Similarity reductions of a nonlinear model for vibrations of beams

Jose Carlos  Camacho; Maria de los Santos  Bruzon

doi:10.3934/proc.2015.0176

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2015, 2015(special): 176-184. doi: 10.3934/proc.2015.0176

Similarity reductions of a nonlinear model for vibrations of beams

Jose Carlos Camacho ^1, and Maria de los Santos Bruzon ^1,

1.	Departament of Mathematics, University of Cdiz, Cadiz, Spain, Spain

Received September 2014 Revised October 2014 Published November 2015

Abstract
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Under a Creative Commons license

In this paper we make a full analysis of the symmetry reductions of this equation by using the classical Lie method of infinitesimals. We consider travelling wave reductions depending on the constants. We present some reductions and explicit solutions.

Keywords: beams, Kirchoff., Similarity reductions, Lie Algebra.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Jose Carlos Camacho, Maria de los Santos Bruzon. Similarity reductions of a nonlinear model for vibrations of beams. Conference Publications, 2015, 2015 (special) : 176-184. doi: 10.3934/proc.2015.0176

References:

[1]	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, New York: Dover, 1972.
[2]	G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation, J. Math. Mech., 18 (1969), 1025-1042.
[3]	J.M. Ball, Initial boundary value problems for an extensible beam, J. Math. Analysis Appl., 42 (1973), 61-90.
[4]	G. W. Bluman and S. Kumei, Symmetries and differential equations, Springer-Verlag, 1989.
[5]	E. Burgreen, Free vibrations of a pinended column with constant distance between pinendes, J. Appl. Mech, 18 (1951), 135-139.
[6]	B. Champagne, W. Hereman, and P. Winternitz, The computer calculation of Lie point symmetries of large systems of differential equations, Com. Phys. Comm., 66 (1991), 319-340.
[7]	R. W. Dickey, Free vibrations and dynamics buckling of extensible beam, J. Math. Analysis Appl. 29 (1970).
[8]	P. A. Djondjorov, Invariant properties of timoshenko beam equations, International Journal of Engineering Science, 33 (1995), 2103-2114.
[9]	J. G. Eisley, Nonlinear vibrations of beams and rectangular plates, Z. angew. Math. Phys., 15 (1964).
[10]	J. Ferreira, R. Benabidallah, and J. E. Mu{\ n}oz Rivera, Asymptotic behaviour for the nonlinear beam equation in a time-dependent domain, Rendiconti di Matematica, Serie VII, 19 (1999), 177-193.
[11]	L. A. Medeiros, On a new class of nonlinear wave equations, J. Math. Appl., 69 (1979), 252-262.
[12]	G. P. Menzala, On classical solutions of a quasilinear hyperbolic equation, Nonlinear Analysis, 3 (1978), 613-627.
[13]	D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation, Nonlinear Analysis, 14(8) (1990), 613-623.
[14]	O. C. Ramos, Regularity property for the nonlinear beam operator, An. Acad. Bras. Ci., 61(1) (1989), 15-24.
[15]	J. E. M. Rivera, Smoothness effect and decay on a class of non linear evolution equation, Ann. Fac. Sc. Toulouse, I(20) (1992), 237-260.
[16]	P. J. Olver, Applications of Lie groups to differential equations, Springer-Verlag, 1986.
[17]	S. K. Woinowsky, The effect of axial force on the vibration of hinged bars, Appl. Mech., 17 (1950), 35-36.

show all references

References:

[1]	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, New York: Dover, 1972.
[2]	G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation, J. Math. Mech., 18 (1969), 1025-1042.
[3]	J.M. Ball, Initial boundary value problems for an extensible beam, J. Math. Analysis Appl., 42 (1973), 61-90.
[4]	G. W. Bluman and S. Kumei, Symmetries and differential equations, Springer-Verlag, 1989.
[5]	E. Burgreen, Free vibrations of a pinended column with constant distance between pinendes, J. Appl. Mech, 18 (1951), 135-139.
[6]	B. Champagne, W. Hereman, and P. Winternitz, The computer calculation of Lie point symmetries of large systems of differential equations, Com. Phys. Comm., 66 (1991), 319-340.
[7]	R. W. Dickey, Free vibrations and dynamics buckling of extensible beam, J. Math. Analysis Appl. 29 (1970).
[8]	P. A. Djondjorov, Invariant properties of timoshenko beam equations, International Journal of Engineering Science, 33 (1995), 2103-2114.
[9]	J. G. Eisley, Nonlinear vibrations of beams and rectangular plates, Z. angew. Math. Phys., 15 (1964).
[10]	J. Ferreira, R. Benabidallah, and J. E. Mu{\ n}oz Rivera, Asymptotic behaviour for the nonlinear beam equation in a time-dependent domain, Rendiconti di Matematica, Serie VII, 19 (1999), 177-193.
[11]	L. A. Medeiros, On a new class of nonlinear wave equations, J. Math. Appl., 69 (1979), 252-262.
[12]	G. P. Menzala, On classical solutions of a quasilinear hyperbolic equation, Nonlinear Analysis, 3 (1978), 613-627.
[13]	D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation, Nonlinear Analysis, 14(8) (1990), 613-623.
[14]	O. C. Ramos, Regularity property for the nonlinear beam operator, An. Acad. Bras. Ci., 61(1) (1989), 15-24.
[15]	J. E. M. Rivera, Smoothness effect and decay on a class of non linear evolution equation, Ann. Fac. Sc. Toulouse, I(20) (1992), 237-260.
[16]	P. J. Olver, Applications of Lie groups to differential equations, Springer-Verlag, 1986.
[17]	S. K. Woinowsky, The effect of axial force on the vibration of hinged bars, Appl. Mech., 17 (1950), 35-36.

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