1

THE TECHNICAL UNIVERSITY OF CLUJ-NAPOCAFACULTY OF CONSTRUCTIONS

THE VIBRATION OF FLAT PLATES WITH VARIOUS BOUNDARY CONDITIONS

PhD Thesis

Eng. Marius Şerban Fetea

SCIENTIFIC ADVISOR

PROF.DR.ENG. IACOB BORŞ

2

-2009-

CONTENTS

INTRODUCTION. THE OBJECT AND STRUCTURE OF THE THESIS.

CHAPTER 1.

GENERAL PROBLEMS IN THE STATIC CALCULUS OF FLAT RECTANGULAR PLATES

CHAPTER 2.

THE FUNCTIONS OF VIBRATION SHAPES OF BEAMS HAVING DIFFERENT BOUNDARY CONDITIONS

CHAPTER 3.

PROBLEMS OF THE DYNAMIC CALCULUS OF FLAT PLATES AND BOUNDARY CONDITIONS

CHAPTER 4.

DETERMINING THE NORMAL VIBRATION MODES OF THE PLATES WITH DIFFERENT BOUNDARY CONDITIONS

CHAPTER 5.

COMPARING THE RESULTS. FINAL CONCLUSIONS. PERSONAL CONTRIBUTION

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INTRODUCTION. THE OBJECT AND STRUCTURE OF THE THESIS

The rectangular flat plates, as well as the the angular ones in general, often intervene as strength elements in the structures of civil and industrial constructions, their actual shape and support mode being imposed by different conditions in the exploitation of the buildings, such as the lay-out of some technological appliances, pipe crossing, embrasures in the stairs and so on.

The results of the scientific research regarding the flat plates, as well as the practical importance related to the knowledge of their way of behaviour in different loading and support situations within some structures (machines, buildings, equipments) are emphasized in numerous treaties, books, scientific papers published throughout the centuries.

As for the historic record of the problem related to the study of flat plates, the first results were out for publishing at the end of the 18th century, the beginning of the 19th century, having Chladni E, Strehlke, Konig, R, Tanaka S, Rayleigh L, Ritz W and later on Gontkevich V, Timoshenko S, Leissa as pioneers. Each of the above mentioned authors have had significant contributions regarding the development of methods in order to solve the plates and establish some rigurous solutions of their differential equations of equilibrium.

The making of constructions, machines and different high-perfomance appliances, whose functioning should take place in safety conditions, have required theoretical studies of rich complexity, as well as practicalexperiments, within which the problem of their free and forced vibrations represent an important category in the respective theme of research. The importance of studying the vibrations of different deformable material systems (elastic systems in constructions, technological equipments, mobile or stationary machines and equipments), whose structures take in types of plates different in terms of shape, loading mode and boundary conditions characterised by forced or free vibration motions and carried on to the structure itself, has been made obvious by the system’s degradation in time.

By means of the dynamic analysis of plates there has been an emphasis on the complexity of the notion of dynamic calculus, which has the following as main working stages:

- establishing the dynamic model considered for the first time- determining the normal vibration modes (self pulsations and vectors, respectively the functions of

vibration modes)- determining the dynamic response in displacements and sectional stresses- checking stability and strength conditions Through a careful analysis of field literature regarding the results obtained for the static and dynamic

calculus of flat plates with different boundary conditions, there are to be noted different authors’ concerns to elaborate a more exact calculus method that ensures an economical projection in safety conditions.

There has also been noted that most of the complications coming up in solving the flat plates are bound to the existence of free edges, these difficulties being reflected by the impossibility of finding some functions to describe the state of in-plate stress, with the rigurous consideration of the static conditions on a free edge.

The object of the thesis consists of the study of the free vibration of rectangular flat plates with different support conditions. The analysed flat plates are thin, elastic and isotropic with stiffness at bending and meet the availability conditions of Kirchhoff hypotheses. The suggested calculus method is an adaptation of the variational method Galerkin-Vlasov from the static calculus of dynamic plates and has been elaborated in such a way that the calculi necessary for determining the dynamic characteristics of the plates are made on the basis of some programmes designed by the author himself.

