86
C. Ozoegwu and P. Eberhard
frequency-domain approach was used to construct stability lobe diagrams that agree
well with those based on impact tests [29].
One problem is clear from studying the above-reviewed works which are based
on time-domain methods. The very complicated modeling and stability analysis of
regenerative thin-walled workpiece milling is mostly done manually. After timedomain discretization, the three major stages for stability analysis of regenerative
chatter of either a milling tool or a milled thin-walled workpiece are chatter state interpolation, monodromy matrix construction, and stability identification/eigenvalue
analysis. Traditionally, the first two stages are done manually on a case-by-case
basis while only the last stage is computerized, and when the degree of interpolation changes, the steps are repeated needing the last stage to be re-computerized.
This work proposes to upgrade the fully general and programmed method in [30] by,
among other things, incorporating modal reduction of elastic thin-walled workpieces
so as to avoid all the stages of case-by-case manual symbolical and numerical analyses
in chatter stability identification. A major motivation is to create a fully computerized framework for the first systematic study of the parametric effect of chatter states
interpolation order on the computational precision of stability lobes identification.
This aim can hardly be achieved on the basis of a partly computerized case-by-case
approach because of the astronomically rising difficulty of symbolic analysis with
rising interpolation order. Another major motivation is to improve the expediency
of practical application since any ordinary industrial operator of thin-walled workpiece milling could make a simulation for precise selection of productive chatter-free
process parameters.
Following this introductory section, Sect. 2 contains the theoretical model and the
stability criteria of the studied system. In Sect. 3, a generalized monodromy matrix
of reduced system is described. The general system is then computed for automatic
study of the parametric effects of interpolation order on computational efficiency and
for automatic simulations of stability diagrams in Sect. 4. Conclusions are presented
in Sect. 5.
2 Model and Stability Condition
The force that excites the regenerative response of flexible tool/rigid workpiece model
have been typically given on the basis of the nonlinear cutting force model as
F(t) = H(t)(z(t) − z(t − τ )),
(1)
where z(t) ∈ R
n d is the state of the milling tool, H(t) ∈ R
n d ×n d is the specific force
variation matrix, and n d is the number of directions of regenerative response of the
tool. Assuming a non-helix tool, the axial component of the cutting force is ignored;
therefore, the regenerative responses of the tool occur in the feed and feed-normal
directions. This means that n d = 2. The matrix-valued function H(t) is typically
given, see [31], by
C. Ozoegwu and P. Eberhard
frequency-domain approach was used to construct stability lobe diagrams that agree
well with those based on impact tests [29].
One problem is clear from studying the above-reviewed works which are based
on time-domain methods. The very complicated modeling and stability analysis of
regenerative thin-walled workpiece milling is mostly done manually. After timedomain discretization, the three major stages for stability analysis of regenerative
chatter of either a milling tool or a milled thin-walled workpiece are chatter state interpolation, monodromy matrix construction, and stability identification/eigenvalue
analysis. Traditionally, the first two stages are done manually on a case-by-case
basis while only the last stage is computerized, and when the degree of interpolation changes, the steps are repeated needing the last stage to be re-computerized.
This work proposes to upgrade the fully general and programmed method in [30] by,
among other things, incorporating modal reduction of elastic thin-walled workpieces
so as to avoid all the stages of case-by-case manual symbolical and numerical analyses
in chatter stability identification. A major motivation is to create a fully computerized framework for the first systematic study of the parametric effect of chatter states
interpolation order on the computational precision of stability lobes identification.
This aim can hardly be achieved on the basis of a partly computerized case-by-case
approach because of the astronomically rising difficulty of symbolic analysis with
rising interpolation order. Another major motivation is to improve the expediency
of practical application since any ordinary industrial operator of thin-walled workpiece milling could make a simulation for precise selection of productive chatter-free
process parameters.
Following this introductory section, Sect. 2 contains the theoretical model and the
stability criteria of the studied system. In Sect. 3, a generalized monodromy matrix
of reduced system is described. The general system is then computed for automatic
study of the parametric effects of interpolation order on computational efficiency and
for automatic simulations of stability diagrams in Sect. 4. Conclusions are presented
in Sect. 5.
2 Model and Stability Condition
The force that excites the regenerative response of flexible tool/rigid workpiece model
have been typically given on the basis of the nonlinear cutting force model as
F(t) = H(t)(z(t) − z(t − τ )),
(1)
where z(t) ∈ R
n d is the state of the milling tool, H(t) ∈ R
n d ×n d is the specific force
variation matrix, and n d is the number of directions of regenerative response of the
tool. Assuming a non-helix tool, the axial component of the cutting force is ignored;
therefore, the regenerative responses of the tool occur in the feed and feed-normal
directions. This means that n d = 2. The matrix-valued function H(t) is typically
given, see [31], by