Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack

Получено точное решение для описания колебаний шарнирно закрепленной по внешнему краю круговой пластины с ограничением по вращению, которая ослаблена круговой трещиной. Рассчитаны частоты шести первых мод колебаний пластины для различных значений упругих характеристик шарнирного защемления, радиуса...

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Datum:	2015
Hauptverfasser:	Bhaskara Rao, L., Kameswara Rao, C.
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Sprache:	English
Veröffentlicht:	Інститут проблем міцності ім. Г.С. Писаренко НАН України 2015
Schriftenreihe:	Проблемы прочности
Schlagworte:	Научно-технический раздел
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/173398
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Zitieren:	Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack / L. Bhaskara Rao,C. Kameswara Rao // Проблемы прочности. — 2015. — № 6. — С. 95-107. — Бібліогр.: 44 назв. — англ.

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spelling	irk-123456789-1733982020-12-03T01:27:25Z Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack Bhaskara Rao, L. Kameswara Rao, C. Научно-технический раздел Получено точное решение для описания колебаний шарнирно закрепленной по внешнему краю круговой пластины с ограничением по вращению, которая ослаблена круговой трещиной. Рассчитаны частоты шести первых мод колебаний пластины для различных значений упругих характеристик шарнирного защемления, радиуса круговой трещины и степени ослабления пластины трещиной. Наличие трещины моделируется в виде фиктивного упругого закрепления пластины по линии трещины. Показано, что ослабление пластины трещиной приводит к снижению собственной частоты колебаний на 30%. Предполагается, что использование результатов точного решения является перспективным при оценке влияния трещины на колебания круговой пластины с внешним шарнирным закреплением и ограничением по вращению и верификации данных, полученных приближенными численными методами, включая метод конечных элементов. Отримано точний розв’язок для опису коливань шарнірно закріпленої по зовнішньому краю кругової пластини з обмеженням по обертанню, яка послаблена круговою тріщиною. Розраховано частоти шести перших мод коливань пластини для різних значень пружних характеристик шарнірного затиснення, радіуса кругової тріщини та міри послаблення пластини тріщиною. Наявність тріщини моделюється у вигляді фіктивного пружного закріплення пластини по лінії тріщини. Показано, що послаблення пластини тріщиною призводить до зниження власної частоти коливань на 30%. Припускається, що використання результатів точного розв’язку є перспективним при оцінці впливу тріщини на коливання кругової пластини із зовнішнім шарнірним закріпленням і обмеженням по обертанню та верифікації даних, що отримані наближеними числовими методами, включаючи з метод скінченних елементів. 2015 Article Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack / L. Bhaskara Rao,C. Kameswara Rao // Проблемы прочности. — 2015. — № 6. — С. 95-107. — Бібліогр.: 44 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173398 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Научно-технический раздел Научно-технический раздел
spellingShingle	Научно-технический раздел Научно-технический раздел Bhaskara Rao, L. Kameswara Rao, C. Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack Проблемы прочности
description	Получено точное решение для описания колебаний шарнирно закрепленной по внешнему краю круговой пластины с ограничением по вращению, которая ослаблена круговой трещиной. Рассчитаны частоты шести первых мод колебаний пластины для различных значений упругих характеристик шарнирного защемления, радиуса круговой трещины и степени ослабления пластины трещиной. Наличие трещины моделируется в виде фиктивного упругого закрепления пластины по линии трещины. Показано, что ослабление пластины трещиной приводит к снижению собственной частоты колебаний на 30%. Предполагается, что использование результатов точного решения является перспективным при оценке влияния трещины на колебания круговой пластины с внешним шарнирным закреплением и ограничением по вращению и верификации данных, полученных приближенными численными методами, включая метод конечных элементов.
format	Article
author	Bhaskara Rao, L. Kameswara Rao, C.
author_facet	Bhaskara Rao, L. Kameswara Rao, C.
author_sort	Bhaskara Rao, L.
title	Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack
title_short	Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack
title_full	Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack
title_fullStr	Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack
title_full_unstemmed	Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack
title_sort	analysis of vibration natural frequencies of rotationally restrained and simply supported circular plate with weakened interior circle due to an angular crack
publisher	Інститут проблем міцності ім. Г.С. Писаренко НАН України
publishDate	2015
topic_facet	Научно-технический раздел
url	http://dspace.nbuv.gov.ua/handle/123456789/173398
citation_txt	Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack / L. Bhaskara Rao,C. Kameswara Rao // Проблемы прочности. — 2015. — № 6. — С. 95-107. — Бібліогр.: 44 назв. — англ.
