ÇİFT SİMETRİLİ DEĞİŞKEN KESİTLİ ÇUBUKLARIN EKSENEL TİTREŞİMLERİ

Yıl 2022, Cilt: 8 Sayı: 2, 307 - 321, 31.12.2022

Aydın Özmutlu

https://doi.org/10.34186/klujes.1183046

Öz

Bu çalışmada çift simetrili değişken kesitli çubukların taşıma matrisi yöntemi ile eksenel titreşimleri araştırılmıştır. Ele alınan problemdeki, çubuk açıklık ortasına göre simetrik ve her iki ucundan basit mesnetlidir. Konik çubuk için küresel koordinatlarda yazılan hareket denkleminin degişkenlerine ayırma yöntemi ile Bessel fonksiyonları cinsinden kapalı çözümü yapılmıştır. Simetrik çubuğun 1’inci ve 2’nci bölgesi için çubuk uçlarında durum vektörleri yazılmış ve her iki bölge için taşıma matrisi türetilmiştir. Simetrik çubuk için toplam taşıma matrisi yazılıp sınır koşulları uygulanarak titreşim denklemine ulaşılmıştır. Bu denklemin çözümünden serbest titreşim frekansları ve mod şekilleri farklı koniklik oranları için belirlenmiştir. Koniklik oranın ve simetrinin eksenel titreşim üzerine olan etkisi ortaya konmuştur.

Anahtar Kelimeler

Konik çubuk, Eksenel titreşim, Taşıma matrisi, Küresel Bessel fonksiyonları

AXIAL VIBRATIONS OF DOUBLY SYMMETRIC TAPERED BARS

Öz

In this study, axial vibrations of bi-symmetrical variable cross-section bars were investigated using the transfer matrix method. The bar in the problem under examination is presumptively symmetrical about its mid-span and simply supported at both ends. The doubly symmetric tapered bar's equation of motion, written in spherical coordinates, is solved in closed form employing the separation of variables, and the solution is expressed in terms of Bessel functions. State vectors are obtained at the ends of the symmetrical bar for the first and second domains, and the transfer matrix is computed for both domains. The vibration equation is obtained by writing the total transfer matrix for the symmetrical bar and applying the boundary conditions. From the solution of this equation, free vibration frequencies and mode shapes are determined for different taper ratios. The effect of taper ratio and symmetry on axial vibration has been demonstrated.

