An Energy-Preserving Description of Nonlinear Beam Vibrations in Modal Coordinates

Abstract Conserved quantities are identified in the equations describing large-amplitude free vibrations of beams projected onto their linear normal modes. This is achieved by writing the geometrically exact equations of motion in their intrinsic, or Hamiltonian, form before the modal transformation. For nonlinear free vibrations about a zero-force equilibrium, it is shown that the finite-dimensional equations of motion in modal coordinates are energy preserving, even though they only approximate the total energy of the infinite-dimensional system. For beams with constant follower forces, energy-like conserved quantities are also obtained in the finite-dimensional equations of motion via Casimir functions. The duality between space and time variables in the intrinsic description is finally carried over to the definition of a conserved quantity in space, which is identified as the local cross-sectional power. Numerical examples are used to illustrate the main results.

[1]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[2]  Bauchau Olivier,et al.  Flexible Multibody Dynamics , 2010 .

[3]  Alessandro Macchelli,et al.  Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach , 2004, SIAM J. Control. Optim..

[4]  Ali H. Nayfeh,et al.  Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control , 2008 .

[5]  P. Frank Pai,et al.  Highly Flexible Structures : Modeling, Computation, and Experimentation , 2007 .

[6]  Dewey H. Hodges,et al.  Nonlinear Composite Beam Theory , 2006 .

[7]  J. C. Simo,et al.  Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics , 1992 .

[8]  Dewey H. Hodges,et al.  Geometrically Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams , 2004 .

[9]  G. A. Hegemier,et al.  A nonlinear dynamical theory for heterogeneous, anisotropic, elasticrods , 1977 .

[10]  Arjan van der Schaft,et al.  Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[11]  K. Lynch.  Nonholonomic Mechanics and Control , 2004, IEEE Transactions on Automatic Control.

[12]  David J. Wagg,et al.  Nonlinear Vibration with Control , 2010 .

[13]  J. C. Simo,et al.  Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms , 1995 .

[14]  T. Grundy,et al.  Progress in Astronautics and Aeronautics , 2001 .

[15]  Moti Karpel,et al.  Intrinsic models for nonlinear flexible-aircraft dynamics using industrial finite-element and loads packages , 2012 .

[16]  E. Reissner,et al.  On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory , 1973 .

[17]  A. J. Chen,et al.  Finite Element Method in Dynamics of Flexible Multibody Systems: Modeling of Holonomic Constraints and Energy Conserving Integration Schemes , 2000 .

[18]  J. Kazdan.  Perturbation of complete orthonormal sets and eigenfunction expansions. , 1971 .

[19]  Alessandro Astolfi,et al.  On feedback equivalence to port controlled Hamiltonian systems , 2005, Syst. Control. Lett..

[20]  O. Bauchau,et al.  On the design of energy preserving and decaying schemes for flexible, nonlinear multi-body systems , 1999 .

[21]  Dewey H. Hodges,et al.  Modeling Beams With Various Boundary Conditions Using Fully Intrinsic Equations , 2010 .

[22]  Dewey H. Hodges,et al.  Flight Dynamics of Highly Flexible Aircraft , 2008 .

[23]  Andrew Y. T. Leung,et al.  A symplectic Galerkin method for non-linear vibration of beams and plates , 1995 .

[24]  Bogdan I. Epureanu,et al.  An Intrinsic Description of the Nonlinear Aeroelasticity of Very Flexible Wings , 2011 .

[25]  Rafael Palacios,et al.  Nonlinear normal modes in an intrinsic theory of anisotropic beams , 2011 .

[26]  Christophe Pierre,et al.  Normal Modes for Non-Linear Vibratory Systems , 1993 .

[27]  Dewey H. Hodges,et al.  Validation Studies for Aeroelastic Trim and Stability Analysis of Highly Flexible Aircraft , 2010 .

[28]  Joseba Murua,et al.  Structural and Aerodynamic Models in Nonlinear Flight Dynamics of Very Flexible Aircraft , 2010 .

[29]  M. Crisfield,et al.  Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.