series	Проблемы прочности
work_keys_str_mv	AT bhaskararaol analysisofvibrationnaturalfrequenciesofrotationallyrestrainedandsimplysupportedcircularplatewithweakenedinteriorcircleduetoanangularcrack AT kameswararaoc analysisofvibrationnaturalfrequenciesofrotationallyrestrainedandsimplysupportedcircularplatewithweakenedinteriorcircleduetoanangularcrack
first_indexed	2025-07-15T10:03:46Z
last_indexed	2025-07-15T10:03:46Z
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fulltext	UDC 539.4 Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack L. Bhaskara Rao a,1 and C. Kameswara Rao b,2 a School of Mechanical and Building Sciences, VIT University, Chennai, Tamil Nadu, India b Nalla Narsimha Reddy Engineering College, Ranga Reddy, Hyderabad, India 1 bhaskarbabu_20@yahoo.com 2 chellapilla95@gmail.com ÓÄÊ 539.4 Àíàëèç ñîáñòâåííûõ ÷àñòîò êîëåáàíèé øàðíèðíî çàêðåïëåííîé êðóãîâîé ïëàñòèíû ñ îãðàíè÷åíèåì ïî âðàùåíèþ, îñëàáëåííîé êðóãîâîé òðåùèíîé Ë. Áõàñêàðà Ðàî à , ×. Êàìåñâàðà Ðàî á à Òåõíîëîãè÷åñêèé èíñòèòóò Âåëëîðå, ×åííàé, Òàìèëíàä, Èíäèÿ á Íàëëà Íàðñèìõà Ðåääè òåõíè÷åñêèé êîëëåäæ, Ðàíãàðåääè, Õàéäàðàáàä, Èíäèÿ Ïîëó÷åíî òî÷íîå ðåøåíèå äëÿ îïèñàíèÿ êîëåáàíèé øàðíèðíî çàêðåïëåííîé ïî âíåøíåìó êðàþ êðóãîâîé ïëàñòèíû ñ îãðàíè÷åíèåì ïî âðàùåíèþ, êîòîðàÿ îñëàáëåíà êðóãîâîé òðåùèíîé. Ðàññ÷èòàíû ÷àñòîòû øåñòè ïåðâûõ ìîä êîëåáàíèé ïëàñòèíû äëÿ ðàçëè÷íûõ çíà÷åíèé óïðó- ãèõ õàðàêòåðèñòèê øàðíèðíîãî çàùåìëåíèÿ, ðàäèóñà êðóãîâîé òðåùèíû è ñòåïåíè îñëàáëåíèÿ ïëàñòèíû òðåùèíîé. Íàëè÷èå òðåùèíû ìîäåëèðóåòñÿ â âèäå ôèêòèâíîãî óïðóãîãî çàêðåïëå- íèÿ ïëàñòèíû ïî ëèíèè òðåùèíû. Ïîêàçàíî, ÷òî îñëàáëåíèå ïëàñòèíû òðåùèíîé ïðèâîäèò ê ñíèæåíèþ ñîáñòâåííîé ÷àñòîòû êîëåáàíèé íà 30%. Ïðåäïîëàãàåòñÿ, ÷òî èñïîëüçîâàíèå ðåçóëüòàòîâ òî÷íîãî ðåøåíèÿ ÿâëÿåòñÿ ïåðñïåêòèâíûì ïðè îöåíêå âëèÿíèÿ òðåùèíû íà êîëåáàíèÿ êðóãîâîé ïëàñòèíû ñ âíåøíèì øàðíèðíûì çàêðåïëåíèåì è îãðàíè÷åíèåì ïî âðàùå- íèþ è âåðèôèêàöèè äàííûõ, ïîëó÷åííûõ ïðèáëèæåííûìè ÷èñëåííûìè ìåòîäàìè, âêëþ÷àÿ ìåòîä êîíå÷íûõ ýëåìåíòîâ. Êëþ÷åâûå ñëîâà: êðóãîâàÿ ïëàñòèíà, ÷àñòîòà, îãðàíè÷åíèå ïî âðàùåíèþ, îñëàáëåí- íàÿ ïëàñòèíà. N o t a t i o n D – flexural rigidity K R1 – rotational spring stiffness at the outer edge K R2 – rotational spring stiffness in the cracked region R11 – non-dimensional rotational flexibility parameter at the outer edge R22 – non-dimensional rotational flexibility parameter in the cracked region k – non-dimensional frequency parameter Introduction. Vibration of circular plates is of great importance in structural design for dynamical loads [1–6]. The problem of free vibrations of uniform isotropic circular plates with classical boundary conditions and internal strengthening has been studied in © L. BHASKARA RAO, C. KAMESWARA RAO, 2015 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 95 detail [7–9]. Cawley and Adams [10] analyzed cracked circular plates by suitably diminishing the values of elastic coefficients of the respective finite element at the nodal location of the crack. Markström and Storákers [11] used the method of unravelling the nodes of the respective finite elements along the face of the crack. Aggarwala and Ariel [12] computed the natural frequencies of cracked circular plates with simply supported boundary conditions by utilizing the second kind of homogeneous Fredholm integral. Leissa et al. [13] studied the influence of V-notches of different angles and depths on the dynamic behavior of circular plates that are completely free at their boundaries. McGee et al. [14] investigated the free vibrations of thin circular plates with free or rigidly constrained V-notches. Using the Ritz method, Yuan and Dickinson [15] investigated the free vibrations of circular, annular and sector plates, which were partially supported along