Abstract
Flexure hinges are joints typically used in the design and manufacturing of compliant mechanisms, especially when small dimensions do not allow for conventional mechanical devices. In this paper, a closed-form solution is proposed for a nonlinear stiffness model used to describe the static displacements obtained on a flexure hinge of elementary geometry as a function of applied loads. A comparison with the most widely used linear model demonstrates the effectiveness of the proposed nonlinear approach, highlighting the advantages of its use in its scope of application. The obtained results are verified by finite element (FE) simulations, taken as a reference of the actual behavior assumed for the joints studied.
Issue Section:
Research Papers
Keywords:
compliant mechanisms,
kinematics,
dynamics,
and control of mechanical systems,
mechanism synthesis and analysis,
microscale mechanisms and robotics,
theoretical and computational kinematics
Topics:
Bending (Stress),
Geometry,
Hinges,
Stiffness,
Displacement,
Stress,
Errors,
Compliant mechanisms,
Simulation
References
1.
Howell
, L. L.
, 2001
, Compliant Mechanisms
, 1st ed., John Wiley & Sons
, Canada
.2.
Kota
, S.
, Joo
, J.
, Li
, Z.
, Rodgers
, S. M.
, and Sniegowski
, J.
, 2001
, “Design of Compliant Mechanisms: Applications to MEMS
,” Analog Integr. Circuits Signal Process.
, 29
(1-2
), pp. 7
–15
. 3.
Wang
, P.
, and Xu
, Q.
, 2017
, “Design of a Flexure-Based Constant-Force xy Precision Positioning Stage
,” Mech. Mach. Theory
, 108
, pp. 1
–13
. 4.
Ghafarian
, M.
, Shirinzadeh
, B.
, Das
, T. K.
, Al-Jodah
, A.
, and Wei
, W.
, 2018
, “Design of a Novel Parallel Monolithic 6-Dof Compliant Micromanipulation Mechanism
,” IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)
, Auckland, New Zealand
, July 9–12
, IEEE, pp. 997
–1002
.5.
Chen
, N.
, and Tian
, C.
, 2021
, “Design, Modeling and Testing of a 3-Dof Flexible Piezoelectric Thin Sheet Nanopositioner
,” Sens. Actuators A
, 323
, p. 112660
. 6.
Wang
, G.
, Yan
, Y.
, Ma
, J.
, and Cui
, J.
, 2019
, “Design, Test and Control of a Compact Piezoelectric Scanner Based on a Compound Compliant Amplification Mechanism
,” Mech. Mach. Theory
, 139
, pp. 460
–475
. 7.
Wang
, F.
, Liang
, C.
, Tian
, Y.
, Zhao
, X.
, and Zhang
, D.
, 2014
, “Design of a Piezoelectric-Actuated Microgripper With a Three-Stage Flexure-Based Amplification
,” IEEE/ASME Trans. Mechatron.
, 20
(5
), pp. 2205
–2213
. 8.
Zubir
, M. N. M.
, and Shirinzadeh
, B.
, 2009
, “Development of a High Precision Flexure-Based Microgripper
,” Precis. Eng.
, 33
(4
), pp. 362
–370
.9.
Das
, T. K.
, Shirinzadeh
, B.
, Al-Jodah
, A.
, Ghafarian
, M.
, and Pinskier
, J.
, 2021
, “A Novel Compliant Piezoelectric Actuated Symmetric Microgripper for the Parasitic Motion Compensation
,” Mech. Mach. Theory
, 155
, p. 104069
.10.
Tseytlin
, Y.
, 1972
, Elastic Kinematical Devices, Mashinostroenie, Leningrad (in Russian).11.
Quinn
, T.
, 1992
, “The Beam Balance as an Instrument for Very Precise Weighing
,” Meas. Sci. Technol.
, 3
(2
), p. 141
. 12.
Li
, C.
, Wang
, N.
, Yue
, F.
, and Zhang
, X.
, 2021
, “Optimization of Translational Flexure Joints Using Corrugated Units Under Stress Constraints
,” ASME J. Mech. Rob.
