We present a compliance matrix for a single-bent leaf flexure (SBLF) that shows the relationships between the deformations and the six-axis loads applied to the SBLF. Higher-order beam theory that considers the variable shear and warping effect is considered in bending. The partially restrained warping at the junction between elements is also considered in torsion. The total strain energy is calculated, and the complete compliance matrix is derived by using Castigliano’s second theorem. Sensitivity analyses over the compliance elements are performed and verified via finite element analysis (FEA). The results show that the derived compliance elements are in good agreement with FEA, with errors of less than 7.6%. We suggest that theoretical compliance elements considering variable shear and warping in bending and partially restrained warping in torsion give highly accurate design equations representing the compliant mechanism of the SBLF. The present work could be used in a modal analysis of a single-bent leaf flexure.
Flexure guides are used in precision engineering, especially in nanoscanner devices, because of their smooth, elastic, and no-friction characteristics. Flexure guides provide short or moderate ranges of travel due to their elastic deformation characteristics [
The compliance matrix is often used to express the load-deformation relationship for a linearly elastic system. It has been applied in many previous studies in the precision field [
In this study, we analyzed and derived a compliance matrix that expresses the relationship between the deformations and applied loads of the SBLF by using Castigliano’s second theorem. The HBT of Levinson is applied in the bending analysis. The partially restrained warping at the junction between elements is introduced, and the full and free warping is also investigated in the torsional analysis. The theoretical results are verified by finite element analysis (FEA) results. The good agreement between the results of the two methods validates the accuracy of the theoretical equations.
Figure
(a) Schematic diagram of a SBLF. (b) Example of the planar scanner using SBLF.
Castigliano’s second theorem was used to find the translational and rotational deformations of the SBLF. Thus, the deformations due to loads are defined by the partial derivative of the total strain energy (SE) with respect to the loads as follows:
The extensional SE (
In the bending and shear analyses, the shear deformations due to the transverse loads are usually neglected in Euler-Bernoulli beam theory (EBT). Accordingly, the EBT can predict the deflection accurately only for a thin beam (ratio
A partially restrained warping torsion analysis at the joint between elements 1 and 2 is presented in [
The torsion constant
The total SE (
From (
The relationship between the translational and rotational displacements and the applied forces and moments at the free end of the SBLF is formulated as follows:
In this study, FEA was conducted by using Pro-Mechanica commercial software (Wildfire 5, PTC Corp., MA, USA) to verify the results obtained by theory and to validate the reliability of the calculated results. The default values of the SBLF were chosen as follows: length
Table
Comparison between theory and the FEA at the default values of the SBLF.
Displacements | FEA | Theory | Error (%) | ||
---|---|---|---|---|---|
Shear and Warp | No shear and No warp | S and W | No S and No W | ||
|
1.16210 | 1.16466 | 1.16183 | 0.22 |
|
|
0.44183 | 0.46480 | 0.46451 | 4.94 | 4.88 |
|
0.21102 | 0.20825 | 0.22556 | 1.31 | 6.45 |
|
0.00239 | 0.00233 | 0.00251 | 2.19 | 5.04 |
|
0.00210 | 0.00209 | 0.00227 | 0.47 | 7.43 |
|
0.00673 | 0.00697 | 0.00697 | 3.43 | 3.43 |
In the research of the reference [
Theoretical and FEA results with various lengths
|
FEA | Theory | Error (%) |
|
FEA | Theory | Error (%) |
|
FEA | Theory | Error (%) |
---|---|---|---|---|---|---|---|---|---|---|---|
(a) Deflection |
|||||||||||
5 | 0.13819 | 0.14550 |
|
0.25 | 8.77255 | 9.29028 |
|
2 | 2.28306 | 2.32366 |
|
6.5 | 0.30485 | 0.31934 |
|
0.325 | 3.97715 | 4.22907 |
|
2.6 | 1.74683 | 1.78743 |
|
8 | 0.57071 | 0.59507 |
|
0.4 | 2.17394 | 2.26869 |
|
3.2 | 1.41120 | 1.45229 |
|
9.5 | 0.95908 | 0.99619 |
|
0.475 | 1.30778 | 1.35502 |
|
3.8 | 1.18188 | 1.22298 |
|
11 | 1.49324 | 1.54623 |
|
0.55 | 0.84862 | 0.87301 |
|
4.4 | 1.01548 | 1.05621 |
|
12.5 | 2.19646 | 2.26869 |
|
0.625 | 0.58257 | 0.59507 |
|
5 | 0.88936 | 0.92946 |
|
14 | 3.09203 | 3.18708 |
|
0.7 | 0.41772 | 0.42366 |
|
5.6 | 0.79058 | 0.82988 |
|
15.5 | 4.20323 | 4.32493 |
|
0.775 | 0.31008 | 0.31227 |
|
6.2 | 0.71120 | 0.74957 |
|
17 | 5.55339 | 5.70573 |
|
0.85 | 0.23676 | 0.23676 |
|
6.8 | 0.64607 | 0.68343 |
|
18.5 | 7.16579 | 7.35301 |
|
0.925 | 0.18508 | 0.18377 |
|
7.4 | 0.59170 | 0.62802 |
|
20 | 9.06374 | 9.29028 |
|
1 | 0.14756 | 0.14550 |
|
8 | 0.54565 | 0.58091 |
|
|
|||||||||||
(b) Deflection |
|||||||||||
5 | 0.05369 | 0.05809 |
|
0.25 | 3.55785 | 3.71567 |
|
2 | 0.90181 | 0.92903 |
|
6.5 | 0.11959 | 0.12759 |
|
0.325 | 1.62113 | 1.69129 |
|
2.6 | 0.69073 | 0.71464 |
|
8 | 0.22485 | 0.23785 |
|
0.4 | 0.86939 | 0.90720 |
|
3.2 | 0.55848 | 0.58064 |
|
9.5 | 0.37879 | 0.39827 |
|
0.475 | 0.51931 | 0.54178 |
|
3.8 | 0.46800 | 0.48896 |
|
11 | 0.59062 | 0.61825 |
|
0.55 | 0.33455 | 0.34901 |
|
4.4 | 0.40226 | 0.42229 |
|
12.5 | 0.86949 | 0.90720 |
|
0.625 | 0.22798 | 0.23785 |
|
5 | 0.35238 | 0.37161 |
|
14 | 1.22458 | 1.27453 |
|
0.7 | 0.16225 | 0.16931 |
|
5.6 | 0.31329 | 0.33180 |
|
15.5 | 1.66503 | 1.72963 |
|
0.775 | 0.11953 | 0.12477 |
|
6.2 | 0.28185 | 0.29969 |
|
17 | 2.19995 | 2.28192 |
|
0.85 | 0.09057 | 0.09458 |
|
6.8 | 0.25604 | 0.27324 |
|
18.5 | 2.83849 | 2.94080 |
|
0.925 | 0.07025 | 0.07339 |
|
7.4 | 0.23449 | 0.25109 |
|