their boundaries or had sectorial cut-outs. Van Rens et al. [16] studied the behavior of perforated plates by tuning the properties of the homogenized plate to be equal to the global properties of the same plate. Huang et al. [17] examined the effect of cracks on the dynamic characteristics of a vibrating circular plate with free edge and a radial crack initiated from the boundary by using an optical system called the AF-ESPI method with out-of-plane displacement. Krawczuk et al. [18] presented a finite element elastoplastic model of a plate having a through crack. Li et al. [19] studied the vibrational power flow characteristics of circular plates having a surface crack at the periphery. Shi et al. [20] computed the eigenfrequency of a cracked circular clamped baffled plate in contact with water and obtained NAVMI factor for each mode of vibration using the method of iteration. Kim and Jung [21] obtained the eigenfrequencies of free edge-rhombus plates with V-notches using a similar method of iteration. Demir and Mermertas [22, 23] studied the natural frequencies of annular plates with radial periodic through cracks and angular cracks by using sector finite elements. Recently, Huang and Leissa [24] studied the influence of side cracks on the eigenfrequencies and mode shapes of plates of rectangular shape with simply supported or completely free boundary conditions. Liu et al. [25] studied the crack influence on the dynamic characteristics of a perforated plate in contact with water using the ANSYS software. Utilizing the Mindlin thick plate theory, Huang et al. [26] implemented the Ritz method to accurately determine the frequencies and nodal patterns of cracked rectangular plates. Using the Rayleigh–Ritz method, Si et al. [27–29] studied the free vibration behavior of a completely clamped cracked circular plate, a baffled cracked circular plate, and a cracked rectangular plate, respectively, considering the influence of water on one side. Chen et al. [30] studied the vibration characteristics of a cantilever rectangular plate with side crack using Ritz method. Chen et al. [31, 32] performed mathematical analysis and numerical study of true and spurious eigenequations for free vibration of plates using the imaginary-part boundary element method (BEM). Lee et al. [33] performed the analytical study and numerical experiments of true and spurious eigensolutions of free vibration of circular plates using the real-part BEM. Lee et al. [34–37] performed the analysis of a circular plate with multiple holes by using indirect BIEMs, the multipole Trefftz method, the direct BIEM and the addition theorem, respectively. However, weakening of a plate may be induced by the presence of partial cracks and internal notching. Only two papers authored by Wang [38] and Yu [39] dealing with free vibration characteristics of circular plates with internal and having simply supported or clamped boundary conditions, as well as movable or free edge conditions, respectively, could be found in the available literature In both studies, the weakened portion was modeled as a hinge with an appropriately computed rotational restraint parameter depending upon the characteristics of the crack present in the plate. Even though the circular symmetry of the plate permits for a substantial interpretation of the problem, very often difficulties escalate due to the intricacy of the respective boundary condition involved. Here the complexity is due to realistic situations where boundary conditions are L. Bhaskara Rao and C. Kameswara Rao 96 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 complex and do not fall under the usual classical boundary conditions. Using the appropriate elastic linear and rotational restraints [40–43], solutions for these complex boundary conditions can be obtained. In a recent study of Bhaskara Rao and Kameswara Rao [44], the vibrations of circular plates weakened along the interior circle and with elastic edge restrictions against translation were analyzed. As per authors’ knowledge, no other research paper addressing the non-classical complex boundary conditions, such as rotationally restrained and simply supported condition at the edge of plate, has yet been published. Therefore, the aim of this paper is to study the effect of weakening of a rotationally restrained and simply supported along the outer edge thin circular plate along a concentric circle due to an angular crack by using the exact method of solution approach. Here, the angular crack is modeled as a hinge with a rotational restraint parameter, while the natural frequencies of a circular plate for different values of the rotational restraint parameter along the plate edge, the radius of the weakened circle, Poisson’s ratio and the rotational restraint with hinge of the cracked region are obtained for the further use in the design of cracked and weakened circular plates, which are applied in engineering, e.g., in the design of doors and hatches used in aircraft and spacecraft, as well as in vibration control and structural design. 1. Analytical Formulations. The plate is rotationally restrained and simply supported at the outer edge, i.e., at a radius of R from the center, as shown in Fig. 1. The radius of the weakened circle because of crack is considered as bR, where b is only a fraction of 1. Here, all lengths are normalized with respect to R, i.e., the radius of the outer region is 1, while the radius of the inner cracked region is b. The subscript I corresponds to the outer region b r� �1, and subscript II to the inner region 0� �r b. Here, h, �, R, and E represent the plate thickness, Poisson’s ratio, radius, and elastic modulus, respectively. The general form of the lateral displacement of vibration of a classical plate can be expressed as w u r n ei t� ( )cos( ) ,� � where (r, �) are polar coordinates, n is the number of modal diameters, w is transverse displacement, � is frequency, and t is time. The function u r( ) is a linear combination of the Bessel functions J krn ( ), Y krn ( ), I krn ( ), and K krn ( ), and k R D� ( ) ,/��2 1 4 where k is the square root of non-dimensional frequency [2] and � is the density. General solution for regions I and II are as follows: u r C J kr C Y kr C I kr C K krI n n n n( ) ( ) ( ) ( ) ( ),� � � �1 2 3 4 (1) u r C J kr C I krII n n( ) ( ) ( ),� �5 6 (2) Analysis of Vibration Natural Frequencies ... ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 97 Fig. 1. A circular plate with rotationally restrained and simply supported edge and an angular crack. where r designates the distance measured from the center of plate, whose maximum value is R. Considering a simply supported and rotationally restrained outer edge, the boundary conditions at outer edge are M r K w r r r R I ( , ) ( , ) ,� � � 1 (3) w rI ( , ) ,� � 0 (4) where the bending moment is defined as M r D R w r r r w r r r w r r I I I ( , ) ( , ) ( , ) ( , ) � � � � � � � � 2 2 2 2 1 1 �2 � � � � � � � � � � � � � � . (5) Equations (3) and (5) yield the following expression: � � � � � 2 2 2 2 2 1 1w r r r w r r r w rI I I( , ) ( , ) ( , ) � � � � � � � � � � � � � � � � � K w r r R I 1 � ( , ) . (6) Equations (4) and (6) can be expressed as �� u r u r n u r R u rI I I I( ) [ ( ) ( )] ( ),� 2 11 (7) u rI ( ) ,� 0 (8) where R K R DR11 1� is the normalized spring constant K R1 of the rotational spring stiffness. At r b� , the plate, except for the slope, is continuous in terms of displacement, moment, and