Quasi-periodic harmonic balance method for rubbing self-induced vibrations in rotor–stator dynamics

A quasi-periodic harmonic balance method (HBM) coupled with a pseudo arc-length continuation algorithm is developed and used for the prediction of the steady-state dynamic behaviour of rotor–stator contact problems. Quasi-periodic phenomena generally involve two incommensurable fundamental frequencies, and at present, the HBM has been adapted to deal with cases where those frequencies are known. The problem here is to improve the procedure in order to be able to deal with cases where one of the two fundamental frequencies is a priori unknown, in order to be able to reproduce self-excited phenomena such as the so-called quasi-periodic partial rub. Considering the proposed developments, the unknown fundamental frequency is automatically determined during calculation and an automatic harmonic selection procedure gives both accuracy and performance improvements. The application is based on a Jeffcott rotor model, and results obtained are compared with traditional time-marching solutions. The modified quasi-periodic HBM appears one order of magnitude faster than transient simulations while providing very accurate results.     

Stochastic averaging method for estimating first-passage statistics of stochastically excited Duffing–Rayleigh–Mathieu system

The first-passage statistics of Duffing-Rayleigh- Mathieu system under wide-band colored noise excitations is studied by using stochastic averaging method. The motion equation of the original system is transformed into two time homogeneous diffusion Markovian processes of amplitude and phase after stochastic averaging. The diffusion process method for first-passage problem is used and the corresponding backward Kolmogorov equation and Pontryagin equation are constructed and solved to yield the conditional reliability function and mean first-passage time with suitable initial and boundary conditions. The analytical results are confirmed by Monte Carlo simulation.     

A symbolic algorithm for the automatic computation of multitone-input harmonic balance equations for nonlinear systems

A symbolic algorithm is developed for the automatic generation of harmonic balance equations for multitone input for a class of nonlinear differential systems with polynomial nonlinearities. Generalized expressions are derived for the construction of balance equations for a defined multitone signal form. Procedures are described for determining combinations for a given output frequency from the desired set obtained from box truncated spectra and their permutations to automate symbolic algorithm. An application of method is demonstrated using the well-known Duffing–Van der Pol equation. Then the obtained analytical results are compared with numerical simulations to show the accuracy of the approach. The computation times for both the numerical solutions of equations versus the number of frequency components and the symbolic generation of the equations versus the power of nonlinearity are also investigated.     

Analysis and synthesis of oscillator systems described by perturbed single well Duffing equations

The paper presents an analysis as well as a synthesis of oscillator systems described by single well Duffing equations under polynomial perturbations of fourth degree. It is proved that such a system can have a unique hyperbolic limit cycle. An analytical condition has been obtained for the arising of a limit cycle and an equation giving the parameters of this limit cycle. There has been proposed a method for the synthesis of oscillator systems of the considered type, having preliminarily assigned properties. The synthesis consists of an appropriate choice of the perturbation coefficients in such a way that the oscillator equation should have a preliminary assigned limit cycle. Both the analysis and the synthesis are performed with the help of the Melnikov function.     

2:1 and 1:1 frequency-locking in fast excited van der Pol–Mathieu–Duffing oscillator

The frequency-locking area of 2:1 and 1:1 resonances in a fast harmonically excited van der Pol–Mathieu–Duffing oscillator is studied. An averaging technique over the fast excitation is used to derive an equation governing the slow dynamic of the oscillator. A perturbation technique is then performed on the slow dynamic near the 2:1 and 1:1 resonances, respectively, to obtain reduced autonomous slow flow equations governing the modulation of amplitude and phase of the corresponding slow dynamics. These equations are used to determine the steady state responses, bifurcations and frequency-response curves. Analysis of quasi-periodic vibrations is carried out by performing multiple scales expansion for each of the dependent variables of the slow flows. Results show that in the vicinity of both considered resonances, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. It was also shown that entrained vibrations with moderate amplitude can be obtained in a small region near the 1:1 resonance. Numerical simulations are performed to confirm the analytical results.     

