Suppressing Regenerative Chatter Instability by Means of Targeted ...

ENOC 2011, 24-29 July 2011, Rome, Italy
Suppressing Regenerative Chatter Instability by Means of Targeted Energy Transfers
Harsheeta Surampalli * , Young S. Lee * , Tamás Kalmár-Nagy **
* Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces,
NM 88003, USA
** Department of Aerospace Engineering, Texas A&M University, College Station, TX 77845, USA
Summary. We study the dynamics of targeted energy transfers in suppressing regenerative chatter instability in a single-degree-offreedom
machine tool model by introducing an ungrounded nonlinear energy sink (NES). Studying variations of a stability boundary
with respect to the NES parameters, we show that Hopf bifurcation associated with chatter instability is affected only by the NES
mass and damping coefficient, but not by the NES nonlinear stiffness. Three mechanisms for chatter suppression are identified; that
is, recurrent burst-outs and suppressions, and partial and complete suppressions of instabilities. By means of a numerical
continuation method through DDEBIFTOOL, we explore topological changes in the bifurcation structure with respect to NES
parameters; furthermore, utilizing the method of Galerkin projection and complexification-averaging technique, we analytically
investigate those suppression mechanisms for regenerative instabilities. Robustness of suppression in terms of domains of attraction
will be explored.
Introduction
Kalmár-Nagy et al. [1] studied a nonlinear dynamics of a single-degree-of-freedom machine tool model (cf. Fig. 1a),
where the equation of motion can be written as a delay-differential equation due to regenerative effects. Taylor-seriesexpanding
and retaining the regenerative force up to the cubic order, stability boundaries where chatter instability
occurs can be calculated in the domain of chip thickness and turning speed (Fig. 1b). It was also shown with the help of
Center Manifold Theorem that the limit cycle oscillations (LCO) due to chatter instability develop as being subcritical.
Existence of such subcritical LCOs, which would involve large-amplitude vibrations of a machine tool in practice, can
be detrimental in precision machining.
(a) (b) (c)
Figure 1. Schematic of a turning process and stability boundary
In order to attenuate or even eliminate such regenerative instability, an ungrounded nonlinear energy sink (NES) was
utilized (Fig. 1c). Various applications of NES as a passive broadband boundary controller have been studied; in
particular, it was demonstrated that aeroelastic instabilities or LCOs can be efficiently and robustly suppressed by the
action of NES involving vigorous nonlinear modal interactions (or targeted energy transfers – TETs) to prevent LCO
triggering (See, for example, [2]). The objective of this work is to seek for similar performance of an NES but possibly
involving nonlinear dynamics peculiar to time-delayed system.
Bifurcation Analysis and Suppression Mechanisms
After establishing the equations of motion for the 2-DOF coupled oscillators (cf. Fig. 1c), the conditions for stability
boundary (i.e., a set of Hopf bifurcation points) were computed with respect to NES parameters. As indicated in Fig.
2a, the most critical parameter for enhancing chatter stability is the NES mass, where stability enhancement is discussed
in terms of the amount of chip thickness that a workpiece can be processed without causing any chatter under small
disturbance. On the other hand, the other two NES parameters (damping factor and nonlinear coefficient) have nonsignificant
influence on shifting the stability boundary upward. Now, performing numerical continuation by means of
DDEBIFTOOL [3], bifurcation of LCO was investigated. Figure 2(b) depicts a typical bifurcation diagram when an
NES is applied. First of all, it is noted that the occurrence of Hopf bifurcation is delayed up to about 30% of the chip
thickness, where complete elimination of chatter instability is guaranteed under small disturbances. Then, the LPC (or
saddle-node) bifurcation point renders the unstable LCO branch generated by subcritical Hopf bifurcation to be stable,
which appears as a partial suppression mechanism. Finally, the Neimark-Sacker bifurcations in the middle branch

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.