Chatter Stability with Orthogonal Rotation

The classical model of regenerative chatter is a simple harmonic oscillator excited by a vibration-dependent force with constant delay. This results in a delay-differential equation whose stability analysis gives rise to a chart demarcating the nondimensionalized depth-of-cut and the rotational speed . The well-developed theory available in [1, 2] is adapted for a general overlap factor and process damping . This Demonstration studies the effect of natural frequency and structural damping ratio on the - space and chatter frequency at the threshold of stability. Process damping is important at low rotational speeds and high lobe numbers, where the lobed borderline of stability is completely separated from the asymptotic borderline.


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Chatter vibration in machining processes is an extremely complex phenomenon; field troubleshooting relies on rigorous data collection and diagnosis by specialists. It also entails expensive downtime and high scrap costs. Chatter is a well-known case of self-excited regenerative vibration, where energy is extracted from the machining process itself. This Demonstration provides a basic insight into the simplest case, that of orthogonal turning. Here, a useful output of the theoretical analysis is the stability chart that demarcates the space of depth of cut and rotational speed into stable and unstable regimes. The stability chart is constructed through a sequence of lobes, each of which prescribes an unstable region.
How the tool and the workpiece interact is modeled as a single-degree-of-freedom system with given resonance frequency and damping ratio . The cutting stiffness is the force per unit of uncut chip thickness. The nondimensionalized depth of cut is defined as the ratio of the cutting stiffness to the machine stiffness. With as the depth of cut, as the specific energy, and as the structural stiffness, is given by . In the literature, is sometimes referred to as the stiffness ratio.
Process damping [1, 2] refers to energy dissipation by the rubbing of the tool flank on the wavy workpiece surface. It is significant at low rotational speeds with a high lobe number. If ignored, the deduced stability chart conflicts with the shop practice of reducing rotational speed to eliminate chatter. That is, if the process is unstable at a higher rotational speed, without process damping, it is necessarily unstable at a lower speed. Another important consideration is the overlap factor, usually assumed to be unity in the literature, that captures the influence of vibration position in the previous rotation.
The workpiece, rotating at a constant rate of rotations per minute, takes the time duration to complete one rotation. Therefore, . The vibration of the tool as a function of time is denoted by . With as the diameter of the workpiece, the cutting speed of the operation is ; is the overlap factor. Process damping is incorporated through the coefficient . With reference to Eq. 4.22 in [1], . Though has the units of length, its relation to the flank wear land length is reported in [1] to be . The governing differential equation with this nomenclature is then given by
With this formulation, the - space has to be demarcated into stable or unstable regimes. The quantity can be suitably absorbed into and therefore ignored in the subsequent stability analysis. The state-space method in [3] is a robust way to compute the stability lobes since it employs a free variable with fixed bounds.
Here, the free variable , , is related to the fractional vibrational cycle in one complete rotation of the workpiece and represents the angular chatter frequency. The real and imaginary parts of are zero at the threshold of stability (TOS). These are given by
These equations have to be solved simultaneously with , where . Each value of gives rise to a stability lobe. The union of all the unstable zones for each gives the complete unstable region in - space, the complement of which is the stable region. At TOS, is given by
The chatter frequency and workpiece rotational speed at TOS are given by
and represents the total number of lobes. when , but when , is finite and given by greatest integer less than , obtained by simultaneous solution of and . Also, is given by
The intersections of the lobes can be solved to give the lobed borderline of stability (LBS). The lobes are first restricted to bounds determined from . Then, using root-finding, the bounds corresponding to the intersection of the and lobes are found by simultaneous solution of the equations
The asymptotic borderline of stability (ABS), governed purely by structural damping, is given by . The stable region is indicated by the presence of the word "stable". The number of lobes for given is also displayed.
By varying , , , and , you can study the effect of all the variables. The and sliders allow rapid zooming in and out to study the stable/unstable regions more closely. The charts have been corroborated with data on p. 139 in [1] and bookmarked in the Demonstration. The chatter frequency at the threshold of stability is shown as a varying color on the LBS. At low , the LBS curve can be seen to move away from the ABS curve completely due to process damping. Its influence on chatter frequency can also be seen clearly.
[1] Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge: Cambridge University Press, 2012.
[2] Y. Altintas, M. Eynian, and H. Onozuka, "Identification of Dynamic Cutting Force Coefficients and Chatter Stability with Process Damping," CIRP Annals, 57, 2008 pp. 371–374. dx.doi.org/10.1016/j.cirp.2008.03.048.
[3] S-G. Chen, A. G. Ulsoy, and Y. Koren, "Computational Stability Analysis of Chatter in Turning," Journal of Manufacturing Science and Engineering, 119(4), 1997 pp. 457–460. dx.doi.org/10.1115/1.2831174.
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