20 | 3.58976 | 3.71567 |
|
1 | 0.05557 | 0.05809 |
|
8 | 0.21624 | 0.23226 |
|
|
|||||||||||
(c) Deflection |
|||||||||||
5 | 0.02203 | 0.02113 |
|
0.25 | 1.60255 | 1.58055 |
|
2 | 0.52645 | 0.51994 |
|
6.5 | 0.05270 | 0.05127 |
|
0.325 | 0.74212 | 0.73006 |
|
2.6 | 0.37023 | 0.36566 |
|
8 | 0.10341 | 0.10146 |
|
0.4 | 0.40469 | 0.39777 |
|
3.2 | 0.28218 | 0.27863 |
|
9.5 | 0.17912 | 0.17672 |
|
0.475 | 0.24617 | 0.24151 |
|
3.8 | 0.22569 | 0.22270 |
|
11 | 0.28479 | 0.28210 |
|
0.55 | 0.16164 | 0.15831 |
|
4.4 | 0.18640 | 0.18372 |
|
12.5 | 0.42541 | 0.42263 |
|
0.625 | 0.11239 | 0.10988 |
|
5 | 0.15750 | 0.15497 |
|
14 | 0.60595 | 0.60338 |
|
0.7 | 0.08169 | 0.07972 |
|
5.6 | 0.13538 | 0.13290 |
|
15.5 | 0.83137 | 0.82938 |
|
0.775 | 0.06152 | 0.05992 |
|
6.2 | 0.11792 | 0.11542 |
|
17 | 1.10665 | 1.10569 |
|
0.85 | 0.04770 | 0.04636 |
|
6.8 | 0.10379 | 0.10126 |
|
18.5 | 1.43675 | 1.43737 |
|
0.925 | 0.03788 | 0.03675 |
|
7.4 | 0.09214 | 0.08956 |
|
20 | 1.82663 | 1.82945 |
|
1 | 0.03071 | 0.02973 |
|
8 | 0.08239 | 0.07975 |
|
Variation of deflection
Variation of deflection
The comparison results among the fully restrained, partially restrained, and free warping in torsion analysis of SBLF were investigated in [
Theoretical and FEA results with various lengths
|
FEA | Theory | Error (%) |
|
FEA | Theory | Error (%) |
|
FEA | Theory | Error (%) |
---|---|---|---|---|---|---|---|---|---|---|---|
(a) Rotation |
|||||||||||
5 | 0.00114 | 0.00108 |
|
0.25 | 0.01795 | 0.01762 |
|
2 | 0.00568 | 0.00561 |
|
6.5 | 0.00152 | 0.00145 |
|
0.325 | 0.00833 | 0.00815 |
|
2.6 | 0.00404 | 0.00397 |
|
8 | 0.00189 | 0.00183 |
|
0.4 | 0.00455 | 0.00445 |
|
3.2 | 0.00312 | 0.00306 |
|
9.5 | 0.00226 | 0.00221 |
|
0.475 | 0.00277 | 0.00270 |
|
3.8 | 0.00254 | 0.00248 |
|
11 | 0.00263 | 0.00258 |
|
0.55 | 0.00183 | 0.00178 |
|
4.4 | 0.00213 | 0.00208 |
|
12.5 | 0.00301 | 0.00296 |
|
0.625 | 0.00127 | 0.00123 |
|
5 | 0.00183 | 0.00179 |
|
14 | 0.00338 | 0.00334 |
|
0.7 | 0.00093 | 0.00090 |
|
5.6 | 0.00161 | 0.00156 |
|
15.5 | 0.00375 | 0.00371 |
|
0.775 | 0.00070 | 0.00068 |
|
6.2 | 0.00143 | 0.00138 |
|
17 | 0.00412 | 0.00409 |
|
0.85 | 0.00054 | 0.00052 |
|
6.8 | 0.00128 | 0.00124 |
|
18.5 | 0.00450 | 0.00447 |
|
0.925 | 0.00043 | 0.00042 |
|
7.4 | 0.00116 | 0.00112 |
|
20 | 0.00487 | 0.00485 |
|
1 | 0.00035 | 0.00034 |
|
8 | 0.00106 | 0.00102 |
|
|
|||||||||||
(b) Rotation |
|||||||||||
5 | 0.00089 | 0.00084 |
|
0.25 | 0.01605 | 0.01583 |
|
2 | 0.00540 | 0.00533 |
|
6.5 | 0.00126 | 0.00122 |
|
0.325 | 0.00744 | 0.00732 |
|
2.6 | 0.00376 | 0.00371 |
|
8 | 0.00163 | 0.00159 |
|
0.4 | 0.00407 | 0.00399 |
|
3.2 | 0.00285 | 0.00281 |
|
9.5 | 0.00200 | 0.00197 |
|
0.475 | 0.00248 | 0.00243 |
|
3.8 | 0.00228 | 0.00224 |
|
11 | 0.00237 | 0.00235 |
|
0.55 | 0.00163 | 0.00159 |
|
4.4 | 0.00188 | 0.00185 |
|
12.5 | 0.00274 | 0.00272 |
|
0.625 | 0.00114 | 0.00111 |
|
