Statistical basin of attraction in time-delayed cutting dynamics: Modelling and computation

doi:10.1016/j.physd.2020.132779

Physica D: Nonlinear Phenomena

Volume 416, February 2021, 132779

https://doi.org/10.1016/j.physd.2020.132779 Get rights and content

Highlights

•
Basin of attraction is generalized to be statistical for time-delayed systems.
•
Time delayed state is approximated by a Fourier series aligned on a straight line.
•
A safe basin with no probability of chatter occurrence is found.
•
State-dependent intermittent control can steer co-existing dynamics.
•
Cutting safety is improved by state-dependent intermittent control.

Abstract

This paper proposes a novel concept of the statistical basin of attraction to analyse the multiple stability in nonlinear time-delayed dynamical systems and shows how they can be computed. This concept has been applied to the cutting dynamics, which has been extensively investigated by the authors. Due to the nonlinearity and non-smoothness of tool-workpiece interactions, the cutting dynamics always exhibit large-amplitude chatter entering a linearly stable zone, making the area below stability boundaries unsafe for high material removal rates. Meanwhile, a thorough investigation of the multiple stability in the cutting dynamics is hampered by infinite-many dimensions introduced by time delays, which induce difficulties in computation and visualization of the conventional basin of attraction. To address this issue, infinite-many dimensional time-delayed states are approximated by a Fourier series aligned on a straight line, and the coefficients of the basis functions and the cutting process are used to construct the statistical basin of attraction. Inside the statistical basin of attraction, a safe basin with no probability of chatter occurrence exists. These findings are instrumental in designing a new state-dependent intermittent control to guide the cutting dynamics towards the safe basins. It is also seen that the state-dependent intermittent control is efficient in improving the cutting safety and shrinking the unsafe zones, even when the targeted basin for the control is larger than the real safe basin.

Introduction

Many physical and engineering systems, including neural dynamics [1], vibration absorber [2], [3], human balancing [4], epidemics [5], laser modulation [6], internet [7], and others are governed by delayed differential equations (DDEs). The primary characteristic of time-delayed systems is their infinite dimensions defined in Banach space, resulting in difficulties of their linear stability analysis [8]. The analysis becomes even more challenging when multiple delays are involved in making the characteristic equations transcendental, where sophisticated numerical algorithms are necessary for those cases [9]. Typical practical examples include grinding [10] and internet dynamics [11]. Nonlinear analysis of the DDEs is also a challenging task so that perturbation methods, including centre manifold reduction and the method of multiple scales, have been appropriately modified, and a novel perturbation incremental method has been proposed to deal with the delayed terms [12], [13]. When parameter values are beyond the vicinity of stability boundaries, where the local perturbation methods are not applicable, bifurcation analysis is normally performed by numerical continuation methods.

The popular and widely used path-following software, AUTO [14], can only handle ordinary differential equations (ODEs), and DDE-BIFTOOL can be used for the numerical bifurcation analysis of DDEs. If non-smoothness is also involved in systems with time delays, despite some initial attempts (e.g., [8]), there is practically no numerical bifurcation tool available until now. Moreover, the classical approach to the Basin of Attraction (BoA) [15] is very computationally intensive, even for a low dimensional system. Initial conditions (ICs) become critical for long-term dynamics requiring the BoA to relate co-existing attractors and the ICs going towards them [16]. Computational effort and difficulty in the visualization of BoA increase exponentially with respect to the rise of systems dimension [17]. Also, for the DDEs with infinite-many dimensions, there is no consent in the definition of BoA up to now.

Regenerative cutting is a typical example of nonlinear and non-smooth time-delayed dynamical systems having all the properties described above. Newly generated workpiece surface will be re-cut by successive tool passes, making the current cutting force related to the previous tool displacement [18]. This kind of regenerative phenomenon in many cutting operations, including turning [19], milling [20], grinding [21], and drilling [22], can be modelled by time delays, which introduce infinite-many “lobes” curves in stability diagrams. When regeneration occurs on several surfaces, multiple delays are involved making the linear stability analysis complicated so that numerical algorithms are usually adopted [23]. When cutting tools leave the workpiece surface during chatter, non-smoothness hampers numerical continuation algorithms, and thus global bifurcation is generally studied by direct numerical integration. Also, local perturbation analysis reveals that periodic chatter generated on the stability boundaries are often of subcritical properties, introducing large-amplitude chatter into the linearly stable region to co-exist with stationary cutting [24]. This multiple stability is deemed as unsafe since it introduces uncertainties in long-term cutting dynamics [25]. However, the unsafe zones (UZs) near the stability boundaries usually have high material removal rate [26] and worth exploring to achieve high-efficient cutting despite the risk of incurring chatter.

