Free Vibration Analysis of Functionally Graded Rotating Double Tapered Beam by using Euler-bernoulli Beam Theory and Differential Transform Method

Document Type : Original Article

Authors

• farzad ebrahimi ¹
• Samanaeh Dashti ²

¹ ikiu

² -

10.22067/fum-mech.v27i1.34525

Abstract

In this study, the free vibration analysis of a functionally graded rotating double tapered beam is performed. The analysis is based on Euler-Bernoulli beam theory. The Material properties of the beam vary continuously in the thickness direction according to the power-law function. The governing differential equation of motion is derived using the Hamilton’s principle. Natural frequencies are obtained using differential transformation (DTM) technique. The effects of the taper ratios, nondimensional rotational speed, nondimensional hub radius and material volume fraction index on the natural frequencies are discussed. Numerical results are tabulated in several tables and figures. To verify the present analysis, the results of this study are compared with the available results from the existing literature. It is shown that the natural frequencies of a functionally graded rotating double tapered Euler-Bernoulli beam can be obtained with high accuracy by using DTM. It was observed that nondimensional rotational speed, height taper ratio and power-law exponent significantly affect the natural frequency. The effects of hub radius and breadth taper ratio on the natural frequencies are negligible.

Keywords

• Free Vibration
• functionally graded material
• Differential Transform Method
• Tapered Euler- Bernoulli beam

---

References

1. Sankar, B.V., "An elasticity solution for functionally graded beams", Composites Science and Technology, Vol. 61, pp. 689–696, (2001).

2. Şimşek, M. "Static analysis of a functionally graded beam under a uniformly distributed load by Ritz method", Int. J. Eng. Appl. Sci., Vol. 1, pp.1-11, (2009).

3. Aydogdu M. and Taskin, V., "Free vibration analysis of functionally graded beams with simply suportededges", Material and Design, Vol. 28, No. 5, pp. 1651-1656, (2007).

4. Şimşek, M., "Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load", Composite Structures, Vol. 92, No. 10, pp. 2532-2546, (2010).

5. Şimşek, M., "Fundamental Frequency Analysis of Functionally Graded Beams by Using Different Higher-Order Beam Theories", Nuclear Engineering and Design, Vol. 240, No. 4, pp. 697-705, (2010).

6. Sina, S.A., Navazi, H.M and Hadadpour, H., "An analytical method for free vibration analysis of functionally graded beams", Material and Design, Vol. 30, No. 3, pp.741-747, (2009)

7. Li, X-F. "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams", Journal of Sound and Vibration, Vol. 318, No. 4, pp. 1210-1229, (2008).

8. Şimşek, M. and T. Kocatürk. "Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load", Composite Structures, Vol. 90, No. 4, pp. 465-473, (2009).

9. Alshorbagy, A.E., Altaher, M.A. and Mahmoud, F.F., "Free vibration characteristics of a functionally beam using inite element method", Applied Mathematical Modelling, Vol. 35, No. 1, pp. 412-425, (2011).

10. Zhong Z. and Yu T., "Analytical solution of cantilever functionally graded beam", Composites Sciences and Technology, Vol. 67, No. 3-4, pp. 481-488, (2007).

11. Daneshmehr, A.R., Hadi, A., and Mehrian, S.N., "Investigation of Elastoplastic Functionally Graded Euler-Bernoulli Beam Subjected to Distribute Transverse Loading", Journal of Basic and Applied Scientific Research, Vol. 2 , No.10, pp. 10628-34, (2012).

12. Kadoli, R., Akhtar, K., and Ganesan, N., "Static analisis of functionally graded beam using higher order shear deformation theory", Applied Mathematical Modeling, Vol. 32, No. 12, pp. 2509-2525, (2008).

13. Bazoune, A., "Effect of tapering on natural frequencies of rotating beams", Shock and Vibration,

Vol. 14, No. 3, pp. 169-179, (2007).

14. Zeng, H., and Bert, C.W., "Vibration analysis of a tapered bar by differential transformation", Journal of Sound and Vibration, Vol. 242, pp. 737–739, (2001).

15. Shahba, A., Attarnejad, R., TavanaieMarvi, M., Hajilar, S., "Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions", Composite Part B: Engineering, Vol. 42, pp.801–808, (2011).

16. Bazoune, A., Khulief, Y.A. and Stephen, N.G., "Further results for modal character- istics of rotating tapered timoshenko beams", Journal of Sound and Vibration, Vol. 219, pp. 157–174, (1999).

17. Huang, Y., Yang, L.E., Luo, Q.Z., "Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section", Composite Part B: Engineering, Vol. 42, pp. 801-8, (2013).

18. Zhou, J.K., "Differential Transformation and its application for Electrical Circuits", Huazhong University Press, Wuhan China, (1986), (in Chinese).

19. Malik, M. and Dang, H.H.,"Vibration analysis of contin- uons system by differeftial transformation", Appl Math Comput, Vol. 96, pp. 17-26, (1998).

20. Ozdemir, O. and Kaya, M.O., "Flapwise bending vibration analysis of a rotating taperded cantilever Bernoulli-Euler beam by differential transform method", Journal of Sound and Vibration , Vol. 289, pp. 413-420, (2006).

21. Ozgumus, O.O. and Kaya, M.O., "Flapwise bending vibration analysis of double tapered rotating Timoshenko beam", Archive of Applied Mechanics, Vol. 78, pp. 379-392, (2008).

22. Banerjee, J.R., "Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method", Journal of Sound and Vibration, Vol. 233, pp. 857-875, (2000).

23. Attarnaejad, R. and Shahba, A., "Application of differential transform method in free vibration Analysis of rotating non-prismatic beams", World Appl. Sci. J., Vol. 5, pp. 441–448, (2008).

24. Piovan, M.T. and Sampaio, R., "A study on the dynamics of rotating beams with functionally graded properties", Journal of Sound and Vibration, Vol. 327, pp. 134-143, (2009).

25. Shahba, A. and Rajasekaran, S., "Free vibration and stability of Euler-Bernoulli beams made of axialliy functionally graded material", Applied Mathematical Modelling, Vol. 36, No. 7, pp.

3094-3111, (2012).

26. Rajasekaran, S., "Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods", Applied Matematical Modelling, Vol. 37, pp. 4440-4463, (2013).

27. Suddoung, K., Charoensuk, J., and Wattanasakulpong, N., "Vibration response of stepped FGM beams with elastically end constraints using differential transformation", Applied Acoustics, Vol. 77, pp. 20-28, (2014).

28. Nguyen, D.K., and Gan, B.S., "Large deflections of tapered functionally graded beams subjected to end forces", Applied Mathematical Modelling, Vol. 38, No. 11, pp. 3054-3066, (2013).

29. Nguyen, D.K., "Large displacement response of tapered cantilever beams made of axially functionally graded material", Composite Part B: Engineering, Vol. 55, pp. 298-305, (2013).

30. Pradhan, K.K. and Chakraverty, S., "Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh – Ritz method", Composites Part B: Engineering, Vol. 51, pp. 175-184, (2013).

31. Pradhan, K.K., and Chakraverty, S., "Effects of different shear deformation theories on free vibration of functionally graded beams", International Journal of Mechanical Sciences, Vol. 82, pp.149-160, (2014).

32. Hodges, D.H. and Dowell, E.H. "Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades", NASA TN D-7818, (1974).

33. Abdel-Halim Hassan, I.H., "On solving some eigenvalue-problems by using a differential transformation", Appl. Math. Comput., Vol. 127, pp. 1–22, (2002).