We present the exact constraint design of a two degrees of freedom cross-flexure-based stage that combines a large workspace to footprint ratio with high vibration mode frequencies. To maximize unwanted vibration mode frequencies the mechanism is an assembly of optimized parts. To ensure a deterministic behavior the assembled mechanism is made exactly constrained. We analyze the kinematics of the mechanism using three methods; Grüblers criterion, opening the kinematic loops, and with a multibody singular value decomposition method. Nine release-flexures are implemented to obtain an exact constraint design. Measurements of the actuation force and natural frequency show no bifurcation, and load stiffening is minimized, even though there are various errors causing nonlinearity. Misalignment of the exact constraint designs does not lead to large stress, it does however decrease the support stiffness significantly. We conclude that designing an assembled mechanism in an exactly constrained manner leads to predictable stiffnesses and modal frequencies.

References

References
1.
Eastman
,
F. S.
,
1935
, “
Flexure Pivots to Replace Knife Edges and Ball Bearings
,” Engineering Experiment Station Bulletin 86, University of Washington.
2.
Hale
,
L. C.
,
1999
, “
Principles and Techniques for Designing Precision Machines
,” Ph. D. thesis, University of California, Livermore, CA.
3.
Soemers
,
H. M. J. R.
,
2010
,
Design Principles for Precision Mechanisms
,
T-Pointprint
,
Enschede
.
4.
Haringx
,
J. A.
,
1949
, “
The Cross-Spring Pivot as a Constructional Element
,”
Appl. Sci. Res., Sect. A
,
1
(
1
), pp.
313
332
.10.1007/BF02120338
5.
Jones
,
R. V.
,
1956
, “
A Parallel-Spring Cross-Movement for an Optical Bench
,”
J. Sci. Instrum.
,
33
(
7
), pp.
279
280
.10.1088/0950-7671/33/7/411
6.
Paros
,
J. M.
, and
Weisbord
,
L.
,
1965
, “
How to Design Flexure Hinges
,”
Mach. Des.
,
37
(
25
), pp.
151
156
.
7.
Eijk
,
J. V.
,
1985
, “
On the Design of Plate Spring Mechanism
,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
8.
Jones
,
R. V.
,
1988
,
Instruments and Experiences, Papers on Measurement and Instrument Design
,
Wiley
,
New York
.
9.
Smith
,
S. T.
,
2000
,
Flexures: Elements of Elastic Mechanisms
,
Taylor & Francis
,
London, England
.
10.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
Wiley
,
New York.
11.
Zelenika
,
S.
, and
de Bona
,
F.
,
2002
, “
Analytical and Experimental Characterisation of High-Precision Flexural Pivots Subjected to Lateral Loads
,”
Precis. Eng.
,
26
(
4
), pp.
381
388
.10.1016/S0141-6359(02)00149-6
12.
Henein
,
S.
,
Spanoudakis
,
P.
,
Droz
,
S.
,
Myklebust
,
L. I.
, and
Onillon
,
E.
,
2003
, “
Flexure Pivot for Aerospace Mechanisms
,”
Proceedings of the 10th ESMATS/ESA
, Vol.
10
, pp.
285
288
.
13.
Awtar
,
S.
, and
Slocum
,
A. H.
,
2007
, “
Constraint-Based Design of Parallel Kinematic XY Flexure Mechanisms
,”
ASME J. Mech. Des.
,
129
(
8
), pp.
816
830
.10.1115/1.2735342
14.
Hopkins
,
J. B.
, and
Culpepper
,
M. L.
,
2010
, “
Synthesis of Multi-Degree of Freedom, Parallel Flexure System Concepts Via Freedom and Constraint Topology (FACT)—Part I: Principles
,”
Precis. Eng.
,
34
(
4
), pp.
259
270
.10.1016/j.precisioneng.2009.06.008
15.
Brouwer
,
D. M.
,
Meijaard
,
J. P.
, and
Jonker
,
J. B.
,
2013
, “
Large Deflection Stiffness Analysis of Parallel Prismatic Leaf-Spring Flexures
,”
Precis. Eng.
,
37
(
3
), pp.
505
521
.10.1016/j.precisioneng.2012.11.008
16.
Wiersma
,
D. H.
,
Boer
,
S. E.
,
Aarts
,
R. G. K. M.
, and
Brouwer
,
D. M.
,
2012
, “
Large Stroke Performance Optimization of Spatial Flexure Hinges
,”
Proceedings of the 1st Biennial International Conference on Dynamics for Design
, No. DETC2012-70502.
17.
Folkersma
,
K. G. P.
,
Boer
,
S. E.
,
Brouwer
,
D. M.
,
Herder
,
J. L.
, and
Soemers
,
H. M. J. R.
,
2012
, “
A 2-DOF Large Stroke Flexure-Based Positioning Mechanism
,”
Proceedings of the 36th Mechanisms and Robotics Conference
, No. DETC2012-70377.
18.
Blanding
,
D.
,
1999
,
Exact Constraint Machine Design Using Kinematic Principles
,
ASME Press
,
New York
.
19.
Brouwer
,
D. M.
,
Boer
,
S. E.
,
Meijaard
,
J. P.
, and
Aarts
,
R. G. K. M.
,
2013
, “
Optimization of Release Locations for Small Stress Large Stiffness Flexure Mechanisms
,”
Mech. Mach. Theory
,
64
, pp.
230
250
.10.1016/j.mechmachtheory.2013.01.007
20.
Awtar
,
S.
,
Shimotsu
,
K.
, and
Sen
,
S.
