Free Vibration Analysis of FG Straight-Axis Beams Based on the First-Order Shear Deformation Theory

Öz

In this work, the free vibration characteristics of functionally graded (FG) beams with straight-axis are investigated based on the first-order shear deformation theory (FSDT). It is assumed that the material properties change only through the thickness of the beam. The effects of the coefficient of variation, length/thickness ratios and boundary conditions on the free vibration behavior of the beams are also examined in a parametric manner. The equations of motion, governing the free vibration behavior of these beams are obtained using the principle of minimum total energy based on the Timoshenko’s beam assumption. These ordinary differential equations (ODEs) obtained in the canonical form are solved numerically by the Complementary Functions Method (CFM). The calculated natural vibration frequencies are compared with the results of the existing studies in the literature and shown to be in agreement with them.

Anahtar Kelimeler

• Functionally graded materials,Complementary functions method,Free vibration analysis

Birinci Mertebe Kayma Deformasyon Teorisine Dayalı FD Düz Eksenli Kirişlerin Serbest Titreşim Analizi

Öz

Bu çalışmada, birinci mertebe kayma deformasyon teorisine dayalı doğru eksenli fonksiyonel derecelenmiş (FD) malzemeli kirişlerin serbest titreşim özellikleri incelenmiştir. Malzeme özelliklerinin sadece kiriş kalınlığı boyunca değiştiği kabulü yapılmıştır. Malzeme değişim katsayısının, uzunluk/kalınlık oranlarının ve sınır koşullarının kirişlerin serbest titreşim davranışı üzerindeki etkileri de parametrik olarak incelenmiştir. Bu kirişlerin serbest titreşim davranışını idare eden hareket denklemleri, Timoshenko kiriş varsayımına dayalı minimum toplam enerji ilkesi kullanılarak elde edilmiştir. Kanonik halde elde edilen bu adi diferansiyel denklemler Tamamlayıcı Fonksiyonlar Yöntemi (TFY) ile sayısal olarak çözülmüştür. Hesaplanan doğal titreşim frekansları, literatürdeki mevcut çalışmaların sonuçları ile karşılaştırılmış ve bunlarla uyum içerisinde olduğu gösterilmiştir.

Anahtar Kelimeler

• Fonksiyonel derecelenmiş malzeme,Tamamlayıcı fonksiyonlar yöntemi,Serbest titreşim analizi

Kaynakça

1. 1. Aydoğdu, M., Taşkın, V., 2007. Free Vibration Analysis of Functionally Graded Beams with Simply Supported Edges. Materials & Design, 28(5), 1651-1656.
2. 2. Li, X.F., 2008. A Unified Approach for Analyzing Static and Dynamic Behaviors of Functionally Graded Timoshenko and Euler– bernoulli Beams. Journal of Sound and Vibrations, 318, 1210-1229.
3. 3. Sina, S.A., Navazi, H.M., Haddadpour, H., 2009. An Analytical Method for Free Vibration Analysis of Functionally Graded Beams. Materials and Design, 30(3), 741-747.
4. 4. Şimşek, M., 2010. Vibration Analysis of a Functionally Graded Beam Under a Moving Mass by Using Different Beam Theories. Composite Structures, 92, 904-917.
5. 5. Alshorbagy, A.E., Eltaher, M.A., ve Mahmoud, F.F., 2011. Free Vibration Characteristics of a Functionally Graded Beam by Finite Element Method. Applied Mathematical Modelling, 35, 412-425.
6. 6. Anandrao, K.S., Gupta, R.K., Ramachandran, P., Rao, G.V., 2012. Free Vibration Analysis of Functionally Graded Beams. Defence Science Journal, 62(3), 139-146.
7. 7. Thai, H.T., Vo, T.P., 2012. Bending and Free Vibration of Functionally Graded Beams Using Various Higher-order Shear Deformation Beam Theories. International Journal of Mechanical Sciences, 62(1), 57-66.
8. 8. Pradhan, K.K., Chakraverty, S., 2013. Free Vibration of Euler and Timoshenko Functionally Graded Beams by Rayleigh–ritz Method. Composites: Part B, 51, 175-184.

9. 9. Nguyen, T.K., Vo, T.P., Thai, H.T., 2013.Static and Free Vibration of Axially Loaded Functionally Graded Beams Based on the First-order Shear Deformation Theory. Composites: Part B, 55, 147-157.
10. 10. Su, H., Banerjee, J.R., Cheung, C.W., 2013. Dynamic Stiffness Formulation and Free Vibration Analysis of Functionally Graded Beams. Composite Structures 106, 854–862.
11. 11. Pradhan, K.K., Chakraverty, S., 2014. Effects of Different Shear Deformation Theories on Free Vibration of Functionally Graded Beams. International Journal of Mechanical Sciences, 82, 149-160.
12. 12. Su, H., Banerjee, J.R., 2015. Development of Dynamic Stiffness Method for Free Vibration of Functionally Graded Timoshenko Beams. Computers and Structures 147, 107-116.
13. 13. Jing, L.L., Ming, P.J., Zhang, W.P., Fu, L.R., Cao, Y.P., 2016. Static and Free Vibration Analysis of Functionally Graded Beams by Combination Timoshenko Theory and Finite Volume Method. Composite Structures, 138, 192-213.
14. 14. Avcar, M., Alwan, H.H.A., 2017. Free Vibration of Functionally Graded Rayleigh Beam. International Journal of Engineering & Applied Sciences (IJEAS), 9(2), 127-137.
15. 15. Lee, J.W., Lee, J.Y., 2017. Free Vibration Analysis of Functionally Graded Bernoulli-euler Beams Using an Exact Transfer Matrix Expression. International Journal of Mechanical Sciences, 122, 1-17.
16. 16. Kahya, V., Turan, M., 2017. Finite Element Model for Vibration and Buckling of Functionally Graded Beams Based on the First-order Shear Deformation Theory. Composites Part B: Engineering, 109, 108-115.
17. 17. Turan, M., Kahya, V., 2018. Fonksiyonel Derecelendirilmiş Kirişlerin Serbest Titreşim Analizi. Karadeniz Fen Bilimleri Dergisi, 8(2), 119-130, DOI: 10.31466/kfbd.453833
18. 18. Banerjee, J.R., Ananthapuvirajah, A,. 2018. Free Vibration of Functionally Graded Beams and Frameworks Using the Dynamic Stiffness Method. Journal of Sound and Vibration, 422, 34-47.
19. 19. Çelebi, K., Yarımpabuç, D., Tütüncü, N., 2018. Free Vibration Analysis of Functionally Graded Beams Using Complementary Functions Method. Arch. Appl. Mech., 88(5), 729-739.
20. 20. Aslan, T.A., Noori, A.R., Temel, B., 2018. Dynamic Response of Viscoelastic Tapered Cycloidal Rods. Mechanics Research Communications, 92, 8–14.
21. 21. Noori, A.R., Aslan, T.A., Temel B., 2018. An Efficient Approach for In-plane Free and Forced Vibrations of Axially Functionally Graded Parabolic Arches With Nonuniform Cross Section. Composite Structures, 200(15), 701-710.