The paper is structured on 5 chapters and aims at bringing the author’s own contributions to the vibration of rectangular flat plates, the obtained results being highlighted in chapters 4 and 5.

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CHAPTER 1

GENERAL PROBLEMS IN THE STUDY OF THE STATIC CALCULUS OF FLAT RECTANGULAR PLATES

In this chapter, there is a presentation of the fundamental relations used in the calculus of thin rectangular flat plates as part of the ones with the most frequent practical use.

Due to the fact that the rigurous theory of plates constitutes a difficult three-dimensional elasticity problem [6], [7], [22], [62], [85], [89], [91], a series of simplifying hypotheses are accepted, on the basis of which a less rigurous calculus theory is formulated, but by means of which satisfying results can be achieved. They are known as Kirchhoff hypotheses.

Establishing the static response of a plate requires the determination of the displacement function on

Oz , ),( yxw axis, of the stresses and strains in each point of the respective plate. As a consequence, the paper presents the results obtained through studies made by numerous authors [19],

[20], [92], [84], [85], [87], displaying the fundamental relations between stress and strain, the relations between strains and displacements, the expression of the plate’s differential equation of equilibrium, respectively its integration with different boundary conditions on the edge.

All these relations are established after considering that the medial plane of the studied plates is the one before bending. As a consequence, during the strain of bending, the points situated in the medial plane will suffer small displacements, perpendicular to this. The total of all these displacements form the deformed medial surface of the plate.

It is also presented the fact that, in order to get the corresponding solutions for the boundary problem of the fourth-order differential equation of flat plates, there have to be some boundary conditions on the edges of the plates, according to the support mode. Therefore, the solutions of the differential equation of equilibrium will depend on certain integration constants that are determined by imposing two boundary conditions on each edge.

The main methods of analitical, variational and numerical calculus of the flat plates are also presented within chapter 1. Among the already known analytical methods, there is a presentation of the simple trigonometric series method (Levy), applied to the flat plates simply supported on two opposite edges, its other two edges having arbitrary support conditions, and the double trigonometric series (Navier), for which the solution of the plates’ equation of equilibrium will have to satisfy not only the equation with partial derivatives of the plate, but also the boundary conditions.

Determining the solution of the plate’s differential equation with partial derivatives that satisfies all the kinematic and static conditions on the considered boundary cannot always be rigurously achieved. As a result, the variational methods [19], [39], [84], [108] represent in many cases, an effective tool in obtaining the solution of the differential or partial derivative equation that can rigurously satisfy the differential equation and partially the static boundary conditions.

Solving the differential equations using the variational methods consists in the replacement of the unknown function that satisfies both the differential equation and the boundary conditions, with an approximate analytical expression chosen in such a way as to approximate the sought out function as well as possible, meaning that the deviation from the real value of the function should be minimum.

Among the variational methods having a general character, the following are presented in the paper: - the method of minimum square deviation- the method of orthogonalizing - the method of minimum potential energyAnd as particular processes of the general variational methods there are:- the Ritz method,

- the Galerkin method,- the Trefftz method,- the Kantorovici method,- the Trefftz method ,

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- the collocation method,If the solutions cannot be acquired through the analytical or variational methods because the volume and

methodology are heavy, it is indicated to draw on the numerical methods in determining them. There are types of flat plates to which one cannot get exact or approximate solutions of the differential

equation by direct integration, or approximate solutions by using the variational methods due to the edge shape, the loading mode and the boundary conditions imposed by the support mode. In these cases, the numerical methods are the most effective way of getting the approximate solutions.

The numerical methods presented in the paper are:- the finite difference method,- the finite element method.

In the case of finite differences, the general calculus relations are presented, by which not only the differential equations with partial derivatives are replaced by algebric systems, but also the way of determining the numerical values of the sizes that characterize the strain in the nodes of the network.