Global bifurcation analysis of Rayleigh–Duffing oscillator through the composite cell coordinate system method

In this paper, a new conception of composite cell coordinate system is presented by dividing the continuous state space into the cell state space with different scales. For a dynamical system, attractors, basins of attraction, basin boundaries, saddles, and invariant manifolds can be easily obtained, and any region of the state space can be refined by this method. The global bifurcations, such as crisis and metamorphosis, of the Rayleigh?CDuffing oscillator are studied by the composite cell coordinate system method. According to the sudden changes in shapes of the chaotic attractor and the chaotic saddle, we find that three types of crises can all occur, including boundary crisis, interior crisis, and attractor emerging crisis. In addition, the basin boundary metamorphoses, such as fractal-Wada, Wada-Wada, and Wada-fractal, are analyzed through observing the shapes of basin boundaries. These results demonstrate the efficiency and validity of this method in analyzing dynamical systems.     

An efficient Galerkin averaging-incremental harmonic balance method for nonlinear dynamic analysis of rigid multibody systems governed by differential–algebraic equations

Nonlinear Dynamics - An efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method is developed for steady-state nonlinear dynamic analysis of index-3 differential algebraic...     

Creation–annihilation process of limit cycles in the Rayleigh–Duffing oscillator

The present paper examines the creation?Cannihilation process of limit cycles in the Rayleigh?CDuffing oscillator with negative linear damping and negative linear stiffness. It is obtained by the perturbation method, in which the number of limit cycles in the Rayleigh?CDuffing oscillator varies with the linear damping and stiffness. Numerical simulations are performed in order to confirm the analytically obtained creation?Cannihilation process of limit cycles. Moreover, we compare the process of limit cycles in the Rayleigh?CDuffing oscillator to that of limit cycles in the van der Pol?CDuffing oscillator. The difference in these oscillator is only in nonlinear forces, which causes a qualitative difference in the creation?Cannihilation processes.     

The codimension-two bifurcation for the recent proposed SD oscillator

In this paper, the complicated nonlinear dynamics at the equilibria of SD oscillator, which exhibits both smooth and discontinuous dynamics depending on the value of a parameter α, are investigated. It is found that SD oscillator admits codimension-two bifurcation at the trivial equilibrium when α=1. The universal unfolding for the codimension-two bifurcation is also found to be equivalent to the damped SD oscillator with nonlinear viscous damping. Based on this equivalence between the universal unfolding and the damped system, the bifurcation diagram and the corresponding codimension-two bifurcation structures near the trivial equilibrium are obtained and presented for the damped SD oscillator as the perturbation parameters vary.     

Bifurcation of periodic solutions and invariant tori for a four-dimensional system

In this paper, a four-dimensional system of autonomous ordinary differential equations depending on a small parameter is considered. Suppose that the unperturbed system is composed of two planar systems: one is a Hamiltonian system and another system has a focus. By using the Poincaré map and the integral manifold theory, sufficient conditions for the existence of periodic solutions and invariant tori of the four-dimensional system are obtained. An application of our results to a nonlinearly coupled Van der Pol–Duffing oscillator system is given.     

Bifurcation analysis for non-local design of a hybrid observer for the impulsive Goodwin’s oscillator

Nonlinear Dynamics - The impulsive Goodwin’s oscillator is a mathematical model capturing the dynamics arising in a closed-loop system, where a third-order linear time-invariant plant is...     

Bifurcation of a predator–prey model with disease in the prey

In this paper, a predator–prey model with disease in the prey is considered. Assume that the predator eats only the infected prey, and the incidence rate is nonlinear. We study the dynamics of the model in terms of local analysis of equilibria and bifurcation analysis of a boundary equilibrium and a positive equilibrium. We discuss the Bogdanov–Takens bifurcation near the boundary equilibrium and the Hopf bifurcation near the positive equilibrium; numerical simulation results are given to support the theoretical predictions.     

Solve a contact problem by optimization method—to calculate the stresses for the softwheel in a harmonic gear

It is difficult to solve the contact problem by usual finite element program. In this paper, we express the contact problem as an optimization problem. In this form we do not need to know all boundary condition in advance. We only need to know the constraint conditions. This method is especially good for solving contact problem. Using this method, we calculate the stresses of the softwheel in the harmonic gear given by Shanghai Jiaotong University, and the results are in good agreement with the experimental results.     

