Nonlinear Dynamics

, Volume 85, Issue 2, pp 739–750

# Nonlinear analysis of a 2-DOF piecewise linear aeroelastic system

• Tamás Kalmár-Nagy
• Rudolf Csikja
• Tarek A. Elgohary
Original Paper

## Abstract

We study the dynamics of a two- degree-of-freedom (pitch and plunge) aeroelastic system where the aerodynamic forces are modeled as a piecewise linear function of the effective angle of attack. Stability and bifurcations of equilibria are analyzed. We find border collision and rapid bifurcations. Bifurcation diagrams of the system were calculated utilizing MATCONT and Mathematica. Chaotic behavior with intermittent switches about the two nontrivial equilibria was also observed.

## Keywords

Aeroelasticity Piecewise linear system Limit cycle oscillation Bifurcation Hybrid system Chaos

## Notes

### Acknowledgments

The authors are grateful to the reviewers whose suggestions helped to improve the paper. This research was partially supported by OTKA-84060. This research was partially supported by the Hungarian Scientific Research Fund OTKA-84060.

## References

1. 1.
Abdelkefi, A., Vasconcellos, R., Nayfeh, A.H., Hajj, M.R.: An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system. Nonlinear Dyn. 71(1–2), 159–173 (2013)
2. 2.
Alighanbari, H., Hashemi, S.: Derivation of odes and bifurcation analysis of a two-dof airfoil subjected to unsteady incompressible flow. Int. J. Aerosp. Eng. (2009)Google Scholar
3. 3.
Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Pergamon Press Ltd, New York (1966)
4. 4.
Antali, M., Stepan, G.: Discontinuity-induced bifurcations of a dual-point contact ball. Nonlinear Dyn. pp. 1–18 (2015)Google Scholar
5. 5.
Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications, vol. 2. Springer, London (2008)
6. 6.
Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MATCONT: a matlab package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141–164 (2003)
7. 7.
Dowell, E., Edwards, J., Strganac, T.W.: Nonlinear aeroelasticity. J. Aircr. 40(5), 857–874 (2003)
8. 8.
Elgohary, T.: Nonlinear Analysis of a Two DOF Piecewise Linear Aeroelastic System. Master’s thesis, Texas A&M University (2010)Google Scholar
9. 9.
Freire, E., Ponce, E., Ros, J.: Limit cycle bifurcation from center in symmetric piecewise-linear systems. Int. J. Bifurcation Chaos 9(5), 895–907 (1999)
10. 10.
Gantmacher, F.R.: The Theory of Matrices, vol. 2. Chelsea Publishing Company, New York (1959)
11. 11.
Gilliatt, H.C., Strganac, T.W., Kurdila, A.J.: Nonlinear aeroelastic response of an airfoil. In: Proceedings of the 35th Aerospace Sciences Meeting and Exhibit. AIAA 97-459, Reno, NV (1997)Google Scholar
12. 12.
Gilliatt, H.C., Strganac, T.W., Kurdila, A.J.: An investigation of internal resonance in aeroelastic systems. Nonlinear Dyn. 31, 1–22 (2003)
13. 13.
Hayashi, H., Ishizuka, S., Hirakawa, K.: Transition to chaos via intermittency in the onchidium pacemaker neuron. Phys. Lett. A 98(8–9), 474–476 (1983)
14. 14.
Hilborn, R.C.: Chaos and Nonlinear Dynamics : An Introduction for Scientists and Engineers. Oxford University Press, Oxford (2000)
15. 15.
Jeffries, C., Perez, J.: Observation of a pomeau-manneville intermittent route to chaos in a nonlinear oscillator. Phys. Rev. A 26(4), 2117–2122 (1982)
16. 16.
Kalmár-Nagy, T., Wahi, P., Halder, A.: Dynamics of a hysteretic relay oscillator with periodic forcing. SIAM J. Appl. Dyn. Syst. 10, 403–422 (2011)
17. 17.
Kriegsmann, G.: The rapid bifurcation of the Wien bridge oscillator. IEEE Trans. Circuits Syst. 34(9), 1093–1096 (1987)
18. 18.
Lee, B.H.K., Price, S.J., Wong, Y.S.: Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos. Prog. Aerosp. Sci. 35(3), 205–334 (1999)
19. 19.
Leine, R.I.: Bifurcations of equilibria in non-smooth continuous systems. Phys. D-Nonlinear Phenom. 223, 121–137 (2006)
20. 20.
Llibre, J., Novaes, D.D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn. 82(3), 1159–1175 (2015)
21. 21.
