Suppressing Regenerative Chatter Instability by Means of Targeted ...
Suppressing Regenerative Chatter Instability by Means of Targeted ...
Suppressing Regenerative Chatter Instability by Means of Targeted ...
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ENOC 2011, 24-29 July 2011, Rome, Italy<br />
<strong>Suppressing</strong> <strong>Regenerative</strong> <strong>Chatter</strong> <strong>Instability</strong> <strong>by</strong> <strong>Means</strong> <strong>of</strong> <strong>Targeted</strong> Energy Transfers<br />
Harsheeta Surampalli * , Young S. Lee * , Tamás Kalmár-Nagy **<br />
* Department <strong>of</strong> Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces,<br />
NM 88003, USA<br />
** Department <strong>of</strong> Aerospace Engineering, Texas A&M University, College Station, TX 77845, USA<br />
Summary. We study the dynamics <strong>of</strong> targeted energy transfers in suppressing regenerative chatter instability in a single-degree-<strong>of</strong>freedom<br />
machine tool model <strong>by</strong> introducing an ungrounded nonlinear energy sink (NES). Studying variations <strong>of</strong> a stability boundary<br />
with respect to the NES parameters, we show that Hopf bifurcation associated with chatter instability is affected only <strong>by</strong> the NES<br />
mass and damping coefficient, but not <strong>by</strong> the NES nonlinear stiffness. Three mechanisms for chatter suppression are identified; that<br />
is, recurrent burst-outs and suppressions, and partial and complete suppressions <strong>of</strong> instabilities. By means <strong>of</strong> a numerical<br />
continuation method through DDEBIFTOOL, we explore topological changes in the bifurcation structure with respect to NES<br />
parameters; furthermore, utilizing the method <strong>of</strong> Galerkin projection and complexification-averaging technique, we analytically<br />
investigate those suppression mechanisms for regenerative instabilities. Robustness <strong>of</strong> suppression in terms <strong>of</strong> domains <strong>of</strong> attraction<br />
will be explored.<br />
Introduction<br />
Kalmár-Nagy et al. [1] studied a nonlinear dynamics <strong>of</strong> a single-degree-<strong>of</strong>-freedom machine tool model (cf. Fig. 1a),<br />
where the equation <strong>of</strong> motion can be written as a delay-differential equation due to regenerative effects. Taylor-seriesexpanding<br />
and retaining the regenerative force up to the cubic order, stability boundaries where chatter instability<br />
occurs can be calculated in the domain <strong>of</strong> chip thickness and turning speed (Fig. 1b). It was also shown with the help <strong>of</strong><br />
Center Manifold Theorem that the limit cycle oscillations (LCO) due to chatter instability develop as being subcritical.<br />
Existence <strong>of</strong> such subcritical LCOs, which would involve large-amplitude vibrations <strong>of</strong> a machine tool in practice, can<br />
be detrimental in precision machining.<br />
(a) (b) (c)<br />
Figure 1. Schematic <strong>of</strong> a turning process and stability boundary<br />
In order to attenuate or even eliminate such regenerative instability, an ungrounded nonlinear energy sink (NES) was<br />
utilized (Fig. 1c). Various applications <strong>of</strong> NES as a passive broadband boundary controller have been studied; in<br />
particular, it was demonstrated that aeroelastic instabilities or LCOs can be efficiently and robustly suppressed <strong>by</strong> the<br />
action <strong>of</strong> NES involving vigorous nonlinear modal interactions (or targeted energy transfers – TETs) to prevent LCO<br />
triggering (See, for example, [2]). The objective <strong>of</strong> this work is to seek for similar performance <strong>of</strong> an NES but possibly<br />
involving nonlinear dynamics peculiar to time-delayed system.<br />
Bifurcation Analysis and Suppression Mechanisms<br />
After establishing the equations <strong>of</strong> motion for the 2-DOF coupled oscillators (cf. Fig. 1c), the conditions for stability<br />
boundary (i.e., a set <strong>of</strong> Hopf bifurcation points) were computed with respect to NES parameters. As indicated in Fig.<br />
