Abstract

Although energy-based methods have advantages over the Newtonian methods for kinetostatic modeling, the geometric nonlinearities inherent in deflections of compliant mechanisms preclude most of the energy-based theorems. Castigliano’s first theorem and the Crotti–Engesser theorem, which do not require the problem being solved to be linear, are selected to construct the energy-based kinetostatic modeling framework for compliant mechanisms in this work. Utilization of these two theorems requires explicitly formulating the strain energy in terms of deflections and the complementary strain energy in terms of loads, which are derived based on the beam constraint model. The kinetostatic modeling of two compliant mechanisms are provided to demonstrate the effectiveness of the explicit formulations in this framework derived from Castigliano’s first theorem and the Crotti–Engesser theorem.

Issue Section:
Technical Brief
Keywords:
compliant mechanism, nonlinear deflection, energy-based method
Topics:
Compliant mechanisms, Deflection, Modeling, Theorems (Mathematics), Stress

References

1.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
Wiley
,
New York
.
2.
Hussain
,
S.
,
Jamwal
,
P. K.
,
Kapsalyamov
,
A.
, and
Ghayesh
,
M. H.
,
2021
, “
Numerical Framework and Design Optimization of An Intrinsically Compliant 3-DOF Parallel Robot
,”
ASME J. Comput. Inf. Sci. Eng.
,
21
(
2
), p.
021008
.
3.
Turkkan
,
O. A.
, and
Su
,
H. -J.
,
2016
, “
Das-2d: a Concept Design Tool for Compliant Mechanisms
,”
Mech. Sci.
,
7
(
2
), pp.
135
–
148
.
4.
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
.
5.
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
.
6.
Dunning
,
A. G.
,
Tolou
,
N.
,
Pluimers
,
P. P.
,
Kluit
,
L. F.
, and
Herder
,
J. L.
,
2012
, “
Bistable Compliant Mechanisms: Corrected Finite Element Modeling for Stiffness Tuning and Preloading Incorporation
,”
ASME J. Mech. Des.
,
134
(
8
), p.
084502
.
7.
Zhang
,
H.
,
Zhu
,
B.
, and
Zhang
,
X.
,
2019
, “
Origami Kaleidocycle-Inspired Symmetric Multistable Compliant Mechanisms
,”
ASME J. Mech. Rob.
,
11
(
1
), p.
011009
.
8.
Li
,
Z.
, and
Bai
,
S.
,
2020
, “
Nonlinear Stiffness Analysis of Spring-Loaded Inverted Slider Crank Mechanisms With a Unified Model
,”
ASME J. Mech. Rob.
,
12
(
3
), p.
031011
.
9.
Li
,
Q.
,
Xu
,
L.
,
Chen
,
Q.
, and
Chai
,
X.
,
2019
, “
Analytical Elastostatic Stiffness Modeling of Overconstrained Parallel Manipulators Using Geometric Algebra and Strain Energy
,”
ASME J. Mech. Rob.
,
11
(
5
), p.
051014
.
10.
Awtar
,
S.
, and
Sen
,
S.
,
2010
, “
A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation
,”
ASME J. Mech. Des.
,
132
(
8
), p.
081009
.
11.
Nishiwaki
,
S.
,
Frecker
,
M. I.
,
Min
,
S.
, and
Kikuchi
,
N.
,
1998
, “
Topology Optimization of Compliant Mechanisms Using the Homogenization Method
,”
Int. J. Num. Methods Eng.
,
42
(
3
), pp.
535
–
559
.
12.
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
.
13.
Saxena
,
A.
, and
Ananthasuresh
,
G. K.