shear, whereas the continuity requirement [38] at the interface of regions I and II can be formulated as u b u bI II( ) ( ),� (9) bu b u b bu b u bI I II II�� ( ) ( ) ( ) ( ),� � (10) b u b n u b b u b nI I II 2 2 2 21 2 1 2�� ( ) [ ( ) ] ( ) ( ) [ (� � � �� ) ] ( ).u bII (11) The moment is set to be proportional to the difference of the slopes b u b bu b n u b b R u b uII II II I II 2 2 2 22�� ( ) [ ( ) ( )] [ ( ) (� b )], (12) where R K R DR22 2� is the normalized spring constant and K R2 is the rotational spring stiffness, which is utilized for modeling the rotational restraint created by the angular crack at r b� . Damage detection experiments can establish the values of rotational stiffness of the crack by matching the theoretically estimated frequency with the one obtained from the experimental studies. Evidently, the stiffness coefficient value K R2 depends on the crack depth and inclination angle. The non-trivial solutions to Eqs. (7)–(12) are sought. Eventually, Eqs. (1), (2) and (7)–(12) yield the following dependencies: 98 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 L. Bhaskara Rao and C. Kameswara Rao k R P k P k n J k Cn 2 4 2 11 1 2 2 2 2 1[( ) ] ( )� �� k R Q k Q k n Y k Cn 2 4 2 11 1 2 2 2 2 2[( ) ] ( )� � � � � � � � � � � � � � � � � � � k R R k R k n I k Cn 2 4 2 11 1 2 2 2 2 3[( ) ] ( )� � � � � � � � � � � � � � � � � k R S k S k n K k Cn 2 4 2 11 1 2 2 2 2 4[( ) ] ( )� � 0, (13) [ ( )] [ ( )] [ ( )] [ ( )] ,J k C Y k C I k C K k Cn n n n1 2 3 4 0� � � � (14) [ ( )] [ ( )] [ ( )] [ ( ) ]J kb C Y kb C I kb C K kb Cn n n n1 2 3 4� � � �[ ( )] [ ( )] ,J kb C I kb Cn n5 6 0 (15) bk P k P bk J kb C bk Q k Q b n 2 2 1 2 1 2 2 1 4 2 2 4 2 �� ( ) k Y kb Cn 2 2 2 ( ) � � � � � � � � � � �� bk R k R bk I kb C bk S k Sn 2 2 1 2 3 2 2 1 4 2 2 4 2 � � ( ) bk K kb Cn 2 4 2 ( ) � � � � � � �� bk P k P bk J kb C bk R k Rn 2 2 1 2 5 2 2 1 4 2 2 4 2 � � ( ) bk I kb Cn 2 6 2 0( ) , � � � � � � � (16) b k P k b k n P C 2 3 3 2 2 2 1 1 8 2 3 4 1 2� � � � � � � � � � � � � � � � �( ( ) )� � � � � � � � � � � � � � � � � � � � � b k Q k b k n Q C 2 3 3 2 2 2 1 8 2 3 4 1 2( ( ) )� � 2 � � � � � � � � � � � � � � � � � � � b k R k b k n R C 2 3 3 2 2 2 1 8 2 3 4 1 2( ( ) )� � 3 � � � � � � � � � � � � � � � � � � � b k S k b k n S 2 3 3 2 2 2 1 8 2 3 4 1 2( ( ) )� � � �C4 � �� b k P k b k n P C 2 3 3 2 2 2 1 8 2 3 4 1 2( ( ) )� � 5 � ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 99 Analysis of Vibration Natural Frequencies ... � � � � � � � � � � � � � � � � � � b k R k b k n R 2 3 3 2 2 2 1 8 2 3 4 1 2( ( ) )� � � �C6 0, (17) b kR P C b kR Q C b kR R 2 22 1 1 2 22 1 2 2 22 1 2 2 2 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � C b kR S C3 2 22 1 4 2 �� b k P kb bR P b k n J kbn 2 2 2 22 1 2 2 2 4 2 2 ( ) ( )� � � � C5 � � � �� b k R kb bR R b k n I kbn 2 2 2 22 1 2 2 2 4 2 2 ( ) ( )� � � � �C6 0, (18) where P J k J kn n1 1 1� �( ) ( ), P J k J kn n2 2 2� � �( ) ( ), P J k J kn n3 3 3� �( ) ( ), Q Y k Y kn n1 1 1� �( ) ( ), Q Y k Y kn n2 2 2� � �( ) ( ), Q Y k Y kn n3 3 3� �( ) ( ), R I k I kn n1 1 1� � �( ) ( ), R I k I kn n2 2 2� � �( ) ( ), R I k I kn n3 3 3� � �( ) ( ), S K k K kn n1 1 1� � �( ) ( ), S K k K kn n2 2 2� � �( ) ( ), K K k K kn n3 3 3� � �( ) ( ), �� P J kb J kbn n1 1 1( ) ( ), � � � �P J kb J kbn n2 2 2( ) ( ), � � �P J kb J kbn n3 3 3( ) ( ), � � �Q Y kb Y kbn n1 1 1( ) ( ), � � � �Q Y kb Y kbn n2 2 2( ) ( ), � � �Q Y kb Y kbn n3 3 3( ) ( ), � � � �R I kb I kbn n1 1 1( ) ( ), � � � �R I kb I kbn n2 2 2( ) ( ), � � � �R I kb I kbn n3 3 3( ) ( ), � � � �S K kb K kbn n1 1 1( ) ( ), � � � �S K kb K kbn n2 2 2( ) ( ), � � � �S K kb K kbn n3 3 3( ) ( ). 2. Results and Discussion. The above equations are used to obtain the exact characteristic equation by eliminating the coefficients of C1 , C2 , C3 , C4 , C5 , and C6. For a non-trivial solution, the determinant of [ ]C 6 6� must be equal to zero. The frequency parameter k can be obtained by solving the characteristic equation via a simple root search method for a given set of n, �, R11 , R22 , and b. Using the mathematica software with symbolic competencies, the exact solutions to this problem are obtained. The numerical result of frequency parameter k obtained from analysis is presented in a pictographic format. Poisson’s ratio employed here is 0.3. The frequency values for various magnitudes of R22 and a constant R11 (R11 2� ) are tabulated in Table 1. The values of frequency parameter k for n� 5 modes with R22 0� , 2, 4, 6, 8, 10, 25, 50, 100, and 1016 and R11 2� are obtained. For b� 1 and R22 0� , the plate frequencies can be seen to be the same as those of the plate with no weakening crack. For a given set of values of b and �, the first frequency for a value of n� 0, the modal frequency converges to that of the plate with no weakening as R22 is increased starting from a value of 0. When � � 0.3, the first six frequencies of the plate with no weakening are obtained as 2.60309 (n� 0), 3.97077 (n� 1), 5.24447 (n� 2), 6.47023 100 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 L. Bhaskara Rao and C. Kameswara Rao ( ),n� 3 7.66549 (n� 4), and 8.8391 (n� 5). Noteworthy is that the fundamental frequency of the plate weakened along an internal concentric circle and resting on the elastically restrained edge against rotation and simply supported edge occurs at n� 0 mode. The variation of the plate fundamental frequency for variable values of the weakened circle radius and the rotational restraint parameter of hinge is presented in Fig. 2. Here, frequency decreases with R22 , and is the lowest for R22 0� , which is a case of a frictionless circular hinge. The internal weakening decreases the fundamental frequency by 2.60309, which makes it lower than that of a plate with no weakening by less than 15% (14.662%). For a given value of R22, the frequency k decreases from 2.60309 to 2.38445, then increases to 2.58649 and finally drops to 2.22145 as the radius b of the weakened circle varies from 0 to 1. The local maximum frequency 2.60309 occurs at b� 0.7. This is the optimal location, where the plate requires to be notched (as a closed hatch). The internal weakening has a minute effect, i.e., it decreases the fundamental frequency by less than 9% (8.4%)] when 0 0 4� �b . . Moreover, it has a more pronounced effect on the fundamental frequency, i.e., decreases the fundamental frequency by less than 15%, when b� 0.7. The frequency values for various magnitudes of R22 and a constant R11 (R11 10� ) are tabulated in Table 1. The first frequencies (k) of n� 5 modes with R22 0� , 2, 4, 6, 8, 10, 25, 50, 100, and 1016 and R11 2� are obtained. For b� 1 and R22 0� , the plate frequencies are same as those of the plate with no weakening. For the given values of b and �, the first frequency of n� 0 mode converges to that of a plate with no weakening as R22 increases from 0. When � � 0.3, the first six frequencies of the plate with no weakening are: 2.95829 (n� 0), 4.30618 (n� 1), 5.55404 ( ),n� 2 6.75602 (n� 3), 7.93028 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 101 Analysis of Vibration Natural Frequencies ... T a b l e 1 Fundamental Frequency Parameter k for a Circular Hinge with � � 0 3. , R22 0� , and Different Values of R11 b 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R11 � � 2 2.60312 2.56339 2.47855 2.40828 2.38445 2.41551 2.49588 2.58649 2.58137 2.43418 2.22145 R11 � �10 2.95826 2.91071 2.81960 2.76548 2.78082 2.86027 2.94944 2.91290 2.71201 2.45934 2.22145 R11 � �1000 3.19296 