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Journal of Computational and Nonlinear Dynamics | Volume 12 | Issue 1 | research-article
Research Papers

Targeted Energy Transfers for Suppressing Regenerative Machine Tool Vibrations

[+] Author and Article Information
Amir Nankali

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: nankali@umich.edu

Young S. Lee

Department of Mechanical
and Aerospace Engineering,
New Mexico State University,
Las Cruces, NM 88003
e-mail: younglee@nmsu.edu

Tamás Kalmár-Nagy

Department of Fluid Mechanics,
Faculty of Mechanical Engineering,
Budapest University of Technology
and Economics,
Budapest 1111, Hungary
e-mail: kalmarnagy@ara.bme.hu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 4, 2016; final manuscript received July 24, 2016; published online September 1, 2016. Assoc. Editor: Brian Feeny.

J. Comput. Nonlinear Dynam 12(1), 011010 (Sep 01, 2016) (11 pages) Paper No: CND-16-1055; doi: 10.1115/1.4034397 History: Received February 04, 2016; Revised July 24, 2016
Article
References
Figures
Tables

We study the dynamics of targeted energy transfers in suppressing chatter instability in a single-degree-of-freedom (SDOF) machine tool system. The nonlinear regenerative (time-delayed) cutting force is a main source of machine tool vibrations (chatter). We introduce an ungrounded nonlinear energy sink (NES) coupled to the tool, by which energy transfers from the tool to the NES and efficient dissipation can be realized during chatter. Studying variations of a transition curve with respect to the NES parameters, we analytically show that the location of the Hopf bifurcation point is influenced only by the NES mass and damping coefficient. We demonstrate that application of a well-designed NES renders the subcritical limit cycle oscillations (LCOs) into supercritical ones, followed by Neimark–Sacker and saddle-node bifurcations, which help to increase the stability margin in machining. Numerical and asymptotic bifurcation analyses are performed and three suppression mechanisms are identified. The asymptotic stability analysis is performed to study the domains of attraction for these suppression mechanisms which exhibit good agreement with the bifurcations sets obtained from the numerical continuation methods. The results will help to design nonlinear energy sinks for passive control of regenerative instabilities in machining.
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References

Figures

Grahic Jump Location
Fig. 1

Machine tool model with an ungrounded NES attached

+Machine tool model with an ungrounded NES attached
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Machine tool model with an ungrounded NES attached
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Fig. 2

(a) Stability chart and (b) the eigenfrequency range for the LCO at each lobe for a regenerative SDOF machine tool model

+(a) Stability chart and (b) the eigenfrequency range for the LCO at each lobe for a regenerative SDOF machine tool model
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(a) Stability chart and (b) the eigenfrequency range for the LCO at each lobe for a regenerative SDOF machine tool model
Grahic Jump Location
Fig. 3

The minimum eigenfrequency ωmin (a) with respect to the mass ratio ϵ and the damping factor ζ1; (b) versus ζ1 (for different ϵ's) and versus ϵ (for different ζ1's)

+The minimum eigenfrequency ωmin (a) with respect to the mass ratio ϵ and the damping factor ζ1; (b) versus ζ1 (for different ϵ's) and versus ϵ (for different ζ1's)
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The minimum eigenfrequency ωmin (a) with respect to the mass ratio ϵ and the damping factor ζ1; (b) versus ζ1 (for different ϵ's) and versus ϵ (for different ζ1's)
Grahic Jump Location
Fig. 4

Stability chart (a) and the eigenfrequency range (b) by varying the mass ratio ϵ for ζ1=0.1 and C = 0.5 [24]

+Stability chart (a) and the eigenfrequency range (b) by varying the mass ratio ϵ for ζ1=0.1 and C = 0.5 [24]
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Stability chart (a) and the eigenfrequency range (b) by varying the mass ratio ϵ for ζ1=0.1 and C = 0.5 [24]
Grahic Jump Location
Fig. 5

Typical tool displacements (left) [24] and their wavelet transforms (right) for the three suppression mechanisms. From top to bottom, each of the rows represents the first-, second-, and third-suppression mechanisms, respectively.

+Typical tool displacements (left) [24] and their wavelet transforms (right) for the three suppression mechanisms. From top to bottom, each of the rows represents the first-, second-, and third-suppression mechanisms, respectively.
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Typical tool displacements (left) [24] and their wavelet transforms (right) for the three suppression mechanisms. From top to bottom, each of the rows represents the first-, second-, and third-suppression mechanisms, respectively.
Grahic Jump Location
Fig. 6

Bifurcation diagram for the tool and NES amplitudes obtained by DDEBIFTOOL [11], with truncated nonlinear terms (Ω=2.6; ϵ=0.2,ζ1=0.1,C=0.5): H, LPC, and NS denote HOPF, limit point cycle, and Neimark–Sacker bifurcation points, respectively [24]

+Bifurcation diagram for the tool and NES amplitudes obtained by DDEBIFTOOL [11], with truncated nonlinear terms (Ω=2.6; ϵ=0.2,ζ1=0.1,C=0.5): H, LPC, and NS denote HOPF, limit point cycle, and Neimark–Sacker bifurcation points, respectively [24]
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Bifurcation diagram for the tool and NES amplitudes obtained by DDEBIFTOOL [11], with truncated nonlinear terms (Ω=2.6; ϵ=0.2,ζ1=0.1,C=0.5): H, LPC, and NS denote HOPF, limit point cycle, and Neimark–Sacker bifurcation points, respectively [24]
Grahic Jump Location
Fig. 7

