External Periodic Force Control of a Single-Degree-of-Freedom Vibroimpact System

Abstract

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.

1. Introduction

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 [1]. Scholars put forward a lot of modified control methods [2, 3] based on the OGY method. Although small perturbation of the system parameters for the control of chaos has been confirmed, the method requires detailed information about the target trajectory and brings a lot of inconvenience to the practical application. Therefore the domestic and foreign researchers put the traditional control theory and chaotic motion characteristics used in chaos control and present a lot of chaos control method, such as delayed feedback control [4, 5], periodic parameter perturbation control [6], continuous feedback control [7], pulse feedback control [8], and adaptive control [9], and so forth. There are also examples of control on the vibroimpact system [10, 11].

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.

2. Mechanics Model of the Vibroimpact System

Figure 1 shows a single-degree-of-freedom mechanical model of vibroimpact system. Oscillator 𝑀 is connected to the left side support by the spring with stiffness 𝐾 and the damper with damping 𝐶. In the harmonic excitation 𝐹sin(Ω𝑇+𝜃), motion of the oscillator is in the horizontal direction. 𝑋 is the displacement of motion. When the oscillator is in the equilibrium position, the gap between the rigid constraints on the right side is 𝐷. Considering the collision as the rigid collision and 𝑅 as the coefficient of restitution, the differential equation of motion of system is 𝑀𝑋̈+𝐶𝑋̇+𝐾𝑋=𝐹sin(Ω𝑇+𝜃)(𝑋<𝐵). (1)

Figure 1 

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

The shock equation of system is 𝑋̇₊=−𝑅𝑋̇₋(𝑋=𝐵), (2) where 𝑋̇₋ and 𝑋̇₊ represent the impacting mass velocities of approach and departure at the instant of impacting, respectively.

After the dimensionless transformation, when 𝑥<𝑑, the differential equations of motion of the system between two collisions are as follows: 𝑥̈+2𝜁𝑥̇+𝑥=sin(𝜔𝑡+𝜃). (3) In which, the nondimensional quantities are 𝑥=𝑋𝐾𝐹𝐶,𝜁=2√𝑀𝐾,𝜔=Ω𝑀𝐾,𝑡=𝑇𝐾𝑀,𝑑=𝐷𝐾𝐹. (4)

When 𝑥=𝑑, shock equation of the system at collision transient is given by 𝑥̇₊=−𝑅𝑥̇₋, (5) where 𝑥̇₋ and 𝑥̇₊ represent the impacting mass velocities of approach and departure at the instant of impacting, respectively.

By (1), the general solution of the system between two collisions between is 𝑥=𝑒^−𝜁𝑡𝑎cos𝜔_𝑑𝑡+𝑏sin𝜔_𝑑𝑡()(),+𝐴sin𝜔𝑡+𝜃+𝐵cos𝜔𝑡+𝜃 (6) where 𝜔_𝑑=1−𝜁². 𝐴 and 𝐵 are amplitude constants. By the initial conditions of the system: 𝑥(𝑡₀)=𝑥₀ and 𝑥̇(𝑡₀)=𝑥̇₀, let 𝑡₀=0; the integral constants 𝐴 and 𝐵 can determined 𝑎=𝑥₀𝑥̇−𝐴sin𝜃−𝐵cos𝜃,𝑏=⁰+𝜁𝑥₀+(𝐵𝜔−𝐴𝜁)sin𝜃−(𝐴𝜔+𝐵𝜁)cos𝜃𝜔_𝑑,𝐴=1−𝜔²1−𝜔²²+(2𝜁𝜔)²,𝐵=−2𝜁𝜔1−𝜔²²+(2𝜁𝜔)². (7)

Periodic motion of the system under certain parameter conditions can be expressed as 𝑛−𝑝, 𝑛 represents a force cycle number, and 𝑝 represents the number of collisions. Considering the 𝑛−1 periodic motion, the collision instantaneous dimensionless time 𝑡=0, and next collision time 𝑡_𝑧=2𝑛𝜋/𝜔, then boundary conditions of the 𝑛−1 periodic motion are 𝑥(0)=𝑥(𝑡_𝑧)=𝑑 and 𝑥̇(0)=−𝑅𝑥(𝑡_𝑧). With (6) applying to the boundary conditions, the existence conditions of periodic motion system are −𝐻tan𝜃−1+tan²𝜃−𝐻²1+tan²𝜃≤1, (8) where 𝜃=arctan(−𝐻₂/𝐻₁), 𝑐=cos𝜔_𝑑𝑡_𝑧, 𝑒=𝑒^{−𝜁𝑡𝑧}, 𝑠=sin𝜔_𝑑𝑡_𝑧, 𝑑𝐻=𝐻₁𝜁+Re𝜔_𝑑+𝜔𝑠+𝜁𝑐𝑒𝑠_𝑑+Re𝜔_𝑑(),𝐻𝑐−𝜁𝑠𝑒𝑐−1₁=𝐴𝜁+Re𝜔_𝑑()𝜔𝑠+𝜁𝑐−𝐵𝜔1+𝑅𝑒𝑠+𝐴_𝑑+Re𝜔_𝑑),𝐻𝑐−𝜁𝑠(𝑒𝑐−1₂=𝐵𝜁+Re𝜔_𝑑()𝜔𝑠+𝜁𝑐−𝐴𝜔1+𝑅𝑒𝑠+𝐵_𝑑+Re𝜔_𝑑𝑐−𝜁𝑠(𝑒𝑐−1). (9)

