3, based on which equation (1) is written for this particular case. The notation of IV-order equation in form (1) makes it possible, applying the factorization method, to derive its solution in the form of the sum of solutions W=W 1 +W 2 to the following two II-order equations Equations (3) can be written in a generalized form where S=ρH 2 =ρ(1-µρ) 4 and 4 2 4 By replacing the variable ρ=(1/µ)(1-e -x ), equation (4) can be written in the following form then it is possible, in the interval x=0÷1, at D 01 =0.21; C 0 = =0.2484, to obtain a satisfactory match between the D and D 1 functions. In this case, as noted in work [16], it becomes possible to derive accurate solutions to equations (6) using the symmetry method, which take the following form: the following: With a general solution (9) and its derivative (11), one can consider a series of problems on the oscillations of a plate of variable thickness of the parabolic profile under different boundary conditions. A plate with a point support is a particular case of the circular plate, fixed in the center, at ρ=ρ 1 if ρ 1 →0. The technique of moving from a general solution to the solution for such a plate is outlined in the following chapter 5.

Transforming a general solution to the form that meets the conditions on the support
Let the closing along the inner contour ρ=ρ 1 of the circular plate be rigid, then the boundary conditions should be fulfilled Given that W ρ =W x x ρ , these conditions while moving to variable x, will be rewritten in the following form There are two cases if x 1 =0. In the first case, if a plate is solid and there is no support in its center, then, at x=0, the plate deflection should be finite; therefore, in (9), it is necessary to adopt B=0 and B 1 =0 because functions Y 0 (αx), Y 1 (αx), K 0 (βx), K 1 (βx) at x→0 tend to infinity. We shall, therefore, derive a general solution for a solid plate without a support in the center, which takes the following form This case will not be considered here.
In the second case, when a plate is supported in the center (Fig. 1), it is obvious that B≠0 and B 1 ≠0 and, therefore, it is necessary, when meeting conditions (12), to take into consideration the behavior of the Bessel functions in the vicinity of the point x=0. After introducing (9) and (11) to (12), we obtain, at Taking into consideration the decomposition of Bessel functions into the power series [17], we shall obtain, for the limits included in (13): where γ=0.5772157 is the Euler constant.
The limit (15), given (17) and decompositions (17) and expression (16), additionally, taking into consideration the decompositions for J 0 (αx), I 0 (βx) in the limit will equal zero, that is ( ) Taking into consideration the results from (18) to (20), we shall obtain, for a point support, a single boundary condition instead of two conditions (12) ( ) Constant B is to be determined from (21) ( ) Following the substitution of (23) in (8) and (10), we obtain ( ) where 2 0 . L x C = + Thus, the resulting equations (24) that meet conditions (21) make it possible to build a frequency equation, based only on the conditions at the edge ρ=ρ 2 .

Deriving a frequency equation, computing the frequencies, and building the oscillation shapes of the plate with a free contour
If the contour ρ=ρ 2 is free, then, in the event of an asymmetrical deformation, the radial bending momenta and transverse forces on the contour are zero, that is After the transition to variable x=-ln (1-µρ), by completing the required transformations and a series of simplifications, we shall obtain, instead of (25), at ν=1/3, the following conditions The substitution of (24) in (26) at x=x 2 leads to two equations relative to the desired constants A and А 1 It is easy to determine the amplitude coefficients from the system of equations (27) 1 3 The oscillation shapes W, according to (24), will be determined from the following expression Thus, equation (28) makes it possible to determine the natural frequencies of a free plate of variable thickness with a rigid point support. Expression (30) defines the functions of the plate deflections.
To illustrate the effectiveness of the constructed problem-solving algorithm, a circular plate was chosen, whose ratio of the limiting edge thicknesses h(0)/h(ρ 2 )=10.8 at ρ 2 =0.5, which corresponds to µ=1.39127.
Based on the obtained estimation ratios, we determined the frequency numbers of plate λ i as the solutions to the corresponding frequency equation (28) ( Table 1).
In order to construct the first three oscillation shapes, one determines, using expression (30), after substituting the frequency numbers λ j ( j=1, 2, 3) in it, the amplitude coefficients A/A 1 . A graphic representation of the deflections for the first three oscillation shapes is shown in Fig. 2-4.  Note that, based on the classical theory of plates, dangerous radial stresses act in the zone of maximum deflections. It is known that the distribution of stresses on a support has a special character.

