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1.
Based on the analysis of a two-degree-of-freedom plastic impact oscillator, we introduce a three-dimensional map with dynamical variables defined at the impact instants. The non-linear dynamics of the vibro-impact system is analyzed by using the Poincaré map, in which piecewise property and singularity are found to exist. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of map is generated via the grazing contact of two masses and corresponding instability of periodic motions. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. Simulations of the free flight and sticking solutions are carried out, and regions of existence and stability of different impact motions are therefore presented in (δω) plane of dimensionless clearance δ and frequency ω. The influence of non-standard bifurcations on dynamics of the vibro-impact system is elucidated accordingly.  相似文献   

2.
A two-degree-of-freedom plastic impact oscillator is considered. Based on the analysis of sticking and non-sticking impact motions of the system, we introduce a three-dimensional impact Poincaré map with dynamical variables defined at the impact instants. The plastic impacts complicate the dynamic responses of the impact oscillator considerably. Consequently, the piecewise property and singularity are found to exist in the three-dimensional map. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after impact, and the singularity of the map is generated via the grazing contact of two masses and the instability of their corresponding periodic motions. The nonlinear dynamics of the plastic impact oscillator is analyzed by using the Poincaré map. The simulated results show that the dynamic behavior of this system is very complex under parameter variation, varying from different types of sticking or non-sticking periodic motions to chaos. Suppressing bifurcation and chaotic-impact motions is studied by using an external driving force, delay feedback and damping control law. The effectiveness of these methods is demonstrated by the presentation of examples of suppressing bifurcations and chaos for the plastic impact oscillator.  相似文献   

3.
A two-degree-of-freedom impact oscillator is considered. The maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Impacts between the mass and the stops are described by an instantaneous coefficient of restitution. Dynamics of the system is studied with special attention to periodic-impact motions and bifurcations. Period-one double-impact symmetrical motion and transcendental impact Poincaré map of the system is derived analytically. Stability and local bifurcations of the period-one double-impact symmetrical motions are analyzed by using the impact Poincaré map. The Lyapunov exponents in the vibratory system with impacts are calculated by using the transcendental impact map. The influence of the clearance and excitation frequency on symmetrical double-impact periodic motion and bifurcations is analyzed. A series of other periodic-impact motions are found and the corresponding bifurcations are analyzed. For smaller values of clearance, period-one double-impact symmetrical motion usually undergoes pitchfork bifurcation with decrease in the forcing frequency. For large values of the clearance, period-one double-impact symmetrical motion undergoes Neimark–Sacker bifurcation with decrease in the forcing frequency. The chattering-impact vibration and the sticking phenomena are found to occur in the region of low forcing frequency, which enlarges the adverse effects such as high noise levels, wear and tear and so on. These imply that the dynamic behavior of this system is very rich and complex, varying from different types of periodic motions to chaos, even chattering-impacting vibration and sticking. Chaotic-impact motions are suppressed to minimize the adverse effects by using external driving force, delay feedback and feedback-based method of period pulse.  相似文献   

4.
Two typical vibro-impact systems are considered. The periodic-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map associated with 1:4 strong resonance is obtained. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed. The results from simulation illustrate some interesting features of dynamics of the vibro-impact systems. Some complicated bifurcations, e.g., tangent, fold and Neimark–Sacker bifurcations of period-4 orbits are found to exist near the 1:4 strong resonance points of the vibro-impact systems.  相似文献   

5.
An impact oscillator with a frictional slider is considered. The basic function of the investigated system is to overcome the frictional force and move downwards. Based on the analysis of the oscillatory and progressive motions of the system, we introduce an impact Poincaré map with dynamical variables defined at the impact instants. The nonlinear dynamics of the impact system with a frictional slider is analyzed by using the impact Poincaré map. The stability and bifurcations of single-impact periodic motions are analyzed, and some information about the existence of other types of periodic-impact motions is provided. Since the system equilibrium is moving downwards, one way to monitor the progression rate is to calculate its progression in a finite time. The simulation results show that in a finite time, the largest progression of the system is found to occur for period-1 multi-impact motions existing in the regions of low forcing frequencies. Secondly, the progression of the period-1 single-impact motion with peak-impact velocity is also distinct enough. However, it is important to note, that the largest progression for period-1 multi-impact motion existing at a low forcing frequency is not an optimal choice for practical engineering applications. The greater the number of the impacts in an excitation period, the more distinct the adverse effects such as high noise levels and wear and tear caused by impacts. As a result, the progression of the period-1 single-impact motion with the peak-impact velocity is still optimal for practical applications. The influence of parameter variations on the oscillatory and progressive motions of the impact-progressive system are elucidated accordingly, and feasible parameter regions are provided.  相似文献   

