Periodic Motions and Global Bifurcations of a Two-Degree-of-Freedom System With Plastic Vibro-Impact

a b s t r a c tThis paper investigates the passive control of chatter instability in turning processes using a vibro-impact nonlinear energy sink (NES). The workpiece is assumed to be rigid and the tool is flexible. A dynamical model including a nonlinear cutting law is presented and the stability lobes diagram is obtained. The behavior of the system with the vibro-impact NES is investigated using an asymptotic analysis. A control mechanism by successive beating is revealed, similarly to the strongly modulated response in the case of NES with cubic stiffness. It is shown that such a response regime may be beneficial for chatter mitigation. An original experimental procedure is proposed to verify the sizing of the vibro-impact NES. An experimental setup is developed with a vibro-impact NES embedded on the lathe tool and the results are analyzed and validated.

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“…Using (46), supposing that ð∂h=∂ðΔt 1 ÞÞ ð0;0;0Þ a0 and ð∂g=∂ðΔt 2 ÞÞ ð0;0;0;0Þ a 0 the derivatives are expressed by [37,39] …”

Section: Asymptotic Analysismentioning

confidence: 99%

Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink

Gourc

Seguy

Michon

et al. 2015

Journal of Sound and Vibration

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“…Therefore, from (14), (20), (34), and (35), the first kind of collision periodic solution of system (6) is obtained…”

Section: Periodic Solutionmentioning

confidence: 99%

“…In order to verify the results of theoretical analysis, the suitable physical parameters and coefficient of restitution are selected to solve the periodic solutions according to (20) and (32). From (17), (18), and (20), we know A 2 = 0, B 2 = 0 and When the parameter value is As can be seen in Figures 5(a), (c), the phase diagram and the velocity at the collision point have a momentary jump, indicating that upper pendulum and the rigid wall collided.…”

Section: Numerical Simulationmentioning

confidence: 99%

“…According to the periodic solution, there are many results of analytical methods. Furthermore, the existence conditions of periodic solutions of single-degree-of-freedom collision systems with unilateral single-impact and bilateral single-impact were obtained in [19][20], which provided a theoretical analytical method for studying periodic solutions of impact systems. However, its expression was so complex that it is difficult to generalize to practical application engineering.…”

Section: Introductionmentioning

confidence: 99%

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Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain

2019

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In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for unilateral double-collision will be discussed in high-dimensional non-smooth asymmetric system. Firstly, the impact laws are classified in order to detect the existence of periodic solutions of the system. The impact recovery matrix is introduced to transform the impact laws of high-dimensional system into matrix. Furthermore, by use of modal analysis and matrix theory, an invertible transformation is constructed to obtain the parameter conditions for the existence of the impact periodic solution, which simplifies the calculation and can be easily extended to high-dimensional non-smooth system. Hence, the range of physical parameters and the restitution coefficients is calculated theoretically and non-smooth analytic expression of the periodic solution is given, which provides ideas for the study of approximate analytical solutions of high-dimensional non-smooth system. Finally, numerical simulation is carried out to obtain the impact periodic solution of the system with small angle motion.

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“…In recent years, vibro-impact problems have become a new subject in nonlinear dynamics. Subjects of recent research include singularity [Chatterjee and Mallik 1996;Whiston 1992]; inelastic vibro-impacts [Luo et al 2001]; high codimension bifurcation [Wen 2001;Luo and Xie 2003;Xie and Ding 2005]; Hopf bifurcations [Padmanabhan and Singh 1995;Luo and Chen 2005;Ding et al 2004;Luo 2004a]; and quasiperiodic impacts [BlazejczykOkolewska 2001;Luo 2004b;Cone and Zadoks 1995]; and so on. Dynamics and bifurcations of a class of single-degree-of-freedom self-excited oscillators with an impact damper were studied by Chatterjee and Mallik [1995].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamic characteristics of a vibro-impact system under harmonic excitation

Jian

2006

JOMMS

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Dynamical behaviors of a two-degree-of-freedom (TDOF) vibro-impact system are investigated. The theoretical solution of periodic-one double-impact motion is obtained by differential equations, periodicity and matching conditions, and the Poincaré map is established. The dynamics of the system are studied with special attention to Hopf bifurcations of the impact system in nonresonance, weak resonance, and strong resonance cases. The Hopf bifurcation theory of maps in ‫ޒ‬ 2 -strong resonance is applied to reveal the existence of Hopf bifurcations of the system. The theoretical analyses are verified by numerical solutions. The evolution from periodic impacts to chaos in nonresonance, weak resonance, and strong resonance cases, is obtained by numerical simulations. The results show that dynamical behavior of the system in the strong resonance case is more complicated than that of the nonresonance and weak resonance cases.

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Periodic Motions and Global Bifurcations of a Two-Degree-of-Freedom System With Plastic Vibro-Impact

Cited by 23 publications

References 18 publications

Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink

Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink

Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain

Nonlinear dynamic characteristics of a vibro-impact system under harmonic excitation

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