Graphical Abstract Figure
Abstract
A kinetostatic model of an asymmetrical double-stepped beam under axial loading is developed. The beam is composed of two thick segments and three thin segments where a thick segment is between two thin segments, and the longitudinal axis of the thick segment is not colinear with that of the thin segment. The kinetostatic model based on the beam constraint model (BCM) is capable of predicting accurate bending and near-buckling behaviors of the beam. A virtual rigid link neighboring the noncolinear segments is introduced in the BCM to broaden the applicability of the BCM. An analytical formula to represent the critical load of a symmetrical double-stepped beam under axial loading is derived and the value calculated by the formula agrees with the limit load of the asymmetrical double-stepped beam within the elastic range. The investigated beam has potential applications in displacement amplifiers and robotic grippers.
References
1.
Qi
, K.-Q.
, Xiang
, Y.
, Fang
, C.
, Zhang
, Y.
, and Yu
, C.-S.
, 2015
, “Analysis of the Displacement Amplification Ratio of Bridge-Type Mechanism
,” Mech. Mach. Theory
, 87
, pp. 45
–56
. 2.
Kim
, J.-J.
, Choi
, Y.-M.
, Ahn
, D.
, Hwang
, B.
, Gweon
, D.-G.
, and Jeong
, J.
, 2012
, “A Millimeter-Range Flexure-Based Nano-Positioning Stage Using a Self-Guided Displacement Amplification Mechanism
,” Mech. Mach. Theory
, 50
, pp. 109
–120
. 3.
Liu
, P.
, and Yan
, P.
, 2016
, “A New Model Analysis Approach for Bridge-Type Amplifiers Supporting Nano-Stage Design
,” Mech. Mach. Theory
, 99
, pp. 176
–188
. 4.
Kim
, J.
, Dorin
, P.
, and Wang
, K. W.
, 2020
, “Vibration Energy Harvesting Enhancement Exploiting Magnetically Coupled Bistable and Linear Harvesters
,” Smart Mater. Struct.
, 29
(6
), p. 065006
. 5.
Lin
, J.-T.
, Shuvra
, P. D.
, Yang
, J. A.
, McNamara
, S.
, Walsh
, K.
, and Alphenaar
, B.
, 2020
, “Buckled Beam Mechanical Memory Using an Asymmetric Piezoresistor for Readout
,” J. Micromech. Microeng.
, 30
(7
), p. 075006
. 6.
Dong
, L.
, and Lakes
, R.
, 2013
, “Advanced Damper With High Stiffness and High Hysteresis Damping Based on Negative Structural Stiffness
,” Int. J. Solids Struct.
, 50
(14–15
), pp. 2416
–2423
. 7.
Wu
, Z.
, Zhu
, B.
, Wang
, R.
, and Zhang
, X.
, 2022
, “Design of Mechanical Metamaterial for Energy Absorption Using a Beam With a Variable Cross-Section
,” Mech. Mach. Theory
, 176
, p. 105027
. 8.
Xu
, Q.
, and Li
, Y.
, 2011
, “Analytical Modeling, Optimization and Testing of a Compound Bridge-Type Compliant Displacement Amplifier
,” Mech. Mach. Theory
, 46
(2
), pp. 183
–200
. 9.
Chen
, X.
, Xie
, Z.
, Tan
, H.
, and Tai
, K.
, 2023
, “Design and Mechanical Modeling of High-Magnification and Low-Parasitic Displacement Microgripper With Three-Stage Displacement Amplification
,” Mech. Mach. Theory
, 190
, p. 105463
. 10.
Howell
, L. L.
, 2002
, Compliant Mechanisms
, John Wiley & Sons
, New York
.11.
Frisch-Fay
, R.
, 1962
, Flexible Bars
, Butterworths
, Washington, DC
.12.
Hussein
, H.
, and Younis
, M. I.
, 2020
, “Analytical Study of the Snap-Through and Bistability of Beams With Arbitrarily Initial Shape
,” ASME J. Mech. Rob.
, 12
(4
), p. 041001
. 13.
Deng
, G.
, Wang
, N.
, Zhou
, C.
, and Li
, J.
, 2018
, “A Simplified Analysis Method for the Piezo jet Dispenser With a Diamond Amplifier
,” Sensors
, 18
(7
), p. 2115
. 14.
Wu
, H.
, Lai
, L.
, Zhang
, L.
, and Zhu
, L.
, 2022
, “A Novel Compliant XY Micro-Positioning Stage Using Bridge-Type Displacement Amplifier Embedded With Scott-Russell Mechanism
,” Precis. Eng.
, 73
, pp. 284
–295
. 15.
Ma
, H.-W.
, Yao
, S.-M.
, Wang
, L.-Q.
, and Zhong
, Z.
, 2006
, “Analysis of the Displacement Amplification Ratio of Bridge-Type Flexure Hinge
,” Sens. Actuators, A
, 132
(2
), pp. 730
–736
. 16.
Venkiteswaran
, V. K.
, and Su
, H.
, 2018
, “A Versatile 3R Pseudo-Rigid-Body Model for Initially Curved and Straight Compliant Beams of Uniform Cross Section
,” ASME J. Mech. Des.
, 140
(9
), p. 092305
. 17.
Holst
, G. L.
, Teichert
, G. H.
, and Jensen
, B. D.
, 2011
, “Modeling and Experiments of Buckling Modes and Deflection of Fixed-Guided Beams in Compliant Mechanisms
,” ASME J. Mech. Des.
, 133
(5
), p. 051002
. 18.
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
. 19.
Wu
, K.
, and Zheng
, G.
, 2022
, “Solutions to Large Beam-Deflection Problems by Taylor Series and Padé Approximant for Compliant Mechanisms
,” Mech. Mach. Theory
, 177
, p. 105033
. 20.
Xiao
, J.
, Chen
, Y.
, Lu
, X.
, Xu
, B.
, Chen
, X.
, and Xu
, J.
, 2019
, “Three Dimensional Buckling Beam Under Cylindrical Constraint
,” Int. J. Mech. Sci.
, 150
, pp. 348
–355
. 21.
Awtar
, S.
, Slocum
, A. H.
, and Sevincer
, E.
, 2007
, “Characteristics of Beam-Based Flexure Modules
,” ASME J. Mech. Des.
, 129
(6
), pp. 625
–639
. 22.
Awtar
, S.
, and Senm
, S.
, 2010
, “A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation
,” ASME J. Mech. Des.
, 132
(8
), p. 081009
. 23.
Chen
, G.
, and Ma
, F.
, 2015
, “Kinetostatic Modeling of Fully Compliant Bistable Mechanisms Using Timoshenko Beam Constraint Model
,” ASME J. Mech. Des.
, 137
(2
), p. 022301
. 24.
Ma
, F.
, and Chen
, G.
, 2016
, “Modeling Large Planar Deflections of Flexible Beams in Compliant Mechanisms Using Chained Beam-Constraint-Model
,” ASME J. Mech. Rob.
, 8
(2
), p. 021018
. 25.
Ma
, F.
, Chen
, G.
, and Wang
, H.
, 2020
, “Large-Stroke Constant-Force Mechanisms Utilizing Second Buckling Mode of Flexible Beams: Evaluation Metrics and Design Approach
,” ASME J. Mech. Des.
, 142
(10
), p. 103303
. 26.
Radgolchin
, M.
, and Moeenfard
, H.
, 2017
, “A Constraint Model for Beam Flexure Modules With an Intermediate Semi-Rigid Element
,” Int. J. Mech. Sci.
, 122
, pp. 167
–183
. 27.
Bilancia
, P.
, Baggetta
, M.
, Hao
, G.
, and Berselli
, G.
, 2021
, “A Variable Section Beams Based Bi-BCM Formulation for the Kinetostatic Analysis of Cross-Axis Flexural Pivots
,” Int. J. Mech. Sci.
, 205
, p. 106587
. 28.
Ma
, F.
, Chen
, G.
, and Hao
, G.
, 2018
, “Determining the Range of Allowable Axial Force for the Third-Order Beam Constraint Model
,” Mech. Sci.
, 9
(1
), pp. 71
–79
. 29.
Li
, Y.
, Bi
, S.
, and Zhao
, C.
, 2019
, “Analytical Modeling and Analysis of Rhombus-Type Amplifier Based on Beam Flexures
,” Mech. Mach. Theory
, 139
, pp. 195
–211
. 30.
Wang
, T.
, Yu
, J.
, and Zhao
, H.
, 2023
, “A Family of Sweat Pore-Inspired Compliant Cellular Expansion Mechanisms
,” Mech. Mach. Theory
, 183
, p. 105251
. 31.
Wu
, K.
, and Zheng
, G.
, 2022
, “Theoretical Analysis on Nonlinear Buckling, Post-Buckling of Slender Beams and Bi-Stable Mechanisms
,” ASME J. Mech. Rob.
, 14
(3
), p. 031015
. 32.
Sepahi
, O.
, Forouzan
, M. R.
, and Malekzadeh
, P.
, 2010
, “Post-Buckling Analysis of Variable Cross-Section Cantilever Beams Under Combined Load via Differential Quadrature Method
,” KSCE J. Civil Eng.
, 14
(2
), pp. 207
–214
. 33.
Smaś
, P.
, 2007
, “Design of Clamped Columns for Maximizing the Axial Displacement at Buckling
,” Struct. Multidiscipl. Optim.
, 33
(3
), pp. 229
–241
. 34.
Zhu
, J.
, Hao
, G.
, Liu
, T.
, and Li
, H.
, 2023
, “Design of an Over-Constraint Based Nearly-Constant Amplification Ratio Compliant Mechanism
,” Mech. Mach. Theory
, 186
, p. 105347
. 35.
Simão
, P. D.
, Girão Coelho
, A. M.
, and Bijlaard
, F. S. K.
, 2012
, “Stability Design of Crane Columns in Mill Buildings
,” Eng. Struct.
, 42
, pp. 51
–82
. 36.
Chi
, I.
, Chang
, P.
, Tran
, N. D. K.
, and Wang
, D.-A.
, 2012
, “Kinetostatic Modeling of Planar Compliant Mechanisms With Flexible Beams, Linear Sliders, Multinary Rigid Links, and Multiple Loops
,” ASME J. Mech. Rob.
, 13
(5
), p. 051002
. 37.
Han
, Q.
, Huang
, X.
, and Shao
, X.
, 2017
, “Nonliear Kinetostatic Modeling of Double-Tensural Fully–Compliant Bistable Mechanisms
,” Int. J. Non-Linear Mech.
, 93
, pp. 41
–46
. 38.
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
. 39.
Hao
, G.
, 2015
, “Extended Nonlinear Analytical Models of Compliant Parallelogram Mechanisms: Third-Order Models
,” Trans. Canadian Soc. Mech. Eng.
, 39
(1
), pp. 71
–83
. 40.
Ma
, F.
, and Chen
, G.
, 2019
, “Influence of Non-Ideal Fixed-end Constraints on Kinetostatic Behaviors of Compliant Bistable Mechanisms
,” Mech. Mach. Theory
, 133
, pp. 267
–277
. 41.
Wang
, F.
, Liang
, C.
, Tian
, Y.
, Zhao
, X.
, and Zhang
, D.
, 2015
, “Design of Piezoelectric-Actuated Microgripper With a Three-Stage Flexure-Based Amplification
,” IEEE/ASME Trans. Mechatron.
, 20
(5
), pp. 2205
–2213
. 42.
Szufnarowski
, F.
, and Schneider
, A.
, 2011
, “Force Control of a Piezoelectric Actuator Based on a Statistical System Model and Dynamic Compensation
,” Mech. Mach. Theory
, 46
(10
), pp. 1507
–1521
. 43.
Ghafarirad
, H.
, Rezaei
, S. M.
, Abdullah
, A.
, Zareinejad
, M.
, and Saadat
, M.
, 2011
, “Observer-Based Sliding Mode Control With Adaptive Perturbation Estimation for Micropositioning Actuators
,” Precis. Eng.
, 35
(2
), pp. 271
–281
. 44.
Chi
, I. T.
, Chanthasopeephan
, T.
, and Wang
, D.-A.
, 2022
, “Design of a Parallel Gripper Based on Topology Synthesis and Evolutionary Optimization
,” ASME J. Mech. Rob.
, 14
(2
), p. 021008
. 
