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A single-degree-of-freedom mechanical model of vibro-impact system is established. Bifurcation and chaos in the system are revealed with the time history diagram, phase trajectory map, and Poincaré map. According to the bifurcation and chaos of the actual vibro-impact system, the paper puts forward external periodic force control strategy. The method of controlling chaos by external periodic force feedback controller is developed to guide chaotic motions towards regular motions. The stability of the control system is also analyzed especially by theory. By selecting appropriate feedback coefficients, the unstable periodic orbits of the original chaotic orbit can be stabilized to the stable periodic orbits. The effectiveness of this control method is verified by numerical simulation.

In the field of nonlinear, along with people understanding the nature of chaos, how to control chaos and chaos synchronization has been a hot topic studied by researchers. In the 1990s, Ott et al. proposed the OGY chaos control method [

Vibroimpact system as a typical nonsmooth dynamical system generally exists in practical engineering. Because of the frequent collision, the system has strong nonlinearity and discontinuities compared with a smooth nonlinear system, presents more complex nonlinear phenomena, and causes hazards on the safe operation of the system. Because of the collision interface differential discontinuities, the original method applied to continuous system can not be used for such system.

This paper puts forward a sine periodic force feedback controller based on the periodic external force feedback control strategy and analyzes the stability of control theory. When selecting the appropriate feedback coefficients, the chaotic orbits can be controlled onto the stable periodic orbits. A single-degree-of-freedom vibroimpact system is transformed into a form of Poincaré map for numerical simulation. The results of numerical simulation show that the method is effective in practical engineering, so it has certain practical significance.

Figure

Mechanics model of a single-degree-of-freedom vibroimpact system.

The shock equation of system is

After the dimensionless transformation, when

When

By (

Periodic motion of the system under certain parameter conditions can be expressed as

At the same time, theory fixed point of the

Define the following section:

The single-degree-of-freedom mechanical model of vibroimpact system, with system parameters

Bifurcation diagram of the system.

The excitation frequency

Poincaré map, phase portrait, and time course diagram of the system with

Poincaré map, phase portrait, and time course diagram of the system with

Poincaré map, phase portrait, and time course diagram of the system with

Poincaré map, phase portrait, and time course diagram of the system with

Poincaré map, phase portrait, and time course diagram of the system with

Poincaré map, phase portrait, and time course diagram of the system with

The paper chooses the sine driving force for the periodic force excitation. Periodic force is easy to produce and control the external driving force in the actual project, so the sine driving force is used to suppress the bifurcation and chaotic motion of single-degree-of-freedom vibroimpact system. Based on the principle of parameter perturbation, periodic force excitation method can stabilize the chaotic motion by applying disturbance directly into the system. An unstable periodic motion of the chaotic system can produce resonance with external periodic force. The system can be from its unstable limit cycle to a stable limit cycle by resonating with external driving signal. So the chaos is controlled.

Periodic force excitation can be expressed as

As the previous analysis, when

Poincaré map, phase portrait, and time course diagram of the controlled system with

Poincaré map, phase portrait, and time course diagram of the controlled system with

Based on a single-degree-of-freedom vibroimpact system as the research object, bifurcation and chaos have been researched with the system parameters changing. By adopting an external periodic force excitation method for suppressing its chaotic behavior, it delayed the occurrence of fault. Because this method does not change the original system parameters, it is easy to implement in engineering. This method is not limited to this kind of mechanical system with clearance collision and can also be used in other similar nonlinear system.

The authors gratefully acknowledge the support of Program for National Natural Science Foundation of China (51275082), New Century Excellent Talents in University (NCET-08-0103), the Research Fund for the Doctoral Program of Higher Education (20100042110013), the Fundamental Research Funds for the Central Universities of China (N110403008), Natural Science Foundation of Liaoning Province of China (201102071), Shenyang City Science Projects (F11-264-1-46), and the Science and Technology Research Projects of Education Department of Liaoning Province of China (L2012068).