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September  2011, 10(5): 1447-1462. doi: 10.3934/cpaa.2011.10.1447

Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam

Emine Kaya 1, , Eugenio Aulisa 1, , Akif Ibragimov 1, and  Padmanabhan Seshaiyer 2,

1. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042
2. 

Department of Mathematical Sciences, George Mason University, Fairfax VA, 22030

Received  May 2009 Revised  November 2010 Published  April 2011

In this work we consider the dynamical response of a non-linear beam with viscous damping, perturbed in both the transverse and axial directions. The system is modeled using coupled non-linear momentum equations for the axial and transverse displacements. In particular we show that for a class of boundary conditions (beam clamped at the extremes) and uniformly distributed load, there exists a non-uniform equilibrium state. Different models of damping are considered: first, third and fifth order dissipation terms. We show that in all cases in the presence of the damping forces, the excited beam is stable near the equilibrium for any perturbation. An energy estimate approach is used in order to identify the space in which the solution of the perturbed system is stable.
Keywords: Euler-Bernoulli beam, Stability analysis, non-linear partial differential equations.
Mathematics Subject Classification: Primary: 49K20, 70K20; Secondary: 74K1.
Citation: Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447
References:
[1]

A. A. Alqaisia and M. N. Hamdan, Bifurcation and chaos of an immersed cantilever beam in a fluid and carrying an intermediate mass, Journal of Sound and Vibration, 253 (2002), 859-888. doi: 10.1006/jsvi.2001.4072.  Google Scholar

[2]

A. Andrianov and A. Hermans, A VELFP on infinite, finite and shallow water, 17-th International workshop on water waves and floating bodies, Cambridge, UK, aPEIL (2002), 14-17. Google Scholar

[3]

E. Aulisa, A. Ibragimov, Y. Kaya and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam,, Accepted in the, ().   Google Scholar

[4]

E. Aulisa, A. Cervone, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition approach for studying coupled flow application, Communications in Computational Physics, 6 (2009), 319-341. doi: 10.4208/cicp.2009.v6.p319.  Google Scholar

[5]

E. Aulisa, S. Manservisi, and P. Seshaiyer, A computational domain decomposition approach for solving coupled flow-structure-thermal interaction problems, Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 13-31. Google Scholar

[6]

E. Aulisa, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition methodology for solving coupled problems in fluid-structure-thermal interaction, Proceedings of ECCM 2006, Lisbon, Portugal (2006). Google Scholar

[7]

R. W. Dickey, Dynamic stability of equilibrium states of the extendible beam, Proceedings of the American Mathematical Society, 41 (1973), 94-102. doi: 10.1090/S0002-9939-1973-0328290-8.  Google Scholar

[8]

L. C. Evans, "Partial Differential Equations," AMS, 1998.  Google Scholar

[9]

D. A. Evensen, Nonlinear vibrations of beams with various boundary conditions, AIAA Journal, 6 (1968), 370-372. doi: 10.2514/3.4506.  Google Scholar

[10]

L. Ferguson, E. Aulisa, P. Seshaiyer, Computational modeling of highly flexible membrane wings in micro air vehicles, Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference Newport, RI (2006). Google Scholar

[11]

D. G. Gorman, I. Trendafilova, A. J. Mulholland and J. Horacek, Analytical modeling and extraction of the modal behavior of a cantilever beam in fluid interaction, Journal of Sound and Vibration, (2007), 231-245. doi: 10.1016/j.jsv.2007.07.032.  Google Scholar

[12]

A. E. Green and J. E. Adkins, "Large Elastic Deformations," Clarendon Press (Oxford), 1970. Google Scholar

[13]

H. W. Haslach, J. D. Humphrey, Dynamics of biological soft tissue or rubber: Internally pressurized spherical membranes surrounded by a fluid, Int J Nonlin Mech, 39 (2004), 399-420. Google Scholar

[14]

J. D. Humphrey, "Cardiovascular Solid Mechanics," Springer, 2002. Google Scholar

[15]

A. I. Ibragimov and P. Koola, The dynamics of wave carpet, P. 2288, OCEAN 2003 MTS/IEEE, proceedings. Google Scholar

[16]

