Suppressing Regenerative Chatter Instability by Means of Targeted ...

ENOC 2011, 24-29 July 2011, Rome, Italy
indicate quasiperiodic responses or recurrent burst-outs and suppressions of chatter instability. Typical time responses
are depicted in Fig. 3, where instantaneous frequency (as wavelet transform spectra) and modal energy exchange are
considered. Indeed, all the three suppression mechanisms studied in aeroelastic applications [2] can be observed, and
therefore, similar interpretation of dynamics can be made.
(a) (b)
Figure 2. Effects of NES: (a) Shift-up of stability boundary (i.e., Hopf bifurcation point); (b) bifurcation diagram of the tool and NES
displacements for 2.6 and the NES parameters 0.2, 1
0.1, C 0.5 (H, LPC and NS denote Hopf, limit point cycle,
Neimark-Sacker bifurcation points, respectively; DDEBIFTOOL was implemented).
Figure 3. Typical responses for recurrent burst-outs and suppressions of chatter instability for the NES parameters in Fig. 2b. Zero
initial conditions except for the tool displacement of 0.5 were used.
Concluding Remarks
Suppressing chatter instabilities by means of targeted energy transfers was studied by applying an ungrounded
nonlinear energy sink to a single-DOF machine tool model with truncated regenerative nonlinearities. Three
suppression mechanisms were identified, which are closely related with the topological structure of bifurcation
phenomena. An example for the suppression mechanism involving recurrent burst-outs and suppressions of chatter
instability was presented, where the instantaneous frequency and energy evidenced nonlinear modal interactions (i.e., a
series of 1:1 transient resonance captures and escapes from resonance) between the tool and the NES.
Acknowledgment
This work was supported in part by National Science Foundation of United States, Grant Numbers CMMI-0928062 (YL)
and CMMI-0846783 (TK).
References
[1] Kalmár-Nagy T., Stépán G., Moon F.C. (2001) Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations. Nonlinear
Dynamics 26:121-142.
[2] Lee Y.S., Vakakis A.F., Bergman L.A. McFarland D.M., Kerschen G. (2007) Suppression of Aeroelastic Instability by Means of Broadband Passive
Targeted Energy Transfers, Part I: Theory. AIAA Journal 45(3): 693-711.
[3] Engelborghs K., Luzyanina T., Roose D. (2002) Numerical Bifurcation Analysis of Delay Differential Equations using DDE-BIFTOOL. ACM
Transactions on Mathematical Software 28(1): 1-21.