6.
A methodology for the local singularity of non-smooth dynamical systems is systematically presented in this paper, and a periodically forced, piecewise linear system is investigated as a sample problem to demonstrate the methodology. The sliding dynamics along the separation boundary are investigated through the differential inclusion theory. For this sample problem, a perturbation method is introduced to determine the singularity of the sliding dynamics on the separation boundary. The criteria for grazing bifurcation are presented mathematically and numerically. The grazing flows are illustrated numerically. This methodology can be very easily applied to predict grazing motions in other non-smooth dynamical systems. The fragmentation of the strange attractors of chaotic motion will be presented in the second part of this work.  相似文献   

7.
Hopf-flip bifurcations of vibratory systems with impacts   总被引:2,自引:1,他引:1  
Two vibro-impact systems are considered. The period n single-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. Stability and local bifurcations of single-impact periodic motions are analyzed by using the Poincaré maps. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. It is found that near the point of codim 2 bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Period doubling bifurcation of period one single-impact motion is commonly existent near the point of codim 2 bifurcation. However, no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of Hopf bifurcation associated with period two fixed points. The results from simulation shows that there exists an interest torus doubling bifurcation occurring near the value of Hopf-flip bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transit to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems.  相似文献   

8.
Forced vibro-impact dynamics of the two heavy mass particle motions, in vertical plane, along rough circle with Coulomb’s type friction and one, one side impact limiter is considered in combinations of applied analytical and numerical methods. System of two differential double equations, each for one of two heavy mass particle motions along same rough circle are composed with corresponding initial conditions as well as impact conditions. By use software package tools differential double equations are numerically integrated for obtaining phase portrait of phase trajectory branches for different mass particles initial kinetic states. By series of the phase trajectory branches for each mass particle motion between two impacts or between impact and alternation of the Coulomb’s friction force direction, two phase trajectory graphs of the system vibro-impact non-linear dynamics are composed. Different software tools are used as helping tools for calculate time moments of the series of the impacts between mass particles, as well as positions of the impacts, necessary for calculations of the impact velocities of the mass particles before and after impacts. Some comparison between forced and free vibro-impact dynamics of the two heavy mass particles in vertical plane, along rough circle with Coulomb’s type friction and one, one side impact limiter is done. Trigger of coupled one side singularities in phase portraits are identified.  相似文献   

9.
Systems constituted by moving components that make intermittent contacts with each other can be modelled by a system of ordinary differential equations containing piecewise linear terms. We consider a soft impact bilinear oscillator for which we obtain bifurcation diagrams, Lyapunov coefficients, return maps and phase portraits of the response. Besides Lyapunov coefficients diagrams, bifurcation diagrams are represented in terms of both non-dimensional time instants of contact (when the mass impacts the obstacle) and of portions of contact duration (the percentage-time interval when the material point is inside the obstacle) vs. non-dimensional external force frequency (or amplitude). The second kind of diagrams is needed because the contact duration (or the complementary flight time duration) are quantities that can easily be measured in an experiment aiming at confirming the validity of the present model. Lyapunov coefficients are evaluated converting the piecewise linear system of ordinary differential equations into a map, the so-called impact map, where time and velocity corresponding to a given impact are evaluated as functions of time and velocity corresponding to the previous impact. Thus, the usual methods related to this last map are used. The trajectories are represented in terms of return maps (all points in the time-velocity plane involved in the impact events) and phase portraits (the trajectory-itself in the displacement-velocity plane). In the bifurcation diagrams, transition between different responses is evidenced and a perfect correlation between chaotic (periodic) attractors and positive (negative) values of the maximum Lyapunov coefficient is found.  相似文献   

10.
11.
This paper investigates the existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibro-impact system. The criterion for existence of grazing period-n motion is presented. A local analysis based on the discontinuity-mapping approach is employed to derive a normal form Poincaré mapping near the grazing trajectory. Based on the above approach, a condition of stability can be formulated, such that a grazing trajectory will be discontinuous if the condition is unfulfilled. The predicted grazing bifurcations are in agreement with the numerical results. In particular, comparison of the grazing bifurcation diagrams of the normal form Poincaré mapping and the simulation diagrams of the original differential equation illustrates the validity of the discontinuity-mapping approach.  相似文献   