R. A. Ibrahim, Nonlinear vibrations of suspended cables, Part III: Random excitation and interaction with fluid flow, Applied Mechanics Reviews, 57 (2004), 515-549. doi: 10.1115/1.1804541.  Google Scholar

[17]

J. E. Lagnese, Modelling and stabilization of nonlinear plates, International Series of Numerical Mathematics, 100 (1991), 247-264. Google Scholar

[18]

C. L. Lou and D. L. Sikarskie, Nonlinear Vibration of beams using a form-function approximation, ASME Journal of Applied Mechanics, 42 (1975), 209-214. doi: 10.1115/1.3423520.  Google Scholar

[19]

C. Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates, Computers and Structures, 3 (1973), 163-174. Google Scholar

[20]

J. Padovan, Nonlinear vibrations of general structures, Journal of Sound and Vibration, 72 (1980), 427-441. Google Scholar

[21]

J. Peradze, A numerical alghorithm for Kirchhoff-Type nonlinear static beam, Journal of Applied Mathematics, in Press (2009). doi: 10.1155/2009/818269.  Google Scholar

[22]

J. N. Reddy, Finite element modeling of structural vibrations: A review of recent advances,, The Shock Vibration Digest, 11 (): 25.   Google Scholar

[23]

J. N. Reddy, An introduction to Nonlinear Finite Element Analysis, Oxford University, 2004. doi: 10.1093/acprof:oso/9780198525295.001.0001.  Google Scholar

[24]

D. L. Russel, A comparison of certain dissipation mechanisms via decoupling and projection techniques, Quart. Appl. Math., XLIX (1991), 373-396.  Google Scholar

[25]

P. Seshaiyer and J. D. Humphrey, A sub-domain inverse finite element characterization of hyperelastic membranes, including soft tissues, ASME J Biomech Engr., 125 (2003), 363-371. doi: 10.1115/1.1574333.  Google Scholar

[26]

W. Shyy, Y. Lian, J. Tang, D. Viieru and H. Liu, Aerodynamics of Low Reynolds Number Flyers, Cambridge University Press, 2007. doi: 10.1017/CBO9780511551154.  Google Scholar

[27]

G. Singh, G. V. Rao and N. G. R. Iyengar, Reinvestigation of large amplitude free vibrations of beams using finite elements, Journal of Sound and Vibration, 143 (1990), 351-355. Google Scholar

[28]

H. Wagner and V. Ramamurti, Beam vibrations-A review, The Shock and Vibration Digest, 9 (1977), 17-24. Google Scholar

[29]

O. C. Zienkiewicz and R. L. Taylor, "The Finite Element Method," McGraw-Hill, 1993,  Google Scholar

show all references

References:
[1]

A. A. Alqaisia and M. N. Hamdan, Bifurcation and chaos of an immersed cantilever beam in a fluid and carrying an intermediate mass, Journal of Sound and Vibration, 253 (2002), 859-888. doi: 10.1006/jsvi.2001.4072.  Google Scholar

[2]

A. Andrianov and A. Hermans, A VELFP on infinite, finite and shallow water, 17-th International workshop on water waves and floating bodies, Cambridge, UK, aPEIL (2002), 14-17. Google Scholar

[3]

E. Aulisa, A. Ibragimov, Y. Kaya and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam,, Accepted in the, ().   Google Scholar

[4]

E. Aulisa, A. Cervone, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition approach for studying coupled flow application, Communications in Computational Physics, 6 (2009), 319-341. doi: 10.4208/cicp.2009.v6.p319.  Google Scholar

[5]

E. Aulisa, S. Manservisi, and P. Seshaiyer, A computational domain decomposition approach for solving coupled flow-structure-thermal interaction problems, Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 13-31. Google Scholar

[6]

E. Aulisa, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition methodology for solving coupled problems in fluid-structure-thermal interaction, Proceedings of ECCM 2006, Lisbon, Portugal (2006). Google Scholar

[7]

R. W. Dickey, Dynamic stability of equilibrium states of the extendible beam, Proceedings of the American Mathematical Society, 41 (1973), 94-102. doi: 10.1090/S0002-9939-1973-0328290-8.  Google Scholar