When applying the finite difference method, one generally aims at solving the following types of problems:- problems of static equilibrum that come up in case one wants to determine the elastic behaviour of the

deformable solid, a situation in which the unknown functions are time-independent, - problems of self-values,- problems of propagation, a case in which the unknown functions are time-dependent, one being able to

apply the finite element method in order to determine the dinamic response. The finite element method represents a numerical method of solving the equations with partial derivatives,

through which different physical systems having a finite number of degrees of freedom are modelled. Applying the finite difference method leads to the reduction of the equations with partial derivatives to an algebric system of equations, acquiring a discrete physical system having a finite number of degrees of freedom. The finite element method can be expressed in a direct, variational or rezidual formulation.

The expression under the direct form of the finite element method is based on the use of the displacementmethod in matrix formulation [18], [23], [60], [63], [64], [113].

The variational formulation of the finite element method is based on minimizing the potential energy of the deformable solid, being applied only when there is no given functional.

In determining the in-plate state of stress, deformation and strains, the finite element method represents the most used method in practical calculus that recquire a big and heavy work volume. In order to apply the finite element method [7], [18], [23], [64], the plate is discretized in finite elements having different shapes, all the

references in the paper having the finite quadrangle element on its basis. Knowing the node coordinates ii yx , ,

the discretization network is defined, enabling the quadrangle finite elements to result from the grouping of every

four nodes. Each node represents three degrees of freedom: the displacement w along z axis and rotations

yx , .

The fundamental concept of the method is based on the idea according to which any continuous size can be approximated by a discrete model, acquired from composing some continuous functions, defined on a finite number of elements. When applying the method, the following can be chosen as unknown: displacements, strains, or part of the displacements and strains. According to this choice, the formulation is in displacements, respectively strains or mixed. The most used formulation is the one in displacements, this also being presented in the thesis.

Following the application algorithm of the method described in the thesis, it is noted that the formulation of a problem in the finite element method is achieved in two stages:

A first stage consists in studying the finite elements, as to the choice of degrees of freedom, the local interpolation function and the determination of expressions that characterize the state of strain on the finite element (deformities, stresses, energy).

A second presented stage is given by the study of the structural assembled system, aiming at determining the unknown node displacements. They have to satisfy conditions of continuity, these being imposed both on and among the elements. By choosing the approximating function in the shape of an algebric polynom, the continuity on

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the element is achieved. To satisfy the conditions of compatibility among elements, the displacement function and eventually a part of its derivatives have to be continuous.

The solution obtained from the finite element method converges towards the exact solution, when a division in less and less finite elements is used, respectively if the interpolation functions on the element are chosen such as to be able to represent the displacements corectly and contain terms that lead to expressions of the strains and deformities capable of representing the homogenous state of stress on the element.

The conditions of continuity and convergence are satisfied if the interpolation polynom is a complete polynom having a degree equal at least to the highest derivative order that comes up in the relations between deformities and displacements.

With the determined node displacements, using the matrix expressions established in [7], [62], [92] one can determine the state of in-plate stress, deformation and strains.

CHAPTER 2

THE FUNCTIONS OF VIBRATION SHAPES OF BEAMS HAVING DIFFERENT BOUNDARY CONDITIONS

Within chapter 2, the problem of transverse vibration of straight beams is presented. We start from the case of a homogeneous beam with different conditions at the ends. By applying the priciple of D’Alembert for aninfinitesimal beam element, the equation of free transverse vibrations of the beam,

02

2

4

4

t

u

x

uEI ,

its particular solutions

txytxu , ,

and the equation of normal vibration modes are obtained,

044

4

xyadx

xyd,

where the notation

4a is adopted.

The equation of the normal vibration modes, along with the homogeneous boundary conditions describe a type Sturm-Liouville problem, [7], [12], [69]. Solving type Sturm –Liuoville problem allows the determination of

inherent values , and the functions of the vibration shapes of beams xy . Knowing the inherent values i gives

self-pulsation beam values of i for i =1,2,3,...

EIEIa i

ii 2.

