Abstract

Movement in compliant mechanisms is achieved, at least in part, via deformable flexible members, rather than using articulating joints. These flexible members are traditionally modeled using finite element analysis (FEA)-based models. In this article, an alternative strategy for modeling compliant cantilever beams is developed with the objectives of reducing computational expense and providing accuracy with respect to design optimization solutions. The method involves approximating the response of a flexible beam with an n-link/m-joint pseudo-rigid-body dynamic model (PRBDM). Traditionally, static pseudo-rigid-body models (PRBMs) have shown an approximation of compliant elements using two or three revolute joints (2R/3R-PRBM). In this study, a more general nR-PRBDM model is developed. The first n resonant frequencies of the PRBDM are matched to exact or FEA solutions to approximate the response of the compliant system and compared with existing PRBMs. PRBDMs can be used for co-design studies of flexible structural members and are capable of modeling large deflections of compliant elements. We demonstrate PRBDMs that show dynamically accurate response for a random geometry cantilever beam by matching the steady-state and frequency response, with dynamical response accuracies up to 10% using a 5R-PRBDM.

References

References
1.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
John Wiley & Sons
.
2.
Jensen
,
B. D.
, and
Howell
,
L. L.
,
2003
, “
Identification of Compliant Pseudo-Rigid-Body Four-Link Mechanism Configurations Resulting in Bistable Behavior
,”
ASME J. Mech. Des.
,
125
(
4
), pp.
701
708
. 10.1115/1.1625399
3.
Chen
,
G.
,
Wilcox
,
D. L.
, and
Howell
,
L. L.
,
2009
, “
Fully Compliant Double Tensural Tristable Micromechanisms (DTTM)
,”
J. Micromech. Microeng.
,
19
(
2
), p.
025011
. 10.1088/0960-1317/19/2/025011
4.
Lyon
,
S.
,
Erickson
,
P.
,
Evans
,
M.
, and
Howell
,
L.
,
1999
, “
Prediction of the First Modal Frequency of Compliant Mechanisms Using the Pseudo-Rigid-Body Model
,”
ASME J. Mech. Des.
,
121
(
2
), pp.
309
313
. 10.1115/1.2829459
5.
Yu
,
Y.-Q.
,
Howell
,
L. L.
,
Lusk
,
C.
,
Yue
,
Y.
, and
He
,
M.-G.
,
2005
, “
Dynamic Modeling of Compliant Mechanisms Based on the Pseudo-Rigid-Body Model
,”
ASME J. Mech. Des.
,
127
(
4
), pp.
760
765
. 10.1115/1.1900750
6.
Midha
,
A.
,
Howell
,
L. L.
, and
Norton
,
T. W.
,
2000
, “
Limit Positions of Compliant Mechanisms Using the Pseudo-Rigid-Body Model Concept
,”
Mech. Mach. Theory.
,
35
(
1
), pp.
99
115
. 10.1016/S0094-114X(98)00093-7
7.
Kimball
,
C.
, and
Tsai
,
L.-W.
,
2002
, “
Modeling of Flexural Beams Subjected to Arbitrary End Loads
,”
Trans.-Am. Soc. Mech. Eng. J. Mech. Des.
,
124
(
2
), pp.
223
235
.
8.
Su
,
H.-J.
,
2009
, “
A Pseudorigid-Body 3R Model for Determining Large Deflection of Cantilever Beams Subject to Tip Loads
,”
ASME J. Mech. Rob.
,
1
(
2
), p.
021008
. 10.1115/1.3046148
9.
Venkiteswaran
,
V. K.
, and
Su
,
H.-J.
,
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
. 10.1115/1.4040628
10.
Chen
,
G.
,
Xiong
,
B.
, and
Huang
,
X.
,
2011
, “
Finding the Optimal Characteristic Parameters for 3R Pseudo-Rigid-Body Model Using An Improved Particle Swarm Optimizer
,”
Precision Eng.
,
35
(
3
), pp.
505
511
. 10.1016/j.precisioneng.2011.02.006
11.
Chudnovsky
,
V.
,
Mukherjee
,
A.
,
Wendlandt
,
J.
, and
Kennedy
,
D.
,
2006
, “
Modeling Flexible Bodies in SimMechanics
,”
MatLab Digest
,
14
(
3
), pp.
1
11
.
12.
Fathy
,
H. K.
,
Reyer
,
J. A.
,
Papalambros
,
P. Y.
, and
Ulsov
,
A.
,
2001
, “
On the coupling between the plant and controller optimization problems
,”
Proceedings of the 2001 American Control Conference. (Cat. No. 01CH37148)
, Vol.
3
,
IEEE
,
New York
, pp.
1864
1869
.
13.
Allison
,
J. T.
, and
Herber
,
D. R.
,
2014
, “
Multidisciplinary Design Optimization of Dynamic Engineering Systems
,”
AIAA J.
,
52
(
4
), pp.
691
710
. 10.2514/1.J052182
14.
Herber
,
D. R.
, and
Allison
,
J. T.
,
2019
, “
Nested and Simultaneous Solution Strategies for General Combined Plant and Control Design Problem
,”
ASME J. Mech. Des.
,
141
(
1
), p.
011402
. 10.1115/1.4040705
15.
Ljung
,
L.
,
1999
, “
System Identification
,”
Wiley Encyclopedia of Electrical and Electronics Engineering
, pp.
1
19
.
16.
COMSOL AB.
COMSOL Multiphysics v. 5.3
. Accessed June 10, 2019.
17.
Herrera-May
,
A. L.
,
Aguilera-Cortés
,
L. A.
,
Plascencia-Mora
,
H.
,
Rodríguez-Morales
,
Á. L.
, and
Lu
,
J.
,
2011
, “
Analytical Modeling for the Bending Resonant Frequency of Multilayered Microresonators With Variable Cross-Section
,”
Sensors
,
11
(
9
), pp.
8203
8226
. 10.3390/s110908203
18.
Chilan
,
C. M.
,
Herber
,
D. R.
,
Nakka
,
Y. K.
,
Chung
,
S.-J.
,
Allison
,
J. T.
,
Aldrich
,
J. B.
, and
Alvarez-Salazar
,
O. S.
,
2017
, “
Co-Design of Strain-Actuated Solar Arrays for Spacecraft Precision Pointing and Jitter Reduction
,”
AIAA J.
,
55
(
9
), pp.
3180
3195
. 10.2514/1.J055748
19.
Vedant
, and
Allison
,
J. T.
,
2019
, “
Pseudo-Rigid Body Dynamic Modeling of Compliant Members for Design
,”
ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 2A: 45th Design Automation Conference
,
Anaheim, CA
,
Aug. 18–21
, p.
V02AT03A013
. 10.1115/DETC2019-97881
20.
Vedant
, and
Allison
,
J. T.
,
2019
, “
Multifunctional Structures for Attitude Control
,”
ASME Smart Materials, Adaptive Structures and Intelligent Systems
,
Louisville, KY
,
Sept. 9–11
, p.
V001T03A005
. 10.1115/SMASIS2019-5565.
21.
Kalthof
,
R.
,
2014
, “
Multibody Dynamics Modeling of Flexible Aircraft Flight Dynamics
.”,
Master thesis
,
TU Delft
.
22.
Howell
,
L. L.
,
Midha
,
A.
, and
Norton
,
T.
,
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
. 10.1115/1.2826843
23.
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
. 10.1016/S0020-7462(00)00076-7
24.
Lee
,
Y. H.
,
Vedant
,
Ewoldt
,
R. H.
, and
Allison
,
J. T.
,
2019
, “
Strain-Actuated Solar Arrays for Spacecraft Attitude Control Assisted by Viscoelastic Damping
,”
Proceedings of the 13th World Congress of Structural and Multidisciplinary Optimization
,
Beijing, China
,
A230788
.
25.
Wen
,
J. T.
, and
Murphy
,
S.
,
1991
, “
Stability Analysis of Position and Force Control for Robot Arms
,”
IEEE Trans. Automat. Control
,
36
(
3
), pp.
365
371
. 10.1109/9.73573
26.
Shamir
,
T.
,
1990
, “
The Singularities of Redundant Robot Arms
,”
Int. J. Robot. Res.
,
9
(
1
), pp.
113
121
. 10.1177/027836499000900105
27.
Vedant
,
PRBDM code repository
. [Online]. https://github.com/VedantFNO/PRBDM_JMD.
28.
Smith
,
N. A.
, and
Tromble
,
R. W.
,
2004
, “
Sampling Uniformly From the Unit Simplex
,”
Johns Hopkins University
,
Technical Report No. 29
.
29.
The MathWorks.
Multistart algorithm.
Accessed on June 10, 2019.
30.
Ugray
,
Z.
,
Lasdon
,
L.
,
Plummer
,
J.
,
Glover
,
F.
,
Kelly
,
J.
, and
Martí
,
R.
,
2007
, “
Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization
,”
INFORMS J. Comput.
,
19
(
3
), pp.
328
340
. 10.1287/ijoc.1060.0175
31.
Glover
,
F.
,
1998
, “A Template for Scatter Search and Path Relinking,”
Artificial Evolution
,
H. J.
Lutton
,
E.
Ronald
,
E.
Schoenauer
, and
M. S.
Dominique
, eds.,
Springer Berlin Heidelberg
, pp.
1
51
.
32.
The MathWorks.
Paretosearch Algorithm
. Accessed June 10, 2019.
33.
Fleischer
,
M.
,
2003
, “
The Measure of Pareto Optima Applications to Multi-Objective Metaheuristics
,”
International Conference on Evolutionary Multi-Criterion Optimization
,
C. M.
Fonseca
,
P. J.
Fleming
,
E. T.
Zitzler
, and
L. D.
Kalyanmoy
, eds.,
Springer
,
Berlin, Heidelberg
, pp.
519
533
.
34.
Custódio
,
A. L.
,
Madeira
,
J. A.
,
Vaz
,
A. I. F.
, and
Vicente
,
L. N.
,
2011
, “
Direct Multisearch for Multiobjective Optimization
,”
SIAM J. Optim.
,
21
(
3
), pp.
1109
1140
. 10.1137/10079731X
35.
Plunkett
,
R.
,
1963
, “
Natural Frequencies of Uniform and Non-Uniform Rectangular Cantilever Plates
,”
J. Mech. Eng. Sci.
,
5
(
2
), pp.
146
156
. 10.1243/JMES_JOUR_1963_005_020_02
You do not currently have access to this content.