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Nonlinear Dynamic and Stability Analysis of an Edge Cracked Rotating Flexible Structure

Milad Azimi

Aerospace Research Institute, (Ministry of Science, Research and Technology) Tehran, Iran

E-mail Address: azimi.m@ari.ac.ir

Corresponding author.

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Samad Moradi

Islamic Azad University, North Tehran Branch, Tehran, Iran

E-mail Address: s.moradi@iau-tnb.ac.ir

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https://doi.org/10.1142/S0219455421500917Cited by:12 (Source: Crossref)

Abstract

The Homotopy Perturbation Method (HPM) is applied to investigate the nonlinear free and forced vibration behavior of the rotating cracked beam. Nonlinear governing equations considering the two-dimensional (2D) large flexible structure in a rotating reference frame considering centrifugal forces are obtained by the Lagrangian approach and the Assumed Mode Method (AMM). The crack is modeled as an elastic nonlinear massless rotational spring, which divides the beam into two parts. The Rayleigh–Ritz method is used to discretize the governing system of equations of the motion. Stability analysis along with bifurcation and phase portrait represents the different behavior of the system, depending on the variations of base angular velocity, crack location, and stiffness. Moreover, it is shown that as the rotational speed increases, a tensile force appears along the neutral axis, stiffening the cracked structure, which results in shifting the backbone to the right and highly affects the nonlinear features of the system. The results obtained through a comparative study of the HPM with first-order approximation and numerical simulations (Runge–Kutta algorithm) demonstrate an accurate and effective solution for structures with nonlinear dynamics.

