Nondegenerate continuation problems for the excitation response of nonlinear beam structures

Mehdi Saghafi, Harry Dankowicz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper investigates the dynamics of a slender beam subjected to transverse periodic excitation. Of particular interest is the formulation of nondegenerate continuation problems that may be analyzed numerically, in order to explore the parameterdependence of the steady-state excitation response, while accounting for geometric nonlinearities. Several candidate formulations are presented, including finite-difference (FD) and finiteelement (FE) discretizations of the governing scalar, integropartial differential boundary-value problem (BVP), as well as of a corresponding first-order-in- space, mixed formulation. As an example, a periodic BVP-obtained from a Galerkin-type, FE discretization with continuously differentiable, piecewisepolynomial trial and test functions, and an elimination of Lagrange multipliers associated with spatial boundary conditions- is analyzed to determine the beam response via numerical continuation using a MATLAB-based software suite. In the case of an FE discretization of the mixed formulation with continuous, piecewise-polynomial trial and test functions, it is shown that the choice of spatial boundary conditions may render the resultant index-1, differential-algebraic BVP equivariant under a symmetry group of state-space translations. The paper demonstrates several methods for breaking the equivariance in order to obtain a nondegenerate continuation problem, including a projection onto a symmetry-reduced state space or the introduction of an artificial continuation parameter. As is further demonstrated, an orthogonal collocation discretization in time of the BVP gives rise to ghost solutions, corresponding to arbitrary drift in the algebraic variables. This singularity is resolved by using an asymmetric discretization in time.

Original languageEnglish (US)
Title of host publication9th International Conference on Multibody Systems, Nonlinear Dynamics, and Control
PublisherAmerican Society of Mechanical Engineers
ISBN (Print)9780791855966
DOIs
StatePublished - Jan 1 2013
EventASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2013 - Portland, OR, United States
Duration: Aug 4 2013Aug 7 2013

Publication series

NameProceedings of the ASME Design Engineering Technical Conference
Volume7 A

Other

OtherASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2013
CountryUnited States
CityPortland, OR
Period8/4/138/7/13

ASJC Scopus subject areas

  • Modeling and Simulation
  • Mechanical Engineering
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design

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