共查询到20条相似文献,搜索用时 31 毫秒
1.
Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1:2 resonance case 总被引:1,自引:0,他引:1
A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each
other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The
Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and
the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce
the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1:2 resonance
is obtained. Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point
of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of
codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion,
but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact
periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are
also discussed.
The project supported by the National Natural Science Foundation of China (10572055, 50475109) and the Natural Science Foundation
of Gansu Province Government of China (3ZS051-A25-030(key item)) The English text was polished by Keren Wang. 相似文献
2.
Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting
an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies
of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems,
associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact
periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of
a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to
reduce the Poincaré map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of
two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong
resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of
period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact
systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T
in, T
on and T
out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact
motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the
vibro-impact systems exhibit very complicated quasi-periodic impact motions.
The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of
Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang. 相似文献
3.
考虑一类具有对称性的三自由度碰撞振动系统.系统的庞加莱映射在一定条件下存在对称不动点,对应于系统的对称周期运动.根据对称性导出庞加莱映射P是另外一个隐式虚拟映射Q的二次迭代.推导了庞加莱映射对称不动点的解析表达式.根据映射不动点的稳定性及分岔理论,映射P的对称不动点发生内伊马克沙克-音叉(Neimark--Saker-pitchfork)分岔对应于映射Q发生内伊马克沙克-倍化(Neimark--Sakerflip)分岔.利用隐式虚拟映射Q,通过对范式作两参数开折分析,研究了映射P的对称不动点在内伊马克沙克-音叉分岔点附近的局部动力学行为.碰撞振动系统在这个余维二分岔点附近的局部动力学行为可能表现为投影后的庞加莱截面上的单一对称不动点、一对共轭不动点、单一对称拟周期吸引子以及一对共轭拟周期吸引子.数值模拟得到了内伊马克沙克-音叉分岔点附近的各种可能情况.内伊马克沙克-分岔和音叉分岔互相作用可能产生新的结果:对称不动点虽然首先分岔为两个共轭不动点,但是这两个共轭不动点是不稳定的,最终收敛到同一个对称拟周期吸引子. 相似文献
4.
This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called ‘‘time-2τ2 map’’ of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T1 (Hopf invariant circles), tori 2T1 and tori 2T2 depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms’ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.The project supported by the National Natural Science Foundation of China (10472096)The English text was polished by Ron Marshall. 相似文献
5.
Van der Pol??s equation with extended delay feedback is investigated as a neutral differential-difference equation. Normal forms near codimension two bifurcations, including Hopf?Cpitchfork and Hopf?CHopf bifurcation, are determined by the method of multiple scales. Through analyzing the associated amplitude equations, we obtain the detailed bifurcation sets and find some interesting phenomena such as quasi-periodic oscillations and strange attractor, which are confirmed by several numerical simulations. 相似文献
6.
强共振情况下冲击成型机的亚谐与Hopf分岔 总被引:4,自引:0,他引:4
通过理论分析与数值仿真研究了双质体冲击振动成型机的周期运动在强共振条件下的亚谐分岔与Hopf分岔,证实了此系统的1/1周期运动在强共振(λ0^4=1)条件下可以分岔为稳定的4/4周期运动及概周期运动.讨论了冲击映射的奇异性,分析了冲击振动系统的“擦边”运动对强共振条件下周期运动及全局分岔的影响。 相似文献
7.
甘春标 《Acta Mechanica Sinica》2004,20(5):558-566
A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system‘s energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system‘s parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions. 相似文献
8.
9.
IntroductionRotor-bearings systems applied widely in industry are nonlinear dynamic systems of multi-degree-of-freedom.Synchronous vibration is its typical motion under unavoidable unbalance.Subharmonic,quasi-periodic and chaotic vibrations,caused by the … 相似文献
10.
In this paper, the dynamics of a generalized two-neuron model with self-connections and distributed delays are investigated, together with the stability of the equilibrium. In particular, the conditions under which the Hopf bifurcation occurs at the equilibrium are obtained for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. Explicit algorithms for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [20]. Some numerical simulations are given to illustrate the effectiveness of the results found. The obtained results are new and they complement previously known results.This work was supported by the National Natural Science Foundation of China under Grants 60574043 and 60373067, the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053. 相似文献
11.
Bifurcation of the electromechanically coupled subsynchronous torsional oscillating system with hysteretic behavior 总被引:1,自引:0,他引:1
In subsynchronous resonance (SSR) systems where shaft systems of turbine-generator sets are coupling with electric networks,
Hopf bifurcation will occur under certain conditions. Some singularity phenomena may generate when the hysteretic behavior
of couplings in the shaft systems in considered. In this paper, the intrinsic multiple-scale harmonic balance method is extended
to the nonlinear autonomous system with the non-analytic property, and the dynamic complexities of the system near the Hopf
bifurcation point are analyzed.
