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.
- NACA Wing with over 260,000 DOFs.
- Chain of coupled nonlinear oscillators.
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.
- Cantilever pipe conveying a stationary fluid.
- Cantilever pipe conveying a non-stationary fluid.
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
- complex Shaw-Pierre type example is studied.
Furthermore SSM-Theory is employed to analyse the dynamics of the Charney-DeVore model in
- Charney DeVore model.
A benchmark example that compares the results of an analytical computation of a 1D SSM to the resulting coefficients from SSMTool is presented in
Internally resonant systems
The following set of examples introduces and documents the use of SSM-theory for systems with internal resonances
- An axially moving beam with a 1:3 internal resonance.
- Three oscillators with a 1:1:1 internal resonance
- Forced response of two oscillators with a 2:1 internal resonance.
- Bifurcations and invariant tori of two oscillators with a 2:1 internal resonance.
- Von Karman beam with a 1:3 internal resonance.
- Von Karman shell with 1:2 internal resonance.
- Timoshenko beam with internal resonance.
- Stretching of a prismatic beam with 1:3 internal resonance
Systems with algebraic constraints
The following examples of dynamical systems with algebraic differential equations are presented here in the context of SSM analysis
- A frequency-divider Beam
- The equation for large deflections of a pendulum.
- A mass with three DOFs constrained in a cube.
- A mass with three DOFs constrained in a sphere.
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
- Example of a prismatic beam discretized via a Galerkin Projection.
- The forced response for a Bernoulli Beam under parametric resonance.
- Stability of a Bernoulli Beam under parametric resonance.
Systems under parametric and external excitation
The following systems are subject to direct excitation in combination with resonant parametric driving.
- A Bernoulli Beam which exhibits an isolated response due to the presence of multiple resonances.
- Coupled Duffing Oscillators which exhibit an isola with a loop.
- A set of parametric amplifiers where the forced response bifurcates into a loop.
- Coupled self excited oscillators with cubic nonlinear damping and stiffness characteristics.
Computation with semi-intrusive routine
- Semi-Intrusive computation of the van Karman Shell example.
- Semi-Intrusive computation of the NACA Wing example.
Computation with non-intrusive routine
- Non-Intrusive computation of the van Karman Shell example.
- Non-Intrusive computation of the NACA Wing example.
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:
- Non-Intrusive computation of the COMSOL model of a beam.
- Non-Intrusive computation of the COMSOL model of a MEMS device.
- Non-Intrusive computation of the COMSOL model of a plate.
Forced Response Surfaces, Ridges and Trenches
- FRS of a van Karman Plate.
- Ridges and Trenches of a van Karman Plate.
- FRS of a cantilever beam.
- FRS of a van Karman Shell.
- Ridges and Trenches of a van Karman Shell using the L2 norm.
- Ridges and Trenches of a van Karman Shell using the infinity norm.