Ekrem TÜFEKÇİ, Osman Yaşar DOĞRUER

Çember eksenli sabit kesitli çubukların düzlem dışı serbest titreşimleri

Çubuklar, bütün yapı elemanları için en yaygın ve en basit eleman olarak kullanılmaktadır. Çubukların analizi, bütün yüzyıl boyunca araştırmacıların ilgilendiği bir konu olmuştur. Bu çalışmada, mühendislik çalışmalarında önemli bir yeri bulunan, çember eksenli sabit kesitli çubukların düzlem dışı titreşimleri, kayma deformasyonu ve hem eğilme hem de burulmadan kaynaklanan dönme eylemsizliği etkileri dikkate alınarak incelenmiştir. Diferansiyel denklem takımının kesin çözümü, başlangıç değerleri yöntemi kullanılarak elde edilmiştir. Çubuğun belli bir eğrilik ve uzunluk kombinasyonunda meydana gelen ve burulmadan eğilme moduna geçiş olarak bilinen olay modların rezonans frekanslarında ani artışla karakterize edilir. Eğri eksenli düzlemsel çubukların düzlem dışı titreşimlerinde, çubuk eğriliğine etki eden mod geçişi olarak adlandırılan bu olay incelenmiş ve verilen grafiklerde gösterilmiştir. Farklı narinlik oranları ve farklı eksen eğrisi açıklık değerlerinde, çubuğun ilk beş modu için boyutsuz frekans katsayıları belirlenmiştir. Elde edilen sonuçlar, narin ve sığ çubuklarda da olduğu gibi, kayma deformasyonu ve hem eğilme hem de burulmadan kaynaklanan dönme eylemsizliği etkilerinin rezonans frekansları üzerinde öneme haiz bir etkisi olduğunu göstermiştir. Burulmadan kaynaklanan dönme eylemsizliği etkisi, eğri eksenli düzlem çubukların düzlem dışı titreşimleri konusunda en önemli etki olarak karşımıza çıkmaktadır. Konu ile ilgili literatürde verilen örnekler çözülerek, sonuçlar tablolarda verilmiştir.

Out-of-plane free vibration of a circular arch with uniform cross-section

Arches have long been widely used as structural elements in many mechanical, aerospace and civil engineering applications such as spring design, brake shoes within drum brakes, tire dynamics, piping systems, turbo-machinery blades, curved wires in missile guidance floated gyroscopes, aerospace structures, stiffeners in aircraft structures, arch bridges, curved girder bridges, long span roof structures and earthquake resistant structures. Hence, the dynamic behaviour of arches has been of interest to many researchers since the nineteenth century. In general, the in-plane and out-of-plane vibrations of a planar arch are coupled. However, based on the Bernoulli-Euler hypothesis, if the cross-section of an arch is uniform and doubly symmetric, i.e., the shear center and centroid coincide, and then the in-plane and out-of-plane vibrations are uncoupled. However the out-of-plane bending and torsional responses will still be coupled. It is often difficult and sometimes impossible to find a general closed-form solution for the vibration problem of an arch, since the governing differential equations possess variable coefficients. The exact solution of the governing equations exists only for a circular beam of uniform cross-section. The previous studies are based upon the classical theory in which either rotatory inertia or shear deformation are taken into account. Timoshenko beam theory considers the effects of shear deformation and rotatory inertia due to both flexural and torsional vibrations and provides a better approximation to the actual arch behaviour. Many techniques have been considered in the papers on out-of-plane vibrations of arches. The Ritz method with different types of trial functions has often been applied in determining the natural frequencies of arches. With the advancement of computer technology and several programs, the finite element method has been used widely to solve for more general geometry and a number of curved elements have been developed. If the behaviour of the arch is non-planar, usual finite element or finite difference model becomes very complicated. In this study, free out-of-plane vibrations of a circular arch with uniform cross-section are investigated by taking into account the effects of transverse shear and rotatory inertia due to both flexural and torsional vibrations. The governing differential equations for out-of-plane vibration of uniform circular beams are solved exactly by using the initial value method. The solution does not depend on the boundary conditions. The same solution procedure is also used to obtain the results of other cases in which each effect is considered individually in order to assess its importance. The frequency coefficients are obtained for the first five modes of arches with various slenderness ratios and opening angles. The results show that the flexural and torsional rotatory inertia and shear deformation have very important effects on resonance frequencies, even if slender shallow arches are considered. It is concluded that the torsional rotatory inertia effect is the most significant effect to be included in the analysis. A phenomenon known as transition of modes from torsional into flexural is characterized by the sharp increment in resonance frequencies of modes that occurs at certain combinations of curvature and length of the arch. This increase in mode frequency is accompanied by a significant change in the mode shapes. In this study, the analysis of the transition phenomenon in vibrational behaviour of a shallow circular arch with uniform cross-section is also presented by using the exact solution of the governing equations. The mode transition phenomenon is shown in figures. Vibration problems for circular beams that have been analyzed in the literature are solved and the results are compared in tables. The comparison shows good agreement between the results. The main purpose of this paper is to present the exact solution to the governing differential equations of out-of-plane vibrations for a circular arch with uniform cross-section. The effects of shear deformation and rotatory inertia due to the flexural and torsional vibrations are taken into account. But the warping deformation of the cross-section is neglected. The initial value method is used in order to solve the governing differential equations. The solution does not depend on the boundary conditions. The variations of the frequency coefficients with respect to the opening angle are presented for a certain slenderness ratio and several boundary conditions. The examples given in the literature are solved and the results are compared.

