Abstract
While nonlinear-elastic materials demonstrate potential in enhancing the performance of compliant mechanisms, their behavior still needs to be captured in a generalized mechanical model. To inform new designs and functionality of compliant mechanisms, a better understanding of nonlinear-elastic materials is necessary and, in particular, their mechanical properties that often differ in tension and compression. In the current work, a beam-based analytical model incorporating nonlinear-elastic material behavior is defined for a folding compliant mechanism geometry. Exact equations are derived capturing the nonlinear curvature profile and shift in the neutral axis due to the material asymmetry. The deflection and curvature profile are compared with finite element analysis along with stress distribution across the beam thickness. The analytical model is shown to be a good approximation of the behavior of nonlinear-elastic materials with tension–compression asymmetry under the assumptions of the von Kármán strain theory. Through a segmentation approach, the geometries of a semicircular arc and folding compliant mechanism design are defined. The deflection of the folding compliant mechanism due to an applied tip load is then evaluated against finite element analysis and experimental results. The generalized methods presented highlight the utility of the model for designing and predicting the behavior of other compliant mechanism geometries and different nonlinear-elastic materials.
Graphical Abstract Figure
References
1.
Jovanova
, J.
, Nastevska
, A.
, and Frecker
, M.
, 2019
, “Tailoring Energy Absorption With Functional Grading of a Contact-Aided Compliant Mechanism
,” Smart Mater. Struct.
, 28
(8
), p. 084003
. 2.
Pernette
, E.
, Henein
, S.
, Magnani
, I.
, and Clavel
, R.
, 1997
, “Design of Parallel Robots in Microrobotics
,” Robotica
, 15
(4
), pp. 417
–420
. 3.
Saggere
, L.
, and Kota
, S.
, 1999
, “Static Shape Control of Smart Structures Using Compliant Mechanisms
,” AIAA J.
, 37
(5
), pp. 572
–578
. 4.
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
. 5.
Speich
, J.
, and Goldfarb
, M.
, 2000
, “A Compliant-Mechanism-Based Three Degree-of-Freedom Manipulator for Small-Scale Manipulation
,” Robotica
, 18
(1
), pp. 95
–104
. 6.
Frecker
, M. I.
, Ananthasuresh
, G. K.
, Nishiwaki
, S.
, Kikuchi
, N.
, and Kota
, S.
, 1997
, “Topological Synthesis of Compliant Mechanisms Using Multi-criteria Optimization
,” ASME J. Mech. Des.
, 119
(2
), pp. 238
–245
. 7.
Howell
, L. 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
. 8.
Moon
, Y.-M.
, Trease
, B. P.
, and Kota
, S.
, 2002
, “Design of Large-Displacement Compliant Joints
,” Proceedings of the ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 5: 27th Biennial Mechanisms and Robotics Conference, ASME
, Montreal, Quebec, Canada
, Sept. 29–Oct. 2
, ASME, Vol. 5, pp. 65
–76
.9.
Ahuett-Garza
, H.
, Chaides
, O.
, Garcia
, P. N.
, and Urbina
, P.
, 2014
, “Studies About the Use of Semicircular Beams as Hinges in Large Deflection Planar Compliant Mechanisms
,” Precis. Eng.
, 38
(4
), pp. 711
–727
. 10.
Saxena
, A.
, and Kramer
, S. N.
, 1998
, “A Simple and Accurate Method for Determining Large Deflections in Compliant Mechanisms Subjected to End Forces and Moments
,” ASME J. Mech. Des.
, 120
(3
), pp. 392
–400
. 11.
Howell
, L. L.
, 2001
, Compliant Mechanisms
, 1st ed., John Wiley & Sons, Inc.
, New York
.12.
Pedersen
, C. B. W.
, Buhl
, T.
, and Sigmund
, O.
, 2001
, “Topology Synthesis of Large-Displacement Compliant Mechanisms
,” Int. J. Numer. Methods Eng.
, 50
(12
), pp. 2683
–2705
. 13.
Kumar
, P.
, Saxena
, A.
, and Sauer
, R. A.
, 2019
, “Computational Synthesis of Large Deformation Compliant Mechanisms Undergoing Self and Mutual Contact
,” ASME J. Mech. Des.
, 141
(1
), p. 012302
. 14.
Saxena
, A.
, 2005
, “Topology Design of Large Displacement Compliant Mechanisms With Multiple Materials and Multiple Output Ports
,” Struct. Multidiscipl. Optim.
, 30
(6
), pp. 477
–490
. 15.
Zeng
, X.
, Hurd
, C.
, Su
, H.-J.
, Song
, S.
, and Wang
, J.
, 2020
, “A Parallel-Guided Compliant Mechanism With Variable Stiffness Based on Layer Jamming
,” Mech. Mach. Theory
, 148
, p. 103791
. 16.
Najmon
, J. C.
, Dehart
, J.
, Wood
, Z.
, and Tovar
, A.
, 2018
, “Cellular Helmet Liner Design Through Bio-inspired Structures and Topology Optimization of Compliant Mechanism Lattices
,” SAE Int. J. Transp. Saf.
, 6
(3
), pp. 217
–235
. 17.
Kumar
, P.
, Sauer
, R. A.
, and Saxena
, A.
, 2021
, “On Topology Optimization of Large Deformation Contact-Aided Shape Morphing Compliant Mechanisms
,” Mech. Mach. Theory
, 156
, p. 104135
. 18.
Liu
, L.
, Xing
, J.
, Yang
, Q.
, and Luo
, Y.
, 2017
, “Design of Large-Displacement Compliant Mechanisms by Topology Optimization Incorporating Modified Additive Hyperelasticity Technique
,” Math. Probl. Eng.
, 2017
, p. 4679746
. 19.
Bruns
, T. E.
, and Tortorelli
, D. A.
, 2001
, “Topology Optimization of Non-linear Elastic Structures and Compliant Mechanisms
,” Comput. Methods Appl. Mech. Eng.
, 190
(26–27
), pp. 3443
–3459
. 20.
Howell
, L. 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
. 21.
Venkiteswaran
, V. K.
, and Su
, H. J.
, 2016
, “Pseudo-Rigid-Body Models for Circular Beams Under Combined Tip Loads
,” Mech. Mach. Theory
, 106
, pp. 80
–93
. 22.
Zirbel
, S. A.
, Tolman
, K. A.
, Trease
, B. P.
, and Howell
, L. L.
, 2016
, “Bistable Mechanisms for Space Applications
,” PLoS One
, 11
(12
), p. e0168218
. 23.
Hargrove
, B.
, Nastevska
, A.
, Frecker
, M.
, and Jovanova
, J.
, 2022
, “Pseudo Rigid Body Model for a Nonlinear Folding Compliant Mechanism
,” Mech. Mach. Theory
, 176
, p. 105017
. 24.
Jin
, M.
, Zhu
, B.
, Mo
, J.
, Yang
, Z.
, Zhang
, X.
, and Howell
, L. L.
, 2020
, “A CPRBM-Based Method for Large-Deflection Analysis of Contact-Aided Compliant Mechanisms Considering Beam-to-Beam Contacts
,” Mech. Mach. Theory
, 145
, p. 103700
. 25.
Banerjee
, A.
, Bhattacharya
, B.
, and Mallik
, A. K.
, 2008
, “Large Deflection of Cantilever Beams With Geometric Non-linearity: Analytical and Numerical Approaches
,” Int. J. Non Linear Mech.
, 43
(5
), pp. 366
–376
. 26.
Chen
, G.
, Ma
, F.
, Hao
, G.
, and Zhu
, W.
, 2019
, “Modeling Large Deflections of Initially Curved Beams in Compliant Mechanisms Using Chained Beam Constraint Model
,” ASME J. Mech. Rob.
, 11
(1
), p. 011002
. 27.
Jeong
, S.
, and Yoo
, H. H.
, 2017
, “Flexibility Modeling of a Beam Undergoing Large Deflection Using the Assumed Mode Method
,” Int. J. Mech. Sci.
, 133
, pp. 611
–618
. 28.
Tsiatas
, G. C.
, and Babouskos
, N. G.
, 2017
, “Linear and Geometrically Nonlinear Analysis of Non-uniform Shallow Arches Under a Central Concentrated Force
,” Int. J. Non Linear Mech.
, 92
, pp. 92
–101
. 29.
Li
, D. K.
, and Li
, X. F.
, 2016
, “Large Deflection and Rotation of Timoshenko Beams With Frictional End Supports Under Three-Point Bending
,” C. R. Méch.
, 344
(8
), pp. 556
–568
. 30.
Shvartsman
, B. S.
, 2013
, “Analysis of Large Deflections of a Curved Cantilever Subjected to a Tip-Concentrated Follower Force
,” Int. J. Non Linear Mech.
, 50
, pp. 75
–80
. 31.
Mohyeddin
, A.
, and Fereidoon
, A.
, 2014
, “An Analytical Solution for the Large Deflection Problem of Timoshenko Beams Under Three-Point Bending
,” Int. J. Mech. Sci.
, 78
, pp. 135
–139
. 32.
Eshghinejad
, A.
, and Elahinia
, M.
, 2015
, “Exact Solution for Bending of Shape Memory Alloy Beams
,” Mech. Adv. Mater. Struct.
, 22
(10
), pp. 829
–838
. 33.
Auricchio
, F.
, Taylor
, R. L.
, and Lubliner
, J.
, 1997
, “Shape-Memory Alloys: Macromodelling and Numerical Simulations of the Superelastic Behavior
,” Comput. Methods Appl. Mech. Eng.
