A NURBS based Galerkin projection method for the numerical computation of nonlinear normal modes using invariant manifolds

A NURBS based Galerkin projection method for the numerical computation of nonlinear normal modes using invariant manifolds

Ponsioen, S.L.

Tiso, P. (mentor)
Rozza, G. (mentor)

Mechanical, Maritime and Materials Engineering

Precision and Microsystems Engineering

Engineering Dynamics

2015-11-02

In this report a NURBS based Galerkin projection method will be synthesised for the numerical computation of nonlinear normal modes using invariant manifolds. The computed nonlinear normal modes will be used to create a reduced order model that describes the single mode dynamical behaviour of a nonlinear vibrational dynamical system. The Galerkin projection method uses non-uniform rational B-splines to build up a finite dimensional solution space in which a solution is sought for the nonlinear partial differential equations describing the geometry of the manifolds. The manifolds pass through a stable equilibrium point of the dynamical system and are tangent to a plane representing the eigenspace of the system linearised around that equilibrium, which graphically emphasises the relation between the well known linear normal modes and nonlinear normal modes. By applying the isoparametric concept, the NURBS functions are used to represent the geometry of the invariant manifolds, which completely parametrises the manifolds by two parameters. The method will be tested on a two degree of freedom mass-spring system with two cubic nonlinear springs. The NURBS based Galerkin projection method accurately predicts the shapes of the invariant manifolds compared to an asymptotic approach, which only gives a local approximate solution for the geometry of the invariant manifolds. The solution of the Galerkin based method is valid over the whole predefined physical domain, and the accuracy of the solution can be increased by increasing the amount of basis functions that are used to describe the two dimensional surfaces.

nonlinear normal modes
invariant manifolds
model order reduction
non-uniform rational B-splines
Galerkin projection method

http://resolver.tudelft.nl/uuid:cb95be38-0fe8-4885-afd1-2c003e9d5874

2017-10-01

52.00100, 4.37188

Student theses

master thesis

(c) 2015 Ponsioen, S.L.