2a, the most critical parameter for enhancing chatter stability is the NES mass, where stability enhancement is discussed<br />
in terms <strong>of</strong> the amount <strong>of</strong> chip thickness that a workpiece can be processed without causing any chatter under small<br />
disturbance. On the other hand, the other two NES parameters (damping factor and nonlinear coefficient) have nonsignificant<br />
influence on shifting the stability boundary upward. Now, performing numerical continuation <strong>by</strong> means <strong>of</strong><br />
DDEBIFTOOL [3], bifurcation <strong>of</strong> LCO was investigated. Figure 2(b) depicts a typical bifurcation diagram when an<br />
NES is applied. First <strong>of</strong> all, it is noted that the occurrence <strong>of</strong> Hopf bifurcation is delayed up to about 30% <strong>of</strong> the chip<br />
thickness, where complete elimination <strong>of</strong> chatter instability is guaranteed under small disturbances. Then, the LPC (or<br />
saddle-node) bifurcation point renders the unstable LCO branch generated <strong>by</strong> subcritical Hopf bifurcation to be stable,<br />
which appears as a partial suppression mechanism. Finally, the Neimark-Sacker bifurcations in the middle branch
ENOC 2011, 24-29 July 2011, Rome, Italy<br />
indicate quasiperiodic responses or recurrent burst-outs and suppressions <strong>of</strong> chatter instability. Typical time responses<br />
are depicted in Fig. 3, where instantaneous frequency (as wavelet transform spectra) and modal energy exchange are<br />
considered. Indeed, all the three suppression mechanisms studied in aeroelastic applications [2] can be observed, and<br />
therefore, similar interpretation <strong>of</strong> dynamics can be made.<br />
(a) (b)<br />
Figure 2. Effects <strong>of</strong> NES: (a) Shift-up <strong>of</strong> stability boundary (i.e., Hopf bifurcation point); (b) bifurcation diagram <strong>of</strong> the tool and NES<br />
displacements for 2.6 and the NES parameters 0.2, 1<br />
0.1, C 0.5 (H, LPC and NS denote Hopf, limit point cycle,<br />
Neimark-Sacker bifurcation points, respectively; DDEBIFTOOL was implemented).<br />
Figure 3. Typical responses for recurrent burst-outs and suppressions <strong>of</strong> chatter instability for the NES parameters in Fig. 2b. Zero<br />
initial conditions except for the tool displacement <strong>of</strong> 0.5 were used.<br />
Concluding Remarks<br />
<strong>Suppressing</strong> chatter instabilities <strong>by</strong> means <strong>of</strong> targeted energy transfers was studied <strong>by</strong> applying an ungrounded<br />
nonlinear energy sink to a single-DOF machine tool model with truncated regenerative nonlinearities. Three<br />
suppression mechanisms were identified, which are closely related with the topological structure <strong>of</strong> bifurcation<br />
phenomena. An example for the suppression mechanism involving recurrent burst-outs and suppressions <strong>of</strong> chatter<br />
instability was presented, where the instantaneous frequency and energy evidenced nonlinear modal interactions (i.e., a<br />
series <strong>of</strong> 1:1 transient resonance captures and escapes from resonance) between the tool and the NES.<br />
Acknowledgment<br />
This work was supported in part <strong>by</strong> National Science Foundation <strong>of</strong> United States, Grant Numbers CMMI-0928062 (YL)<br />
and CMMI-0846783 (TK).<br />
References<br />
[1] Kalmár-Nagy T., Stépán G., Moon F.C. (2001) Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations. Nonlinear<br />
Dynamics 26:121-142.<br />
[2] Lee Y.S., Vakakis A.F., Bergman L.A. McFarland D.M., Kerschen G. (2007) Suppression <strong>of</strong> Aeroelastic <strong>Instability</strong> <strong>by</strong> <strong>Means</strong> <strong>of</strong> Broadband Passive<br />
<strong>Targeted</strong> Energy Transfers, Part I: Theory. AIAA Journal 45(3): 693-711.<br />
[3] Engelborghs K., Luzyanina T., Roose D. (2002) Numerical Bifurcation Analysis <strong>of</strong> Delay Differential Equations using DDE-BIFTOOL. ACM<br />
Transactions on Mathematical S<strong>of</strong>tware 28(1): 1-21.