To this end, a well defined BoA for DDEs is an important prerequisite, particularly as there is no widely accepted one available up to now. When time delays appear in systems with feedback control, some studies artificially switched off the time-delayed control at the very beginning, so that no functional ICs are required for the delayed terms, and there is no difference between the ODEs and DDEs in the definition of BoA [27], [28], [29]. Alternatively, one can approximate the functional ICs by using various elementary functions. The simplest approximation is the constant function used by Ji [30], who studied the BoA of two co-existing periodic attractors in a delayed nonlinear oscillator. An improved version is a linear polynomial employed by Wang et al. [31] in the study of a delayed Duffing oscillator, where the coefficients of the polynomial were used to construct the BoA. This approximation is then developed by Zheng and Sun [32], who used second-order polynomials for the BoA of a time-delayed Duffing system. Besides, the Fourier series is also a good candidate for the approximation of functional ICs, especially when the delayed system has oscillatory properties. Thus Hu [33] used the coefficients of two sinusoidal functions to construct the BoA of a Duffing system with delayed velocity feedback. It is noteworthy that all the above studies used no more than three basis functions for the BoA of DDEs, no matter what elementary functions were employed.

An accurate approximation of the functional ICs should involve higher-order basis functions, but it introduces too many undetermined coefficients for the computation and visualization of the BoAs. To illustrate, Leng et al. [34] employed 20 orthogonal functions in the study of a two-dimensional delayed Hopfield neuronal network, which yielded an accurate approximation but compelled them to use the basin stability instead of the BoA. Thus they can only find the probabilities of each attractors’ appearance instead of relating the long-term dynamics to the given ICs. Yan et al. [35], [36] encountered the same dilemma in the studies of cutting with multiple stability, i.e., the unsafe cutting. To reveal the relationship between the chatter occurrence and the workpiece’s characteristics, they constructed a simple BoA by using the amplitude and frequency of each basis functions [35]. When they tried to improve the approximation accuracy by using a random combination of the basis functions based on Monte Carlo simulation, building a BoA with too many undefined coefficients became impossible. Hence, they used only the basin stability [36].

Based on BoAs, a specified attractor can be enhanced by an intermittent control directionally guiding the ICs in others’ basins towards it without affecting the properties of the original nonlinear systems [37]. To illustrate, Liu et al. [38] switched periodic orbits of a Duffing oscillator and an impact oscillator by an intermittent control, which activates a small impulse towards the basin of the targeted attractor when the systems’ state approaches the basin’s boundary. Similarly, Yadav et al. [39] used a state-dependent intermittent linear augmentation to destroy an unwanted attractor, which made a bistable system monostable at a targeted chaotic attractor. This method is also applicable in the unsafe cutting, i.e., the UZs can be shrunk, and the cutting safety can be improved with the stationary cutting enhanced by intermittent control. Yan et al. [35] introduced a sudden impact to quench the chatter, finding that an impact directed opposite to the tool’s bending velocity should be applied when it begins to increase from zero towards its maximum. Later on in [36] an intermittent control to disturb the large-amplitude chatter was introduced, finding that a large threshold for switching off the disturbance is critical for the control effect. Besides, linear velocity feedback is more effective than cubic velocity feedback and a spindle speed variation.

As discussed above, there are significant challenges with approximation accuracy of the ICs and computation/visualization for non-smooth dynamical systems with delays. This paper proposes some novel ideas, which are applied to the analysis of a nonlinear time-delayed regenerative cutting process to address these issues. In Section 2, a new concept of statistical basin of attraction is proposed for the analysis and visualization of multiple stability in time-delayed systems. Then Section 2 revisits the cutting multiple stability before the statistical basin of attraction applied to relate the probability of chatter occurrence with chatter marks and the tool’s initial bending velocity. A new state-dependent intermittent control is introduced in Section 4 to guide the cutting dynamics to a targeted safe basin, in which the chatter has no chance to occur so that the cutting safety is significantly improved. Finally, some conclusions are drawn in Section 5.