12.
The bifurcations of periodic motions of nonlinear systems are studied, which can lead to a state in which the change in the sign of the control parameter ceases to affect the character of the nonsymmetric motion of the system. Results of the investigation of the concrete systems (piecewise connected and with power-type nonlinearity) are given, illustrating the mechanism of the appearance of a malfunction.  相似文献   

13.
In this note we investigate the influence of structural nonlinearity of a simple cantilever beam impacting system on its dynamic responses close to grazing incidence by a means of numerical simulation. To obtain a clear picture of this effect we considered two systems exhibiting impacting motion, where the primary stiffness is either linear (piecewise linear system) or nonlinear (piecewise nonlinear system). Two systems were studied by constructing bifurcation diagrams, basins of attractions, Lyapunov exponents and parameter plots. In our analysis we focused on the grazing transitions from no impact to impact motion. We observed that the dynamic responses of these two similar systems are qualitatively different around the grazing transitions. For the piecewise linear system, we identified on the parameter space a considerable region with chaotic behaviour, while for the piecewise nonlinear system we found just periodic attractors. We postulate that the structural nonlinearity of the cantilever impacting beam suppresses chaos near grazing.  相似文献   

14.
In this paper we consider two classes of one dimensional piecewise smooth continuous maps that have been derived as normal forms for grazing bifurcations of piecewise smooth dynamical systems. These maps are linear on one side of the phase space and nonlinear on the other side. The case of nonlinear parts with negative coefficients has been studied previously and it is proved that period-adding scenarios are generic in this case. In contrast to this result, in our analytical and numerical results, the period-adding scenarios are not observed when the nonlinear parts have positive coefficients. Furthermore, our results suggest that the typical bifurcation scenario is period doubling cascade leading to chaos in this case, which is similar to that of the smooth logistic map.  相似文献   

15.
This paper presents some new ideas to understand the strange attractor fragmentation caused by grazing in non-smooth dynamic systems. The sufficient and necessary conditions for grazing bifurcations in non-smooth dynamic systems are presented. The initial sets of grazing mapping are introduced and the corresponding initial grazing manifolds are discussed. The grazing-induced fragmentation of strange attractors of chaotic motions in non-smooth dynamical systems is presented. The mathematical theory for such a fragmentation of strange attractors should be further developed.  相似文献   

16.
The criterion for grazing motions in a dry-friction oscillator is obtained from the local theory of non-smooth dynamical systems on the connectable and accessible domains. The generic mappings for such a dry-friction oscillator are also introduced. The sufficient and necessary conditions for grazing at the final states of mappings are expressed. The initial and final switching sets of grazing mapping, varying with system parameters, are illustrated for the grazing parametric characteristics. The initial and grazing, switching manifolds in the switching sets are defined through grazing mappings. Finally, numerical illustrations of grazing motions are very easily carried out with help of the analytical predictions. This paper provides a comprehensive investigation of grazing motions in the dry-friction oscillator for a better understanding of the grazing mechanism of such a discontinuous system. The investigation based on the local singularity theory is more intuitive and efficient than the discontinuous mapping techniques.  相似文献   

17.
We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.  相似文献   

18.
In this work we consider the homoclinic bifurcations of expanding periodic points. After Marotto, when homoclinic orbits to expanding periodic points exist, the points are called snap-back-repellers. Several proofs of the existence of chaotic sets associated with such homoclinic orbits have been given in the last three decades. Here we propose a more general formulation of Marotto’s theorem, relaxing the assumption of smoothness, considering a generic piecewise smooth function, continuous or discontinuous. An example with a two-dimensional smooth map is given and one with a two-dimensional piecewise linear discontinuous map.  相似文献   

19.
The criterion for grazing motions in a friction-induced oscillator with a time-varying transport belt is obtained. The three mappings for such a friction-induced oscillator are introduced for analytical prediction of the grazing motions. The sufficient and necessary conditions of grazing are expressed from mappings. With system parameter variations, the initial and final switching sets of grazing mapping are illustrated. Numerical illustrations of grazing motions are carried out from analytical predictions. This investigation provides a technique for how to determine the grazing on the time-varying boundary in discontinuous dynamical systems.  相似文献   

20.
There exist many types of external excitations that make the damping oscillator with impact have complex dynamics. In this study, both external impulsive excitation and impact are considered to construct a vibro-impact system. The fixed time pulse (impulsive excitation) and the state pulse (impact) lead to the complex and interesting dynamics. The conditions of the existence and stability of four kinds of periodic solutions are investigated, and the bifurcations of period-(1, 0) and period-(1, 1) solutions are analytically studied. Numerical simulations on periodic solutions and bifurcation diagrams are shown by the illustrative example.  相似文献   

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