5 | 0.00158 | 0.00155 |
|
14 | 0.00312 | 0.00310 |
|
0.7 | 0.00083 | 0.00081 |
|
5.6 | 0.00136 | 0.00133 |
|
15.5 | 0.00349 | 0.00348 |
|
0.775 | 0.00063 | 0.00061 |
|
6.2 | 0.00118 | 0.00116 |
|
17 | 0.00386 | 0.00385 |
|
0.85 | 0.00049 | 0.00047 |
|
6.8 | 0.00104 | 0.00101 |
|
18.5 | 0.00423 | 0.00423 |
|
0.925 | 0.00039 | 0.00037 |
|
7.4 | 0.00092 | 0.00090 |
|
20 | 0.00460 | 0.00461 |
|
1 | 0.00032 | 0.00030 |
|
8 | 0.00082 | 0.00080 |
|
|
|||||||||||
(c) Rotation |
|||||||||||
5 | 0.00328 | 0.00348 |
|
0.25 | 0.05405 | 0.05573 |
|
2 | 0.01365 | 0.01393 |
|
6.5 | 0.00432 | 0.00453 |
|
0.325 | 0.02462 | 0.02537 |
|
2.6 | 0.01047 | 0.01072 |
|
8 | 0.00535 | 0.00557 |
|
0.4 | 0.01320 | 0.01361 |
|
3.2 | 0.00848 | 0.00871 |
|
9.5 | 0.00638 | 0.00662 |
|
0.475 | 0.00789 | 0.00813 |
|
3.8 | 0.00711 | 0.00733 |
|
11 | 0.00741 | 0.00766 |
|
0.55 | 0.00508 | 0.00523 |
|
4.4 | 0.00612 | 0.00633 |
|
12.5 | 0.00845 | 0.00871 |
|
0.625 | 0.00346 | 0.00357 |
|
5 | 0.00537 | 0.00557 |
|
14 | 0.00948 | 0.00975 |
|
0.7 | 0.00246 | 0.00254 |
|
5.6 | 0.00477 | 0.00498 |
|
15.5 | 0.01051 | 0.01080 |
|
0.775 | 0.00182 | 0.00187 |
|
6.2 | 0.00430 | 0.00449 |
|
17 | 0.01153 | 0.01184 |
|
0.85 | 0.00138 | 0.00142 |
|
6.8 | 0.00391 | 0.00410 |
|
18.5 | 0.01256 | 0.01289 |
|
0.925 | 0.00107 | 0.00110 |
|
7.4 | 0.00358 | 0.00377 |
|
20 | 0.01359 | 0.01393 |
|
1 | 0.00084 | 0.00087 |
|
8 | 0.00330 | 0.00348 |
|
Variation of rotation
In summary, the relationship between the deformation and the applied loads for the SBLF is presented in compliance matrix form. The variable shear deformation and warping effect were considered, and partial restraint was introduced in the analysis. All results were verified by FEA, with strong agreement between the two methods. The errors were lower than 7.6%. These results demonstrate the high accuracy and reliability of the proposed theoretical equations.
In this study, a compliance matrix that expresses the relationship between the deformations and applied loads of the SBLF was analyzed and derived by using Castigliano’s second theorem. In bending analysis, higher-order beam theory was applied wherein variable shear deformation and the warping were considered in the calculated formulas for the shear forces. At the joint of two elements of the SBLF, the partially restrained warping in torsion was analysed with consideration of the warping restraint factor. The theoretical results were verified by FEA at both the default and the sensitive values. The results indicate that there is strong agreement between the two methods, with errors below 7.6%. This suggests the accuracy of the proposed theoretical equations and that they can be used in the precision machine design.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Nghia-Huu Nguyen and Moo-Yeon Lee equally contributed to this work.
This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657).