Abstract

Machine tool contacts must be represented accurately for reliable prediction of machine behavior. In structural optimization problems, contact constraints are represented as an additional minimization problem based on computational contact mechanics theory. An accurate contact constraint representation is challenging for structural optimization problems: (i) “No penetration” rule dictated by Hertz-Signorini-Moreau (HSM) conditions at contacts is satisfied by varying the contact stiffness during a finite element (FE) solution without control of a user which causes increased contact stiffness “erroneously” to avoid penetration of contacting node pairs in an FE solution; and (ii) the reliability of solutions varies according to the chosen computational contact method. This paper is devoted to the topology optimization of machine tools with contact constraints. A hybrid approach is followed that combines the computational contact problem framework and an obtained stable contact stiffness function (analytically or experimentally). According to the proposed method, the existing optimization problem in FE literature is restated in a reliable form for machine tool applications. To avoid the existing computational challenges and reliability problems, contact forces are directly mapped onto an FE model used in the restated topology optimization problem with the help of proposed method. In this study, the existing and the proposed methods for contact are investigated by means of the solid isotropic material with penalization model (SIMP) algorithm for topology optimization. The effectiveness of the proposed method is demonstrated by comparing the experimental measurements on a prototype machine tool manufactured according to the optimization solutions of the proposed method and those of a conventional machine tool.

References

References
1.
Altintas
,
Y.
,
Brecher
,
C.
,
Weck
,
M.
, and
Witt
,
S.
,
2005
, “
Virtual Machine Tool
,”
CIRP Ann.
,
54
(
2
), pp.
115
138
. 10.1016/S0007-8506(07)60022-5
2.
Abele
,
E.
,
Altintas
,
Y.
, and
Brecher
,
C.
,
2010
, “
Machine Tool Spindle Units
,”
CIRP Ann.
,
59
(
2
), pp.
781
802
. 10.1016/j.cirp.2010.05.002
3.
Altintas
,
Y.
,
Verl
,
A.
,
Brecher
,
C.
,
Uriarte
,
L.
, and
Pritschow
,
G.
,
2011
, “
Machine Tool Feed Drives
,”
CIRP Ann.
,
60
(
2
), pp.
779
796
. 10.1016/j.cirp.2011.05.010
4.
Rivin
,
E. I.
,
Agapiou
,
J.
,
Brecher
,
C.
,
Clewett
,
M.
,
Erickson
,
R.
,
Huston
,
F.
,
Kadowaki
,
Y.
,
Lenz
,
E.
,
Moriwaki
,
T.
,
Pitsker
,
A.
, and
Shimizu
,
S.
,
2000
, “
Tooling Structure: Interface Between Cutting Edge and Machine Tool
,”
CIRP Ann.
,
49
(
2
), pp.
591
634
. 10.1016/S0007-8506(07)63457-X
5.
Ertürk
,
A.
,
Özgüven
,
H. N.
, and
Budak
,
E.
,
2007
, “
Effect Analysis of Bearing and Interface Dynamics on Tool Point FRF for Chatter Stability in Machine Tools by Using a new Analytical Model for Spindle–Tool Assemblies
,”
Int. J. Mach. Tools Manuf.
,
47
(
1
), pp.
23
32
. 10.1016/j.ijmachtools.2006.03.001
6.
Matthias
,
W.
,
Özşahin
,
O.
,
Altintas
,
Y.
, and
Denkena
,
B.
,
2016
, “
Receptance Coupling Based Algorithm for the Identification of Contact Parameters at Holder–Tool Interface
,”
CIRP J. Manuf. Sci. Technol.
,
13
, pp.
37
45
. 10.1016/j.cirpj.2016.02.005
7.
Özşahin
,
O.
,
Ertürk
,
A.
,
Nevzat Özgüven
,
H.
, and
Budak
,
E.
,
2009
, “
A Closed-Form Approach for Identification of Dynamical Contact Parameters in Spindle–Holder–Tool Assemblies
,”
Int. J. Mach. Tools Manuf.
,
49
(
1
), pp.
25
35
. 10.1016/j.ijmachtools.2008.08.007
8.
Holkup
,
T.
,
Cao
,
H.
,
Kolář
,
P.
,
Altintas
,
Y.
, and
Zelený
,
J.
,
2010
, “
Thermo-Mechanical Model of Spindles
,”
CIRP Ann.
,
59
(
1
), pp.
365
368
. 10.1016/j.cirp.2010.03.021
9.
Zaeh
,
M. F.
,
Oertli
,
T.
, and
Milberg
,
J.
,
2004
, “
Finite Element Modelling of Ball Screw Feed Drive Systems
,”
CIRP Ann.
,
53
(
1
), pp.
