Non-periodic motions of a Jeffcott rotor with non-linear elastic restoring forces

Abstract

The conditions that give rise to non-periodic motions of a Jeffcott rotor in the presence of non-linear elastic restoring forces are examined. It is well known that non-periodic behaviours that characterise the dynamics of a rotor are fundamentally a consequence of two aspects: the non-linearity of the hydrodynamic forces in the lubricated bearings of the supports and the non-linearity that affects the elastic restoring forces in the shaft of the rotor. In the present research the analysis was restricted to the influence of the non-linearity that characterises the elastic restoring forces in the shaft, adopting a system that was selected the simplest as possible. This system was represented by a Jeffcott rotor with a shaft of mass that was negligible respect to the one of the disk, and supported with ball bearings. In order to check in a straightforward manner the non-linearity of the system and to confirm the results obtained through theoretical analysis, an investigation was carried out using an experimental model consisting of a rotating disk fitted in the middle of a piano wire pulled taut at its ends but leaving the tension adjustable. The adopted length/diameter ratio was high enough to assume the wire itself was perfectly flexible while its mass was negligible compared to that of the disk. Under such hypotheses the motion of the disk centre can be expressed by means of two ordinary, non-linear and coupled differential equations. The conditions that make the above motion non-periodic or chaotic were found through numerical integration of the equations of motion. A number of numerical trials were carried out using a 4th order Runge-Kutta routine with adaptive stepsize control. This procedure made it possible to plot the trajectories of the disk centre and the phase diagrams of the component motions, taken along two orthogonal coordinate axes, with their projections of the Poincaré sections. On the basis of the theoretical results obtained, the conditions that give rise to non-periodic motions of the experimental rotor were identified

Rotating shaft's non-linear response statistics under biaxial random excitation, by path integration

“…Numerous studies have been done to study non-linear effects in rotor dynamics [8], [9], [16]- [20]. This paper considers a Jeffcott rotor with non-linear restoring force.…”

“…The non-linear version of the restoring forces studied in this paper, is assumed to be given as in [9]:…”

“…It is worth noting that in this paper the shaft is assumed to be perfectly balanced. As mentioned above, 𝑋 and 𝑌 are being non-dimensionalized displacements, representing actual radial shaft transverse displacements 𝑢 and 𝑣 by scaling, 𝑋 = 𝑢/𝐿, 𝑌 = 𝑣/𝐿 with 𝐿 being the half distance between two shaft supports [9].…”

“…Jeffcott type rotating machinery can be found in a wide range of industrial areas: from automotive to airspace and marine engines, see e.g. [3], [6]- [9]. The primary feature of Jeffcott type rotating machinery design is to deliver stability under random and even extreme type of external excitations.…”

Dynamics of a Rigid Rotor Linear/Nonlinear Bearings System Subject to Rotating Unbalance and Base Excitations

“…Adiletta et al (1996a) presented theoretical and experimental results from an analysis of a rigid rotor supported on short journal bearings and subject to mass imbalance. Later Adiletta et al (1996b) studied a Jeffcott rotor with nonlinear restoring forces due to supporting ball bearings. Bearing restoring forces are represented by the cubic nonlinearity of bearing deflections.…”