Nonlinear Analysis of Rigid Rotor System Supported by 2 Lobes Journal Bearing in Comparison With Circular Journal Bearing

The effect of 2 lobes journal bearing parameters such as L/D ratio and pad preload on the bifurcation of the rigid rotor is investigated in comparison with the circular bearing. Nonlinear bearing force in the equation of motion is obtained by solving Reynolds equation using the finite difference method. Shooting method and Floquet multiplier analysis are employed to obtain limit cycles and their stability. The results show that, for some bearing parameters, multiple limit cycles coexist at a specific shaft rotational speed range. Comparing with the circular bearing of same L/D ratio, the 2 lobes bearing without pad preload decreases the onset speed of instability and also decreases speed range from the onset speed of instability (Hopf) point to the limit point of the bifurcation (saddle-node) in the subcritical bifurcation case. Increasing the pad preload only increases the onset speed of instability significantly in the small L/D ratio case. For both circular and 2 lobes bearing, increasing the L/D ratio decreases the onset speed of instability and tends to change the type of the bifurcation from supercritical to subcritical.

An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices

In this work, we revisit a criterion, originally proposed in Friesecke & Pego (Friesecke & Pego 2004 Nonlinearity 17 , 207–227. ( doi:10.1088/0951715/17/1/013 )), for the stability of solitary travelling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-travelling frame, as well as at that of the Floquet problem arising when considering the travelling wave as a periodic orbit modulo shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the travelling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

Intermediate-Frequency Oscillation Behavior of One-Cycle Controlled SEPIC Power Factor Correction Converter via Floquet Multiplier Sensitivity Analysis

In this paper, we investigate the intermediate-frequency oscillation in a SEPIC power-factor-correction (PFC) converter under one-cycle control. The converter operates in continuous conduction mode (CCM). A systematic method is proposed to analyze the bifurcation behavior and explain the inherent physical mechanism of the intermediate-frequency oscillation. Based on the nonlinear averaged model, the approximate analytical expressions of the nominal periodic equilibrium state are calculated with the help of Galerkin approach. Then, the stability of the system is judged by the Floquet theory and the Floquet multiplier movement of the monodromy matrix is analyzed to reveal the underlying mechanism of the intermediate-frequency oscillation behavior. In addition, Floquet multiplier sensitivity is proposed to facilitate the selection of key parameters with respect to system stability so as to guide the optimal design of the system. Finally, PSpice circuit experiments are performed to verify the above theoretical and numerical ones.

Dynamics on tumor immunotherapy model with periodic impulsive infusion

A periodic pulse differential equation model of tumor immunotherapy is established by considering the periodic and transient behavior of infusing immune cells. Using comparison theorem and Floquet multiplier theory of the impulsive differential equation, the boundedness of the model solution, the existence and stability of the free-tumor periodic solution are given. Furthermore, the persistence of the system is analyzed. Numerical simulations are carried to confirm the main theorems.

Assessing the stability of human locomotion: a review of current measures

Falling poses a major threat to the steadily growing population of the elderly in modern-day society. A major challenge in the prevention of falls is the identification of individuals who are at risk of falling owing to an unstable gait. At present, several methods are available for estimating gait stability, each with its own advantages and disadvantages. In this paper, we review the currently available measures: the maximum Lyapunov exponent ( λ S and λ L ), the maximum Floquet multiplier, variability measures, long-range correlations, extrapolated centre of mass, stabilizing and destabilizing forces, foot placement estimator, gait sensitivity norm and maximum allowable perturbation. We explain what these measures represent and how they are calculated, and we assess their validity, divided up into construct validity, predictive validity in simple models, convergent validity in experimental studies, and predictive validity in observational studies. We conclude that (i) the validity of variability measures and λ S is best supported across all levels, (ii) the maximum Floquet multiplier and λ L have good construct validity, but negative predictive validity in models, negative convergent validity and (for λ L ) negative predictive validity in observational studies, (iii) long-range correlations lack construct validity and predictive validity in models and have negative convergent validity, and (iv) measures derived from perturbation experiments have good construct validity, but data are lacking on convergent validity in experimental studies and predictive validity in observational studies. In closing, directions for future research on dynamic gait stability are discussed.

A Method to Determine the Periodic Solution Based on Observed State Information and Nonlinear Analysis of Hydrodynamic Bearing-Rotor System

In this paper, a numerical method is presented to determine the periodic response of hydrodynamic bearing-rotor system. The observed state information of the system is used to solve inversely the Jacobian matrix, and to trace the periodic response with the change of the control parameter. Jacobian matrix obtained is used to calculate the Floquet multiplier, so the stability of the periodic response can be determined by Floquet theory. The proposed method is applied to a rotor system with the elliptical bearing supports to solve the periodic response and determine its nonlinear stability. Validity of this method is illustrated by comparing numerical results with the traditional method.