By replacing the suggested solutions for the differential equation of normal vibration modes, its characteristic equation with real and complex roots can be obtained. The general solution of the differential equation of normal vibration modes that is obtained is

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axKaxKshaxKchaxKxy sincos 4321 ,

In any beam section the sizes ),(xy ),(' xy ),(xy ),(xy are defined, where the function and the first

derivative of the general equation of normal vibration modes represent the displacement, respectively the rotation for a certain vibration mode, and the second and third-order derivatives are sizes proportional to the bending

moment, respectively the shearing force. The constants that interfere in their expressions ,,,, 4321 KKKK are

determined from the boundary conditions that have to be fulfilled by the medial fiber of the beam. The normal vibration modes for several/more combinations of conditions at the ends of the beam are

determined. The following cases are studied:

- simply supported beam, having the shape function )(xHy ii ,

- doubly-clamped beam, having the shape function )(xGy ii ,

- overhang beam, having the shape function )(xy ii ,

- clamped -hinged beam, having the shape function )(xJy ii ,

- hinged -free beam, having the shape function )(xFy ii ,

- free at both ends beam, having the shape function, )(xy ii .

CHAPTER 3

PROBLEMS OF THE DYNAMIC CALCULUS OF FLAT PLATES AND OF BOUNDARY CONDITIONS

In chapter 3, references are made to the actual study of the problems, both the authors and their achieved results being mentioned. There is a unitary presentation of the methods to solve the flat plates with differentboundary conditions and different edge ratios, even for some certain types of plates with one free edge. For the case of rectangular flat plates having two opposite edges simply supported and the other two arbitrary supported, the most often used calculus method is the Levy series method.

The values of self-pulsation parameters determined by different authors for the viewed normal vibration modes using the analytical, variational and numerical methods of calculus are presented in tables, for the following types of plates:

rectangular flat plate simply supported on boundary rectangular flat plate simply supported on two opposite edges and clamped on the other two rectangular flat plate clamped on one edge and simply supported on the other three rectangular flat plate simply supported on two opposite edges, clamped and free on the other two rectangular flat plate simply supported on three edges and free on the fourth rectangular flat plate clamped on all edges rectangular flat plate having two opposite edges clamped and the other two simply supported and freerectangular flat plate clamped on three edges and simply supported on the fourth .In the case of these types of plates there is a presentation of the values of self-pulsation parameters

determined by different authors. The fact is noted that for the same type of plate studied by more authors using different calculus methods, the achieved results are very close. Also, in the case of flat plates having two opposite edges simply supported there is a presentation of equation systems formed out of imposing boundary conditions and the characterisitc equations given by solving the determinant.

The energetical methods applied within this chapter are more general calculus methods of flat plates, by means of which one can achieve satisfying results of the pulsation parameters even for the cases where free edge conditions occur. These methods can be extended even to the cases of plates having different shapes and thickness,

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the choice of shape functions that satisy at least the kinematic conditions and approximate the normal vibration modes being but necessary.

In the case in which one aims at determining the fundamental pulsation of plates with very high precision , the application of Raleigh method is recommended.

The Ritz method, based on the principle of minimum potential energy leads to better approximations than the Rayleigh method, because it has been noted to allow one to determine the shape pulsations and function with a good precision for the superior vibration modes. Still, even if the precision of determining the first and the second vibration mode is big when applying this method, this gradually deteriorates for the superior vibration modes.

The Russian scientists, among whom Galerkin, have had significant contributions in the field of elasticity theory, contributing to the development of energetical methods. As in the Ritz method, the solution adopted by Galerkin for the shape functions is some series whose terms each satify all the boundary conditions and contain fourth-order derivatives. The problem is reduced to evaluating the integrals of the chosen shape functions. Following the calculus algorithm, a linear system of equations comes up, which, in order to have solutions different from the common solution makes it is necessary for the system determinant to be null. The frequency equation thus results.

In [82], Szilard presents the application of finite difference method that allows to determine pulsation parameters with an acceptable precision. By the use of the iterative approximation Stodola-Vianello, associated with finite differences, one achieves results close to the the exact value of the problem. The great advantage of this method is given by its great degree of generalization regarding the boundary conditions, the types of loading as well as the possibility of studying plates with variable thickness and unhomogeneous boundary conditions.

As for the application of the finite element method in solving the rectangular flat plates to determine solutions in free and forced vibration problems, it is relatively easy to use. This method is a lot more general and flexible than any other numerical method, one being able to apply it successfully on any type of plate from the point of view of geometry, loading and boundary conditions.

CHAPTER 4

DETERMINING THE NORMAL VIBRATION MODES OF THE PLATES WITH VARIOUS BOUNDARY CONDITIONS

Within this chapter, the object of study is determining the vibration modes, self- pulsations

ij (represented by pulsation parameter ij ) and the vibration functions ),( yxij . There is a unitary presentation of the calculus algorithm of the variational method Galerkin-Vlasov for a number of 7 flat plates having

different boundary conditions and different edge ratios ba .

In determining these normal vibration modes, the variational method Galerkin-Vlasov is adapted, being regarded as a particularity of the Bubnov-Galerkin method for the dynamic infinite-dimensional systems.

7 types of rectangular and square flat plates with different boundary conditions are studied, that is:- plate clamped on the boundary,- plate simply supported on the boundary,- plate simply supported on two opposite edges and clamped on the other two,- plate clamped on two opposie edges and free on the other two,- plate simply supported on two opposite edges, clamped and free on the other two,- plate simply supported on three edges and clamped on the fourth,- plate clamped on three edges and simply supported on the fourth.The study of the 7 rectangular plates started from the normal vibration mode equation of the plates, that

expresses their dynamic equilibrium, choosing for the plate functions the products of the shape functions of the

beams with the same boundary conditions as the plate in x and respectively y direction.The equation of the normal vibration modes

),(),(4 yxyx in which

4 is the double Laplacean operator, together with the boundary conditions, represents a Sturm-Liouville problem, whose solving with the suggested method leads to the charateristics of pulsations and vibration shapes.

The expression of the parameter of specific pulsations obtained by applying the suggested method is

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,

)()(

)()()()()()()(2)()()(

0 0

22

00

2

0

''

0

''

0

2

0

a b

ji

b

jIVj

a

i

b

jj

a

ii

b

j

a

iIVi

ij

dyyYdxxX

dyyYyYdxxXdyyYyYdxxXxXdyyYdxxXxX

in which )(xX i şi )( yY j , are the vibration functions of the beams that have the same boundary conditions as the

plate on directions x respectively y .The use of Galerkin-Vlasov method for determining the normal vibration modes of the plates is reduced to

the evaluation of the integrals defined above.For the studied plates, the pulsation parameters for a minimum number of 3 normal vibration modes and

maximum 9 normal vibration modes are determined. The obtained results regarding the pulsation parameters for the studied flat rectangular plates are subsequently presented.

1. Flat plate clamped on the boundary

For the flat plate clamped on the /boundary we determine the values of the specific pulsations adequate to the first 7 normal vibration modes considered. The values of the parameters of specific pulsations are presented in table 1.

Table 1.Vibration

modeMode(1,1)

Mode(1,2)

Mode(1,3)

Mode(2,1)

Mode(2,2)

Mode(2,3)

Mode(3,1)

ij , 1 36,0036 74,52 129,21 74,52 108,97 165,2 129,21

For the rectangular plate having the edge ratio 2 ba , the value of fundamental pulsation

parameter obtained by applying the suggested method is

05,2411 .

2. Rectangular flat plate simply supported on the boundary

Using the suggested method, we determine the values pulsation parameters adequate to the normal

vibration modes (1,1), (2,1), (3,1) and the plate edge ratios, 1 ba , 5,1 b

a , 2 ba . The

values for the pulsation parameters are presented in table 2.

Table 2.Vibration

ModeMode(1,1)

Mode(2,1)

Mode(3,1)

ij , 1 19.7392 49,34 78,65

ij , 1 ,5 32,07 61,68 111.03

ij , 2 49,34 78,95 128,3

3. Rectangular flat plate simply supported on two opposite edges and clamped on the other two

We determine the pulsation parameters adequate to the first 8 normal vibration modes in the case of the

square flat plate with the edge ratio 1 . The values of parameters of the specific pulsations are presented in table 3.

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Table 3.Vibration Mode

Mode(1,1)

Mode(1,2)

Mode(1,3)

Mode(2,1)

Mode(2,2)

Mode(2,3)

Mode(3,1)

Mode(3,2)

ij ,

1

28,944

70,11 123,16 54,93 97,07 154,15 102,7 144,33

4. Rectangular flat plate clamped on two opposite edges and simply supported on the other two

For the square plate 1 ba , we determine the values of the pulsation parameters, adequate to the

normal vibration modes (1,1), (2,1), (3,1). The values of parameters of the specific pulsations are presented in table 4.

Table 4.Vibration

ModeMode(1,1)

Mode(2,1)

Mode(3,1)

ij , 1 22,37 61,67 102,9

5. Rectangular flat plate simply supported on two opposite edges, clamped and free on the other two

Adequate to the normal vibration modes (1,1), (2,1), (3,1) and the edge ratio 1 ba of the flat plate

and, 5,1 ba , the parameters of the specific pulsations ij are obtained. The values of parameters of the

specific pulsations are presented in table 5. Table 5.

Vibration Mode Mode(1,1)

Mode(2,1)

Mode(3,1)

ij ; 1 13,8 38,38 86,64

ij ; 5,1 17,15 42,38 86,35

6. Rectangular flat plate simply supported on three edges and clamped on the fourth

Adequate to the first 9 normal vibration modes of the square flat plates 1 ba , we determine the

values of the pulsation parameters ij , presented in table 6.

Table 6.Vibration Mode

Mode(1,1)

Mode(1,2)

Mode(1,3)

Mode(2,1)

Mode(2,2)

Mode(2,3)

Mode(3,1)

Mode(3,2)

Mode(3,3)

ij , 1 23,71 58,88 112,25 52,006 86,89 137,73 100,85 135,1 182,9

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7. Rectangular flat plate clamped on three edges and simply supported on the fourth

For the square flat plate we determine the values of the pulsation parameters adequate to the first 7 normal vibration modes, respectively fundament pulsation parameters for the rectangular plates with edge

ratios 5,1b

a , 0,2 . The values the specific pulsation parameters are presented in table 7.

Table 7.Vibration Mode Mode

(1,1)Mode(1,2)

Mode(1,3)

Mode(2,1)

Mode(2,2)

Mode(2,3)

Mode(3,1)

ij ; 1 31,90 63,81 115,68 72,28 104,18 152,31 124,04

ij ; 5,1 48,30 - - - - - -

ij ; 2 73,63 - - - - - -

For each plate, there is also a presentation of the functions of vibration shapes in the nodes of a network considered on the medial surface of the plates, as well as the plate distorsions for at least three normal vibration modes.

It is noted that very good results are obtained in the case of plates with boundary conditions that exclude free edges and less good in the case of plates with one free edge. This is due to the fact that on the free edges, the way of choosing the displacement function doesn’t rigurously satisfy the static boundary conditions.

CHAPTER 5COMPARING THE RESULTS. FINAL CONCLUSIONS. PERSONAL CONTRIBUTIONS

To validate the suggested method, we compare the achieved theoretical results with the ones determined by other authors when applying the trigonometric series method, Rayleigh, Ritz, Galerkin, Rayleigh-Ritz, the finite difference method and the finite element one.

Ther is a presentation of the percentage deviations of the vibration parameters determined by applying the suggested method in comparison to the ones determined by other authors through the use of the analytical, variational and numerical methods.

From the analysis of present data, we note that for the rectangular flat plates considered in the thesis, the percentage deviations are in the limits of high precision.

1. Rectangular flat plate clamped on the boundary

Comparing the values of parameters of specific pulsations, we note that the percentage deviations of the pulsation parameters determined through the suggested method are the slightest in the case of the normal vibration modes (1,1), (2,1), (2,3), (3,1), and the biggest for the modes (1,2), (1,3), (2,2), in comparison to the ones obtained by other authors

For the rectangular plate with the edge ratio 2 of the plate, the authors have calculated the

fundamental pulsation parameter using type Ritz solutions, its value being 994,2311 . For the same type of

plate having the same edge ratio, Iguchi obtained in [47] the value the of fundamental pulsation parameter

56,2411 . The perecentage deviation in the value of the fundamental pulsation parameter determined through

the suggested method in comparison to the value determined by Iguchi in [47], using the Rayleigh-Ritz method is of 2,24 %, respectively 0,23 %, in comparison to the value determined by appplying type Ritz solutions.

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2. The rectangular flat plate simply supported on the boundary

For the rectangular flat plate simply supported on the boundary, the values of the pulsation parametersdetermined by the suggested method are compared with the values obtained by Szilard in [82], when using the finite element method, respectively by Polidor Bratu in [74], who applied the trigonometric series method to solve the plate.

The fundamental pulsation parameters of the flat square plate determined with the suggested method is equal to the one determined by Polidor Bratu [74], when applying the trigonometric series method and by Szilard [82, who applied the finite element method.

For the other normal vibration modes considered, that is (2,1), (3,1), the percenatge deviations of the pulsation parameteres determined by means of appplying the suggested method are small, in comparison to the ones achieved by the authors in [74] and [82]: between a minimum value of 0% and a maximum value of 2,5% for vibration mode (2,1), respectively a minimum deviation of 0,31 % and a maximum one of 0,51 % for vibration mode (3,1).

3. Rectangular flat plate simply supported on two opposite edges and clamped on the other two

The results obtained for the pulsation parameters of the square plate after applying the suggested variational method are compared to: the ones obtained by Iguchi in [47] and Odman in [69] and Polidor Bratu [74], who have used the trigonometric series method for determining pulsation parameters; the ones determined by Szilard [82], when applying the finite element method; Nishimura’s methods in [68], who used the finite difference method for determining the parameter of fundamental pulsation and the ones determined by Janich [49], using the Rayleigh method.

The percentage deviations for the pulsation parameters of the square plate, determined with the suggested method are small, in comparison to the one obtained by the authors in [47], [49], [68], [69], [74]: between 0,01% for the fundamental vibration mode (1,1) and 4,61 % for the normal vibration mode (3,1).

4. Rectangular flat plate clamped on two opposite edges, simply supported and free on the other two

In field literature the results regarding the normal vibration modes for this type of plate are not published, the only references that can be considered are those presented by Leissa [58], where it is stated that for the case of antisymmetric-antisymmetric vibration mode (2,1), the values of parameters are close to the ones determined in the case of the plate clamped on two opposite edges and free on the other two. Using the Rayleigh method, Janich [49]

determined the value 64,2411 for the fundamental vibration parameter.

The percentage deviation of the fundamental pulsation parameter determined through the suggested method is 9,22 %, in comparison to the value of the pulsation parameter determined through Rayleigh method.

5. Rectangular flat plate simply supported on two opposite edges, clamped and free on the other two

It is noted the fact that the maximum percentage deviation recorded for the fundamental pulsation parameter of the square plate determined through the suggested variational method is 8,7 %, in comparison to the value obtained by Fletcher [38], by means of the trigonometric series, and the minimum percentage deviation of the fundamental pulsation parameter is 1,8%, in comparison to the value determined by Janich [49] through Rayleigh method.

For the normal vibration modes (2,1), (3,1), the procentual deviations are determined in comparison to the

values obtained by Fletcher in [38], being 7,97 % for parameter 21 and 4,38 % for parameter 31 .

In the case of the flat rectangular plate with the edge ratio of 5,1 , the percentage deviation of thefundamental pulsation parameter determined through the suggested method is very small, 0,18 %, in comparison to

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the value determined by Timoshenko in [91] and it is very big, 33,74%, in comparison to the value of the pulsation parameter determined by Janich in [49].