Magri, L., Galvanetto, U.: Example of a non-smooth hopf bifurcation in an aero-elastic system. Mech. Res. Commun. 40, 26–33 (2012)
22. 22.
Mahfouz, I.A., Badrakhan, F.: Chaotic behavior of some piecewise-linear systems.1. Systems with set-up spring or with unsymmetric elasticity. J. Sound Vib. 143, 255–288 (1990)
23. 23.
Mahfouz, I.A., Badrakhan, F.: Chaotic behavior of some piecewise-linear systems. 2. Systems with clearance. J. Sound Vib. 143, 289–328 (1990)
24. 24.
Makarenkov, O., Lamb, J.S.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241(22), 1826–1844 (2012)
25. 25.
O’Neil, T., Gilliatt, H.C., Strganac, T.W.: Investigations of aeroelastic response for a system with continuous structural nonlinearities. In: Proceedings of the 37th Structures, Structural Dynamics and Materials Conference. AIAA 96-1390, Salt Lake City, UT (1996)Google Scholar
26. 26.
O’Neil, T., Strganac, T.W.: Aeroelastic response of a rigid wing supported by nonlinear springs. J. Aircr. 35, 616–622 (1998)
27. 27.
Pratap, R., Mukherjee, S., Moon, F.C.: Dynamic behavior of a bilinear hysteretic elastoplastic oscillator 1. Free oscillations. J. Sound Vib. 172, 321–337 (1994)
28. 28.
Pratap, R., Mukherjee, S., Moon, F.C.: Dynamic behavior of a bilinear hysteretic elastoplastic oscillator. 2. Oscillations under periodic impulse forcing. J. Sound Vib. 172, 339–358 (1994)
29. 29.
Price, S.J., Alighanbari, H., Lee, B.H.K.: The aeroelastic response of a 2-dimensional airfoil with bilinear and cubic structural nonlinearities. J. Fluids Struct. 9, 175–193 (1995)
30. 30.
Roberts, I., Jones, D., Lieven, N., Di Bernado, M., Champneys, A.: Analysis of piecewise linear aeroelastic systems using numerical continuation. Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 216(1), 1–11 (2002)
31. 31.
Seiranyan, A.P.: Collision of eigenvalues in linear oscillatory systems. J. Appl. Math. Mech. 58, 805–813 (1994)
32. 32.
Shaw, S.W., Holmes, P.J.: A periodically forced piecewise linear-oscillator. J. Sound Vib. 90, 129–155 (1983)
33. 33.
Sheldahl, R.E., Klimas, P.C.: Aerodynamic characteristics of seven symmetrical airfoil sections through 180-degree angle of attack for use in aerodynamic analysis of vertical axis wind turbines. Tech. rep., Sandia National Laboratories. SAND80-2114 (1981)Google Scholar
34. 34.
Tang, D.M., Dowell, E.H.: Flutter and stall response of a helicopter blade with structural nonlinearity. J. Aircr. 29, 953–960 (1992)
35. 35.
Tang, D.M., Dowell, E.H.: Comparison of theory and experiment for nonlinear flutter and stall response of a helicopter blade. J. Sound Vib. 165, 953–960 (1993)
36. 36.
Thota, P., Dankowicz, H.: Tc-hat (tc): a novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems. SIAM J. Appl. Dyn. Syst. 7(4), 1283–1322 (2008)
37. 37.
Ueda, T., Dowell, E.H.: Flutter analysis using nonlinear aerodynamic forces. J. Aircr. 21, 101–109 (1984)Google Scholar
38. 38.
Vieth, E.: Fitting piecewise linear regression functions to biological responses. J. Appl. Physiol. 67(1), 390–396 (1989)Google Scholar
39. 39.
Woolston, D.S., Runyan, H.L., Andrews, R.E.: An investigation of effects of certain types of structural nonlinearities on wing and control surface flutter. J. Aeronaut. Sci. 24, 57–63 (1957)
40. 40.
Yeh, W.J., Kao, Y.H.: Universal scaling and chaotic behavior of a josephson-junction analog. Phys. Rev. Lett. 49(26), 1888–1891 (1982)
41. 41.
Zhusubaliyev, Z.T., Mosekilde, E.: Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems. World Scientific, River Edge (2003)

## Authors and Affiliations

• Tamás Kalmár-Nagy
• 1
• Rudolf Csikja
• 2
• Tarek A. Elgohary
• 3
1. 1.Department of Fluid Mechanics, Faculty of Mechanical EngineeringBudapest University of Technology and EconomicsBudapestHungary
2. 2.Mathematics Institute, Faculty of Natural SciencesBudapest University of Technology and EconomicsBudapestHungary
3. 3.Aerospace Engineering DepartmentTexas A&M UniversityCollege StationUSA