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.

The goal – cutting force characteristics

“high performance”

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

Unstable limit cycle and bi-stable zone

20/27

Fly-over

• Dombovari, Barton, Wilson

• Stepan, 2010

9/10

Variation of the bi-stable zone

Tobias, Shi

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,…)

Turning(Tobias, Tlusty, 1960)

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)

Dynamic experiment for cutting force

Unsafe/bistable zone identification

Checking the hysteresis loop

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

The experiment

Evaluation of the results

Force characteristics reconstruction

Conclusion

The invers application of the results of the Hopf bifurcation calculation in case of regenerative machine tool vibrations makes it possible to measure the nonlinear cutting force characteristics with cheap accelerometers only in a fast and accurate way.

Thank you for your attention!