Anahtar Kelimeler

Tapered bar, Axial vibration, Transfer matrix, Spherical Bessel functions

Kaynakça

• Elishakoff, I., Eigenvalues of inhomogeneous structures: Unusual closed-form solutions, CRC Press, Boca Raton, Fla., 2005.
• Balduzzi, G., Aminbaghai, M., Sacco, E., Füssl, J., Eberhardsteiner, J., ve Auricchio, F., Non-prismatic beams: A simple and effective Timoshenko-like model, International Journal of Solids and Structures, 90, ss. 236–250, 2016. https://doi.org/10.1016/j.ijsolstr.2016.02.017.
• Vilar, M., Hadjiloizi, D. A., Masjedi, P. K., ve Weaver, P. M., Stress analysis of generally asymmetric non-prismatic beams subject to arbitrary loads, European Journal of Mechanics - A/Solids, 90, s. 104284, 2021. https://doi.org/10.1016/j.euromechsol.2021.104284.
• Andrade, A. ve Camotim, D., Lateral–Torsional Buckling of Singly Symmetric Tapered Beams: Theory and Applications, Journal of Engineering Mechanics, 131, ss. 586–597, 2005. https://doi.org/10.1061/(ASCE)0733-9399(2005)131:6(586).
• Demir, E., Çallioğlu, H., ve Sayer, M., Vibration analysis of sandwich beams with variable cross section on variable Winkler elastic foundation, Science and Engineering of Composite Materials, 20, ss. 359–370, 2013. https://doi.org/10.1515/secm-2012-0151.
• Soltani, M., Atoufi, F., Mohri, F., Dimitri, R., ve Tornabene, F., Nonlocal Analysis of the Flexural-Torsional Stability for FG Tapered Thin-Walled Beam-Columns, Nanomaterials (Basel, Switzerland), 11, 2021. https://doi.org/10.3390/nano11081936.
• Banerjee, J. R. ve Williams, F. W., Exact Bernoulli-Euler dynamic stiffness matrix for a range of tapered beams, International Journal for Numerical Methods in Engineering, 21, ss. 2289–2302, 1985. https://doi.org/10.1002/nme.1620211212.
• Gupta, A. K., Vibration of Tapered Beams, Journal of Structural Engineering, 111, ss. 19–36, 1985. https://doi.org/10.1061/(ASCE)0733-9445(1985)111:1(19).
• Lee, S. Y., Ke, H. Y., ve Kuo, Y. H., Analysis of non-uniform beam vibration, Journal of Sound and Vibration, 142, ss. 15–29, 1990. https://doi.org/10.1016/0022-460X(90)90580-S.
• Rosa, M. A. de ve Auciello, N. M., Free vibrations of tapered beams with flexible ends, Computers & Structures, 60, ss. 197–202, 1996. https://doi.org/10.1016/0045-7949(95)00397-5.
• Hsu, J.-C., Lai, H.-Y., ve Chen, C. K., Free vibration of non-uniform Euler–Bernoulli beams with general elastically end constraints using Adomian modified decomposition method, Journal of Sound and Vibration, 318, ss. 965–981, 2008. https://doi.org/10.1016/j.jsv.2008.05.010.
• Banerjee, J. R., Su, H., ve Jackson, D. R., Free vibration of rotating tapered beams using the dynamic stiffness method, Journal of Sound and Vibration, 298, ss. 1034–1054, 2006. https://doi.org/10.1016/j.jsv.2006.06.040.
• Ece, M. C., Aydogdu, M., ve Taskin, V., Vibration of a variable cross-section beam, Mechanics Research Communications, 34, ss. 78–84, 2007. https://doi.org/10.1016/j.mechrescom.2006.06.005.
• Lee, J. W. ve Lee, J. Y., Free vibration analysis using the transfer-matrix method on a tapered beam, Computers & Structures, 164, ss. 75–82, 2016. https://doi.org/10.1016/j.compstruc.2015.11.007.
• Banerjee, J. R. ve Ananthapuvirajah, A., Free flexural vibration of tapered beams, Computers & Structures, 224, s. 106106, 2019. https://doi.org/10.1016/j.compstruc.2019.106106.
• Çalım, F. F., Deği̇şken Kesi̇tli̇ Timoshenko Ki̇ri̇şi̇ni̇n Serbest Ti̇treşi̇m Anali̇zi̇, Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, 6, ss. 76–82, 2017. https://doi.org/10.28948/ngumuh.297736.
• Magnucki, K., Magnucka-Blandzi, E., Milecki, S., Goliwąs, D., ve Wittenbeck, L., Free flexural vibrations of homogeneous beams with symmetrically variable depths, Acta Mechanica, 232, ss. 4309–4324, 2021. https://doi.org/10.1007/s00707-021-03053-x.
• Abrate, S., Vibration of non-uniform rods and beams, Journal of Sound and Vibration, 185, ss. 703–716, 1995. https://doi.org/10.1006/jsvi.1995.0410.
• Li, Q. S., Exact solutions for free longitudinal vibrations of non-uniform rods, Journal of Sound and Vibration, 234, ss. 1–19, 2000. https://doi.org/10.1006/jsvi.1999.2856.
• Li, Q. S., Exact solutions for free longitudinal vibration of stepped non-uniform rods, Applied Acoustics, 60, ss. 13–28, 2000. https://doi.org/10.1016/S0003-682X(99)00048-1.
• Raj, A. ve Sujith, R. I., Closed-form solutions for the free longitudinal vibration of inhomogeneous rods, Journal of Sound and Vibration, 283, ss. 1015–1030, 2005. https://doi.org/10.1016/j.jsv.2004.06.003.
• Guo, S. ve Yang, S., Free longitudinal vibrations of non-uniform rods, Science China Technological Sciences, 54, ss. 2735–2745, 2011. https://doi.org/10.1007/s11431-011-4534-6.
• Yardimoglu, B. ve Aydin, L., Exact longitudinal vibration characteristics of rods with variable cross-sections, Shock and Vibration, 18, ss. 555–562, 2011. https://doi.org/10.3233/SAV-2010-0561.
• Gan, C., Wei, Y., ve Yang, S., Longitudinal wave propagation in a rod with variable cross-section, Journal of Sound and Vibration, 333, ss. 434–445, 2014. https://doi.org/10.1016/j.jsv.2013.09.010.
• Pillutla, S. H., Gopinathan, S., ve Yerikalapudy, V. R., Free longitudinal vibrations of functionally graded tapered axial bars by pseudospectral method, Journal of Vibroengineering, 20, ss. 2137–2150, 2018. https://doi.org/10.21595/jve.2018.19373.
• Šalinić, S., Obradović, A., ve Tomović, A., Free vibration analysis of axially functionally graded tapered, stepped, and continuously segmented rods and beams, Composites Part B: Engineering, 150, ss. 135–143, 2018. https://doi.org/10.1016/j.compositesb.2018.05.060.
• Todorovska, M. I., Girmay, E. A., Wang, F., ve Rahmani, M., Wave propagation in a doubly tapered shear beam: Model and application to a pyramid‐shaped skyscraper, Earthquake Engineering & Structural Dynamics, 51, ss. 764–792, 2022. https://doi.org/10.1002/eqe.3590.
• Abramowitz, M., ve Stegun, I. A., Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables / edited by Milton Abramowitz and Irene A. Stegun, Dover Publications, New York, 1970.