A flexure strip has constraint characteristics, such as stiffness properties and error motions, that govern its performance as a basic constituent of flexure mechanisms. This paper presents a new modeling approach for obtaining insight into the deformation and stiffness characteristics of general three-dimensional flexure strips that exhibit bending, shear, and torsion deformation. The approach is based on the use of a discretized version of a finite (i.e., nonlinear) strain spatial beam formulation for extracting analytical expressions that describe deformation and stiffness characteristics of a flexure strip in a parametric format. This particular way of closed-form modeling exploits the inherent finite-element assumptions on interpolation and also lends itself for numeric implementation. As a validating case study, a closed-form parametric expression is derived for the lateral support stiffness of a flexure strip and a parallelogram flexure mechanism. This captures a combined torsion–bending dictated geometrically nonlinear effect that undermines the support bearing stiffness when the mechanism moves in the intended degree of freedom (DoF). The analytical result is verified by simulations and experimental measurements.

References

References
1.
Jones
,
R. V.
,
1956
, “
A Parallel-Spring Cross-Movement for an Optical Bench
,”
J. Sci. Instrum.
,
33
(
7
), pp.
279
280
.
2.
Howell
,
L. L.
,
Magleby
,
S. P.
, and
Olsen
,
B. M.
,
2013
,
Handbook of Compliant Mechanisms
,
Wiley
,
Chichester, UK
.
3.
Blanding
,
D. L.
,
1999
,
Exact Constraint: Machine Design Using Kinematic Principles
,
ASME Press
,
New York
.
4.
Awtar
,
S.
,
Slocum
,
A. H.
, and
Sevincer
,
E.
,
2007
, “
Characteristics of Beam-Based Flexure Modules
,”
ASME J. Mech. Des.
,
129
(
6
), pp.
625
639
.
5.
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
.
6.
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
.
7.
Sen
,
S.
,
2013
, “
Beam Constraint Model: Generalized Nonlinear Closed-Form Modeling of Beam Flexures for Flexure Mechanism Design
,” Ph.D. thesis, University of Michigan, Ann Arbor, MI.
8.
Awtar
,
S.
, and
Sen
,
S.
,
2010
, “
A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Load-Displacement Formulation
,”
ASME J. Mech. Des.
,
132
(
8
), p.
081008
.
9.
Sen
,
S.
, and
Awtar
,
S.
,
2013
, “
A Closed-Form Nonlinear Model for the Constraint Characteristics of Symmetric Spatial Beams
,”
ASME J. Mech. Des.
,
135
(
3
), p.
11
.
10.
Rasmussen
,
N. O.
,
Wittwer
,
J. W.
,
Todd
,
R. H.
,
Howell
,
L. L.
, and
Magleby
,
S. P.
,
2006
, “
A 3D Pseudo-Rigid-Body Model for Large Spatial Deflections of Rectangular Cantilever Beams
,”
ASME
Paper No. DETC2006-99465.
11.
Ramirez
,
I. A.
, and
Lusk
,
C. P.
,
2011
, “
Spatial-Beam Large-Deflection Equations and Pseudo-Rigid-Body Model for Axisymmetric Cantilever Beams
,”
ASME
Paper No. DETC2011-47389.
12.
Meijaard
,
J. P.
,
2011
, “
Refinements of Classical Beam Theory for Beams With a Large Aspect Ratio of Their Cross-Sections
,”
IUTAM
Symposium on Dynamics Modeling and Interaction Control in Virtual and Real Environments
, Budapest, Hungary, June 7–11, pp.
285
292
.
13.
Jonker
,
J. B.
, and
Meijaard
,
J. P.
,
1990
, “
SPACAR—Computer Program for Dynamic Analysis of Flexible Spatial Mechanisms and Manipulators
,”
Multibody Systems Handbook
,
W.
Schiehlen
, ed.,
Springer
,
Berlin
, pp.
123
143
.
14.
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
(
1
), pp.
77
97
.
15.
Brouwer
,
D. M.
,
Folkersma
,
K. G. P.
,
Boer
,
S. E.
, and
Aarts
,
R. G. K. M.
,
2013
, “
Exact Constraint Design of a Two-Degree of Freedom Flexure-Based Mechanism
,”
ASME J. Mech. Rob.
