Search

This paper takes an asymmetric support ball bearing-rotor system subjected by unbalanced force and parametric excitation (varying compliance) as the research object. Multi-modes of resonance such as natural frequency resonance region of each order, VC (varying compliance) frequency resonance region, the 1/2 sub-harmonic VC frequency resonance region, and quasi-periodic regions, and the nonlinear characteristics in these regions are analyzed. Besides, the effect of the number of balls of the ball bearing is also considered and the result shows that the parity of this parameter matters a lot. By introducing a definition of the absolute quasi-periodic frequency, the law of the occurrence of the quasi-periodic motion is demonstrated and the possible cause is given to some extent. The work provides a theoretical basis for clarifying the nonlinear characteristics of the bearing-rotor system and suppressing the nonlinear behavior of the system.

Duffing equation is a representative nonlinear equation in practical engineering systems, where the most existing research focuses on local solutions of weakly nonlinear systems. In this paper, we study global bifurcations and chaos of the standard Duffing system by employing the Arnold tongue, and use the basin of attraction to investigate the properties of the Arnold tongue overlap. Our results show that a resonance solution and chaos could coexist, when the parameters are on the Arnold tongue overlap. The phenomenon does not exist in a system described by a weak Duffing equation. Numerical solutions for these bifurcations and chaos are also provided to demonstrate the theoretical results.

The complicated dynamical behavior of a flexible rotor-bearing system is studied in this paper. The unsteady oil-film force model described by three functions is considered. The bifurcation and chaos behaviors were revealed by calculating the maximum Lyapunov exponent of the system. Two new phenomena were found in this system: first, the chaos with two attracting areas which cannot be distinguished from the stable period doubling motion on Poincare section; second, for the flexible rotor system with two unbalanced disks, the response varies in a large extent when the phase angle between the eccentricities of disks is different. The experiments were also carried out. Comparison between experimental and calculated results shows that the significant use of the max Lyapunov exponent in revealing the bifurcation and chaos characteristics of the rotor-bearing system.

The lock-in periodic solutions of the Stuart-Landau equation with a periodic excitation are studied. Using singularity theory, the bifurcation behavior of these solutions with respect to the excitation amplitude and frequency are investigated in detail, respectively. The results show that the universal unfolding with respect to the excitation amplitude possesses codimension 3. The transition sets in unfolding parameter plane and the bifurcation diagrams are plotted under some conditions. Additionally, it has also been proved that the bifurcation problem with respect to frequence possesses infinite codimension. Therefore the dynamical bifurcation behavior is very complex in this case. Some new dynamical phenomena are presented, which are the supplement of the results obtained by Sun Liang et al.

The method of multiple scales is used to construct non-linear normal modes (NNMs) of a class of systems with three double of pure imaginary roots and 1:2:5 dual internal resonance. It is found that the three NNMs associated with dual internal resonances include two uncoupled NNMs as well as a coupled NNM. And the bifurcation problem of the coupled NNM is in two variables, which is greatly different from the bifurcation of the NNMs of systems with single internal resonance. Because no results in singularities can be straightly applied, a practical way is proposed to do singularity analysis for bifurcation of two dimensions. It is also noted that with the variation of the bifurcation parameters, the modes may convert to each other or suddenly emerge and disappear, which give rise to the number of the NNMs more or fewer than the number of the degrees of freedom.

The stability degree of periodic solution of nonlinear nonautonomous system was defined by means of the Floquet theory. A method evaluating the stability degree of periodic solution based on transient response was presented by the aid of the concept of dynamic systems or flows. The critical value of a system was determined by the condition, i. e., its stability degree equals zero. Stable regions of rotor systems with balanced and unbalanced disk supported on lubricated bearings were calculated. The study shows that stable region decreases with the increase of speed for a balanced rotor system and decreases with the increase of unbalance for an unbalanced rotor system. Stable regions of periodic solutions are less than that of equilibrium points under the same systematic conditions.

The nonlinear normal modes (NNMs) associated with integrnal resonance can be classified into two kinds: uncoupled and coupled. The bifurcation problem of the coupled NNM of systems with 1∶2∶5 dual internal resonance is in two variables. The singular analysis of it is presented after separating the two variables by taking advantage of Maple algebra, and some new bifurcation paterns are found. Different from the NNMs of systems with single internal resonance, the number of the NNMs of systems with dual internal resonance may be more or less than the number of the degress of freedom. At last, it is pointed out that bifurcation problems in two variables can be conveniently solved by separating variables as well as using coupling equations.