Condensation models for the water–steam interface and the volume of fluid method

We assess the quantitative capabilities of three condensation models. These models are: (1) Numerical iteration technique; (2) heat flux balance equation; (3) phase field. The numerical iteration technique introduces a mass and energy transfer at the interface, if the temperature of the corresponding cell differs from the saturation value. The second approach solves the heat flux balance equation at the interface, hence, the resolution of the thermal boundary layer around the liquid-vapor interface is necessary to obtain an accurate value for the condensation rate. The third technique is based on a recently derived phase field model for boiling and condensation phenomena. The models were implemented in FLUENT and the interface was tracked explicitly with the volume of fluid (VOF) method. The models were tested on the LAOKOON facility, which measured direct contact condensation in a horizontal duct. The results showed that the phase field model fit best the experimental results.     

Soliton dynamics and interaction in the Bose–Einstein condensates with harmonic trapping potential and time-varying interatomic interaction

Investigated in this paper is the quasi-one-dimensional Gross–Pitaevskii equation, which describes the dynamics of the Bose–Einstein condensates with the harmonic trapping potential and time-varying interatomic interaction. Via the Horita method and symbolic computation, analytic bright N-soliton solution is obtained. One-, two- and three-soliton solutions are analyzed graphically. Based on the limit analysis on the one- and two-soliton solutions, the modulation on the speed of the matter-wave bright solitons is realized. Via the parameters, the interaction between the matter-wave solitons are adjustable. Furthermore, an approach to construct the interference between the matter-wave solitons has been proposed. Finally, investigation on the three-soliton solution verifies our conclusions drawn from the one and two solitons. Our conclusions might be useful in the fields of the control on the matter-wave solitons, atom lasers, and atomic accelerators.     

A method for Lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh–Nagumo model

This work presents a new method to calculate the Lyapunov spectrum of dynamical systems based on the time evolution of initially small disturbed copies (“clones”) of the motion equations. In this approach, it is not necessary to construct the tangent space associated with the time evolution of linearized versions of motion equations, being the Lyapunov exponents directly estimated in terms of the rate of convergence or divergence of these disturbed clones with respect to the fiducial trajectory, there being periodic correction via the Gram–Schmidt Reorthonormalization procedure. The proposed method offers the possibility of partial estimation of the Lyapunov spectrum and can also be applied to nonsmooth dynamics, since the linearization procedure is no longer required. The idea is tested for representative continuous- and discrete-time dynamical systems and validated by means of comparison with the classical method to perform this calculation. To illustrate its applicability in the nonsmooth context, the largest Lyapunov exponent of the FitzHugh–Nagumo neuronal model under discontinuous periodic excitation is calculated taking the amplitude of stimulation as control parameter. This analysis reveals some complex behaviours for this simple neuronal model, which motivates relevant discussions about the possible role of chaos in the cognitive process.     

Evolution Model for Linearized Micropolar Plates by the Fourier Method

In this paper we justify a two-dimensional evolution and eigenvalue model for micropolar plates starting from three-dimensional linearly micropolar elasticity. A small parameter representing the thickness of the plate-like body is introduced in the problem. The asymptotics of the evolution and eigenvalue problems is then developed as this small parameter tends to zero. First the appropriate convergences of the eigenpairs of the three-dimensional problem to the eigenpairs of the two-dimensional eigenvalue problem for micropolar plates is shown. Then these convergences are used in the Fourier method to obtain the convergences of the solution of the three-dimensional evolution problem to the solution of the two-dimensional evolution plate model.      

Optimal Orthotropy for Minimum Elastic Energy by the Polar Method

The polar method is a minimal invariant representation in plane elasticity. A plane orthotropic elastic behaviour is expressed by five polar invariants related to the elastic symmetries. In this paper, considering the orthotropy orientation and the polar invariants as optimisation parameters, we discuss the problem of minimising the elastic energy for a given state of stress. The minimisation with respect to the orientation is solved in order to find the associated optimal elastic energy for given polar invariants. Then, this quantity is minimised with respect to the polar invariants which characterise the magnitude of the anisotropic components of the elastic stiffness tensor. Optimal uncoupled composite laminates corresponding to this optimum are presented for membrane and bending loadings.