,
5
(
4
), p.
041011
.
16.
Wijma
,
W.
,
Boer
,
S. E.
,
Aarts
,
R. G. K. M.
,
Brouwer
,
D. M.
, and
Hakvoort
,
W. B. J.
,
2014
, “
Modal Measurements and Model Corrections of a Large Stroke Compliant Mechanism
,”
Arch. Mech. Eng.
,
61
(
2
), pp.
347
366
.
17.
Cosserat
,
E.
, and
Cosserat
,
F.
,
1909
,
Théorie des Corps Déformables
,
Librairie Scientifique A. Hermann et Fils
,
Paris, France
.
18.
Lang
,
H.
,
Linn
,
J.
, and
Arnold
,
M.
,
2011
, “
Multi-Body Dynamics Simulation of Geometrically Exact Cosserat Rods
,”
Multibody Syst. Dyn.
,
25
(
3
), pp.
285
312
.
19.
Cowper
,
G. R.
,
1966
, “
The Shear Coefficient in Timoshenko's Beam Theory
,”
ASME J. Appl. Mech.
,
33
(
2
), pp.
335
340
.
20.
Goldstein
,
H.
,
Poole
,
C.
, and
Safko
,
J.
,
2001
,
Classical Mechanics
,
3rd ed.
,
Addison-Wesley
,
Boston, MA
.
21.
Jonker
,
J. B.
, and
Meijaard
,
J. P.
,
2013
, “
A Geometrically Non-Linear Formulation of a Three-Dimensional Beam Element for Solving Large Deflection Multibody System Problems
,”
Int. J. Non-Linear Mech.
,
53
, pp.
63
74
.
22.
Reissner
,
E.
,
1973
, “
On One-Dimensional Large-Displacement Finite-Strain Beam Theory
,”
Stud. Appl. Math.
,
52
(
2
), pp.
87
95
.
23.
Irschik
,
H.
, and
Gerstmayr
,
J.
,
2011
, “
A Continuum-Mechanics Interpretation of Reissner's Non-Linear Shear-Deformable Beam Theory
,”
Math. Comput. Modell. Dyn. Syst.
,
17
(
1
), pp.
19
29
.
24.
Simo
,
J. C.
, and
Vu-Quoc
,
L.
,
1986
, “
On the Dynamics of Flexible Beams Under Large Overall Motions—The Plane Case: Part I
,”
ASME J. Appl. Mech.
,
53
(
4
), pp.
849
854
.
25.
Nachbagauer
,
K.
,
Pechstein
,
A. S.
,
Irschik
,
H.
, and
Gerstmayr
,
J.
,
2011
, “
A New Locking-Free Formulation for Planar, Shear Deformable, Linear and Quadratic Beam Finite Elements Based on the Absolute Nodal Coordinate Formulation
,”
Multibody Syst. Dyn.
,
26
(
3
), pp.
245
263
.
26.
Przemieniecki
,
J. S.
,
1968
,
Theory of Matrix Structural Analysis
,
McGraw-Hill
,
New York
.
27.
Awtar
,
S.
,
2004
, “
Synthesis and Analysis of Parallel Kinematic XY Mechanisms
,”
Sc.D. thesis
, Massachusetts Institute of Technology, Cambridge, MA.
28.
Besseling
,
J. F.
,
1974
, “
Non-Linear Analysis of Structures by the Finite Element Method as a Supplement to a Linear Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
3
(
2
), pp.
173
194
.
29.
Boer
,
S. E.
,
Aarts
,
R. G. K. M.
,
Meijaard
,
J. P.
,
Brouwer
,
D. M.
, and
Jonker
,
J. B.
,
2014
, “
A Nonlinear Two-Node Superelement for Use in Flexible Multibody Systems
,”
Multibody Syst. Dyn.
,
31
(
4
), pp.
405
431
.
30.
Timoschenko
,
S.
,
1922
, “
On the Torsion of a Prism, One of the Cross-Sections of Which Remains Plane
,”
Proc. London Math. Soc.
,
20
(
1
), pp.
389
397
.
You do not currently have access to this content.