3.13583 3.03446 2.98881 3.03054 3.13626 3.18937 3.03205 2.74937 2.46594 2.22145 Fig. 2. The fundamental frequency and concentric weakened radius parameter for different R22, R11 2� , � � 0.3, and n � 0. (n� 4), and 9.08575 (n� 5). Noteworthy is that the fundamental frequency of the plate weakened along an internal concentric circle and resting on the elastically restrained edge against rotation and simply supported edge occurs at n� 0 mode. The plate fundamental frequencies varying with the radius of the weakened circle and the elastic rotational restraint of the hinge are presented in Fig. 3. The frequency decreases with R22, and is the lowest for R22 0� , which is a case of a frictionless circular hinge. The internal weakening decreases the fundamental frequency by 2.95829 making it less than that of the plate with no weakening by 25%. For a given value of R22, the frequency k decreases from 2.95829 to 2.76548, then increases to 2.94944 and finally drops to 2.22145 as the radius b of the weakened circle varies from 0 to 1. The local maximum frequency 2.95829 occurs at b� 0.6. This is the optimum location where the plate needs to be notched. For 0 0 3� �b . the internal weakening has a feeble effect on the fundamental frequency, decreasing it by less than 7%. A stronger effect on the fundamental frequency is observed for b� 0.6 – it drops by less than 25%. The plate higher-mode frequencies of plate varying with radius of the weakened circle and elastic rotational restraint of the hinge are presented in Figs. 4–8. The frequencies for various values of R22 and constant R11 (R11 1000� ) are tabulated in Table 1. The first frequencies (k) of n� 5 modes with R22 0� , 2, 4, 6, 8, 10, 102 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 L. Bhaskara Rao and C. Kameswara Rao Fig. 3. The fundamental frequency and concentric weakened radius parameter for different R22 , R11 10� , � � 0.3, and n � 0. Fig. 4. The fundamental frequency and concentric weakened radius parameter for different R22 , R11 10� , � � 0.3, and n �1. 25, 50, 100, and 1016 and R11 1000� are determined. For b� 1 and R22 0� , the plate frequency is equal that with no weakening. For the specified values of b and �, the first frequency of n� 0 mode converges to that with weakening as R22 increases from 0. ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 103 Analysis of Vibration Natural Frequencies ... Fig. 5. The fundamental frequency and concentric weakened radius parameter for different R22 , R11 10� , � � 0.3, and n � 2. Fig. 6. The fundamental frequency and concentric weakened radius parameter for different R22 , R11 10� , � � 0.3, and n � 3. Fig. 7. The fundamental frequency and concentric weakened radius parameter for different R22 , R11 10� , � � 0.3, and n � 4. When � � 0.3, the first six frequencies of the plate with no weakening are: 3.19297 (n� 0), 4.60631 (n� 1), 5.89977 (n� 2), 7.13636 (n� 3), 8.33825 (n� 4), and 9.51625 (n� 5). It is observed that the fundamental frequency of the plate weakened along an internal concentric circle and resting on elastically restrained edge against rotation and simply supported edge occurs at n� 0 mode. The plate fundamental frequency variations with the radius of weakened circle and elastic rotational restraint of the hinge are presented in Fig. 9, where frequency decreases with R22, and is the lowest for R22 0� , which is a case of a frictionless circular hinge. Here, the internal weakening decreases the fundamental frequency 3.19297 that is less than that with no weakening by 31%. For the given value of R22, the frequency decreases from 3.19297 to 2.98881, then increases to 3.18937, and finally drops to 2.22145, as the radius b of the weakened circle varies from 0 to 1. The local maximum frequency 3.19297 occurs at b� 0.6. This is the optimum location where the plate needs to be notched. The internal weakening has a feeble effect on the fundamental frequency (drops it by less than 7%) when 0 0 3� �b . . For the case of b� 0 6. , the fundamental frequency drops drastically (by 31%). In all above-discussed cases, if b� 1, the system is reduced to the case of a circular plate with elastic edge restraints. A fundamental frequency for the case of R22 0� , which models a through circular crack, is tabulated in Table 1. 