Abstract

This paper presents an algorithm to solve for all solutions to the forward problem for large deflections of inextensible end loaded Euler beams, a problem often encountered in compliant mechanism design and analysis. The forward problem is characterized by known end moment and end force (magnitude and direction), and the horizontal, vertical, and rotational deflections of the end of the beam must be found. Previous solutions have relied on the use of numerical solvers, which normally result in finding a single solution, but are unable to find all possible solutions for a given loading condition. The algorithm presented here works by reformulating the problem to have a single unknown, the end angle of the beam. Using this reformulation, a search vector of possible end angles can be used to find all solutions within desired bounds for the rotation of the end of the beam. The results were compared to nonlinear finite element modeling for verification. The results show that the vast majority of possible load conditions result in multiple (at least two) solutions, with larger end forces generally leading to more solutions. This finding suggests that such solutions may be used to design novel multi-stable compliant mechanisms, including the possibility of metamaterials with variable volume.

References

1.
Levien
,
R.
,
2008
, “The Elastica: A Mathematical History”. Technical Report UCB/EECS-2008-103, University of California at Berkeley.
2.
Bisshopp
,
K.
, and
Drucker
,
D. C.
,
1945
, “
Large Deflection of Cantilever Beams
,”
Q. Appl. Math.
,
3
(
3
), pp.
272
275
.
3.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
John Wiley & Sons, Inc
,
New York
.
4.
Shoup
,
T. E.
, and
McLarnan
,
C. W.
,
1971
, “
On the Use of the Undulating Elastica for the Analysis of Flexible Link Mechanisms
,”
J. Eng. Ind.
,
93
(
1
), pp.
263
267
.
5.
Shoup
,
T. E.
,
1972
, “
On the Use of the Nodal Elastica for the Analysis of Flexible Link Devices
,”
J. Eng. Ind.
,
94
(
3
), pp.
871
873
.
6.
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
.
7.
Xu
,
K.
,
Liu
,
H.
, and
Xiao
,
J.
,
2021
, “
Static Deflection Modeling of Combined Flexible Beams Using Elliptic Integral Solution
,”
Int. J. Non-Linear Mech.
,
129
, p.
103637
.
8.
Xu
,
K.
,
Liu
,
H.
,
Yue
,
W.
,
Xiao
,
J.
,
Ding
,
Y.
, and
Wang
,
G.
,
2021
, “
Kinematic Modeling and Optimal Design of a Partially Compliant Four-Bar Linkage Using Elliptic Integral Solution
,”
Mech. Mach. Theory
,
157
, p.
104214
.
9.
Xianheng
,
W.
,
Mu
,
W.
, and
Xinming
,
Q.
,
2023
, “
Sign Problems in Elliptic Integral Solution of Planar Elastica Theory
,”
Eur. J. Mech. - A/Solids
,
100
, p.
105032
.
10.
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
.
11.
Kimball
,
C.
, and
Tsai
,
L.-W.
,
2002
, “
Modeling of Flexural Beams Subjected to Arbitrary End Loads
,”
ASME J. Mech. Des.
,
124
(
2
), pp.
223
235
.
12.
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, Volume 5A: 38th Mechanisms and Robotics Conference, p.
V05AT08A044
.
13.
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
.
14.
Mattiasson
,
K.
,
1981
, “
Numerical Results From Large Deflection Beam and Frame Problems Analysed by Means of Elliptic Integrals
,”
Int. J. Numerical Methods Eng.
,
17
(
1
), pp.
145
153
.
15.
Banerjee
,
A.
,
Bhattacharya
,
B.
, and
Mallik
,
A.
,
2008
, “
Large Deflection of Cantilever Beams With Geometric Non-Linearity: Analytical and Numerical Approaches
,”
Int. J. Non-Linear Mech.
,
43
(
5
), pp.
366
376
.
16.
Chucheepsakul
,
S.
, and
Phungpaigram
,
B.
,
2004
, “
Elliptic Integral Solutions of Variable-Arc-Length Elastica Under an Inclined Follower Force
,”
ZAMM - J. Appl. Math. Mech. / Z. für Angew. Math. Mech.
,
84
(
1
), pp.
29
38
.
17.
Humer
,
A.
,
2011
, “
Elliptic Integral Solution of the Extensible Elastica With a Variable Length Under a Concentrated Force
,”
Acta Mech.
,
222
(
3
), pp.
209
223
.
18.
Yoshiaki
,
G.
,
Tomoo
,
Y.
, and
Makoto
,
O.
,
1990
, “
Elliptic Integral Solutions of Plane Elastica With Axial and Shear Deformations
,”
Int. J. Solids Struct.
,
26
(
4
), pp.
375
390
.
19.
Chen
,
L.
,
2010
, “
An Integral Approach for Large Deflection Cantilever Beams
,”
Int. J. Non-Linear Mech.
,
45
(
3
), pp.
301
305
.
20.
Masters
,
N. D.
, and
Howell
,
L. L.
,
2003
, “
A Self-Retracting Fully Compliant Bistable Micromechanism
,”
J. Microelectromech. Syst.
,
12
(
3
), pp.
273
280
.
21.
Wittwer
,
J. W.
,
Baker
,
M. S.
, and
Howell
,
L. L.
,
2006
, “
Simulation, Measurement, and Asymmetric Buckling of Thermal Microactuators
,”
Sens. Actuat. A
,
128
(
2
), pp.
395
401
.
22.
Shamshirasaz
,
M.
, and
Asgari
,
M. B.
,
2008
, “
Polysilicon Micro Beams Buckling With Temperature-Dependent Properties
,”
Microsyst. Technol.
,
14
(
7
), pp.
975
961
.
23.
Howell
,
L. L.
,
Midha
,
A.
, and
Norton
,
T. W.
,
1996
, “
Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms
,”
ASME J. Mech. Des.
,
118
(
1
), pp.
126
131
.
24.
Dado
,
M. H.
,
2001
, “
Variable Parametric Pseudo-Rigid-Body Model for Large-Deflection Beams With End Loads
,”
Int. J. Non-Linear Mech.
,
36
(
7
), pp.
1123
1133
.
25.
Verotti
,
M.
,
2020
, “
A Pseudo-Rigid Body Model Based on Finite Displacements and Strain Energy
,”
Mech. Mach. Theory
,
149
, p.
103811
.
26.
Vedant
, and
Allison
,
J. T.
,
2020
, “
Pseudo-Rigid-Body Dynamic Models for Design of Compliant Members
,”
ASME J. Mech. Des.
,
142
(
3
), p.
031116
.
27.
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
.
28.
Huxman
,
C.
, and
Butler
,
J.
,
2023
, “
An Analytical Stress–Deflection Model for Fixed-Clamped Flexures Using a Pseudo-Rigid-Body Approach
,”
ASME J. Mech. Rob.
,
15
(
6
), p.
061010
.
29.
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
.
30.
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
.
31.
Ma
,
F.
, and
Chen
,
G.
,
2016
, “
Bi-BCM: A Closed-Form Solution for Fixed-Guided Beams in Compliant Mechanisms
,”
ASME J. Mech. Rob.
,
9
(
1
), p.
014501
.
32.
Ma
,
F.
, and
Chen
,
G.
,
2016
, “
Modeling Large Planar Deflections of Flexible Beams in Compliant Mechanisms Using Chained Beam-Constraint-Model1
,”
ASME J. Mech. Rob.
,
8
(
2
), p.
021018
.
33.
Turkkan
,
O. A.
, and
Su
,
H. -J.
,
2017
, “
A General and Efficient Multiple Segment Method for Kinetostatic Analysis of Planar Compliant Mechanisms
,”
Mech. Mach. Theory
,
112
, pp.
205
217
.
34.
Li
,
R.
, and
Yang
,
Z.
,
2022
, “
Modeling the Nonlinear Deflection of Elliptical-Arc-Fillet Leaf Springs
,”
Mech. Mach. Theory
,
176
, p.
105037
.
35.
Shvartsman
,
B.
,
2007
, “
Large Deflections of a Cantilever Beam Subjected to a Follower Force
,”
J. Sound Vib.
,
304
(
3
), pp.
969
973
.
36.
Nallathambi
,
A. K.
,
Lakshmana Rao
,
C.
, and
Srinivasan
,
S. M.
,
2010
, “
Large Deflection of Constant Curvature Cantilever Beam Under Follower Load
,”
Int. J. Mech. Sci.
,
52
(
3
), pp.
440
445
.
37.
Jin
,
Y.
, and
Su
,
H.-J.
,
2022
, “
Machine Learning Models for Predicting Deflection and Shape of 2D Cantilever Beams
,” International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Paper No. DETC2022–89694.
38.
Zhu
,
J.
, and
Hao
,
G.
,
2024
, “
Modelling of a General Lumped-Compliance Beam for Compliant Mechanisms
,”
Int. J. Mech. Sci.
,
263
, p.
108779
.
39.
Cammarata
,
A.
,
Lacagnina
,
M.
, and
Sequenzia
,
G.
,
2019
, “
Alternative Elliptic Integral Solution to the Beam Deflection Equations for the Design of Compliant Mechanisms
,”
Int. J. Interact. Des. Manuf. (IJIDeM)
,
13
(
2
), pp.
499
505
.
40.
Handral
,
P.
, and
Rangarajan
,
R.
,
2020
, “
An Elastica Robot: Tip-Control in Tendon-Actuated Elastic Arms
,”
Extreme Mech. Lett.
,
34
, p.
100584
.
You do not currently have access to this content.