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A thorough look at the (in)stability of piston-theoretic beams

1 Carnegie Mellon University, 5000 Forbes Avenue Pittsburgh, PA 15213, USA
2 University of Maryland, Baltimore County, 1000 Hilltop Cir, Baltimore, MD, 21250, USA
3 College of Charleston, 66 George St, Charleston, SC 29424, USA

We consider a beam model representing the transverse deflections of a one dimensional elastic structure immersed in an axial fluid flow. The model includes a nonlinear elastic restoring force, with damping and non-conservative terms provided through the flow effects. Three different configurations are considered: a clamped panel, a hinged panel, and a flag (a cantilever clamped at the leading edge, free at the trailing edge). After providing the functional framework for the dynamics, recent results on well-posedness and long-time behavior of the associated solutions are presented. Having provided this theoretical context, in-depth numerical stability analyses follow, focusing both on the onset of flow-induced instability (flutter), and qualitative properties of the post-flutter dynamics across configurations. Modal approximations are utilized, as well as finite difference schemes.
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Keywords extensible beam; nonlinear elasticity; flutter; stability; attractors; modal analysis

Citation: Jason Howell, Katelynn Huneycutt, Justin T. Webster, Spencer Wilder. A thorough look at the (in)stability of piston-theoretic beams. Mathematics in Engineering, 2019, 1(3): 614-647. doi: 10.3934/mine.2019.3.614