289
292
. 10.1016/S0007-8506(07)60700-8
10.
Semm
,
T.
,
Nierlich
,
M. B.
, and
Zaeh
,
M. F.
,
2019
, “
Substructure Coupling of a Machine Tool in Arbitrary Axis Positions Considering Local Linear Damping Models
,”
ASME J. Manuf. Sci. Eng.
,
141
(
7
), p.
071014
. 10.1115/1.4043767
11.
Dadalau
,
A.
,
Groh
,
K.
,
Reuß
,
M.
, and
Verl
,
A.
,
2012
, “
Modeling Linear Guide Systems With CoFEM: Equivalent Models for Rolling Contact
,”
Prod. Eng.
,
6
(
1
), pp.
39
46
. 10.1007/s11740-011-0349-3
12.
Majda
,
P.
,
2012
, “
Modeling of Geometric Errors of Linear Guideway and Their Influence on Joint Kinematic Error in Machine Tools
,”
Precis. Eng.
,
36
(
3
), pp.
369
378
. 10.1016/j.precisioneng.2012.02.001
13.
Law
,
M.
,
Altintas
,
Y.
, and
Srikantha Phani
,
A.
,
2013
, “
Rapid Evaluation and Optimization of Machine Tools With Position-Dependent Stability
,”
Int. J. Mach. Tools Manuf.
,
68
, pp.
81
90
. 10.1016/j.ijmachtools.2013.02.003
14.
Wriggers
,
P.
, and
Zavarise
,
G.
,
2002
,
Computational Contact Mechanics
,
Wiley
,
West Sussex, UK
.
15.
Camuz
,
S.
,
Bengtsson
,
M.
,
Söderberg
,
R.
, and
Wärmefjord
,
K.
,
2019
, “
Reliability-Based Design Optimization of Surface-to-Surface Contact for Cutting Tool Interface Designs
,”
ASME J. Manuf. Sci. Eng.
,
141
(
4
), p.
041006
. 10.1115/1.4042787
16.
Hilding
,
D.
,
Klarbring
,
A.
, and
Petersson
,
J.
,
1999
, “
Optimization of Structures in Unilateral Contact
,”
ASME Appl. Mech. Rev.
,
52
(
4
), pp.
139
160
. 10.1115/1.3098931
17.
Luo
,
Y.
,
Li
,
M.
, and
Kang
,
Z.
,
2016
, “
Topology Optimization of Hyperelastic Structures With Frictionless Contact Supports
,”
Int. J. Solids Struct.
,
81
, pp.
373
382
. 10.1016/j.ijsolstr.2015.12.018
18.
Kikuchi
,
N.
, and
Oden
,
J. T.
,
1988
,
Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods
, Vol.
8
,
Siam
,
Philadelphia, PA
.
19.
Konyukhov
,
A.
, and
Izi
,
R.
,
2015
,
Introduction to Computational Contact Mechanics: A Geometrical Approach
,
John Wiley & Sons
,
West Sussex, UK
.
20.
Konyukhov
,
A.
, and
Schweizerhof
,
K.
,
2009
, “
Incorporation of Contact for High-Order Finite Elements in Covariant Form
,”
Comput. Methods Appl. Mech. Eng.
,
198
(
13–14
), pp.
1213
1223
. 10.1016/j.cma.2008.04.023
21.
Liu
,
F.
, and
Borja
,
R. I.
,
2010
, “
Stabilized Low-Order Finite Elements for Frictional Contact With the Extended Finite Element Method
,”
Comput. Methods Appl. Mech. Eng.
,
199
(
37–40
), pp.
2456
2471
. 10.1016/j.cma.2010.03.030
22.
Kim
,
N.-H.
,
2014
,
Introduction to Nonlinear Finite Element Analysis
,
Springer
,
New York
.
23.
Sanders
,
J. D.
,
Dolbow
,
J. E.
, and
Laursen
,
T. A.
,
2009
, “
On Methods for Stabilizing Constraints Over Enriched Interfaces in Elasticity
,”
Int. J. Numer. Methods Eng.
,
78
(
9
), pp.
1009
1036
. 10.1002/nme.2514
24.
Kim
,
T. Y.
,
Dolbow
,
J.
, and
Laursen
,
T.
,
2007
, “
A Mortared Finite Element Method for Frictional Contact on Arbitrary Interfaces
,”
Comput. Mech.
,
39
(
3
), pp.
223
235
. 10.1007/s00466-005-0019-4
25.
Wriggers
,
P.
, and
Panagiotopoulos
,
P. D.
1999
,”
New Developments in Contact Problems
, Vol.
384
,
Springer
,
Udine
.
26.
Strömberg
,
N.
, and
Klarbring
,
A.
,
2010
, “
Topology Optimization of Structures in Unilateral Contact
,”
Struct. Multidiscip. Optim.
,
41
(
1
), pp.
57
64
. 10.1007/s00158-009-0407-z
27.
Bendsøe
,
M. P.
, and
Sigmund
,
O.
,
1999
, “
Material Interpolation Schemes in Topology Optimization
,”
Arch. Appl. Mech.
,
69
(
9–10
), pp.
635
654
.
28.
Stolpe
,
M.
, and
Svanberg
,
K.
,
2001
, “
An Alternative Interpolation Scheme for Minimum Compliance Topology Optimization
,”
Struct. Multidiscip. Optim.
,
22
(
2
), pp.
116
124
. 10.1007/s001580100129
29.
Eschenauer
,
H. A.
, and
Olhoff
,
N.
,
2001
, “
Topology Optimization of Continuum Structures: A Review
,”
ASME Appl. Mech. Rev.
,
54
(
4
), pp.
331
390
. 10.1115/1.1388075
30.
Bandeira
,
A. A.
,
Wriggers
,
P.
, and
de Mattos Pimenta
,
P.
,
2004
, “
Numerical Derivation of Contact Mechanics Interface Laws Using a Finite Element Approach for Large 3D Deformation
,”
Int. J. Numer. Methods Eng.
,
59
(
2
), pp.
173
195
. 10.1002/nme.867
31.
Tur
,
M.
,
Fuenmayor
,
F. J.
, and
Wriggers
,
P.
,
2009
, “
A Mortar-Based Frictional Contact Formulation for Large Deformations Using Lagrange Multipliers
,”
Comput. Methods Appl. Mech. Eng.
,
198
(
37–40
), pp.
2860
2873
. 10.1016/j.cma.2009.04.007
32.
Facchinei
,
F.
,
Jiang
,
H.
, and
Qi
,
L.
,
1999
, “
A Smoothing Method for Mathematical Programs With Equilibrium Constraints
,”
Math. Prog.
,
85
(
1
), pp.
107
134
. 10.1007/s10107990015a
33.
Fancello
,
E. A.
,
2006
, “
Topology Optimization for Minimum Mass Design Considering Local Failure Constraints and Contact Boundary Conditions
,”
Struct. Multidiscip. Optim.
,
32
(
3
), pp.
229
240
. 10.1007/s00158-006-0019-9
34.
Zhang
,
W.
, and
Niu
,
C.
,
2018
, “
A Linear Relaxation Model for Shape Optimization of Constrained Contact Force Problem
,”
Comput. Struct.
,
200
, pp.
53
67
. 10.1016/j.compstruc.2018.02.005
35.
Galin
,
L. A.
,
2008
,
Contact Problems: the Legacy of LA Galin
, Vol.
155
,
Springer
,
Netherlands
.
36.
Yüksel
,
E.
,
Budak
,
E.
, and
Ertürk
,
A. S.
,
2017
, “
The Effect of Linear Guide Representation for Topology Optimization of a Five-Axis Milling Machine
,”
Proc. CIRP
,
58
, pp.
487
492
. 10.1016/j.procir.2017.03.257
37.
Johnson
,
K. L.
,
1982
, “
One Hundred Years of Hertz Contact
,”
Proc. Inst. Mech. Eng.
,
196
(
1
), pp.
363
378
. 10.1243/PIME_PROC_1982_196_039_02
38.
Altintas
,
Y.
,
2012
,
Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design
,
Cambridge University Press
,
Cambridge
.
39.
de Lacalle
,
N. L.
, and
Mentxaka
,
A. L.
,
2008
,
Machine Tools for High Performance Machining
,
Springer-Verlag
,
London
.
40.
Albayrak
,
M.
,
Yalçın
,
T. K.
,
Erberdi
,
M.
,
Özlü
,
E.
, and
Budak
,
E.
,
2015
, “
Static and Dynamic Analysis of Different Machine Tool Spindles
.”
41.
Popov
,
V. L.
, and
Heß
,
M.
,
2015
,
Method of Dimensionality Reduction in Contact Mechanics and Friction
,
Springer
,
Berlin, Heidelberg
.
42.
Greenwood
,
J. A.
, and
Williamson
,
J. P.
,
1966
, “
Contact of Nominally Flat Surfaces
,”
Proc. R. Soc. London, A
,
295
(
1442
), pp.
300
319
. 10.1098/rspa.1966.0242
43.
Engin
,
S.
, and
Altintas
,
Y.
,
2001
, “
Mechanics and Dynamics of General Milling Cutters.: Part I: Helical end Mills
,”
Int. J. Mach. Tools Manuf.
,
41
(
15
), pp.
2195
2212
. 10.1016/S0890-6955(01)00045-1
You do not currently have access to this content.