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References

1. W. He, X. Mu, L. Zhang and Y. Zou , Modeling and trajectory tracking control for flapping-wing micro aerial vehicles, IEEE/CAA J. Autom. Sin. 8(1) (2020) 148–156. Web of Science, Google Scholar
2. M. Azimi and E. F. Joubaneh , Dynamic modeling and vibration control of a coupled rigid-flexible high-order structural system: A comparative study, Aerosp. Sci. Technol. 102 (2020) 105875. Web of Science, Google Scholar
3. J. Warminski and J. Latalski , Nonlinear control of flexural–torsional vibrations of a rotating thin-walled composite beam, Int. J. Struct. Stab. Dyn. 17(3) (2017) 1740003. Link, Web of Science, Google Scholar
4. Y. Bian, Z. Gao and M. Fan , Nonlinear vibration suppression for flexible arms via modal coupling, Int. J. Struct. Stab. Dyn. 17 (2017) 1771004. Link, Web of Science, Google Scholar
5. W. He, T. Wang, X. He, L.-J. Yang and O. Kaynak , Dynamical modeling and boundary vibration control of a rigid-flexible wing system, IEEE/ASME Trans. Mechatronics 25(2) (2020) 2711–2721. Web of Science, Google Scholar
6. W. He, T. Meng, X. He and C. Sun , Iterative learning control for a flapping wing micro aerial vehicle under distributed disturbances, IEEE Trans. Cybern. 49(4) (2018) 1524–1535. Web of Science, Google Scholar
7. J. Huang, C.-O. Chang and C.-C. Chang , Analysis of structural vibrations of vertical axis wind turbine blades via Hamilton’s principle-part 2: Exact and approximate solutions, Int. J. Struct. Stab. Dyn. 20(9) (2020) 2050099. Link, Web of Science, Google Scholar
8. A. Yashar, N. Ferguson and M. Ghandchi-Tehrani , Simplified modelling and analysis of a rotating Euler-Bernoulli beam with a single cracked edge, J. Sound Vib. 420 (2018) 346–356. Web of Science, Google Scholar
9. J. Tian, J. Su, K. Zhou and H. Hua , A modified variational method for nonlinear vibration analysis of rotating beams including Coriolis effects, J. Sound Vib. 426 (2018) 258–277. Web of Science, Google Scholar
10. H. Kim, H. H. Yoo and J. Chung , Dynamic model for free vibration and response analysis of rotating beams, J. Sound Vib. 332(22) (2013) 5917–5928. Web of Science, Google Scholar
11. L. Li and D. Zhang , Dynamic analysis of rotating axially FG tapered beams based on a new rigid–flexible coupled dynamic model using the B-spline method, Compos. Struct. 124 (2015) 357–367. Web of Science, Google Scholar
12. M. Rafiee, F. Nitzsche and M. Labrosse , Dynamics, vibration and control of rotating composite beams and blades: A critical review, Thin-Walled Struct. 119 (2017) 795–819. Web of Science, Google Scholar
13. S. Hashemi and M. Richard , Natural frequencies of rotating uniform beams with Coriolis effects, J. Vib. Acoust. 123(4) (2001) 444–455. Web of Science, Google Scholar
14. G. Wang and N. M. Wereley , Free vibration analysis of rotating blades with uniform tapers, AIAA J. 42(12) (2004) 2429–2437. Google Scholar
15. J. B. Gunda and R. Ganguli , Stiff-string basis functions for vibration analysis of high speed rotating beams, J. Appl. Mech. 75(2) (2008) 024502. Web of Science, Google Scholar
16. S. Talukdar and S. K. Roy , Free in-plane vibration of a cracked curved beam with fixed ends, Adv. Eng. Des. (2019) 363–376, https://doi.org/10.1007/978-981-13-6469-3_33. Google Scholar
17. Ş. D. Akbaş , Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory, Int. J. Struct. Stab. Dyn. 17(3) (2017) 1750033. Link, Web of Science, Google Scholar
18. A. Ç. Batihan and F. S. Kadioğlu , Vibration analysis of a cracked beam on an elastic foundation, Int. J. Struct. Stab. Dyn. 16(5) (2016) 1550006. Link, Web of Science, Google Scholar
19. Ş D. Akbaş , On post-buckling behavior of edge cracked functionally graded beams under axial loads, Int. J. Struct. Stab. Dyn. 15(4) (2015) 1450065. Link, Web of Science, Google Scholar
20. J. Fernandez-Saez, L. Rubio and C. Navarro , Approximate calculation of the fundamental frequency for bending vibrations of cracked beams, J. Sound Vib. 225(2) (1999) 345–352. Web of Science, Google Scholar
21. K. Mazanoglu and M. Sabuncu , Flexural vibration of non-uniform beams having double-edge breathing cracks, J. Sound Vib. 329(20) (2010) 4181–4191. Web of Science, Google Scholar
22. A. Saito, M. P. Castanier, C. Pierre and O. Poudou , Efficient nonlinear vibration analysis of the forced response of rotating cracked blades, J. Comput. Nonlinear Dyn. 4(1) (2009) 011005. Web of Science, Google Scholar
23. O. A. Bauchau and C. H. Hong , Finite element approach to rotor blade modeling, J. Am. Helicopter Soc. 32(1) (1987) 60–67. Google Scholar
24. F. Bekhoucha, S. Rechak, L. Duigou and J.-M. Cadou , Branch switching at hopf bifurcation analysis via asymptotic numerical method: Application to nonlinear free vibrations of rotating beams, Commun. Nonlinear Sci. Numer. Simul. 22(1–3) (2015) 716–730. Web of Science, Google Scholar
25. O. Thomas, A. Sénéchal and J.-F. Deü , Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams, Nonlinear Dyn. 86(2) (2016) 1293–1318. Web of Science, Google Scholar
26. F. Ebrahimi and M. Mokhtari , Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method, J. Braz. Soc. Mech. Sci. Eng. 37(4) (2015) 1435–1444. Web of Science, Google Scholar
27. F. Ebrahimi and M. Mokhtari , Free vibration analysis of a rotating Mori–Tanaka-based functionally graded beam via differential transformation method, Arab. J. Sci. Eng. 41(2) (2016) 577–590. Web of Science, Google Scholar
28. S. Hosseini, M. Zamanian, S. Shams and A. Shooshtari , Vibration analysis of geometrically nonlinear spinning beams, Mech. Mach. Theory 78 (2014) 15–35. Web of Science, Google Scholar
29. I. Pakar and M. Bayat , Vibration analysis of high nonlinear oscillators using accurate approximate methods, Struct. Eng. Mech. 46(1) (2013) 137–151. Web of Science, Google Scholar
30. H.-L. Zhang , Application of He’s frequency-amplitude formulation to an x1/3 force nonlinear oscillator, Int. J. Nonlinear Sci. Numer. Simul. 9(3) (2008) 297–300. Web of Science, Google Scholar
31. M. Akbarzade and A. Farshidianfar , Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method, Appl. Math. Model. 45 (2017) 393–404. Web of Science, Google Scholar
32. Y. Chen, J. Zhang and H. Zhang , Free vibration analysis of rotating tapered Timoshenko beams via variational iteration method, J. Vib Control 23(2) (2017) 220–234. Web of Science, Google Scholar
33. M. R. Barati and H. Shahverdi , Nonlinear vibration of nonlocal four-variable graded plates with porosities implementing homotopy perturbation and Hamiltonian methods, Acta Mech. 229(1) (2018) 343–362. Web of Science, Google Scholar
34. H. Moeenfard, M. Mojahedi and M. T. Ahmadian , A homotopy perturbation analysis of nonlinear free vibration of Timoshenko microbeams, J. Mech. Sci. Technol. 25(3) (2011) 557. Web of Science, Google Scholar
35. S. Taeprasartsit , Nonlinear free vibration of thin functionally graded beams using the finite element method, J. Vib. Control 21(1) (2015) 29–46. Web of Science, Google Scholar
36. M. C. Luintel and N. Vyas , Dynamic response and stability of a spinning turbine blade subjected to pitching and yawing, Int. J. Dyn. Control 7(4) (2019) 1252–1277. Google Scholar
37. J. Chen and Q.-S. Li , Vibration characteristics of a rotating pre-twisted composite laminated blade, Compos. Struct. 208 (2019) 78–90. Web of Science, Google Scholar
38. S. Khosravi, H. Arvin and Y. Kiani , Vibration analysis of rotating composite beams reinforced with carbon nanotubes in thermal environment, Int. J. Mech. Sci. 164 (2019) 105187. Web of Science, Google Scholar
39. A. A. Yazdi , Nonlinear aeroelastic stability analysis of three-phase nano-composite plates, Mech. Based Des. Struct. Mach. 47(6) (2019) 1–16. Web of Science, Google Scholar
40. K. Manimegalai, C. F. Sagar Zephania, P. Bera, P. Bera, S. Das and T. Sil , Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method, Eur. Phys. J. Plus 134(9) (2019) 462. Web of Science, Google Scholar
41. A. A. Yazdi , Nonlinear flutter of laminated composite plates resting on nonlinear elastic foundations using homotopy perturbation method, Int. J. Struct. Stab. Dyn. 15(5) (2015) 1450072. Link, Web of Science, Google Scholar
42. S. Kitipornchai, L. Ke, J. Yang and Y. Xiang , Nonlinear vibration of edge cracked functionally graded Timoshenko beams, J. Sound Vib. 324(3–5) (2009) 962–982. Web of Science, Google Scholar
43. S. S. Rao , Vibration of Continuous Systems (Wiley Online Library, 2007). Google Scholar