The project supported by the National Natural Science Foundation of China (as a key project) and the State Education Committee
Pre-research Foundation. 相似文献
12.
13.
This paper shows the mechanism of instability and chaos in a cantilevered pipe conveying steady fluid. The pipe under consideration
has added mass or a nozzle at the free end. The Galerkin method is used to transform the original system into a set of ordinary
differential equations and the standard methods of analysis of the discrete system are introduced to deal with the instability.
With either the nozzle parameter or the flow velocity increasing, a route to chaos can be observed very clearly: the pipe
undergoing buckling (pitchfork bifurcation), flutter (Hopf bifurcation), doubling periodic motion (pitchfork bifurcation)
and chaotic motion occurring finally.
The project supported by the National Key Projects of China under grant No. PD9521907 and Science Foundation of Tongji University
under grant No. 1300104010. 相似文献
14.
A kind of 2-dimensional neural network model with delay is considered. By analyzing the distribution of the roots of the characteristic
equation associated with the model, a bifurcation diagram was drawn in an appropriate parameter plane. It is found that a
line is a pitchfork bifurcation curve. Further more, the stability of each fixed point and existence of Hopf bifurcation were
obtained. Finally, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions were determined
by using the normal form method and centre manifold theory.
Foundation item: the National Natural Science, Foundation of China (19831030)
Biography: WEI Jun-jie, Professor, Doctor, E-mail: weijj@hit.edu.cn 相似文献
15.
The bifurcation dynamics of shallow arch which possesses initial deflection under periodic excitation for the case of 1∶2
internal resonance is studied in this paper. The whole parametric plane is divided into several different regions according
to the types of motions; then the distribution of steady state motions of shallow arch on the plane of physical parameters
is obtained. Combining with numerical method, the dynamics of the system in different regions, especially in the Hopf bifurcation
region, is studied in detail. The rule of the mode interaction and the route to chaos of the system is also analysed at the
end.
Project supported by National Natural Science Foundation and National Youth Science Foundation of China 相似文献
16.
碰撞振动系统的一类余维二分岔及T2环面分岔 总被引:9,自引:0,他引:9
建立了三自由度碰撞振动系统的动力学模型及其周期运动的Poincaré映射,当Jacobi矩阵存在两对共轭复特征值同时在单位圆上时,通过中心流形-范式方法将六维映射转变为四维范式映射.理论分析了这种余维二分岔问题,给出了局部动力学行为的两参数开折.证明系统在一定的参数组合下,存在稳定的Hopf分岔和T2环面分岔.数值计算验证了理论结果. 相似文献
17.
The dynamic behavior close to a non-resonant double Hopf bifurcation is analyzed via a frequency-domain technique. Approximate expressions of the periodic solutions are computed using the higher order harmonic balance method while their accuracy and stability have been evaluated through the calculation of the multipliers of the monodromy matrix. Furthermore, the detection of secondary Hopf or torus bifurcations (Neimark–Sacker bifurcation for maps) close to the analyzed singularity has been obtained for a coupled electrical oscillatory circuit. Then, quasi-periodic solutions are likely to exist in certain regions of the parameter space. Extending this analysis to the unfolding of the 1:1 resonant double Hopf bifurcation, cyclic fold and torus bifurcations have also been detected in a controlled oscillatory coupled electrical circuit. The comparison of the results obtained with the suggested technique, and with continuation software packages, has been included. 相似文献
18.
A general theory for the study of degenerate Hopf bifurcation in the presence of symmetry has been carried out only in situations where the normal form equations decouple into phase/amplitude equations. In this paper we prove a theorem showing that in general we expect such degeneracies to lead to secondary torus bifurcations. We then apply this theorem to the case of degenerate Hopf bifurcation with triangular (D3) symmetry, proving that in codimension two there exist regions of parameter space where two branches of asymptotically stable 2-tori coexist but where no stable periodic solutions are present. Although this study does not lead to a theory for degenerate Hopf bifurcations in the presence of symmetry, it does present examples that would have to be accounted for by any such general theory. 相似文献
19.
The topological bifurcation diagrams and the coefficients of bifurcation equation were obtained by C-L method. According to obtained bifurcation diagrams and combining control theory, the method of robust control of periodic bifurcation was presented, which differs from generic methods of bifurcation control. It can make the existing motion pattern into the goal motion pattern. Because the method does not make strict requirement about parametric values of the controller, it is convenient to design and make it. Numerical simulations verify validity of the method. 相似文献