Bifurcation of periodic solutions widely existed in nonlinear dynamical systems is a kind of constrained one in intrinsic quality because its amplitude is always non-negative. Classification of the bifurcations with the type of constraint was discussed. All its six types of transition sets are derived, in which three types are newly found and a method is proposed for analyzing the constrained bifurcation.

The inverse problem for a class of nonlinear evolution equations of dispersive type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution was given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin.

This paper puts forward a complex inner product averaging method for calculating normal form of ODE. Compared with conventional averaging method, the theoretic analytical process has such simple forms as to realize computer program easily. Results can be applied in both autonomous and non-autonomous systems. At last, an example is resolved to verify the method.

Bifurcations with constraints are open problems appeared in research on periodic bifurcations of nonlinear dynamical systems, but the present singularity theory doesn’t contain any analytical methods and results about it. As the complement to singularity theory and the first step to study on constrained bifurcations, here are given the transition sets and persistent perturbed bifurcation diagrams of 10 elementary bifurcation of codimension no more than three.

The1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in1:2 internal resonance were derived by using the direct method of normal form. In the normal forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the4-dimension bifurcation equations were reduced to3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on4-dimension center manifolds.

The existence of solutions for systems of nonlinear impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces is studied. Some existence theorems of extremal solutions are obtained, which extend the related results for this class of equations on a finite interval with a finite number of moments of impulse effect. The results are demonstrated by means of an example of an infinite systems for impulsive integral equations.

Using the monotone iterative method and Monch fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differential equations in Banach spaces are studied. Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained.

The existence of solutions for nonlinear impulsive Hammerstein integral equations with infinite numbers of moments of impulse effect on the infinite intervalR + in Banach spaces is studied. By means of Monch fixed point theorem, an existence theorem of solutions is obtained. The result is demonstrated by means of an infinite system for impulsive integral equations.

Certain deterministic nonlinear systems may show chaotic behavior. We consider the motion of qualitative information and the practicalities of extracting a part from chaotic experimental data. Our approach based on a theorem of Takens draws on the ideas from the generalized theory of information known as singular system analysis. We illustrate this technique by numerical data from the chaotic region of the chaotic experimental data. The method of the singular-value decomposition is used to calculate the eigenvalues of embedding space matrix. The corresponding concrete algorithm to calculate eigenvectors and to obtain the basis of embedding vector space is proposed in this paper. The projection on the orthogonal basis generated by eigenvectors of timeseries data and concrete paradigm are also provided here. Meanwhile the state space reconstruction technology of different kinds of chaotic data obtained from dynamical system has also been discussed in detail.

Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z2-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.

On the basis of the method proposed in [1], the paper gives the method for finding the normal form of nonsemi-simple bifurcation problems. As an example, it analyses the normal form of a general nonlinear dynamical system with the nonsemi-simple double zero eigenvalues, and gives out the expression for the coefficients in the normal form by using those in the original system.

By using coordinate and nearly identical transformations, the strongly nonlinear Duffing system is reduced to normal form in this paper, and then the bifurcation equations with different resonant conditions and their solutions are obtained. The local bifurcation diagrams and the transition sets on unfolding parameter and physical parameter plane are analysized by singularity theory.

A class of bifurcation solutions of almost-periodic (a. p. for short) parametric vibration systems is studied by Liapunov-Schmidt reduction. The bifurcation diagrams and formulas are given.

The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in (α,β) -plane. In this paper, we extended the results of [1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.

In the present paper subharmonic resonance solution of a piecewise linear oscillator with two degrees of freedom is studied. It is shown that in this system there exist a series of subharmonic resonance solutions, among them there are 1/2, 1/3, 1/4, 1/5, 1/6 ... ... subharmonic resonance solutions. The calculated results by the analogy computer and the field experiments in the factory partly verify this theory. Under certain circumstances, the generation of chaotic states of the oscillation is observed in analogy computer solutions.

A new approach is presented in this paper on the basis of dynamic systems theory. This paper presents the form of a generic classification of stable response diagrams for the nonlinear Mathieu equation. In addition, a general method is presented for determining the topological type of the response diagram for a given equation. This method has been successfully applied to Euler dynamic buckling problems. Some new results are obtained.