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Auciello, N.M., De Rosa, M.A., (1994). Free vibrations of circular arches: A review, Journal of Sound and Vibration, 174 , 433-458.
Bickford, W.B., Maganty, S.P., (1986). On the out-of-plane vibrations of thick rings, Journal of Sound and Vibration, 108 , 503-507.
Chidamparam, P., Leissa, A.W., (1993). Vibrations of a Planar Curved Beams, Rings and Arches, Applied Mechanics Reviews, 46, 467–483.
E Silva, J.M.M., Urgueria, A.P.V., (1988). Out-of-plane dynamic response of curved beams-An analytical model, International Journal of Solids and Structures , 24, 271-284.
Howson, W.P., Jamah, A.K., Zhou, J.Q., (1995). Exact natural frequencies for out-of-plane motion of plane structures composed of curved beam members, Computers and Structures, 55, 989-995.
Howson, W.P., Jemah, A.K., (1999). Exact out-of-plane natural frequencies of curved Timeshenko beams, Journal of Engineering Mechanics, 125 , 19-25.
Huang, C.S., Tseng, Y.P., Chang, S.H., (1998). Out-of-plane dynamic responses of non-circular curved beams by numerical Laplace transform, Journal of Sound and Vibration, 215, 407-424.
Irie, T., Yamada, G., Takahashi, I., (1980). The steady state out-of-plane response of a Timoshenko curved beam with internal damping, Journal of Sound and Vibration , 71, 145-156.
Irie, T., Yamada, G., Takahashi, I., (1982). Natural frequencies of out-of-plane vibration of arcs, Journal of Applied Mechanics, 49, 910-913.
Kang, K., Bert, C.W., Stritz, A.G., (1995). Vibration analysis of shear deformable circular arches by the differential quadrature method, Journal of Sound and Vibration, 181, 353-360.
Kawakami, M., Sakiyama, T., Matsuda, H., Morita, C., (1995). In-plane and out-of-plane free vibrations of curved beams with variable sections, Journal of Sound and Vibration, 187, 381-401.
Lee, S.Y., Chao, J.C., (2000). Out-of-plane vibrations of curved non-uniform beams of constant radius, Journal of Sound and Vibration, 238, 443-458.
Laura, P.A.A., Maurizi, M.J., (1987). Recent research on vibrations of arch-type structures, The Shock and Vibration Digest, 19, 6-9.
Love, A.E.H., (1944). A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 4th edition.
Markus, S., Nanasi, T., (1981). Vibration of Curved Beams, The Shock and Vibration Digest, 13, 3-14.
Rubin, M.B., (2000). Cosserat Theories: Shells, Rods and Points, Kluwer Academic Publishers, The Netherlands.
Rubin, M.B., Tufekci, E., (2005). Three-dimensional free vibrations of a circular arch using the theory of a Cosserat point, Journal of Sound and Vibration, 286, 799-816.
Tarnopolskaya, T., De Hoog, F.R., Fletcher, N.H., Thwaites, S., (1996). Asymptotic analysis of the free vibrations of beams with arbitrarily varying curvature and cross-section, Journal of Sound and Vibration, 196, 659-680.
Tarnopolskaya, T., De Hoog, F.R., Fletcher, N.H., (1999). Low-frequency mode transition in the free in-plane vibration of curved beams, Journal of Sound and Vibration, 228 , 69-90.
Timoshenko, S., Goodier, J. N., (1951). Theory of Elasticity, McGraw-Hill Book Co., Tokyo, 2nd edition.
Tufekci, E., Arpacı, A., (1998). Exact solution of in-plane vibrations of circular arches with account taken of axial extension, transverse shear and rotatory inertia effects, Journal of Sound and Vibration, 209 ,845-856.
Tufekci, E., (2001). Exact solution of free in-plane vibration of shallow circular arches, International Journal of Structural Stability and Dynamics, 1, 409-428.
Tufekci, E., (2004). On finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures, Computer Methods in Applied Mechanics and Engineering, 193 , 4067-4068.
Volterra, E., Morell, J.D., (1961). Lowest natural frequency of elastic arc for vibrations outside the plane of initial curvature, Journal of Applied Mechanics, 28 , 624-627.
Wang, T.M., Laskey, A., Ahmad, M., (1984). Natural frequencies for out-of-plane vibrations of continuous curved beams considering shear and rotary inertia, International Journal of Solids and Structures, 20, 257-265.
Yang, Y.-B., Wu, C.-M., Yau, J.-D., (2001). Dynamic response of a horizontally curved beam subjected to vertical and horizontal moving loads, Journal of Sound and Vibration, 242, 519-537.
Wang, T.M., Guilbert, M.P., (1982). Effects of rotatory inertia and shear on natural frequencies of continuous circular curved beams, International Journal of Solids and Structures, 17, 281-289.
Takahashi, S., (1962). Vibration of circular arch (Perpendicular to its plane), Bulletin of the JSME, 6 (24), 674-681.
Suzuki, K., Aida, H., Takahashi, S., (1978). Vibration of curved bars perpendicular to their planes, Bulletin of the JSME, 21 (162), 1685-1695.