At the same time, theory fixed point of the 𝑛−1 periodic motion is given by 𝜃_𝑑=arccos−𝐻tan𝜃−1+tan²𝜃−𝐻²1+tan²𝜃,𝑥_𝑑𝑥̇=𝑑,_𝑑=𝑑−𝐴sin𝜃_𝑑−𝐵cos𝜃_𝑑(1−𝑒𝑐)𝜔_𝑑−𝑒𝑠𝜁𝑒𝑠+𝐴𝜔cos𝜃_𝑑−𝐵𝜔sin𝜃_𝑑. (10)

Define the following section: 𝜎={(𝜃,𝑥,𝑥̇)∈𝑅²×𝑆₁|𝑥=𝑑}. In the paper, we choose the section 𝜎 to establish the Poincaré map 𝜃_𝑘+1=𝜔𝑡+𝜃_𝑘,𝑥_𝑘+1=𝑒^−𝜁𝑡𝑎cos𝜔_𝑑𝑡+𝑏̂sin𝜔_𝑑𝑡+𝐴sin𝜔𝑡+𝜃_𝑘+𝐵cos𝜔𝑡+𝜃_𝑘,𝑥̇_𝑘+1=−𝑅𝑒^−𝜁𝑡𝑏̂𝜔_𝑑−𝑎𝜁cos𝜔_𝑑𝑡−𝑏̂𝜁+𝑎𝜔_𝑑sin𝜔_𝑑𝑡+𝐴𝜔cos𝜔𝑡+𝜃_𝑘−𝐵𝜔sin𝜔𝑡+𝜃_𝑘, (11)(12)(13) where 𝑎=𝑥_𝑘−𝐴sin𝜃_𝑘−𝐵cos𝜃_𝑘,   and  𝑏̂=(𝑥̇_𝑘+𝜁𝑥_𝑘+(𝐵𝜔−𝐴𝜁)sin𝜃_𝑘−(𝐴𝜔+𝐵𝜁)cos𝜃_𝑘)/𝜔_𝑑. In which, initial iteration value of the Poincaré map of the 𝑛−1 periodic motion is 𝜃₀=𝜃_𝑑,𝑥_𝑘+1=𝑥_𝑘=𝑥₀=𝑥_𝑑=𝑑,𝑥̇₀=𝑥̇_𝑑. (14)

3. Chaos and Bifurcation

The single-degree-of-freedom mechanical model of vibroimpact system, with system parameters 𝜁=0.01, 𝑑=0.05, and 𝑅=0.8, has been chosen to be analyzed. The system parameter 𝜔 is taken as the bifurcation parameter. The global bifurcation diagram can be obtained with the 𝜔 changing in the range of [0,6] as shown in Figure 2. As you can see from Figure 2, the system has stable 𝑛−1 periodic motion in a certain range. But in different period of single touch movement, the system will produce double periodic bifurcation, from periodic motion to chaos in the process of transition. But with the increase in 𝜔, the system will degenerate to the periodic motion. The Figures 2(b) and 2(c) are a partial enlargement of Figure 2(a). It can be seen that, when 𝜔=2.5585, the vibroimpact system will be from periodic 1 motion to periodic 2 motion by bifurcation and when 𝜔=2.624, the vibroimpact system will be from periodic 2 motion to periodic 4 motion by double periodic bifurcation. When 𝜔=2.641, the vibroimpact system will be from periodic 4 motion to periodic 8 motion by double periodic bifurcation. When 𝜔=2.644, the vibroimpact system will be from periodic 8 motion to periodic 16 motion by double periodic bifurcation. With the vibration frequency 𝜔 increasing further, the system leads to chaotic motion.

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Figure 2 

Bifurcation diagram of the system.

The excitation frequency 𝜔 takes 2.55, 2.6, 2.635, 2.642, 2.644, and 2.652. Poincaré map, phase portrait, and time course diagram of the system are shown in Figures 3, 4, 5, 6, 7, and 8. The system is, respectively, periodic 1 motion, periodic 2 motion, periodic 4 motion, periodic 8 motion, periodic 16 motion, and chaotic motion.

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Figure 3 

Poincaré map, phase portrait, and time course diagram of the system with 𝜔=2.55.

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Figure 4 

Poincaré map, phase portrait, and time course diagram of the system with 𝜔=2.6.

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Figure 5 

Poincaré map, phase portrait, and time course diagram of the system with 𝜔=2.635.

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Figure 6 

Poincaré map, phase portrait, and time course diagram of the system with 𝜔=2.42.

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Figure 7 

Poincaré map, phase portrait, and time course diagram of the system with 𝜔=2.644.

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Figure 8 

Poincaré map, phase portrait, and time course diagram of the system with 𝜔=2.52.

4. Chaos Control

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 𝑓=ℎsin𝛾𝜔𝜃. (15) In the formula, ℎ and 𝛾 are adjustable parameters.

As the previous analysis, when 𝜔=2.652, the system is in chaotic motion. After the introduction of the external periodic force feedback controller, using the fourth-order Runge-Kutta method of numerical simulation, the Poincaré map, phase portrait, and time course diagram of the controlled system are obtained, as shown in Figures 9 and 10. When 𝛾=0.05 and ℎ=4.6, the system is controlled to periodic 1 motion. When 𝛾=0.05 and ℎ=5.6, the system is controlled to periodic 2 motion. The simulation results show the effectiveness of the method. Because the method does not change the original system parameters, it can be applied to a 2-DOF and multiple-DOF vibroimpact mechanical system.

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Figure 9 

Poincaré map, phase portrait, and time course diagram of the controlled system with 𝛾=0.05 and ℎ=4.6.

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Figure 10 

Poincaré map, phase portrait, and time course diagram of the controlled system with 𝛾=0.05 and ℎ=5.6.

5. Conclusion

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.

Acknowledgments

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).