Equivalent Linearization Analysis Of Geometrically Nonlinear Random Vibrations Using Commercial Finite Element Codes

Two new equivalent linearization implementations for geometrically nonlinear random vibrations are presented. Both implementations are based upon a novel approach for evaluating the nonlinear stiffness within commercial finite element codes and are suitable for use with any finite element code having geometrically nonlinear static analysis capabilities. The formulation includes a traditional force-error minimization approach and a relatively new version of a potential energy-error minimization approach, which has been generalized for multiple degree-of-freedom systems. Results for a simply supported plate under random acoustic excitation are presented and comparisons of the displacement root-mean-square values and power spectral densities are made with results from a nonlinear time domain numerical simulation.Rizzi, Stephen A. and Muravyov, Alexander A.Langley Research CenterLINEARIZATION; ACOUSTIC EXCITATION; FINITE ELEMENT METHOD; RANDOM VIBRATION; OPTIMIZATION; MATHEMATICAL MODELS; POTENTIAL ENERGY; COMPUTERIZED SIMULATION; LOAD DISTRIBUTION (FORCES); SPECTRAL BANDS; ROOT-MEAN-SQUARE ERRORS; ACOUSTIC FATIGUE; DEGREES OF FREEDOM

This report documents two new implementations of equivalent linearization for solving geometrically nonlinear random vibration problems of complicated structures. The implementations are given the acronym ELSTEP, for "Equivalent Linearization using a Stiffness Evaluation Procedure." Both implementations of ELSTEP are fundamentally the same in that they use a novel nonlinear stiffness evaluation procedure to numerically compute otherwise inaccessible nonlinear stiffness terms from commercial finite element programs. The commercial finite element program MSC/NASTRAN (NASTRAN) was chosen as the core of ELSTEP. The FORTRAN implementation calculates the nonlinear stiffness terms and performs the equivalent linearization analysis outside of NASTRAN.

When vibrating with large amplitudes, thin structures experience geometric nonlinearity due to the nonlinear relationship between strains and displacements. Because full-order nonlinear analysis on geometrically nonlinear models are computationally very expensive, the derivation of efficient reduced-order models (ROMs) has always been a topic of interest.In this thesis, nonlinear reduction methods for building ROMs with geometric nonlinearity in the framework of the Finite Element (FE) procedure, are investigated. Three non-intrusive nonlinear reduction methods are specifically investigated and systematically compared. They are: implicit condensation and expansion (ICE), modal derivatives (MD), and the reduction to invariant manifold. Theoretical analysis shows that the first two methods can give reliable results only if a slow/fast assumption between slave and master coordinates holds. On the other hand, reduction to invariant manifolds allows proposing a simulation-free reduction method that can be applied without restricting assumptions on the frequencies of the slave modes.Numerical comparisons and numerous applications to continuous structures discretized with the FE procedure, are given subsequently. For application of the invariant manifold-based method, the computation is based on a direct application of the normal form to the physical space and hence to the nodes of the FE mesh, a method recently developed. The examples show the advantages and drawbacks of each reduction method when deriving ROM, and the results of the theoretical comparison are validated.Finally, the analysis of the dynamics of a system with 1:2 internal resonance and cubic nonlinearity is given in the last part of the thesis. The real normal form of the problem is first derived. Then the solution branches of the problem are investigated and compared to simpler solutions with the dynamics truncated at order two. The divergent behaviour of the hardening/softening characteristics for single-mode reduction is investigated with this more complete model.

Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist René de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfield's first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors' integrated and consistent style and unrivalled engineering approach assures this book's unique position within the computational mechanics literature. Key features: Combines the two previous volumes into one heavily revised text with obsolete material removed, an improved layout and updated references and notations Extensive new material on more recent developments in computational mechanics Easily readable, engineering oriented, with no more details in the main text than necessary to understand the concepts. Pseudo-code throughout makes the link between theory and algorithms, and the actual implementation. Accompanied by a website (www.wiley.com/go/deborst) with a Python code, based on the pseudo-code within the book and suitable for solving small-size problems. Non-linear Finite Element Analysis of Solids and Structures, 2nd Edition is an essential reference for practising engineers and researchers that can also be used as a text for undergraduate and graduate students within computational mechanics.