104 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 L. Bhaskara Rao and C. Kameswara Rao Fig. 8. The fundamental frequency and concentric weakened radius parameter for different R22 , R11 10� , � � 0.3, and n � 5. Fig. 9. The fundamental frequency and concentric weakened radius parameter for different R22 , R11 1000� , � � 0.3, and n � 0. To the best of the authors’ knowledge, the results for a circular plate with rotationally restrained and simply supported boundary presented here, are quite new and, hence, can be compared only with those available for the classical boundary conditions, such as simply supported and clamped plate [38] by setting the rotational restraint with R11 0� and R11 � �, respectively, as well as non-classical boundaries, such as a plate with an edge, which is elastically restrained from translation [44] with R11 � � by setting R11 0� . Here, the internal weakening decreases the fundamental frequency by less than 1% for the clamped plate and less than 1% when b is 0 or 1 for the simply supported plate. Conclusions. The circular plate frequency variation versus the values of elastic rotational restraint, elastic rotational restraint of hinge and the radius of the weakened circle is determined. The crack-induced internal weakening of a plate with elastically restrained rotation and simply supported edge is found to decrease the fundamental frequency by less than 15% for to R11 2� , about 25% for R11 10� and 31% for R11 1000� . It is observed that internal weakening provides a stronger reduction of the fundamental frequency at higher values of elastic rotational restraint: at R11 2� the respective drop is 14.662%, whereas at R11 1000� it is 30.4266%. In addition, frequencies are obtained for different rotational parameters: a clamped boundary is simulated by R11 � �and a simply supported boundary by R11 0� . Accuracy of the results obtained is provided by the closed form of the proposed solution. Ð å ç þ ì å Îòðèìàíî òî÷íèé ðîçâ’ÿçîê äëÿ îïèñó êîëèâàíü øàðí³ðíî çàêð³ïëåíî¿ ïî çîâí³ø- íüîìó êðàþ êðóãîâî¿ ïëàñòèíè ç îáìåæåííÿì ïî îáåðòàííþ, ÿêà ïîñëàáëåíà êðóãî- âîþ òð³ùèíîþ. Ðîçðàõîâàíî ÷àñòîòè øåñòè ïåðøèõ ìîä êîëèâàíü ïëàñòèíè äëÿ ð³çíèõ çíà÷åíü ïðóæíèõ õàðàêòåðèñòèê øàðí³ðíîãî çàòèñíåííÿ, ðàä³óñà êðóãîâî¿ òð³ùèíè òà ì³ðè ïîñëàáëåííÿ ïëàñòèíè òð³ùèíîþ. Íàÿâí³ñòü òð³ùèíè ìîäåëþºòüñÿ ó âèãëÿä³ ô³êòèâíîãî ïðóæíîãî çàêð³ïëåííÿ ïëàñòèíè ïî ë³í³¿ òð³ùèíè. Ïîêàçàíî, ùî ïîñëàáëåííÿ ïëàñòèíè òð³ùèíîþ ïðèçâîäèòü äî çíèæåííÿ âëàñíî¿ ÷àñòîòè êîëèâàíü íà 30%. Ïðèïóñêàºòüñÿ, ùî âèêîðèñòàííÿ ðåçóëüòàò³â òî÷íîãî ðîçâ’ÿçêó º ïåðñïåê- òèâíèì ïðè îö³íö³ âïëèâó òð³ùèíè íà êîëèâàííÿ êðóãîâî¿ ïëàñòèíè ³ç çîâí³øí³ì øàðí³ðíèì çàêð³ïëåííÿì ³ îáìåæåííÿì ïî îáåðòàííþ òà âåðèô³êàö³¿ äàíèõ, ùî îòðè- ìàí³ íàáëèæåíèìè ÷èñëîâèìè ìåòîäàìè, âêëþ÷àþ÷è ç ìåòîä ñê³í÷åííèõ åëåìåíò³â. 1. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw- Hill, New York (1959). 2. A. J. Mcleod and R. E. D. 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Analysis of Vibration Natural Frequencies of Rotationally Restrained and Simply Supported Circular Plate with Weakened Interior Circle Due to an Angular Crack

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