Section snippets

Approximation of the functional delayed terms

Consider the $n$ -dimensional nonlinear time-delayed system in the form as follows: $\frac{d y}{d t} = F (t, y (t), y (t - τ_{1}), y (t - τ_{2}), \dots, y (t - τ_{m})), y \in R^{n \times 1},$ where $0 < τ_{1} < τ_{2} < \dots < τ_{m} = τ$ are $m$ different time delays. Without loss of generality, one can arrange $y (t - τ_{i})$ ( $i = 1, 2, \dots, m$ ) as follows $y (t - τ_{i}) = (\begin{pmatrix} y_{1} (t - τ_{i}) \\ y_{2} (t) \end{pmatrix}), y_{1} = (\begin{pmatrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{l} \end{pmatrix}) \in R^{l \times 1} and y_{2} = (\begin{pmatrix} y_{l + 1} \\ y_{l + 2} \\ ⋮ \\ y_{n} \end{pmatrix}) \in R^{(n - l) \times 1},$ indicating that the time delays appear only in the $l$ preceding elements of $y$ . Specially when $l = 0$ , Eq. (1) becomes ordinary differential equations (ODEs).

When a nonlinear system

Multiple stability and statistical basin of attraction of regenerative cutting dynamics

Next, the concept of SBoA and state-dependent intermittent control described above will be used to explore the multiple stability and enhance the stationary state in regenerative cutting dynamics.

State-dependent intermittent control

Fig. 7 demonstrates to us that all the control of chatter can be cut off once the cutting dynamics enters the safe basin, in which the chatter has no chance to occur. This concept is the state-dependent intermittent control described in Eq. (10), which changes Eq. (12) into $\frac{d y}{d t} = F (y (t), y (t - τ)) + U (s (t)),$ where $U (s (t)) = h (s (t)) u (t)$ . Here $u (t)$ is the control applied out of the safe (targeted) basin, $B_{s} = \{s | P_{c} (s) \leq 0 %\}$ , in which the probability to the stationary cutting is not less than 100%. Based on

Conclusions

To monitor and control the cutting multiple stability in the unsafe zones, this paper has proposed a new concept of statistical basin of attraction. The safe basin with $P_{c} = 0 %$ has been used for the state-dependent intermittent control to improve the cutting safety to decrease the unsafe zones and consequently reduce the probability of chatter occurrence.

The functional initial conditions were approximated by the Fourier series superposed on a straight line. The coefficients of the basis functions

CRediT authorship contribution statement

Yao Yan: Conceptualization, Methodology, Investigation, Writing - original draft. Jian Xu: Supervision, Project administration, Funding acquisition. Marian Wiercigroch: Writing - review & editing. Qing Guo: Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is supported by National Natural Science Foundation of China (Grants No. 11872147, 12072068, 11932015, 11502048, and 11772229), and the Fundamental Research Funds for the Central Universities, China (Grant No. ZYGX2018J078).

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Statistical basin of attraction in time-delayed cutting dynamics: Modelling and computation

Highlights

Abstract

Introduction

Section snippets

Approximation of the functional delayed terms

Multiple stability and statistical basin of attraction of regenerative cutting dynamics

State-dependent intermittent control

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Physica D

Physica D

J. Sound Vib.

Physica D

Commun. Nonlinear Sci. Numer. Simul.

Automatica

Physica D

J. Sound Vib.

Physica D

Procedia IUTAM

Int. J. Non-Linear Mech.

Int. J. Mech. Sci.

Int. J. Mech. Sci.

Int. J. Mech. Sci.

Chaos Solitons Fractals

J. Sound Vib.

Procedia IUTAM

Int. J. Non-Linear Mech.

Physica D

Physica D

Int. J. Mach. Tool Des. Res.

Saturation limits the contribution of acceleration feedback to balancing against reaction delay

J. R. Soc. Interface

A time delay dynamical model for outbreak of 2019-ncov and the parameter identification

J. Inverse Ill-posed Probl.

Desynchronization-based congestion suppression for a star-type internet system with arbitrary dimension

Neurocomputing