Download - Identification of nonlinear characteristics based on bistability in delayed model of cutting
Identification of nonlinear characteristics based on bistability
in delayed model of cutting
G Stepan, Z DombovariDepartment of Applied Mechanics
Budapest University of Technology and Economics
J MunoaIdeko Research Alliance IK4, Danobat Group
Introduction to cutting
Specific amount of material cut within a certain time
wherew – chip width h – chip thicknessv – cutting speedΩ ~ cutting speed
2D
whV .
Cutting force
Introduction to milling
Number of cutting edgesin contact varies periodically with periodequal to the delay between two subsequent cutting edges.
Thus, the resultant cutting force also varies with the same period.
Cutting force characteristics
Linear (Taylor):
Power (Kienzle):
Cubic pol. (Tobias): Exponential (Endres):
nonlinearities?uniqueness?
}
}Shifted lin. (Altintas): {
How to measure/identify?
PreliminariesClassical experiment (Tobias, Shi, 1984)• cutting process is sensitive to large perturbations• self excited vibrations (chatter) “around” stable cutting• important effect of chip thickness on size of unsafe zone
2/17
Mechanical model of turning
τ – time period of revolution
)( hFkxxbxm x )()(
)()( 0
txtx
hthth
)()()()()( 1 txtxktkxtxbtxm
)2/(,/ nn mbmk
A pair of complex conjugate roots at stability limit
Transversality condition
,...2,1,0Re,0122 kwew k
i21
Linear stability & Hopf Bifurcation
18/27
Subcritical Hopf bifurcation
2231
3
22
1inf
3
03
1)(
whh
Fq
Centre manifold reduction, andcalculation of Poincare-Ljapunov constant (PLC)
since
and
0)(),(),(),( 2 ndd wu
19/27
0)()(
)()(
)(
)1()( 32
221
22
dd
nn
uw
)()( 33
221 hhhwtFq
Model of milling
Mechanical model: - number of cutting edges
in contact varies periodically with periodequal to the delay
)()()()()()( 1 txtxtktkxtxbtxm
)()( 11 tktk
High-speedmilling Theory &
experiments: stability chart
(Insperger,Mann, Stepan,Bayly, 2004,
also groupsin Dortmund,Ljubljana,…)
Newtonian impact theory and regenerative effect(Davies, Burns, Dutterer, Pratt,… Insperger, Stépán, 2001 Szalay, Stépán, 2002 – subcr, flip)
Semi-discretization method – Insperger, StépánMulti-frequency method – Merdol, AltintasTime Finite Element method – Bayly, Mann,…Full discretization – Altintas, Balachandran,…
Period-doubling(Corpus, Endres)
Characteristic matrices(Szalai, 2006)
= 0.05… 0.1 … 0.2
Experiments on lenses/islands(Zatarian, Mann, 2008)
Time averaging (basic Fourier component)provides satisfactory stability limits, bifurcations(Tobias, Tlusty, Minis,… 1965…1995, Altintas, Budak – multi DoF, single frequency… 1998),
but the frequency content is rich (Insperger,... 2003)
Differential equation of cutting force characteristics
+ 2=
=
𝑤 (h )=h2
4 (𝛿1(𝜔)𝛿2(𝜔)
(h )+3 3(h)22 )≅ 3
4h2 3 (h)
From the Hopf calculation:
𝐹 ′ ′ ′ (h )− 8𝑤 (h )
h2𝐹 ′ (h )= 0
where we can measure:
Example: size w of bistable zone does not depend on chip thickness h
𝐹 ′ ′ ′ (h )− 80
h2 𝐹′ (h )= 0 Eulerian-type diff. equ,
,
With the boundary conditions
,
, softening
With a typical measured value of
𝛼1=−14
,𝛼2=54
𝐹 (h )= 𝐶1 h3 /4Typical power law