Stability Diagrams

Contents

Consider a mechanical system of the type

$\mathbf{M} \mathbf{\ddot{y}} + \mathbf{C} \mathbf{\dot{y}} + \bigg( \mathbf{K} + \epsilon \mathbf{Q} \cos(\Omega t) \bigg) \mathbf{y} +\mathbf{f}(\mathbf{y}, \dot{\mathbf{y}}) =\mathbf{0}$

Here, $\mathbf{f}$ is a nonlinear function and $\mathbf{Q}$ is a linear matrix. This constitutes a generalised $n$ -dimensional Mathieu equation. The system is parametrically excited with amplitude $\epsilon$ . This excitation can destabilise the trivial fixed point with $\mathbf{y} = 0$ , $\mathbf{\dot{y}}=0$ . The change of stability of this behaviour depending on the excitation frequency $\Omega$ and amplitude $\epsilon$ is commonly documented in a stability diagram. There, regions of instability show up as resonance tongues, which occur when the excitation frequency assumes subharmonic resonances with a mode of the system. The resonance of main interest is the principal parametric resonance where

$\Omega \approx 2 \omega_i$

for some eigenfrequency $\omega_i$ of the mechanical system. By changing either the frequency or amplitude, a bifurcation of the stability type is detected, when crossing the boundary of the resonance tounge. Consequently families of this bifurcation can be continued to get the resonance tongue itself. As damping is decreased, it gets more pointed and closes in to the $\epsilon=0$ axis, as indicated by the dotted line.

To efficiently extract such resonance tongues, we compute the SSM tangent to the $i$ -th modal subspace, to obtain an exact ROM for the dynamics of the system. It serves as the ODE which is analysed using continuation with COCO. Detailed treatment of the theory and algorithm can also be accessed in the related publication (Thurnher, Haller & Jain, 2023).

After setting up the dynamical system, and the SSM object S, the boundary region of stable response can be detected by using the following built-in method:

SD = S.extract_Stability_Diagram(resModes, order, OmegaRange,epRange,'amp', p0,'PD',PlotSD);
        

The input arguments are as follows

Saddle-Node and Period-Doubling Bifurcations

We write the mechanical system in the standard form. By extending the resulting dynamical system

$\mathbf{B}\dot{\mathbf{z}}=\mathbf{Az}+\mathbf{F}(\mathbf{z})+\epsilon\mathbf{G}(\mathbf{z},\phi)$

to an autonomous system of variables $(\mathbf{z}, \tau) \in \mathbf{R}^N \times S^1$ the trivial fixed point $\mathbf{z} =\mathbf{0}$ of the paremtrically excited system can be interpreted as the periodic orbit $(\mathbf{z}, \tau ) = (\mathbf{0}, t \ \textrm{mod} \ 2\pi )$ . Any change of the stability behaviour of this periodic orbit is then given by some bifurcation. At the stability boundary of the principal resonance with $\Omega \approx 2 \omega_i$ nontrivial periodic orbits with response period $T = 4 \pi / \Omega$ emerge. If continuation of $2 \pi / \Omega$ periodic orbits is used then these bifurcations show up as period doubling ('PD') bifurcations. Initially continuing $4 \pi / \Omega$ periodic orbits leads to a saddle node ('SN') bifurcation. The function extract_Stability_Diagram allows to chose between these two options for constructing the stability diagram.

Emergent response

The boundary region of unstable response is at the same time a branch point, from which non-trivial periodic orbits emerge. An illustration of this is shown in the following figure, which shows the response for a 2-dimensional, nonlinear Mathieu equation. The response obtained from the ROM provided by the dynamics on the SSM is verified using collocation on the full dynamical system:

Published with MATLAB® R2023a