,
2000
, “
On An Optimal Property of Compliant Topologies
,”
Struct. Multidiscipl. Optim.
,
19
(
1
), pp.
36
–
49
.
14.
Yamada
,
T.
,
Yamasaki
,
S.
,
Nishiwaki
,
S.
,
Izui
,
K.
, and
Yoshimura
,
M.
,
2011
, “
Design of Compliant Thermal Actuators Using Structural Optimization Based on the Level Set Method
,”
ASME J. Comput. Inf. Sci. Eng.
,
11
(
1
), p.
011005
.
15.
Lobontiu
,
N.
,
Paine
,
J. S. N.
,
Garcia
,
E.
, and
Goldfarb
,
M.
,
2001
, “
Corner-Filleted Flexure Hinges
,”
ASME J. Mech. Des.
,
123
(
9
), pp.
346
–
352
.
16.
Eastwood
,
K. W.
,
Francis
,
P.
,
Azimian
,
H.
,
Swarup
,
A.
,
Looi
,
T.
,
Drake
,
J. M.
, and
Naguib
,
H. E.
,
2018
, “
Design of a Contact-Aided Compliant Notched-tube Joint for Surgical Manipulation in Confined Workspaces
,”
ASME J. Mech. Rob.
,
10
(
1
), p.
015001
.
17.
Timoshenko
,
S.
,
1940
,
Strength of Materials Part 1
,
D. Van Nostrand Company
,
New York
.
18.
Timoshenko
,
S. P.
, and
Young
,
D. H.
,
1945
,
Theory of Structures
,
McGraw-Hill
.
19.
Jennings
,
A.
,
1967
, “
Energy Theorems in Structural Mechanics
,”
J. Eng. Math.
,
1
(
4
), pp.
307
–
326
.
20.
Jensen
,
B. D.
, and
Howell
,
L. L.
,
2003
, “
Identification of Compliant Pseudo-Rigid-Body Mechanism Configurations Resulting in Bistable Behavior
,”
ASME J. Mech. Des.
,
125
(
3
), pp.
701
–
708
.
21.
Aten
,
Q. T.
,
Zirbel
,
S. A.
,
Jensen
,
B. D.
, and
Howell
,
L. L.
,
2011
, “
A Numerical Method for Position Analysis of Compliant Mechanisms With More Degrees of Freedom Than Inputs
,”
ASME. J. Mech. Des.
,
133
(
6
), p.
061009
.
22.
Venkiteswaran
,
V. K.
,
Turkkan
,
O. A.
, and
Su
,
H. -J.
,
2017
, “
Speeding Up Topology Optimization of Compliant Mechanisms With a Pseudorigid-Body Model
,”
ASME J. Mech. Rob.
,
9
(
4
), p.
041007
.
23.
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
.
24.
Chen
,
G.
, and
Zhang
,
A.
,
2011
, “
Accuracy Evaluation of PRBM for Predicting Kinetostatic Behavior of Flexible Segments in Compliant Mechanisms
,”
Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Aug. 28-31
,
Washington, DC
, p.
DETC2011
.
25.
Awtar
,
S.
,
Slocum
,
A. H.
, and
Sevincer
,
E.
,
2007
, “
Characteristics of Beam-Based Flexure Modules
,”
ASME J. Mech. Des.
,
129
(
6
), pp.
625
–
639
.
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.
Bai
,
R.
,
Awtar
,
S.
, and
Chen
,
G.
,
2019
, “
A Closed-form Model for Nonlinear Spatial Deflections of Rectangular Beams in Intermediate Range
,”
Int. J. Mech. Sci.
,
160
(
9
), pp.
307
–
315
.
28.
Chen
,
G.
, and
Bai
,
R.
,
2016
, “
Modeling Large Spatial Deflections of Slender Bisymmetric Beams in Compliant Mechanisms Using Chained Spatial-Beam-Constraint Model
,”
ASME J. Mech. Rob.
,
8
(
4
), p.
041011
.
29.
Liu
,
L.
,
Bi
,
S.
,
Yang
,
Q.
, and
Wang
,
Y.
,
2014
, “
Design and Experiment of Generalized Triple-Cross-Spring Flexure Pivots Applied to the Ultra-Precision Instruments
,”
Rev. Sci. Instrum.
,
85
(
10
), p.
105102
.
Copyright © 2021 by ASME
You do not currently have access to this content.