Bifurcations, symmetries and the notion of fixed subspace

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension [Formula: see text] in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

Numerical and Experimental Investigations on Flexible Multi-bearing Rotor Dynamics

An experimental test rig is built to verify the dynamics of a multi-bearing rotor. It consists of two flexibly coupled shafts and is connected to a motor at one end via a flexible coupling. Each of the shafts is supported at the ends by two hydrodynamic bearings and is attached with two disks with equal and unequal masses, respectively. The mathematical model of the test rig is developed and is simulated numerically. The non-stationary dynamic responses of the system during speed-up with a constant angular acceleration are shown, respectively, by the non-stationary bifurcation diagrams, the selected time flows, and the spectrum cascades. Experiments are then carried out on the test rig. Generally, the numerical results are verified qualitatively by the experiments. Both results indicate that the non-synchronous whirls of the two shafts influence each other when flexibly coupled together. In particular, a new phenomenon is found for the four-bearing rotor system: the pre-existing non-synchronous whirl/whip resulted from the instability of one shaft can activate the onset of oil instability of another shaft. In the theoretical simulation, this phenomenon represents the rapid increase of the non-synchronous whirl orbit, whereas in the experiment, it represents the simultaneous existence of two whirl/whip frequencies in the spectra.

Non-Dimensional Parameters Governing the Onset of Wake-Induced Marine Riser Collision

Wake-induced riser oscillation can lead to riser clashing. Furthermore, the onset of the wake-induced riser oscillation, which is typified by its large amplitude and low frequency, is attributed to the loss of stability of the downstream riser in the wake once the current exceeds a critical value. The loss of stability is mathematically characterised by a stationary bifurcation. Based upon our previous work, further results are presented in the paper on the non-dimensional parameters which govern the critical current speed. These non-dimensional parameters, once computed, can then be used by riser designers to assess potential riser clashing and/or ascertain the minimum riser spacing and top tension required in order to avoid riser clashing.