American Institute of Mathematical Sciences

Advanced
July  2009, 12(1): 23-38. doi: 10.3934/dcdsb.2009.12.23

Vibrations of a nonlinear dynamic beam between two stops

K. T. Andrews 1, , M. F. M'Bengue 1, and  Meir Shillor 1,

1. 

Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, United States, United States, United States

Received  September 2008 Revised  February 2009 Published  May 2009

This work extends the model developed by Gao (1996) for the vibrations of a nonlinear beam to the case when one of its ends is constrained to move between two reactive or rigid stops. Contact is modeled with the normal compliance condition for the deformable stops, and with the Signorini condition for the rigid stops. The existence of weak solutions to the problem with reactive stops is shown by using truncation and an abstract existence theorem involving pseudomonotone operators. The solution of the Signorini-type problem with rigid stops is obtained by passing to the limit when the normal compliance coefficient approaches infinity. This requires a continuity property for the beam operator similar to a continuity property for the wave operator that is a consequence of the so-called div-curl lemma of compensated compactness.
Keywords: Gao beam, dynamic nonlinear beam, Signorini condition., existence and uniqueness, contact.
Mathematics Subject Classification: Primary: 74M15; Secondary: 74H20, 74H25, 74K10, 35L7.
Citation: K. T. Andrews, M. F. M'Bengue, Meir Shillor. Vibrations of a nonlinear dynamic beam between two stops. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 23-38. doi: 10.3934/dcdsb.2009.12.23
[1]

Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783
[2]

Jeongho Ahn, David E. Stewart. A viscoelastic Timoshenko beam with dynamic frictionless impact. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 1-22. doi: 10.3934/dcdsb.2009.12.1
[3]

Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735
[4]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029
[5]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447
[6]

Arnaud Münch, Ademir Fernando Pazoto. Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 197-219. doi: 10.3934/dcdsb.2008.10.197
[7]

Yanfang Li, Zhuangyi Liu, Yang Wang. Weak stability of a laminated beam. Mathematical Control and Related Fields, 2018, 8 (3&4) : 789-808. doi: 10.3934/mcrf.2018035
[8]

María Teresa Cao-Rial, Peregrina Quintela, Carlos Moreno. Numerical solution of a time-dependent Signorini contact problem. Conference Publications, 2007, 2007 (Special) : 201-211. doi: 10.3934/proc.2007.2007.201
[9]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253
[10]

Aslihan Demirkaya, Milena Stanislavova. Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 197-209. doi: 10.3934/dcdsb.2018097
[11]

M. Grasselli, Vittorino Pata, Giovanni Prouse. Longtime behavior of a viscoelastic Timoshenko beam. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 337-348. doi: 10.3934/dcds.2004.10.337
[12]

Quanyi Liang, Kairong Liu, Gang Meng, Zhikun She. Minimization of the lowest eigenvalue for a vibrating beam. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2079-2092. doi: 10.3934/dcds.2018085
[13]

Yanan Li, Zhijian Yang, Fang Da. Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5975-6000. doi: 10.3934/dcds.2019261
[14]

Yi Cheng, Zhihui Dong, Donal O' Regan. Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021230
[15]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213
[16]

Scott W. Hansen, Andrei A. Lyashenko. Exact controllability of a beam in an incompressible inviscid fluid. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 59-78. doi: 10.3934/dcds.1997.3.59
[17]

Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks and Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19
[18]

Maurizio Garrione, Manuel Zamora. Periodic solutions of the Brillouin electron beam focusing equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 961-975. doi: 10.3934/cpaa.2014.13.961
[19]

Roberto Triggiani. The coupled PDE system of a composite (sandwich) beam revisited. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 285-298. doi: 10.3934/dcdsb.2003.3.285
[20]

T. F. Ma, M. L. Pelicer. Attractors for weakly damped beam equations with $p$-Laplacian. Conference Publications, 2013, 2013 (special) : 525-534. doi: 10.3934/proc.2013.2013.525

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy