Capturing & Bounding Nonlinear Modal Energy Transfer for Geometrically Exact Beams using Semidefinite Programming

A recursive way to choose which vibration modes a reduced-order model should keep - judged by the nonlinear energy transfer they capture - together with rigorous bounds on the energy a chosen set of modes can hold.

1. 01

   A systematic, recursive algorithm for selecting the vibration modes used to build reduced-order models of geometrically nonlinear flexible structures.

2. 02

   Modes are chosen by their ability to capture the nonlinear energy transfer between modes, rather than by frequency alone.

3. 03

   An optimization problem that gives rigorous bounds on the time-averaged energy contained in a predefined set of modes.

4. 04

   A precise quantification of the difference in behavior between reduced-order models derived from linear beam theory and those derived using Geometrically Exact Beam Theory.

Abstract

We present a systematic method of selecting vibration modes with which to build reduced-order models of geometrically nonlinear flexible structures. Our approach is a recursive algorithm which selects modes based on their ability to capture the nonlinear energy transfer between vibration modes. Furthermore, we formulate an optimization problem which can give rigorous bounds on the time-averaged energy contained in a predefined set of modes. This enables a precise and rigorous quantification of the difference in behavior between reduced-order models derived from linear beam theory, and those derived using Geometrically Exact Beam Theory.