Biomedical Engineering Reference
In-Depth Information
where: each h
ij
is evaluated from the receptance coupling expression by
including both translational (x) and rotational (q) displacements due to lateral
force (f) and moment (M).
The direct and cross receptances at the tool tip need to be expanded as
well as:
h
h
h
h
h
h
h
h
1
A
11,
ff
A
11,
fM
A
12,
ff
A
12,
fM
2,
MM
2,
fM
A
21,
ff
A
21,
fM
H
11
2
h
h
h
h
h
h
h
h
h
h
h
A
11,
Mf
A
11,
MM
A
12,
Mf
A
12,
MM
2
Mf
2,
ff
A
21,
Mf
A
21,
MM
2,
Mf
2,
MM
2,
ff
h
h
h
h
h
h
1
A
12,
ff
A
12,
fM
A
12,
ff
A
12,
fM
2,
MM
2,
fM
A
22,
ff
A
22,
fM
H
12
2
h
h
h
h
h
h
h
h
h
A
12,
Mf
A
12,
MM
A
12,
Mf
A
12,
MM
2
Mf
2,
ff
A
22,
Mf
A
22,
MM
2,
Mf
2,
MM
2,
ff
The elements that represent the rotational degrees of freedom are complex
to determine and they can be expressed trough rewritten equations with
common FRF terms. After extensive rewriting, rotational degrees of freedom
can be expressed as:
h
h
B
22,
Mf
A
22,
Mf
h
h
B
22,
MM
A
22,
MM
where: β,δ are explained in detail in [40] through a symbolic nonlinear
analytical toolbox and will not be further analyzed here.
When comparing results with previously suggested approaches, it has
been shown that the design has improved accuracy in experiments carried out
with various sizes of end mills, successfully indentifying chatter free
conditions in milling operations.
Developing capability for tool point frequency response prediction for
micro end mills has been covered by reference [41]. In order to predict the
response of the whole system as an assembly, receptance coupling
substructure analysis was used. Response functions of the tool, tool holder and
spindle were coupled showing an integrated system. The model geometry used
is presented in Figure 9.
Receptances of the tool and collet holder were obtained from using
Timoshenko beam models due to their size and fragility, while the spindle
translational and rotational receptances were obtained experimentally.
