Examples

Examples

Contents

Overview

The numerous examples provided in SSMTool serve for both the transparent documentation of peer reviewd research as well as a set of tutorials on how to make use of the features and methods provided in the toolbox. They show how a wide class of dynamical systems such as systems featuring direct or parametric excitation, internal resonances, algebraic constraints, gyroscopic forces and many more can be treated.

They range from small academic examples that aim at uncovering interesting physical phenomena such as bifurcations and invariant tori up to large scale finite element models (FEM), which may feature up to several hundreds of thousand degrees of freedom.

Extracting FRCs on 2D SSMs

In the following examples the standard routines for extracting FRCs of systems without internal resonances for directly excited dynamical systems are used.

Analysis of dynamics from 2D SSMs

In the following examples the behaviour of a pipe that conveys a fluid is analysed using SSM-theory. The fluid-structure interactions lead to gyroscopic and follower forces, the resulting dynamics are analysed via the ROM on the SSM.

The unstable, one-dimensional manifold of the Lorentz system is computed and the reduced dynamics on that manifold are analysed in

As an example of a complex dynamical system, and for comparing the modal and physical computation of SSMs a

Furthermore SSM-Theory is employed to analyse the dynamics of the Charney-DeVore model in

A benchmark example that compares the results of an analytical computation of a 1D SSM to the resulting coefficients from SSMTool is presented in

Benchmark 1D SSM.

Internally resonant systems

The following set of examples introduces and documents the use of SSM-theory for systems with internal resonances

Systems with algebraic constraints

The following examples of dynamical systems with algebraic differential equations are presented here in the context of SSM analysis

Systems featuring parametric excitation

The following examples of dynamical systems which are subject to parametric excitation and parametric resonances are presented here in the context of SSM analysis

Extraction of Stability Diagrams

The following systems feature purely parametric excitation. For them the stability diagram and related forced response is computed

Systems under parametric and external excitation

The following systems are subject to direct excitation in combination with resonant parametric driving.

Computation with semi-intrusive routine

Computation with non-intrusive routine

Using this routine we also may combine SSMTool with FEM software such as COMSOL The following examples use a model in COMSOL, without having explicit access to the model nonlinearities:

Forced Response Surfaces, Ridges and Trenches