6. Rectangular flat plate simply supported on three edges and clamped on the fourth

In the case of this plate the following are noted:- the fundamental pulsation parameter for the flat square plate, determined through the suggested method

presents a minimum procentual deviation of 0,27 %, in comparison to the values obtained by Iguchi in [47] and Polidor Bratu in [74] and a maximum procentual deviation of 2,31 %, in comparison to the value determined by Janich in [49],

- for the normal vibration modes (2,3) and (3,3) of the square plate, the values of the procentual deviation of the parameters of specific pulsations are bigger, of 2,21 %, respectively 2,73 %, in comparison to the ones determined by Iguchi in [47] and Fletcher in [37],

-for the normal vibration modes (1,2), (1,3), (2,1), (2,2) şi (3,1) of the square plate, the procentual deviations of the parameters of specific pulsations determined through the suggested method are smaller, below 1%, in comparison to the parameters of specific pulsations determined by Iguchi in [47].

7. Rectangular flat plate, clamped on three edges and simply supported on the fourth

Comparing the results obtained by the author and the ones given in [32], [44], [49], [52], [55] for the square plate, it is noted that the percentage deviations of vibration parameters vary between 0,06 % for the normal vibration mode (1,1) and 5,2 %, for the normal vibration mode (3,1).

It is also noted that for the normal vibration modes (2,1), (2,2), the percentage deviations are superunity of 1,43 %, respectively 3,35 %.

For the vibration modes (1,2), (1,3), respectively (2,3), the achieved procentual deviations are subunitary.

For the case of the recatngular flat plates with edge ratios 5,1 and 2 , only the fundamental pulsation parameter was determined through the variational method, calculating the procentual deviations of the parameter of fundamental pulsation determined by means of the suggested method, in comparison to the ones determined in [44] and [32].

In the case of rectangular flat plates it is noted that the procentual deviations are very small, below 1%,

that is 0,42% for the rectangular plate with the edge ratio 5,1 , respectively of 0,77% for the rectangular plate

with the edge ratio 2 . By means of a clear, ordered and logical structuring of its chapters, the paper tries to present the dynamic

analysis of rectangular flat plates subjected to free vibrations. The content of the paper has been conceived and achieved in such a way as to emphasize the essential theoretical aspects, along with the adequate physical and mathematical subtilities and the actual problem in the practice of dynamic analysis of rectangular flat plates. The thesis contains not only information from Romanian and worldwide field literature, supported and represented by illustrious proffesors and researchers of the dynamic school in our country, but also personal contributions regarding the determined dynamic characterisitcs that can be a valuable data base within subsequent research.

From the study of field literature it has been noted that there is no data concerning the results obtained by other authors regarding the values of the shape functions, respectively the parameters of the rectangular plate pulsations and the pulsations proper by means of applying the Galerkin-Vlasov variational method, a reason why the results presented in the thesis represent a novelty element brought by the author.

The following are appreciated as the author’s personal contribution to the development of the paper’s subject:

- elaborating the Galerkin-Vlasov variational method in solving the rectangular flat plates with differentboundary conditions,

- the calculus and practical determination of pulsation parameters, adequate to the normal vibration modes considered for different types of plates, which present different edge ratios, concretely for a number of 7 rectangular flat plates, with different boundary conditions;

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- developing a principle of calculating and checking the approximate solutions represented by the vibration parameters;

- determining the values of functions for the rectangular flat plates with different boundary conditions in the nodes of a network for each considered vibration mode,

- making a data base for the 7 studied rectangular flat plates, using the suggested method, containing the pulsation parameters and the values for the functions of the vibration shapes, presented in tables that can be useful in other applications of problems in the dynamics of plates;

- the practical use of the functions of vibration shapes with continuous mass when solving the flat plates, boundary supported or with one or two free edges, using the suggested method;

-checking the correctness of the obtained results with the suggested method by comparing it with the results determined by other authors using different calculation methods and the representation of the percentage deviations obtained.

- making calculus programmes, by means of which we can determine the parameters of specific pulsations for the 7 rectangular flat plates, as well as the values of the functions of specific vibration modes in the nodes of the considered network.

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