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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

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).   Google Scholar

[2]

G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation,, {\em J. Math. Mech.}, 18 (1969), 1025.   Google Scholar

[3]

J.M. Ball, Initial boundary value problems for an extensible beam,, {\em J. Math. Analysis Appl.}, 42 (1973), 61.   Google Scholar

[4]

G. W. Bluman and S. Kumei, Symmetries and differential equations,, Springer-Verlag, (1989).   Google Scholar

[5]

E. Burgreen, Free vibrations of a pinended column with constant distance between pinendes,, {\em J. Appl. Mech}, 18 (1951), 135.   Google Scholar

[6]

B. Champagne, W. Hereman, and P. Winternitz, The computer calculation of Lie point symmetries of large systems of differential equations,, {\it Com. Phys. Comm.}, 66 (1991), 319.   Google Scholar

[7]

R. W. Dickey, Free vibrations and dynamics buckling of extensible beam,, {\em J. Math. Analysis Appl.} {\bf 29} (1970)., 29 (1970).   Google Scholar

[8]

P. A. Djondjorov, Invariant properties of timoshenko beam equations,, {\em International Journal of Engineering Science}, 33 (1995), 2103.   Google Scholar

[9]

J. G. Eisley, Nonlinear vibrations of beams and rectangular plates,, {\em Z. angew. Math. Phys.}, 15 (1964).   Google Scholar

[10]

J. Ferreira, R. Benabidallah, and J. E. Mu{\ n}oz Rivera, Asymptotic behaviour for the nonlinear beam equation in a time-dependent domain,, {\em Rendiconti di Matematica, 19 (1999), 177.   Google Scholar

[11]

L. A. Medeiros, On a new class of nonlinear wave equations,, {\em J. Math. Appl.}, 69 (1979), 252.   Google Scholar

[12]

G. P. Menzala, On classical solutions of a quasilinear hyperbolic equation,, {\em Nonlinear Analysis}, 3 (1978), 613.   Google Scholar

[13]

D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation,, {\em Nonlinear Analysis}, 14 (1990), 613.   Google Scholar

[14]

O. C. Ramos, Regularity property for the nonlinear beam operator,, {\em An. Acad. Bras. Ci.}, 61 (1989), 15.   Google Scholar

[15]

J. E. M. Rivera, Smoothness effect and decay on a class of non linear evolution equation,, {\em Ann. Fac. Sc. Toulouse}, (1992), 237.   Google Scholar

[16]

P. J. Olver, Applications of Lie groups to differential equations,, Springer-Verlag, (1986).   Google Scholar

[17]

S. K. Woinowsky, The effect of axial force on the vibration of hinged bars,, {\em Appl. Mech.}, 17 (1950), 35.   Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,, New York: Dover, (1972).   Google Scholar

[2]

G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation,, {\em J. Math. Mech.}, 18 (1969), 1025.   Google Scholar

[3]

J.M. Ball, Initial boundary value problems for an extensible beam,, {\em J. Math. Analysis Appl.}, 42 (1973), 61.   Google Scholar

[4]

G. W. Bluman and S. Kumei, Symmetries and differential equations,, Springer-Verlag, (1989).   Google Scholar

[5]

E. Burgreen, Free vibrations of a pinended column with constant distance between pinendes,, {\em J. Appl. Mech}, 18 (1951), 135.   Google Scholar

[6]

B. Champagne, W. Hereman, and P. Winternitz, The computer calculation of Lie point symmetries of large systems of differential equations,, {\it Com. Phys. Comm.}, 66 (1991), 319.   Google Scholar

[7]

R. W. Dickey, Free vibrations and dynamics buckling of extensible beam,, {\em J. Math. Analysis Appl.} {\bf 29} (1970)., 29 (1970).   Google Scholar

[8]

P. A. Djondjorov, Invariant properties of timoshenko beam equations,, {\em International Journal of Engineering Science}, 33 (1995), 2103.   Google Scholar

[9]

J. G. Eisley, Nonlinear vibrations of beams and rectangular plates,, {\em Z. angew. Math. Phys.}, 15 (1964).   Google Scholar

[10]

J. Ferreira, R. Benabidallah, and J. E. Mu{\ n}oz Rivera, Asymptotic behaviour for the nonlinear beam equation in a time-dependent domain,, {\em Rendiconti di Matematica, 19 (1999), 177.   Google Scholar

[11]

L. A. Medeiros, On a new class of nonlinear wave equations,, {\em J. Math. Appl.}, 69 (1979), 252.   Google Scholar

[12]

G. P. Menzala, On classical solutions of a quasilinear hyperbolic equation,, {\em Nonlinear Analysis}, 3 (1978), 613.   Google Scholar

[13]

D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation,, {\em Nonlinear Analysis}, 14 (1990), 613.   Google Scholar

[14]

O. C. Ramos, Regularity property for the nonlinear beam operator,, {\em An. Acad. Bras. Ci.}, 61 (1989), 15.   Google Scholar

[15]

J. E. M. Rivera, Smoothness effect and decay on a class of non linear evolution equation,, {\em Ann. Fac. Sc. Toulouse}, (1992), 237.   Google Scholar

[16]

P. J. Olver, Applications of Lie groups to differential equations,, Springer-Verlag, (1986).   Google Scholar

[17]

S. K. Woinowsky, The effect of axial force on the vibration of hinged bars,, {\em Appl. Mech.}, 17 (1950), 35.   Google Scholar

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