Metafor

ULiege - Aerospace & Mechanical Engineering

User Tools

Site Tools


team:gdeliege:nnm

This is an old revision of the document!


Nonlinear normal modes

Backgournd

Nonlinear normal modes (NNMs) are an extension of linear normal modes to nonlinear systems. This is one of the research topics of the Space Structures & Systems Lab. (Prof. Kerschen). Actually, I knew nothing on the subject until Ludovic Renson told me about it and about the nonlinear equations he wanted to solve with finite elements. The idea to be faced with an unusual set of PDEs was too much of a temptation and I decided to implement a simple case in my old code as a test.

Problem description

Rigorous definitions of NNMs can be found in references [1-2]. Here, let us consider a discrete mechanical systems with $N$ degrees of freedom. A pair of state variables $u=x_k$ and $v=y_k$ is arbitrarily chosen among the displacements $x_i(t)$ and velocities $y_i(t)$, $i\in\{1,...,N\}$. The time variable can be eliminated and the state equations are rewritten as a set of 2N-2 partial differential equations whose unknowns are now space variables $X_i(u,v)$ and velocities $Y_i(u,v)$, $i\neq k$. These equations are solved in a two-dimensional invariant manifold in phase space.

Find $X_i\in V$ et $Y_i\in V$, $i=\{1, \ldots, 5\}$, such that $$ \begin{eqnarray*} \int_\Omega (Y'_i+\tau_{\rm e}\, \vec{v}\cdot\nabla Y'_i)\, (\vec{v}\cdot\nabla X_i-Y_i) \;{\rm d}\Omega = 0\;, && \forall Y'_i \in V \;, \\ \int_\Omega (X'_i+\tau_{\rm e}\, \vec{v}\cdot\nabla X'_i)\, (\vec{v}\cdot\nabla Y_i-f_i) \;{\rm d}\Omega = 0\;, && \forall X'_i \in V \;, \end{eqnarray*} $$ with $$ \begin{equation*} V = \{ f(x,y)\in H^1(\Omega):\, f(0,0)=0\} \;. \end{equation*} $$

team/gdeliege/nnm.1439453930.txt.gz · Last modified: 2016/03/30 15:22 (external edit)

Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki