Statics analysis and optimization design for a fixed - guided beam flexure

n (mm)
M-PRBM FEM Error % PRBM FEM Error %
AL
7075-T6
12 210.94 214.14 1.52 16.732 16.974 1.45
13 202.67 211.71 4.46 17.415 17.734 1.83
14 195.29 197.68 1.22 18.072 17.994 0.43
15 188.67 193.31 2.46 18.706 18.454 1.35
Titanium
Alloys Ti-13
heat treated
12 265.75 260.08 2.13 21.079 19.644 6.81
13 255.32 220.48 13.65 21.940 21.722 0.99
14 246.03 215.59 12.37 22.768 23.067 1.31
15 237.69 209.32 11.94 23.567 24.145 2.45
Stainless steel
17-7TH 1050
12 354.67 292.41 18.38 28.132 24.52 13.71
13 340.76 280.99 18.36 29.281 25.249 14.63
14 328.36 275.33 16.99 30.386 27.381 10.79
15 317.23 273.71 14.58 31.453 29.837 6.08
Carbon steel
4130 Q&T 800
12 358.24 292.41 17.55 28.415 24.033 14.57
13 344.19 284.75 16.44 29.576 25.395 13.27
14 331.67 278.46 15.20 30.692 27.484 9.55
15 320.42 274.24 13.55 31.769 29.435 6.42
and aluminum are 24.15 mm and 18.71 mm, re-
spectively. The maximum displacements of the
carbon and stainless steels are approximate val-
ues, and are higher than those of the titanium
and aluminum. The averaged errors between the
PRBM and FEM are 11.30%, 11.56%, 2.89% and
1.27% for carbon steel, stainless steel, titanium,
and aluminum, respectively. There are large dif-
ferences between the two aforementioned meth-
ods for carbon steel, stainless steel. There are
good agreements between them for titanium and
aluminum. Similar to the calculated results of
the maximum stresses, the elastic modulus of
aluminum is smaller than that of the other ma-
terials. This leads to the displacement of the
aluminum obtains the smallest values.
Case 2: h = 0.5
Similar to the calculated method of the h =
0.4 mm. Calculation results for h = 0.5 mm are
presented in Tab. 8, and plotted in Fig. 10 and
Fig. 11.
The data in Fig. 10 shows that maximum
stress distributions of four different materials
are plotted against the beam width. Similar to
the calculated results of the maximum stresses
show in Fig. 8. All of the stresses also decrease
when the beam width increase. The averaged er-
rors of the carbon steel, stainless steel, titanium,
and aluminum are 16.44%, 12.02%, 14.02%, and
3.54%, respectively. The maximum stresses of
those are 320 N/mm
2
, 317 N/mm
2
, 238 N/mm
2
,
and 197 N/mm
2
, respectively. The maximum
stresses of the titanium and aluminum lower
than of the carbon and stainless steel. The car-
bon steel, stainless steel, and titanium have large
averaged errors while the aluminum is nearly ac-
curacy. This may be explained similarly to the
calculated results of the maximum stresses with
h =0.4 mm. The data in Fig. 11 shows that cal-
culated displacements in the x−direction profile
of four different materials are drawn against the
beam width. The maximum displacements in
the x−direction of titanium and aluminum are
32.94 mm and 26.14 mm, respectively while car-
bon and stainless steels of this value are 44.4 mm
and 43.96 mm, respectively. The averaged er-
rors of the carbon steel, stainless steel, titanium
and aluminum between the PRBM and FEM are
14.56%, 15.21%, 4.83%, and 2.96%, respectively.
Similarly, as Fig. 9, the displacement of the alu-
minum obtains the smallest values.
Case 3: h = 0.6
c© 2020 Journal of Advanced Engineering and Computation (JAEC) 135
VOLUME: 4 | ISSUE: 2 | 2020 | June
Tab. 8: Comparing the difference of maximum stress and displacement in the x-direction values, h = 0.5 mm.
Material w(mm)
Maximum stress (N/mm
2
)
Displacement in the
x−direction (mm)
M-PRBM FEM Error % PRBM FEM Error %
AL 7075-T6
12 188.67 196.62 4.21 23.383 24.548 4.98
13 181.27 191.65 5.73 24.338 25.424 4.46
14 174.68 182.01 4.20 25.257 25.746 1.94
15 168.75 168.75 0.00 26.143 26.007 0.52
Titanium Alloys
Ti-13 heat treated
12 237.69 198.07 16.67 29.459 26.274 10.81
13 228.37 192.45 15.73 30.662 28.747 6.25
14 220.06 191.09 13.16 31.819 31.195 1.96
15 212.6 190.23 10.52 32.936 32.845 0.28
Stainless steel
17-7TH 1050
12 317.23 275.53 13.15 39.316 34.545 12.14
13 304.784 251.85 17.37 40.921 34.696 15.21
14 293.7 242.19 17.54 42.466 35.573 16.23
15 283.74 230.53 18.75 43.957 36.379 17.24
Carbon steel
4130 Q&T 800
12 320.42 276.25 13.79 39.712 34.199 13.88
13 307.85 257.39 16.39 41.333 34.85 15.68
14 296.65 241.38 18.63 42.893 35.91 16.28
15 286.59 238.04 16.94 44.399 38.894 12.40
Fig. 10: Distribution of maximum stress against the
beam width, h = 0.5 mm.
Calculation results for h = 0.6 mm are pre-
sented in Tab. 9, and plotted in Fig. 12 and
Fig. 13.
Maximum stress results drawn in Fig. 12 can
be compared with the data in Fig. 8. The av-
eraged error of titanium is 13.84%, and of the
carbon steel, stainless steel, and aluminum are
4.96%, 4.56%, and 4.5%, respectively. There is
a large difference between the PRBM and FEM
for the titanium while there is a good agree-
ment between them for the other materials. The
maximum stresses of those are 292.5 N/mm
2
Fig. 11: Distribution of displacement in the
x−direction against the beam width,
h = 0.5 mm.
and 289.59 N/mm
2
, 216.98 N/mm
2
and 177.93
N/mm
2
, respectively. Figure 13 is drawn dis-
placements in the x-direction of four materials
such as carbon steel, stainless steel, and alu-
minum and titanium. Those values are calcu-
lated by the PRBM and FEM methods. The
maximum displacements in the x−direction of
carbon and stainless steel are 61.49 mm and
60.35 mm, respectively. Those of non-steel such
as titanium and aluminum are 43.3 mm and 35.5
mm, respectively. The maximum displacements
136
c© 2020 Journal of Advanced Engineering and Computation (JAEC)
VOLUME: 4 | ISSUE: 2 | 2020 | June
Tab. 9: Comparing the difference of maximum stress and displacement in the x-direction values, h = 0.6 mm.
Material w(mm)
Maximum stress (N/mm
2
)
Displacement in the
x−direction (mm)
M-PRBM FEM Error % PRBM FEM Error %
AL 7075-T6
12 172.23 177.93 3.31 30.738 31.467 2.37
13 165.48 174.61 5.52 31.993 31.862 0.41
14 159.46 165.81 3.98 33.201 34.632 4.31
15 154.05 162.01 5.17 34.366 35.498 3.29
Titanium Alloys
Ti-13 heat treated
12 216.98 191.45 11.77 38.724 37.114 4.16
13 208.47 181.42 12.98 40.306 38.011 5.69
14 200.89 170.78 14.99 41.827 38.429 8.12
15 194.08 163.81 15.60 43.295 39.681 8.35
Stainless steel
17-7TH 1050
12 289.59 274.37 5.26 51.682 54.745 5.93
13 278.23 262.11 5.79 53.792 56.392 4.83
14 268.11 260.05 3.01 55.823 59.21 6.07
15 259.02 248.2 4.18 57.782 60.348 4.44
Carbon steel
4130 Q&T 800
12 292.5 278.53 4.78 52.202 55.87 7.03
13 281.03 266.28 5.25 54.334 57.863 6.50
14 270.81 252.42 6.79 56.385 61.355 8.81
15 261.62 253.74 3.01 58.364 61.486 5.35
Fig. 12: Distribution of maximum stress against the
beam width, h = 0.6 mm.
of the titanium and aluminum are approximate
values and less than the carbon and stainless
steels. The averaged errors between the PRBM
and FEM are 6.92%, 5.32%, 6.58%, and 2.6% for
the carbon steel, stainless steel, titanium, and
aluminum, respectively. There are small differ-
ences between two methods for all materials.
Summarily, the errors of maximum stress
and x−direction displacements between the
M-PRBM and FEM are generated by calcu-
lated. The M-PRBM calculates the line while
FEM base on the area. Moreover, materials
Fig. 13: Distribution of displacement in the
x−direction against the beam width,
h = 0.6 mm.
change (elastic modulus E change) greatly af-
fect the accuracy of the maximum stress and the
x−direction displacement between M-PRBM
and FEM. Therefore, M-PRBM theory is only
suitable for materials with elastic modulus simi-
lar to aluminum material. Besides, when h = 0.4
mm and 0.5 mm, the error of maximum stress
and the x−direction displacement increases as
the material has an elastic modulus larger (i.e.
the error is proportional to the elastic modulus
E). However, when h = 0.6 mm, this error does
c© 2020 Journal of Advanced Engineering and Computation (JAEC) 137
VOLUME: 4 | ISSUE: 2 | 2020 | June
not change much when materials are changed,
except for the average error of the maximum
stress for titanium material.
5. Conclusions
In this paper, static analysis and optimization
design are proposed for the fixed-guided beam
flexures. These flexures are intended for driv-
ing the ratchet mechanism. The PRBM theory
for large deformations of fixed-guided parallel
beams flexures are used to calculate the max-
imum stress and the x−direction displacement.
Predicted values from the analytical method are
verified by FEM using ANSYS 18.1 software.
The FEM simulations are established for three
cases of the beams that have the height vary-
ing from 0.4 mm to 0.6 mm with four differ-
ent materials. The configuration of the fixed-
guided beam has been optimized for maximum
x−direction displacement. Because there is a
large error of maximum stress between PRBM
and FEM, so M-PRBM is designed to obtain
the accuracy of the maximum stress value. The
value of y−direction displacement (parasitical
displacement) is very small, so it is neglected.
Compared with the FEM simulations, the M-
PRBM is a better calculation of the maximum
stress than the PRBM. The averaged errors be-
tween the M-PRBM and the FEM simulation
are 3.48% for aluminum, and less than 10.9%
for titanium, carbon steel, and alloy steel. The
M-PRBM is therefore good for the design and
fabrication of the compliant mechanism. Future
work, the proposed method is extended for re-
lated compliant mechanisms.
Acknowledgement
This research is funded by Vietnam National
Foundation for Science and Technology De-
velopment (NAFOSTED) under grant number
107.01-2019.14.
References
[1] Howell, L. L. (2001). Compliant mech-
anisms. Mechanical Engineering Depart-
ment. Brigham Young University.
[2] Yuanqiang, L., & Wangyu, L. (2014). Anal-
ysis of the displacement of distributed com-
pliant parallel-guiding mechanism consid-
ering parasitic rotation and deflection on
the guiding plate. Mechanism and Machine
Theory, 80, 151-165.
[3] Liu, Y., & Xu, Q. (2016). Design of a
compliant constant force gripper mecha-
nism based on buckled fixed-guided beam.
In 2016 International Conference on Manip-
ulation, Automation and Robotics at Small
Scales (MARSS) (pp. 1-6). IEEE.
[4] Zhang, J., Yan, K., & Kou, Z. (2019). De-
sign and Analysis of Flexible Hinge Used for
Unfolding Spacecraft Solar Panels. Journal
of Aerospace Technology and Management,
11.
[5] She, Y., Meng, D., Su, H. J., Song, S.,
& Wang, J. (2018). Introducing mass pa-
rameters to PseudoRigidBody models for
precisely predicting dynamics of compliant
mechanisms. Mechanism and Machine The-
ory, 126, 273-294.
[6] Mattson, C. A., Howell, L. L., & Magleby,
S. P. (2004). Development of commer-
cially viable compliant mechanisms using
the pseudo-rigid-body model: case studies
of parallel mechanisms. Journal of intelli-
gent material systems and structures, 15(3),
195-202.
[7] Zirbel, S. A., Tolman, K. A., Trease, B.
P., & Howell, L. L. (2016). Bistable mech-
anisms for space applications. PloS one,
11(12).
[8] Qi, K. Q., Ding, Y. L., Xiang, Y., Fang, C.,
& Zhang, Y. (2017). A novel 2-DOF com-
pound compliant parallel guiding mecha-
nism. Mechanism and Machine Theory, 117,
21-34.
[9] Liu, P., & Peng, Y. (2017). A modified
pseudo-rigid-body modeling approach for
138
c© 2020 Journal of Advanced Engineering and Computation (JAEC)
VOLUME: 4 | ISSUE: 2 | 2020 | June
compliant mechanisms with fixed-guided
beam flexures. Mechanical Sciences, 8(2),
359.
[10] Kennedy, J. A., Howell, L. L., & Green-
wood, W. (2007). Compliant high-precision
E-quintet ratcheting (CHEQR) mechanism
for safety and arming devices. Precision en-
gineering, 31(1), 13-21.
[11] Pei, X., Yu, J., Zong, G., & Bi, S. (2010).
An effective pseudo-rigid-body method for
beam-based compliant mechanisms. Preci-
sion Engineering, 34(3), 634-639.
[12] Dao, T. P., Ho, N. L., Nguyen, T. T.,
Le, H. G., Thang, P. T., Pham, H. T.,
Do, H. T., Tran, M. D., Nguyen, T.
T. (2017). Analysis and optimization of
a micro-displacement sensor for compli-
ant microgripper. Microsystem Technolo-
gies, 23(12), 5375-5395.
[13] Lofroth, M., & Avci, E. (2019). Develop-
ment of a novel modular compliant gripper
for manipulation of micro objects. Micro-
machines, 10(5), 313.
[14] Gupta, V., Perathara, R., Chaurasiya, A.
K., & Khatait, J. P. (2019). Design and
analysis of a flexure based passive gripper.
Precision Engineering, 56, 537-548.
[15] Chau, N. L., Dao, T. P., & Nguyen, V. T.
T. (2018). Optimal design of a dragonfly-
inspired compliant joint for camera posi-
tioning system of nanoindentation tester
based on a hybrid integration of Jaya-
ANFIS. Mathematical Problems in Engi-
neering, 2018.
[16] Dang, M. P., Dao, T. P., Chau, N. L., & Le,
H. G. (2019). Effective hybrid algorithm of
Taguchi method, FEM, RSM, and teaching
learning-based optimization for multiobjec-
tive optimization design of a compliant ro-
tary positioning stage for nanoindentation
tester. Mathematical Problems in Engineer-
ing, 2019.
[17] Le Chau, N., Le, H. G., & Dao, T. P. (2017).
Robust parameter design and analysis of
a leaf compliant joint for micropositioning
systems. Arabian Journal for Science and
Engineering, 42(11), 4811-4823.
[18] Reddy, J. N. (2006). Theory and analysis of
elastic plates and shells. CRC press.
[19] Venkataraman, P. (2009). Applied op-
timization with MATLAB programming.
John Wiley & Sons.
[20] Juvinall, R. C. and Marshek, K. M. (2017).
Fundamentals of machine component. Pro-
fessor of Mechanical Engineering. Univer-
sity of Michigan.
About Authors
Ngoc Thoai TRAN received his B.S.
degree in mechanical engineering, Can Tho
University, Vietnam, in 2009. He received his
M.S. degree in mechanical engineering, Ho
Chi Minh City University of Technology, in
2013. He is currently a lecturer at Faculty of
Mechanical Engineering, Industrial University
of Ho Chi Minh City, Vietnam. His research
interests include compliant mechanism, assistive
technology and rehabilitation, and optimization
algorithm..
Thanh-Phong DAO is currently an assistant
professor at the Institute for Computational
Science, Ton Duc Thang University, Ho Chi
Minh City, Vietnam. He received his B.S.
degree in mechanical engineering from the Ho
Chi Minh City University of Technology and
Education, Vietnam in 2008. He received his
M.S. and Ph.D. degree in mechanical engineer-
ing from the National Kaohsiung University of
Applied Sciences, Taiwan, ROC, in 2011 and
2015, respectively. His research interests include
compliant mechanism, assistive technology and
rehabilitation, and optimization algorithm.
"This is an Open Access article distributed under the terms of the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium provided the original work is
properly cited (CC BY 4.0)."
139