,
2010
, “
Elastic Averaging in Flexure Mechanisms: A Three-Beam Parallelogram Flexure Case Study
,”
ASME J. Mech. Rob.
,
2
(
4
), p.
041006
.10.1115/1.4002204
21.
Meijaard
,
J. P.
,
Brouwer
,
D. M.
, and
Jonker
,
J. B.
,
2010
, “
Analytical and Experimental Investigation of a Parallel Leaf Spring Guidance
,”
Multibody Syst. Dyn.
,
23
, pp.
77
97
.10.1007/s11044-009-9172-4
22.
Maxwell
,
J. C.
,
1864
, “
On the Calculation of the Equilibrium and Stiffness of Frames
,”
Philos. Mag.
,
27
(
4
), pp.
294
299
10.1080/14786446408643668.
23.
Chebychev
,
P. L.
,
1907
,
Sur les parallélogrammes
,
Oeuvres
,
Tome II
, pp.
85
106
.
24.
Grübler
,
M.
,
1883
, “
Allgemeine Eigenschaften der Zwangläufigen ebenen kinematischen Ketten
,”
Civilingenieur
,
29
, pp.
167
200
.
25.
Kutzbach
,
K.
,
1929
, “
Mechanische Leitungsverzweigung, ihre Gesetze und Anwendungen
,”
Maschinenbau. Betrieb
,
8
, pp.
710
716
.
26.
Besseling
,
J. F.
,
1979
, “
Trends in Solid Mechanics 1979
,”
Proceedings of the Symposium Dedicated to the 65th Birthday
,”
W. T.
Koiter
,
J.
Besseling
, and
A.
van der Heijden
, eds.,
Delft University Press
, pp.
53
78
.
27.
Pellegrino
,
S.
, and
Calladine
,
C. R.
,
1986
, “
Matrix Analysis of Statically and Kinematically Indeterminate Frameworks
,”
Int. J. Solids Struct.
,
22
(4), pp.
409
428
.10.1016/0020-7683(86)90014-4
28.
Angeles
,
J.
, and
Gosselin
,
C.
,
1989
, “
Détermination du degré de liberté des chaînes cinématiques
,”
Transactions de la Société Canadienne de Génie Mécanique
,
12
, pp.
219
226
.
29.
Aarts
,
R. G. K. M.
,
Meijaard
,
J. P.
, and
Jonker
,
J. B.
,
2012
, “
Flexible Multibody Modelling for Exact Constraint Mechatronic Design of Compliant Mechanisms
,”
Multibody Syst. Dyn.
,
27
(
1
), pp.
119
133
.10.1007/s11044-011-9272-9
30.
Nelder
,
J. A.
, and
Mead
,
R.
,
1965
, “
A Simplex Method for Function Minimization
,”
Comput. J.
,
7
(
4
), pp.
308
313
.10.1093/comjnl/7.4.308
31.
Ryu
,
J. W.
,
Gweon
,
D.-G.
, and
Moon
,
K. S.
,
1997
, “
Optimal Design of a Flexure Hinge Based XY Wafer Stage
,”
Precis. Eng.
,
21
(
1
), pp.
18
28
.10.1016/S0141-6359(97)00064-0
32.
Culpepper
,
M. L.
, and
Anderson
,
G.
,
2004
, “
Design of a Low-Cost Nano-Manipulator Which Utilizes a Monolithic, Spatial Compliant Mechanism
,”
Precis. Eng.
,
28
(
4
), pp.
469
482
.10.1016/j.precisioneng.2004.02.003
33.
Yao
,
Q.
,
Dong
,
J.
, and
Ferreira
,
P. M.
,
2007
, “
Design, Analysis, Fabrication, and Testing of a Parallel-Kinematic Micropositioning XY Stage
,”
Int. J. Mach. Tools Manuf.
,
47
(
6
), pp.
946
961
.10.1016/j.ijmachtools.2006.07.007
34.
de Jong
,
B. R.
,
Brouwer
,
D. M.
,
de Boer
,
M. J.
,
Jansen
,
H. V.
,
Soemers
,
H. M. J. R.
, and
Krijnen
,
G. J. M.
,
2010
, “
Design and Fabrication of a Planar Three-DOFs MEMS-Based Manipulator
,”
J. Microelectromech. Syst.
,
19
(
5
), pp.
1116
1130
.10.1109/JMEMS.2010.2067196
35.
Werner
,
C.
,
Rosielle
,
P. C. J. N.
, and
Steinbuch
,
M.
,
2010
, “
Design of a Long Stroke Translation Stage for AFM
,”
Int. J. Mach. Tools Manuf.
,
50
(
2
), pp.
183
190
.10.1016/j.ijmachtools.2009.10.012
36.
Boer
,
S. E.
,
Aarts
,
R. G. K. M.
,
Brouwer
,
D. M.
, and
Jonker
,
J. B.
,
2010
, “
Multibody Modelling and Optimization of a Curved Hinge Flexure
,”
The 1st Joint International Conference on Multibody System Dynamics
,
Lappeenranta
, pp.
1
10
.
37.
Wijma
,
W.
,
Boer
,
S. E.
,
Aarts
,
R. G. K. M.
,
Brouwer
,
D. M.
, and
Hakvoort
,
W. B. J.
,
2013
, “
Modal Measurements and Model Corrections of a Large Stroke Compliant Mechanism
,” ECCOMAS Multibody Dynamics 2013, July 1–4, 2013, University of Zagreb, Croatia, pp.
831
843
.
You do not currently have access to this content.