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
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 ,
has the units of length, its relation to the flank wear land length
is reported in  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  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
. 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
represents the total number of lobes.
, but when
is finite and given by greatest integer less than
, obtained by simultaneous solution of
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
. Then, using root-finding, the bounds
corresponding to the intersection of the
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
is also displayed.
, you can study the effect of all the variables. The
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  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.
 Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design
, Cambridge: Cambridge University Press, 2012.
 Y. Altintas, M. Eynian, and H. Onozuka, "Identification of Dynamic Cutting Force Coefficients and Chatter Stability with Process Damping," CIRP Annals
, 2008 pp. 371–374. dx.doi.org/10.1016/j.cirp.2008.03.048
 S-G. Chen, A. G. Ulsoy, and Y. Koren, "Computational Stability Analysis of Chatter in Turning," Journal of Manufacturing Science and Engineering
(4), 1997 pp. 457–460. dx.doi.org/10.1115/1.2831174