6.3
Application of structural modification the-
ory to milling
Since the geometry, and more specifically the diameter, of the tool holder has a significant effect on the tool tip dynamics, and in turn the stability of the milling operation, a simple method of avoiding chatter would be to choose a tool holder that results in stability lobes at a preferred spindle speed. Using the direct SM method presented above, this section will show that, it is possible to predict the dynamics of a number of different tool holders with a single experimental model. Whilst this is useful, it is a long winded method of optimising the tool holder geometry. However, this section will also show that, since the location of the stability lobes is directly proportional to the natural frequencies of the structure, the inverse method may be used to optimise to the geometry by solving a single equation.
6.3.1
The stability of a milling operation
In order to apply SM theory to the chatter avoidance problem, it is helpful to revisit the average tooth angle approach for the prediction of stability lobe di- agrams. Tlusty’s method (section 3.2.1, Eqns. (3.34) and (3.32)) gives us that
blim = −1 2KsRe[FRFor]Nt∗ where Nt∗ = Nt(ϕe− ϕs) 360 (6.8)
where blim is the stability limit of the axial depth of cut in m, Ks is the specific
force, FRFor is the oriented FRF (a linear combination of the x- and y-direction
FRFs), N∗
t is the average number of teeth in the cut at any one time, Nt is the
number of teeth on the tool, and ϕe and ϕs are the exit and entry angle of the
tool respectively. The stability limit also depends upon the spindle speed of the operation and this is given by
Ω = fc
Nt(L + 2π )
where = 2π − 2 tan−1 Re[FRFor] Im[FRFor]
GEOMETRY
where Ω is the spindle speed in revolutions per second, fc is the valid chatter
frequency range, and L ∈ Z is the lobe number. In order to produce a stability lobe diagram, Ω is calculated for a number of lobes (N = 0, 1, 2, . . .) over the valid chatter frequency range(s) i.e. where Re[FRFor]≤ 0, then each of these lobes are
plotted against blim.
Eq. (6.8) tells us that the stability limit tends to infinity as the real part of the oriented FRF tends to zero. Since the real part of any FRF will tend to zero at resonance, by shifting the resonance frequencies, the peaks in the stability lobes are also shifted. Considering Eq. (6.9) at resonance (where Re[FRFor]= 0),we
have that
Ωbest=
fn
Nt(N + 1)
(6.10) Where Ωbest is the location of the peak in the stability lobes, and fn is the natural
frequency of the oriented FRF in Hz. The oriented FRF is given by:
FRFor = µxFRFx+ µyFRFy (6.11)
where FRFx and FRFy are the frequency response functions is the x-and y-
directions respectively, and µx and µy are the directional orientation factors used
to project the cutting force model onto the surface normal direction.
Hence, the tool tip FRF in both the x- and y-directions are required to predict the stability of a given milling operation. Whilst this has no effect on the appli- cation of the direct SM method to high speed milling, as FRFx and FRFy can be
predicted individually (using Eq. (6.5)) and used to generate the stability lobe diagram. When applying the inverse method, the location of the optimum spin- dle speed (Ωbest) is dependent not on the resonant frequencies of the individual
FRFs but on FRFor, therefore the inverse method in Eq. (6.6) must be updated
6.3. APPLICATION OF STRUCTURAL MODIFICATION THEORY TO MILLING
6.3.2
Inverse structural modification of tool holder geom-
etry
It will now be shown that, because of the properties inherent in any FRF curve, by shifting the resonance of either the x- or y-direction using the inverse method, the resonance of the oriented FRF is also shifted.
Consider an experimental model of some standard tool holder HA, and a partially
constructed numerical model (B(d)) with diameter d unknown. The inverse SM method set out in Eq. (6.6) may be applied to optimise d in terms of the natural frequency of either the x or y direction FRFs. Now, let HxA be an experimental model in the x direction, and let HyAbe an experimental model in the y direction, then the inverse SM equations are given by:
HxC = adj(I + H x AB(d))HxA det(I + HxAB(d)) HyC = adj(I + H y AB(d))H y A det(I + HyAB(d)) (6.12)
where Matrix partitions have been removed for simplicity. However, since the location of the stability lobes depends on FRFor, the equations in Eq. (6.12)
must be combined in the same manner as FRFor, this gives
µxHxC+ µyHyC = µx adj(I + HxAB(d))HxA det(I + HxAB(d)) + µy adj(I + HyAB(d))HyA det(I + HyAB(d)) = det(I + H y AB(d)) det(I + HxAB(d))µx[I + HxAB(d)]−1HxA+ µy[(I + HyAB(d)]−1H y A det(I + HxAB(d)) det(I + HyAB(d)) (6.13) Then, the natural frequencies of the oriented FRF can be set by solving
det(I + HxAB(d)) det(I + HyAB(d)) = 0 (6.14)
which clearly has solutions when either det(I+HxAB(d)) = 0 or det(I+HyAB(d)) = 0. Therefore, the inverse SM method, as set out in Eqs. (6.6) and (6.7) can be applied in either the x- or y-directions in order to reposition the stability lobes.
GEOMETRY
This is a standard property for any structure; when adding two FRFs together, the infinitesimal nature of resonance in either curve will also cause resonance in their sum.
When selecting which direction to modify, it is important to keep in mind that the most dominant mode (the mode with the largest amplitude) will have the lowest stability limit and thus determine the absolute stability limit; therefore, this mode should be optimised. However the magnitude of µx and µy should also
be taken into consideration, as they may change the dominance of the modes.