References

  • 1. Ashley H, Zartarian G (1956) Piston theory: a new aerodynamic tool for the aeroelastician. J Aeronaut Sci 23: 1109–1118.    
  • 2. Ball JM (1973) Initial-boundary value problems for an extensible beam. J Math Anal Appl 42: 61–90.    
  • 3. Ball JM (1973) Stability theory for an extensible beam. J Differ Equations 14: 399–418.    
  • 4. Han SM, Benaroya H, Wei T (1999) Dynamics of transversely vibrating beams using four engineering theories. J Sound Vib 225: 935–988.    
  • 5. Cássia BA, Crippa HR (1994) Global attractor and inertial set for the beam equation. Appl Anal 55: 61–78.    
  • 6. Bociu I, Toundykov D (2012) Attractors for non-dissipative irrotational von karman plates with boundary damping. J Differ Equations 253: 3568–3609.    
  • 7. Bolotin VV (1963) Nonconservative Problems of the Theory of Elastic Stability, Macmillan, New York.
  • 8. Chen SP, Triggiani R (1989) Proof of extensions of two conjectures on structural damping for elastic systems. Pac J Math 136: 15–55.    
  • 9. Chueshov I (2015) Dynamics of Quasi-Stable Dissipative Systems, Springer.
  • 10. Chueshov I, Dowell EH, Lasiecka I, et al. (2016) Nonlinear elastic plate in a flow of gas: recent results and conjectures. Appli Math Optim 73: 475–500.    
  • 11. Chueshov I, Lasiecka I (2008) Long-time behavior of second order evolution equations with nonlinear damping. Mem Am Math Soc 195.
  • 12. Chueshov I, Lasiecka I (2010) Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer-Verlag, New York.
  • 13. Chueshov I, Lasiecka I, Webster JT (2014) Attractors for delayed, nonrotational von karman plates with applications to flow-structure interactions without any damping. Commun Part Diff Eq 39: 1965–1997.    
  • 14. Chueshov I, Lasiecka I, Webster JT (2014) Flow-plate interactions: well-posedness and long-time behavior. Discrete Contin Dyn Syst S 7: 925–965.    
  • 15. Dickey RW (1970) Free vibrations and dynamic buckling of the extensible beam. J Math Anal Appl 29: 443–454.    
  • 16. Dowell EH (1966) Nonlinear oscillations of a fluttering plate. I. AIAA J 4: 1267–1275.    
  • 17. Dowell EH (1967) Nonlinear oscillations of a fluttering plate. II. AIAA J 5: 1856–1862.    
  • 18. Dowell EH (1982) Flutter of a buckled plate as an example of chaotic motion of a deterministic autonomous system. J Sound Vib 85: 333–344.    
  • 19. Jaworski JW, Dowell EH (2008) Free vibration of a cantilevered beam with multiple steps: Comparison of several theoretical methods with experiment. J Sound Vib 312: 713–725.    
  • 20. Dowell EH, Clark R, Cox D A Modern Course in Aeroelasticity (Vol. 3), Dordrecht: Kluwer academic publishers.
  • 21. Dowell EH, McHugh K (2016) Equations of motion for an inextensible beam undergoing large deflections. J Appl Mech 83: 051007.    
  • 22. Eden A, Milani AJ (1993) Exponential attractors for extensible beam equations. Nonlinearity 6: 457–479.    
  • 23. Hansen SW, Lasiecka I (2000) Analyticity, hyperbolicity and uniform stability of semigroups arising in models of composite beams. Math Models Methods Appl Sci 10: 555–580.
  • 24. Holmes PJ (1977) Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis. J Sound Vib 53: 471–503.    
  • 25. Holmes P, Marsden J (1978) Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis. Automatica 14: 367–384.    
  • 26. Howell J, Lasiecka I, Webster JT (2016) Quasi-stability and exponential attractors for a non-gradient system-applications to piston-theoretic plates with internal damping. EECT 5: 567–603.    
  • 27. Howell JS, Toundykov D, Webster JT (2018) A cantilevered extensible beam in axial flow: semigroup well-posedness and postflutter regimes. SIAM J Math Anal 50: 2048–2085.    
  • 28. Huang L, Zhang C (2013) Modal analysis of cantilever plate flutter. J Fluid Struct 38: 273–289.    
  • 29. Kim D, Cossé J, Cerdeira CH, et al. (2013) Flapping dynamics of an inverted flag. J Fluid Mech 736.
  • 30. Lagnese J (1989) Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics.
  • 31. Lagnese JE, Leugering G (1991) Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. J Differ Equations 91: 355–388.    
  • 32. Lasiecka I, Webster JT (2014) Eliminating flutter for clamped von karman plates immersed in subsonic flows. Commun Pure Appl Anal 13: 1935–1969.    
  • 33. Lasiecka I, Webster JT (2016) Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow. SIAM J Math Anal 48: 1848–1891.    
  • 34. Ma TF, Narciso V, Pelicer ML (2012) Long-time behavior of a model of extensible beams with nonlinear boundary dissipations. J Math Anal Appl 396: 694–703.    
  • 35. Païdoussis MP (1998) Fluid-structure interactions: slender structures and axial flow (Vol. 1), Academic press.
  • 36. Russell DL (1993) A general framework for the study of indirect damping mechanisms in elastic systems. J Math Anal Appl 173: 339–358.    
  • 37. Russell DL (1991) A comparison of certain elastic dissipation mechanisms via decoupling and projection techniques. Q Appl Math 49: 373–396.    
  • 38. Semler C, Li GX, Païdoussis MP (1994) The non-linear equations of motion of pipes conveying fluid. J Sound Vib 169: 577–599.    
  • 39. Serry M, Tuffaha A (2018) Static stability analysis of a thin plate with a fixed trailing edge in axial subsonic flow: possio integral equation approach. Appl Math Model 63: 644–659.    
  • 40. Shubov M (2010) Solvability of reduced possio integral equation in theoretical aeroelasticity. Adv Differ Equations 15: 801–828.
  • 41. Tang D, Zhao M, Dowell EH (2014) Inextensible beam and plate theory: computational analysis and comparison with experiment. J Appl Mech 81: 061009.    
  • 42. Vedeneev VV (2013) Effect of damping on flutter of simply supported and clamped panels at low supersonic speeds. J Fluid Struct 40: 366–372.    
  • 43. Vedeneev VV (2012) Panel flutter at low supersonic speeds. J Fluid Struct 29: 79–96.    
  • 44. Webster JT (2011) Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach. Nonlinear Anal: Theory Methods Appl 74: 3123–3136.    
  • 45. Woinowsky-Krieger S (1950) The effect of an axial force on the vibration of hinged bars. J Appl Mech 17: 35–36.
  • 46. Zhao W, Païdoussis MP, Tang L, et al. (2012) Theoretical and experimental investigations of the dynamics of cantilevered flexible plates subjected to axial flow. J Sound Vib 331: 575–587.    

 

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