Variant of method of symmetries in a task about the vibrations of circular plate with a decreasing thickness by law of concave parabola

Trapezon, Kirill

doi:10.20535/2312-1807.2015.20.2.47781

Cited by 3 publications

(7 citation statements)

References 3 publications

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“…It should be noted that because of the flexibility of the symmetry method, there are no rigid restrictions to select the D 1 (x) function. It can be expressed, for example, not only through the Bessel functions [18]. It is owing to the method of symmetry that an analytical solution and a frequency equation for η=2 were obtained when the plate is rigidly clamped.…”

Section: Discussion Of Results Of Solving a Problem For A Solid Variamentioning

confidence: 99%

Analytical solution to the problem about free oscillations of a rigidly clamped circular plate of variable thickness

Trapezon¹,

Trapezon²

2020

EEJET

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This paper reports an analytical solution to one of the problems related to applied mechanics and acoustics, which tackles the analysis of free axisymmetric bending oscillations of a circular plate of variable thickness. A plate rigidly-fixed along the contour has been considered, whose thickness changes by parabola h(ρ)=H0(1+µρ) 2. For the initial assessment of the effect exerted by coefficient µ on the results, the solutions at µ=0 and some µ≠0 have been investigated. The differential equation of the shapes of a variable-thickness plate's natural oscillations, set by the h(ρ) function, has been solved by a combination of factorization and symmetry methods. First, a problem on the oscillations of a rigidly-fixed plate of the constant thickness (µ=0), in which h(1)/h(0)=η=1, was solved. The result was the computed natural frequencies (numbers λ i at i=1...6), the constructed oscillation shapes, as well as the determined coordinates of the nodes and antinodes of oscillations. Next, a problem was considered about the oscillations of a variable-thickness plate at η=2, which corresponds to µ=0.4142. Owing to the symmetry method, an analytical solution and a frequency equation for η=2 were obtained when the contour is rigidly clamped. Similarly to η=1, the natural frequencies were calculated, the oscillation shapes were constructed, and the coordinates of nodes and antinodes of oscillations were determined. Mutual comparison of frequencies (numbers λ i) shows that the natural frequencies at η=2 for i=1...6 increase significantly by (28...19.9) % compared to the case when η=1. The increase in frequencies is a consequence of the increase in the bending rigidity of the plate at η=2 because, in this case, the thickness in the center of both plates remains unchanged, and is equal to h=H 0. The reported graphic dependences of oscillation shapes make it possible to compare visually patterns in the distribution of nodes and antinodes for cases when η=1 and η=2. Using the estimation formulae derived from known ratios enabled the construction of the normalized diagrams of the radial σ r and tangential σ θ normal stresses at η=1 and η=2. Mutual comparison of stresses based on the magnitude and distribution character has been performed. Specifically, there was noted a more favorable distribution of radial stresses at η=2 in terms of strength and an increase in technical resource

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Section: Discussion Of Results Of Solving a Problem For A Solid Variamentioning

confidence: 99%

Analytical solution to the problem about free oscillations of a rigidly clamped circular plate of variable thickness

Trapezon¹,

Trapezon²

2020

EEJET

View full text Add to dashboard Cite

show abstract

“…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] ( )…”

Section: The Original Differential Equation and Its General Solution For A Plate With A Thickness Of H=h 0 (1-μρ)mentioning

confidence: 99%

“…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…”

Section: The Original Differential Equation and Its General Solution For A Plate With A Thickness Of H=h 0 (1-μρ)mentioning

confidence: 99%

“…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:…”

Section: The Original Differential Equation and Its General Solution For A Plate With A Thickness Of H=h 0 (1-μρ)mentioning

confidence: 99%

“…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: (…”

Section: Formulae For Calculating Cyclical Stresses In a Platementioning

confidence: 99%

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Analysis of free oscillations of circular plates with variable thickness based on the symmetry method

Trapezon¹,

Trapezon²

2020

EEJET

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На основi методiв симетрiї i факторизацiї отримано загальне аналiтичне рiшення диференцiального рiвняння четвертого порядку для задачi про вiльнi вiсесиметричнi коливання кругової пластинки зi змiнною товщиною. Законом змiни товщини є увiгнута парабола h=H 0 (1-μρ) 2 , де μ -постiйний коефiцiєнт, який визначає ступiнь угнутостi пластинки. Рiшення представлено у функцiях Беселя нульового та першого порядку вiд дiйсного i уявного аргументу. Розглянута кругова кiльцева пластинка з жорстким закрiпленням внутрiшнього контуру i вiльним зовнiшнiм краєм при трьох значеннях коефiцiєнта μ. Визначено першi три власнi значення задачi (частотнi числа) i власнi функцiї (форми коливань). Показано, що власнi частоти перших трьох форм коливань зi зростанням угнутостi (збiльшенням μ) знижуються в рiзнiй мiрi, яка визначається номером частотного числа λ i (i=1, 2, 3). При μ=1,21417 i μ=1,39127 частоти в порiвняннi з випадком μ=0,5985 знижуються вiдповiдно на (1; 1,3) %, (17,6; 24) %, (22,85; 30, 35) % для λ 1 , λ 2 , λ 3 . Видно, що наявне iстотне падiння частоти на вищих формах коливань (λ 2 , λ 3 ) i незначне при основнiй формi (λ 1 ). Встановлено значення i координати екстремальних прогинiв (пучностей коливань) i орiєнтовнi координати вузлових перерiзiв. Наведенi числовi параметри поряд з частотними показниками є засобом для iдентифiкацiї коливальних властивостей пластинки при її вивченнi на практицi. Побудовано графiчнi залежностi для радiальних σ r i тангенцiальних σ θ циклiчних напружень при основнiй формi для кожного з трьох варiантiв угнутостi параболiчної пластинки. Встановлено, що збiльшення вiдношення крайових товщин, тобто увiгнутостi, призводить до пiдвищення σ r в перерiзах за межами закрiплення. Цi напруження, що дiють вiддалено вiд вiльного краю, наприклад, в закрiпленнi або в районi дiї максимальних σ θ в рiзнiй мiрi перевершують σ θ . Через це цi напруження представляють собою основну небезпеку з точки зору циклiчної мiцностi пластинки при досягненнi σ r руйнiвних значень. Вiдзначено можливiсть за рахунок пiдвищення увiгнутостi параболiчної пластинки забезпечити оптимальне спiввiдношення мiж значеннями σ r в закрiпленнi i σ r , якi дiють за межами вiд закрiплення. Це спiввiдношення, яке дорiвнює приблизно 1, забезпечується у випадку μ=1,39127, що розглянуто в роботiКлючовi слова: кiльцева пластинка, змiнна товщина, метод симетрiй, власне число, напружено-деформований стан

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Analysis of free oscillations of round thin plates of variable thickness with a point support

Trapezon¹,

Trapezon²,

Orlov³

2020

EEJET

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Variant of method of symmetries in a task about the vibrations of circular plate with a decreasing thickness by law of concave parabola

Cited by 3 publications

References 3 publications

Analytical solution to the problem about free oscillations of a rigidly clamped circular plate of variable thickness

Analytical solution to the problem about free oscillations of a rigidly clamped circular plate of variable thickness

Analysis of free oscillations of circular plates with variable thickness based on the symmetry method

Analysis of free oscillations of round thin plates of variable thickness with a point support

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