FE Model Validation

The FE model was validated by considering the benchmark cases of vibration of a rectangular plate, an elliptical membrane, and a tubular cantilever beam. These cases validated the selection of elements and the solution approach from the FE models of the artificial insect- sized wing and the crane fly forewing.

4.4.1.1 Shell Modeling Validation

A benchmark case of the vibrations of a rectangular flat shell was considered to validate the solution approach of the FE model of a shell. The natural frequencies of a FE model of a rectangular thin flat shell were calculated and compared with those from the analytical solution [93]. The FE rectangular shell model was developed as a thin shell with a thickness of 0.0125 mm, a width of 5.6 mm, and a length of 14 mm. These dimensions fairly agreed with the

dimensions of the membrane of the composite material artificial wing. The material properties of the Kapton film, described in Table 2.4, were assigned to the FE model of the rectangular shell. The shell was clamped at one edge and free at the other three edges. The FE model of the rectangular shell was meshed using S4R (4-node general-purpose shell) type of elements. The natural frequencies of the rectangular shell were calculated with a converged mesh grid of 8037 nodes. The variation of the natural frequencies of the FE model and the analytical solution [93] was less than 1.0 % as shown in Table 4.9. Similarly, the first three mode shapes of the

rectangular shell were determined and agreed with those from the analytical solution [93]. The mode shapes of the rectangular shell are shown in Fig. 4.26.

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Table 4.9 Comparison between the natural frequencies of the FE model of the rectangular shell and the analytical solution [93]

Figure 4.26 Mode shapes of the rectangular shell with one fixed end and three free ends. (A and B) First at 15.20 Hz, (C and D) second at 79.62 Hz, and (E and F) third at 94.87 Hz mode shapes of the rectangular flat shell.

4.4.1.2 Membrane Modeling Validation

A benchmark case of the vibrations of an elliptical membrane was considered to validate the solution approach of the FE model of a membrane. The natural frequencies of a FE model of

Mode Analytical Natural frequency (Hz) solution [93] FE model Percent error (%) First 15.32 15.20 0.78 Second 79.75 79.62 0.16 Third 95.60 94.87 0.76

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an elliptical membrane were calculated and compared with those from the analytical solution from Mathieu functions [106]. The FE elliptical membrane model was developed considering a thickness of 0.009 mm, a major radius of 6.93 mm, a minor radius of 1.6 mm, and a uniform pretension of 1000 Pa. These dimensions fairly agreed with those from the artificial insect-sized wing. The material properties of the Kapton film, described in Table 2.4, were assigned to the FE model of the elliptical membrane. The FE model of the elliptical membrane was meshed using M3D6 (6-node quadratic triangular membrane) type of elements. The natural frequencies of the elliptical membrane were calculated with a converged mesh grid of 4485 nodes. The variationof the natural frequencies of the FE model of the elliptical membrane and the analytical solution [106] was less than 1.0 % as shown in Table 4.10. Similarly, the first three mode shapes of the elliptical membrane were determined and agreed with those from the analytical solution [93]. The mode shapes of the elliptical membrane are shown in Fig. 4.27.

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Table 4.10 Comparison between the natural frequencies of the FE model of the elliptical membrane and the analytical solution [106]

Figure 4.27 Mode shapes of the elliptical membrane with a clamped edge. (A and B) First at 163.54 Hz, (C and D) second at 186.25 Hz, and (E and F) third at 210.20 Hz mode shapes of the elliptical

membrane.

4.4.1.3 Beam Modeling Validation

A benchmark case of the vibrations of a cantilever beam was considered to validate the solution approach of a FE model of a beam. The natural frequencies of a FE model of a

Mode Analytical Natural frequency (Hz) solution [106] FE model Percent error (%) First 163.20 163.54 0.21 Second 185.62 186.25 0.34 Third 209.55 210.20 0.31

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cantilever beam with a tubular cross-section were calculated and compared with those from the analytical solution using the Euler-Bernoulli beam theory [6]. The FE beam model was

developed considering a length of 14 mm and a tubular cross-section with an outer diameter of 0.126 mm and an inner diameter of 0.045 mm. These dimensions were comparable to those from the thick veins of the artificial wing. The material properties of the SU-8, described in Table 2.4, were assigned to the FE model of the beam. The FE model of the cantilever beam was meshed using B32 (3-node quadratic beam) type of elements. The natural frequencies of the cantilever beam were calculated with a converged mesh grid of 121 nodes. The variationof the natural frequencies of the FE model of the cantilever beam and the analytical solution [6] was less than 1.0 % as shown in Table 4.11. Similarly, the first three mode shapes of the cantilever beam were determined and agreed with those from the analytical solution [6]. The mode shapes of the cantilever beam are shown in Fig. 4.28.

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Table 4.11 Comparison between the natural frequencies of the FE model of cantilever beam with tubular cross-section and the analytical solution [6]

Figure 4.28 Mode shapes of the cantilever beam with tubular cross-section. (A) First at 171.11 Hz, (B) second at 1096.91 Hz, and (C) third at 3069.30 Hz mode shapes of the cantilever beam with tubular cross-section.