An energy-preserving description of nonlinear beam vibrations in modal coordinates

A. Wynn, Yinan Wang, R. Palacios-Nieto and P. J. Goulart

@article{WWPG:2013,
  author = {A. Wynn and Yinan Wang and R. Palacios-Nieto and P. J. Goulart},
  title = {An energy-preserving description of nonlinear beam vibrations in modal coordinates},
  journal = {Journal of Sound and Vibration},
  year = {2013},
  volume = {332},
  number = {21},
  pages = {5543-5558},
  url = {http://dx.doi.org/10.1016/j.jsv.2013.05.021},
  doi = {10.1016/j.jsv.2013.05.021}
}

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.