, 146
(3–4
), pp. 281
–312
. 34.
Hargrove
, B.
, Nastevska
, A.
, Jovanova
, J.
, and Frecker
, M.
, 2021
, “Shape Memory Modeling of a Nonlinear and Superelastic Compliant Mechanism
,” ASME 2021 Conference on Smart Materials, Adaptive Structures and Intelligent Systems
, Virtual, Online
, Sept. 14–15
.35.
Chen
, L.
, 2010
, “An Integral Approach for Large Deflection Cantilever Beams
,” Int. J. Non Linear Mech.
, 45
(3
), pp. 301
–305
. 36.
Ghuku
, S.
, and Nath Saha
, K.
, 2016
, “A Theoretical and Experimental Study on Geometric Nonlinearity of Initially Curved Cantilever Beams
,” Eng. Sci. Technol. Int. J.
, 19
(1
), pp. 135
–146
. 37.
Van Viet
, N.
, Zaki
, W.
, and Umer
, R.
, 2018
, “Analytical Model for a Superelastic Timoshenko Shape Memory Alloy Beam Subjected to a Loading–Unloading Cycle
,” J. Intell. Mater. Syst. Struct.
, 29
(20
), pp. 3902
–3922
. 38.
Van Viet
, N.
, Zaki
, W.
, and Moumni
, Z.
, 2019
, “A Model for Shape Memory Alloy Beams Accounting for Tensile Compressive Asymmetry
,” J. Intell. Mater. Syst. Struct.
, 30
(18–19
), pp. 18
–19
. 39.
Nastevska
, A.
, Jovanova
, J.
, and Frecker
, M.
, 2020
, “Design of Compliant Joints for Large Scale Structures
,” ASME 2020 Conference on Smart Materials, Adaptive Structures and Intelligent Systems
, Virtual, Online
, Sept. 15
, pp. 1
–10
.40.
Rajagopal
, K. R.
, 2014
, “On the Nonlinear Elastic Response of Bodies in the Small Strain Range
,” Acta Mech.
, 225
(6
), pp. 1545
–1553
. 41.
Janečka
, A.
, Průša
, V.
, and Rajagopal
, K. R.
, 2016
, “Euler–Bernoulli Type Beam Theory for Elastic Bodies With Nonlinear Response in the Small Strain Range
,” Arch. Mech.
, 68
(1
), pp. 3
–25
.42.
Khodabakhshi
, P.
, and Reddy
, J. N.
, 2017
, “A Unified Beam Theory With Strain Gradient Effect and the von Kármán Nonlinearity
,” Z. Angew. Math. Mech.
, 97
(1
), pp. 70
–91
. 43.
Reddy
, J. N.
, and Mahaffey
, P.
, 2013
, “Generalized Beam Theories Accounting for von Kármán Nonlinear Strains With Application to Buckling
,” J. Coupled Syst. Multiscale Dyn.
, 1
(1
), pp. 120
–134
. 44.
Altenbach
, H.
, Belyaev
, A.
, and Palmov
, V. A.
, 2020
, “Plate Theory After von Kármán,” Encyclopedia of Continuum Mechanics
, 1st ed., Springer
, Berlin, Heidelberg
, pp. 2068
–2076
. 45.
Naghdi
, P. M.
, and Vongsarnpigoon
, L.
, 1983
, “A Theory of Shells With Small Strain Accompanied by Moderate Rotation
,” Arch. Ration Mech. Anal.
, 83
(3
), pp. 245
–283
. 46.
Desai
, S. M.
, Sonawane
, R. Y.
, and More
, A. P.
, 2023
, “Thermoplastic Polyurethane for Three-Dimensional Printing Applications: A Review
,” Polym. Adv. Technol.
, 34
(7
), pp. 2061
–2082
. 47.
Płatek
, P.
, Rajkowski
, K.
, Cieplak
, K.
, Sarzyński
, M.
, Małachowski
, J.
, Woźniak
, R.
, and Janiszewski
, J.
, 2020
, “Deformation Process of 3D Printed Structures Made From Flexible Material With Different Values of Relative Density
,” Polymers
, 12
(9
), p. 2120
. 48.
Robertson
, S. W.
, Pelton
, A. R.
, and Ritchie
, R. O.
, 2013
, “Mechanical Fatigue and Fracture of Nitinol
,” Int. Mater. Rev.
, 57
(1
), pp. 1
–36
. 49.
Qi
, H. J.
, and Boyce
, M. C.
, 2005
, “Stress–Strain Behavior of Thermoplastic Polyurethanes
,” Mech. Mater.
, 37
(8
), pp. 817
–839
. 50.
Pham
, N. K.
, and Peraza Hernandez
, E. A.
, 2021
, “Modeling and Design Exploration of a Tensegrity-Based Twisting Wing
,” ASME J. Mech. Rob.
, 13
(3
), p. 031019
. 