[8]

L. C. Evans, "Partial Differential Equations," AMS, 1998.  Google Scholar

[9]

D. A. Evensen, Nonlinear vibrations of beams with various boundary conditions, AIAA Journal, 6 (1968), 370-372. doi: 10.2514/3.4506.  Google Scholar

[10]

L. Ferguson, E. Aulisa, P. Seshaiyer, Computational modeling of highly flexible membrane wings in micro air vehicles, Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference Newport, RI (2006). Google Scholar

[11]

D. G. Gorman, I. Trendafilova, A. J. Mulholland and J. Horacek, Analytical modeling and extraction of the modal behavior of a cantilever beam in fluid interaction, Journal of Sound and Vibration, (2007), 231-245. doi: 10.1016/j.jsv.2007.07.032.  Google Scholar

[12]

A. E. Green and J. E. Adkins, "Large Elastic Deformations," Clarendon Press (Oxford), 1970. Google Scholar

[13]

H. W. Haslach, J. D. Humphrey, Dynamics of biological soft tissue or rubber: Internally pressurized spherical membranes surrounded by a fluid, Int J Nonlin Mech, 39 (2004), 399-420. Google Scholar

[14]

J. D. Humphrey, "Cardiovascular Solid Mechanics," Springer, 2002. Google Scholar

[15]

A. I. Ibragimov and P. Koola, The dynamics of wave carpet, P. 2288, OCEAN 2003 MTS/IEEE, proceedings. Google Scholar

[16]

R. A. Ibrahim, Nonlinear vibrations of suspended cables, Part III: Random excitation and interaction with fluid flow, Applied Mechanics Reviews, 57 (2004), 515-549. doi: 10.1115/1.1804541.  Google Scholar

[17]

J. E. Lagnese, Modelling and stabilization of nonlinear plates, International Series of Numerical Mathematics, 100 (1991), 247-264. Google Scholar

[18]

C. L. Lou and D. L. Sikarskie, Nonlinear Vibration of beams using a form-function approximation, ASME Journal of Applied Mechanics, 42 (1975), 209-214. doi: 10.1115/1.3423520.  Google Scholar

[19]

C. Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates, Computers and Structures, 3 (1973), 163-174. Google Scholar

[20]

J. Padovan, Nonlinear vibrations of general structures, Journal of Sound and Vibration, 72 (1980), 427-441. Google Scholar

[21]

J. Peradze, A numerical alghorithm for Kirchhoff-Type nonlinear static beam, Journal of Applied Mathematics, in Press (2009). doi: 10.1155/2009/818269.  Google Scholar

[22]

J. N. Reddy, Finite element modeling of structural vibrations: A review of recent advances,, The Shock Vibration Digest, 11 (): 25.   Google Scholar

[23]

J. N. Reddy, An introduction to Nonlinear Finite Element Analysis, Oxford University, 2004. doi: 10.1093/acprof:oso/9780198525295.001.0001.  Google Scholar

[24]

D. L. Russel, A comparison of certain dissipation mechanisms via decoupling and projection techniques, Quart. Appl. Math., XLIX (1991), 373-396.  Google Scholar

[25]

P. Seshaiyer and J. D. Humphrey, A sub-domain inverse finite element characterization of hyperelastic membranes, including soft tissues, ASME J Biomech Engr., 125 (2003), 363-371. doi: 10.1115/1.1574333.  Google Scholar

[26]

W. Shyy, Y. Lian, J. Tang, D. Viieru and H. Liu, Aerodynamics of Low Reynolds Number Flyers, Cambridge University Press, 2007. doi: 10.1017/CBO9780511551154.  Google Scholar

[27]

G. Singh, G. V. Rao and N. G. R. Iyengar, Reinvestigation of large amplitude free vibrations of beams using finite elements, Journal of Sound and Vibration, 143 (1990), 351-355. Google Scholar

[28]

H. Wagner and V. Ramamurti, Beam vibrations-A review, The Shock and Vibration Digest, 9 (1977), 17-24. Google Scholar

[29]

O. C. Zienkiewicz and R. L. Taylor, "The Finite Element Method," McGraw-Hill, 1993,  Google Scholar

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