# American Institute of Mathematical Sciences

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

## Similarity reductions of a nonlinear model for vibrations of beams

 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.
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
 [1] Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-15. doi: 10.3934/dcdss.2020066 [2] Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10. [3] Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239 [4] Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453 [5] María Rosa, María S. Bruzón, M. L. Gandarias. A model of malignant gliomas throug symmetry reductions. Conference Publications, 2015, 2015 (special) : 974-980. doi: 10.3934/proc.2015.0974 [6] Vladimir S. Gerdjikov, Rossen I. Ivanov, Aleksander A. Stefanov. Riemann-Hilbert problem, integrability and reductions. Journal of Geometric Mechanics, 2019, 11 (2) : 167-185. doi: 10.3934/jgm.2019009 [7] Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523 [8] Philippe Jaming, Vilmos Komornik. Moving and oblique observations of beams and plates. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020013 [9] Colin Rogers, Tommaso Ruggeri. q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2297-2312. doi: 10.3934/dcdsb.2014.19.2297 [10] Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420 [11] Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023 [12] Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399 [13] Vassilis G. Papanicolaou, Kyriaki Vasilakopoulou. Similarity solutions of a multidimensional replicator dynamics integrodifferential equation. Journal of Dynamics & Games, 2016, 3 (1) : 51-74. doi: 10.3934/jdg.2016003 [14] A. M. Vershik. Polymorphisms, Markov processes, quasi-similarity. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1305-1324. doi: 10.3934/dcds.2005.13.1305 [15] Boran Hu, Zehui Cheng, Zhangbing Zhou. Web services recommendation leveraging semantic similarity computing. Mathematical Foundations of Computing, 2018, 1 (2) : 101-119. doi: 10.3934/mfc.2018006 [16] Palle Jorgensen, Feng Tian. Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2307-2348. doi: 10.3934/dcdss.2019146 [17] Marcio Antonio Jorge da Silva, Vando Narciso. Attractors and their properties for a class of nonlocal extensible beams. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 985-1008. doi: 10.3934/dcds.2015.35.985 [18] Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107 [19] Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39 [20] André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351

Impact Factor:

## Metrics

• PDF downloads (11)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]