Discussion of results of solving the problem about a plate with a point support
The resulting general solution to differential equation (1) makes it possible to consider a series of specific problems on the oscillations of parabolic platescircular and solid ones, with different types of contour fastening. This paper considers the ultimate case of a plate rigidly fixed on the inner contour (ρ=ρ 1 ), whose radius ρ 1 →0. There are two objectives for stating this type of problem. First, there are structures that demonstrate the "point" fastening; therefore, one must have an algorithm to analyze their oscillations. Second, it is often impossible to judge with certainty the scale of the fastening, but, given the relatively cumbersome way of obtaining a rigorous solution to the problem, the assumption about a "point" fastening is made. One can derive a solution for the case of point fastening, which passes the limit to the point support, if, by following conditions (12) and (26), one builds four equations regarding the integration constants A, B, A 1 , B 1 . A rather cumbersome determinant of the constructed system of algebraic equations of the fourth order, equal to zero, is a frequency equation. Further, accepting the condition ρ 1 →0, it is necessary to run a mathematical analysis of the resulting equation, reducing it to the form containing the final terms. This path of transition to a point type of fastening is not productive, as it is still cumbersome because it is based on general boundary conditions. Another way is to transform the general solution (8) to the form that pre-meets the conditions on a rigid support. As a result of the transformations performed, we derived, instead of (8), a general form of the solution W as the sum of solutions (24) for a solid plate with a point support under any conditions on the contour ρ=ρ 2 . This solution contains only two basic integration constants А and А 1 , so, in this paper, the frequency equation (for a free contour case) has been easily derived from the order II determinant. Such a problem-solving scheme is based on the assumption about the independence of frequencies on the magnitude of fastening radius ρ 1 if it is small compared to ρ 2 .
For a plate of constant thickness, it is believed that, if ρ 1 /ρ 2 <0.2, the fastening is of the point type. Based on the calculation results, in particular, from determining the coordinates ρ 0 of the first nodal circles S for five oscillation shapes, it can be assumed that the ratio ρ 1 /ρ 2 <0.2 remains valid for the plate of variable thickness. The basis for this conclusion is the comparison of ρ 0 for the plates of constant thickness [18] and the plate, examined in the given example, whose ratio of limit thicknesses h(0)/h(0,5)=10. 8. The results of the comparison and the relative error δ of deviation in the parameters are given in Table 2. Table 2 Comparison of parameter ρ 0 for the plates of constant thickness and the plate considered in this paper The location of nodal circles, given in Table 1, could in practice serve as a criterion for determining the character of fixation. If a plate is fixed in the center by some diameter, then, by exciting the oscillations in it of one of the shapes (from II to V) and by measuring the radius R 0 of a first (from the edge) nodal circle, we compare the magnitude R 0 /0.5 to ρ 0 , which should be approximately equal. If the relative value of this radius is not much different from, for example, ρ 0 =0.453 for shape III, the fastening should be assumed to be of the point type; the oscillations should be analyzed based on the model of a plate with a point support.
The limitation of the current study stems from the algorithm to produce a general solution that requires lowering the order of the original differential equation. The factorization method, used for this purpose, while not universal, pre-restricts the form of functions at which it holds. These include parabolic functions.
The solution to the problem, reported in our paper, seems to be one of the possible cases of the theory being applied to the practical calculation of variable thickness plates. The theoretical basis of such calculations is the method of symmetry, developed and implemented as a general solution to equation (1).
Given this, in the future it is easy to consider, for this type of variable thickness plates, by directly using expressions (24), the free oscillations of a plate rigidly-or freely fixed along the contour with varying degrees of concaveness, determined from the parameter µ.

1.
A general analytical solution to the IV-order differential equation has been derived for a problem on the free axisymmetric oscillations of a plate whose thickness h=H 0 (1-µρ) 2 . This solution makes it possible to consider a series of problems about oscillations of parabolic platessolid and circular ones, with different types of contour fastening, including a point support in the center.
2. The general solution has been transformed into the form that pre-meets the conditions on a rigid point support. The transform result is a simpler solution, which includes, instead of four desired integration constants, only two. Therefore, the frequency equation could be easily obtained from the second-order determinant under any conditions on the contour.
3. The frequency equation for a free-edged plate at µ= =1.39127 (the ratio of limit thicknesses is 10.8) has produced the first five eigenvalues λ i . For λ i at i=1,2,3 the oscillation shapes have been built as a graphic illustration. We have given the numerical values of amplitude ratios, the coordinates of oscillation antinodes and nodal circles in the form of relative radii for each of the five oscillation shapes (i=1÷5). All numerical oscillation parameters could in practice be used to identify the characteristics and types of plate oscillations.