Displacements and their wavelet transforms for a typical first-suppression mechanism; modal energy exchanges are depicted at the bottom [24]

+Displacements and their wavelet transforms for a typical first-suppression mechanism; modal energy exchanges are depicted at the bottom [24]
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Displacements and their wavelet transforms for a typical first-suppression mechanism; modal energy exchanges are depicted at the bottom [24]
Grahic Jump Location
Fig. 8

LCO surfaces for the system (5) with contact loss conditions (ϵ=0.1, ζ=ζ1=0.1, C=0.5) [24]

+LCO surfaces for the system (5) with contact loss conditions (ϵ=0.1, ζ=ζ1=0.1, C=0.5) [24]
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LCO surfaces for the system (5) with contact loss conditions (ϵ=0.1, ζ=ζ1=0.1, C=0.5) [24]
Grahic Jump Location
Fig. 9

Comparison between direct numerical simulation and analytical approximation (a) and 1:1 resonance capture in the slow-flow phase plane (b) for ζ=0.1,ζ1=0.4,ϵ=0.2,Ω=2.6,p=0.8

+Comparison between direct numerical simulation and analytical approximation (a) and 1:1 resonance capture in the slow-flow phase plane (b) for ζ=0.1,ζ1=0.4,ϵ=0.2,Ω=2.6,p=0.8
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Comparison between direct numerical simulation and analytical approximation (a) and 1:1 resonance capture in the slow-flow phase plane (b) for ζ=0.1,ζ1=0.4,ϵ=0.2,Ω=2.6,p=0.8
Grahic Jump Location
Fig. 10

First-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.7 (a) SIM and super-slow flow and (b) time response

+First-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.7 (a) SIM and super-slow flow and (b) time response
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First-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.7 (a) SIM and super-slow flow and (b) time response
Grahic Jump Location
Fig. 11

Second-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.67; (a) SIM and super-slow flow and (b) time response

+Second-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.67; (a) SIM and super-slow flow and (b) time response
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Second-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.67; (a) SIM and super-slow flow and (b) time response
Grahic Jump Location
Fig. 12

Third-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.65; (a) SIM and super-slow flow configuration and (b) time response

+Third-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.65; (a) SIM and super-slow flow configuration and (b) time response
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Third-suppression mechanism; ζ1=.2,ϵ=.02,ζ=.1,Ω=2.6,p=.65; (a) SIM and super-slow flow configuration and (b) time response
Grahic Jump Location
Fig. 13

No suppression; ζ1=0.2,ϵ=0.02,ζ=0.1,Ω=2.6,p=0.765; (a) SIM and super-slow flow and (b) time response

+No suppression; ζ1=0.2,ϵ=0.02,ζ=0.1,Ω=2.6,p=0.765; (a) SIM and super-slow flow and (b) time response
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No suppression; ζ1=0.2,ϵ=0.02,ζ=0.1,Ω=2.6,p=0.765; (a) SIM and super-slow flow and (b) time response
Grahic Jump Location
Fig. 14

Basins of attraction for the three suppression mechanisms; ε=0.2,ζ=0.1,Ω=2.6

+Basins of attraction for the three suppression mechanisms; ε=0.2,ζ=0.1,Ω=2.6
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Basins of attraction for the three suppression mechanisms; ε=0.2,ζ=0.1,Ω=2.6
Grahic Jump Location
Fig. 15

Analytical (—) and Numerical (- - -) basins of attraction for ϵ=.02. The region for first-, second-, and third-suppression mechanisms are denoted by I, II, and III, respectively. Shilnikov Homoclinic bifurcation is indicated by the dashed–dotted line.

+Analytical (—) and Numerical (- - -) basins of attraction for ϵ=.02. The region for first-, second-, and third-suppression mechanisms are denoted by I, II, and III, respectively. Shilnikov Homoclinic bifurcation is indicated by the dashed–dotted line.
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Analytical (—) and Numerical (- - -) basins of attraction for ϵ=.02. The region for first-, second-, and third-suppression mechanisms are denoted by I, II, and III, respectively. Shilnikov Homoclinic bifurcation is indicated by the dashed–dotted line.
Grahic Jump Location
Fig. 16

Shilnikov Homoclinic bifurcation; ζ1=0.2,ε=0.02,ζ=0.1,Ω=2.6,p=0.755

+Shilnikov Homoclinic bifurcation; ζ1=0.2,ε=0.02,ζ=0.1,Ω=2.6,p=0.755
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Shilnikov Homoclinic bifurcation; ζ1=0.2,ε=0.02,ζ=0.1,Ω=2.6,p=0.755
Grahic Jump Location
Fig. 17

Bifurcation surface and basin of attraction of suppression mechanisms: — ,  analytical;  - - -, numerical; ε=0.2,ζ=0.1;Ω=2.6

+Bifurcation surface and basin of attraction of suppression mechanisms: — ,  analytical;  - - -, numerical; ε=0.2,ζ=0.1;Ω=2.6
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Bifurcation surface and basin of attraction of suppression mechanisms: — ,  analytical;  - - -, numerical; ε=0.2,ζ=0.1;Ω=2.6
Grahic Jump Location
Fig. 18

Bifurcation surface and basin of attraction of suppression mechanisms; ζ1=0.4,ζ=0.1,Ω=2.6

+Bifurcation surface and basin of attraction of suppression mechanisms; ζ1=0.4,ζ=0.1,Ω=2.6
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Bifurcation surface and basin of attraction of suppression mechanisms; ζ1=0.4,ζ=0.1,Ω=2.6

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