, 13
(6
), p. 061006
. 13.
Lin
, J.
, Qi
, C.
, Gao
, F.
, Yue
, Y.
, Hu
, Y.
, and Wei
, B.
, 2023
, “Modeling and Verification for a Three-Degree-of-Freedom Flexure-Based Planar Parallel Micro Manipulator
,” ASME J. Mech. Rob.
, 15
(4
), p. 041006
. 14.
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
. 15.
Howell
, L.
, and Midha
, A.
, 1994
, “A Method for the Design of Compliant Mechanisms With Small-Length Flexural Pivots
,” ASME J. Mech. Des.
, 116
(1
), pp. 280
–290
. 16.
Howel
, L.
, and Midha
, A.
, 1995
, “Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms
,” ASME J. Mech. Des.
, 117
(1
), pp. 156
–165
. 17.
Eastman
, F. S.
, 1937
, “The Design of Flexure Pivots
,” J. Aeronaut. Sci.
, 5
(1
), pp. 16
–21
. 18.
Smith
, S. T.
, Badami
, V. G.
, Dale
, J. S.
, and Xu
, Y.
, 1997
, “Elliptical Flexure Hinges
,” Rev. Sci. Instrum.
, 68
(3
), pp. 1474
–1483
. 19.
Paros
, J.
, 1965
, “How to Design Flexure Hinges
,” Mach. Des.
, 37
, pp. 151
–156
.20.
Lobontiu
, N.
, and Paine
, J. S.
, 2002
, “Design of Circular Cross-Section Corner-Filleted Flexure Hinges for Three-Dimensional Compliant Mechanisms
,” ASME J. Mech. Des.
, 124
(3
), pp. 479
–484
. 21.
Wu
, Y.
, and Zhou
, Z.
, 2002
, “Design Calculations for Flexure Hinges
,” Rev. Sci. Instrum.
, 73
(8
), pp. 3101
–3106
. 22.
Schotborgh
, W. O.
, Kokkeler
, F. G.
, Tragter
, H.
, and van Houten
, F. J.
, 2005
, “Dimensionless Design Graphs for Flexure Elements and a Comparison Between Three Flexure Elements
,” Precis. Eng.
, 29
(1
), pp. 41
–47
. 23.
Yong
, Y. K.
, and Lu
, T. -F.
, 2009
, “Comparison of Circular Flexure Hinge Design Equations and the Derivation of Empirical Stiffness Formulations
,” IEEE/ASME International Conference on Advanced Intelligent Mechatronics
, Singapore
, July 14–17
, IEEE, pp. 510
–515
.24.
Zelenika
, S.
, Munteanu
, M. G.
, and De Bona
, F.
, 2009
, “Optimized Flexural Hinge Shapes for Microsystems and High-Precision Applications
,” Mech. Mach. Theory
, 44
(10
), pp. 1826
–1839
. 25.
Smith
, S. T.
, 2000
, Flexures: Elements of Elastic Mechanisms
, CRC Press
, Boca Raton, FL
.26.
Lobontiu
, N.
, Paine
, J. S.
, O’Malley
, E.
, and Samuelson
, M.
, 2002
, “Parabolic and Hyperbolic Flexure Hinges: Flexibility, Motion Precision and Stress Characterization Based on Compliance Closed-Form Equations
,” Precis. Eng.
, 26
(2
), pp. 183
–192
. 27.
Lobontiu
, N.
, Paine
, J. S.
, Garcia
, E.
, and Goldfarb
, M.
, 2002
, “Design of Symmetric Conic-Section Flexure Hinges Based on Closed-Form Compliance Equations
,” Mech. Mach. Theory
, 37
(5
), pp. 477
–498
. 28.
Lobontiu
, N.
, Garcia
, E.
, and Canfield
, S.
, 2003
, “Torsional Stiffness of Several Variable Rectangular Cross-Section Flexure Hinges for Macro-Scale and MEMS Applications
,” Smart Mater. Struct.
, 13
(1
), p. 12
. 29.
Chen
, G.
, Shao
, X.
, and Huang
, X.
, 2008
, “A New Generalized Model for Elliptical Arc Flexure Hinges
,” Rev. Sci. Instrum.
, 79
(9
), p. 095103
. 30.
Chen
, G.
, Liu
, X.
, Gao
, H.
, and Jia
, J.
, 2009
, “A Generalized Model for Conic Flexure Hinges
,” Rev. Sci. Instrum.
, 80
(5
), p. 055106
. 31.
Linß
, S.
, Erbe
, T.
, and Zentner
, L.
, 2011
, “On Polynomial Flexure Hinges for Increased Deflection and an Approach for Simplified Manufacturing
,” 13th World Congress in Mechanism and Machine Science
, Guanajuato, Mexico
, June 19–23
, p. A11_512.32.
Li
, Q.
, Pan
, C.
, and Xu
, X.
, 2013
, “Closed-Form Compliance Equations for Power-Function-Shaped Flexure Hinge Based on Unit-Load Method
,” Precis. Eng.
, 37
(1
), pp. 135
–145
. 33.
Vallance
, R. R.
, Haghighian
, B.
, and Marsh
, E. R.
, 2008
, “A Unified Geometric Model for Designing Elastic Pivots
,” Precis. Eng.
, 32
(4
), pp. 278
–288
. 34.
Yang
, M.
, Du
, Z.
, Dong
, W.
, and Sun
, L.
, 2018
, “Design and Modeling of a Variable Thickness Flexure Pivot
,” ASME J. Mech. Rob.
, 11
(1
), p. 014502
. 35.
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
. 36.
Zhang
, A.
, and Chen
, G.
, 2013
, “A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms
,” ASME J. Mech. Rob.
, 5
(2
), p. 021006
. 37.
Midha
, A.
, and Kuber
, R.
, 2014
, “Closed-Form Elliptic Integral Solution of Initially-Straight and Initially-Curved Small-Length Flexural Pivots
,” International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Buffalo, NY
, Aug. 17–20
, Vol. 46360, American Society of Mechanical Engineers, p. V05AT08A044.38.
Bisshopp
, K.
, and Drucker
, D.
, 1945
, “Large Deflection of Cantilever Beams
,” Quart. Appl. Math.
, 3
(3
), pp. 272
–275
. 39.
Awtar
, S.
, and Slocum
, A. H.
, 2005
, “Closed-Form Nonlinear Analysis of Beam-Based Flexure Modules
,” International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Long Beach, CA
, Sept. 24–28
, Vol. 47446, pp. 101
–110
.40.
Friedrich
, R.
, Lammering
, R.
, and Heurich
, T.
, 2015
, “Nonlinear Modeling of Compliant Mechanisms Incorporating Circular Flexure Hinges With Finite Beam Elements
,” Precis. Eng.
, 42
, pp. 73
–79
. 41.
Rad
, F. P.
, Vertechy
, R.
, Berselli
, G.
, and Parenti-Castelli
, V.
, 2016
, “Analytical Compliance Analysis and Finite Element Verification of Spherical Flexure Hinges for Spatial Compliant Mechanisms
,” Mech. Mach. Theory
, 101
, pp. 168
–180
.42.
Li
, L.
, Zhang
, D.
, Qu
, H.
, and Wang
, Y.
, 2022
, “Generalized Model and Configuration Design of Multiple-Axis Flexure Hinges
,” Mech. Mach. Theory
, 169
, p. 104677
. 43.
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. 031003
. 44.
Moschini
, S.
, and Claudio Palpacelli
, M.
, 2022
, “Practical Range of Applicability of a Linear Stiffness Model of an Elliptical Flexure Hinge
,” IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications
, Taipei, Taiwan
, Nov. 28–30
, pp. 1
–6
.45.
Moschini
, S.
, and Palpacelli
, M.
, 2023
, “Insights Into Bending Stiffness Modeling of Elementary Flexure Hinges
,” Appl. Sci.
, 13
(17
), p. 9785
. 46.
Villaggio
, P.
, 1997
, Mathematical Models for Elastic Structures
, Cambridge University Press
, Cambridge, MA
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