Diatom-Inspired Structural Adaptation According to Mode Shapes: A Study on 3D Structures and Software Tools
<p>Scanning electron microscopic images of diatom frustules (adapted according to [<a href="#B5-biomimetics-09-00241" class="html-bibr">5</a>]).</p> "> Figure 2
<p>Procedure of the investigation.</p> "> Figure 3
<p>Three dimensional view of (<b>a</b>) hollow cuboid, (<b>b</b>) thick hexagonal prism, (<b>c</b>) cube, (<b>d</b>) truncated pyramid, (<b>e</b>) thin hexagonal prism, (<b>f</b>) curved rectangular duct, and (<b>g</b>) connector square investigated in the present study. The edges and surfaces highlighted in red indicate the defined clamped condition.</p> "> Figure 4
<p>1st (1) and 2nd (2) mode shape of the studied structures (<b>a</b>) hollow cuboid, (<b>b</b>) thick hexagonal prism, (<b>c</b>) cube, (<b>d</b>) truncated pyramid, (<b>e</b>) thin hexagonal prism, (<b>f</b>) curved rectangular duct, and (<b>g</b>) connector square. The coloring represents the normalized vibration amplitude.</p> "> Figure 5
<p>Automatization process of the bio-inspired mode shape adaptation method.</p> "> Figure 6
<p>Created mode shape adaptation components for a (<b>a</b>) beam, (<b>b</b>) shell, and (<b>c</b>) solid model.</p> "> Figure 7
<p>Explanation of the inputs and outputs of the created software components.</p> "> Figure 8
<p>Example of a pre-deformed model creation process with in the Synera environment using the developed software component for beam models.</p> "> Figure 9
<p>Three dimensional view of the undeformed beam (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 0 mm) and the beam pre-deformed according to the 1st mode shape considering different maximum pre-deformations <math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math>. The beam itself is colored in dark grey and the undeformed beam is illustrated with a red dashed line.</p> "> Figure 10
<p>Circular arc dimensions.</p> "> Figure 11
<p>Three dimensional view of the undeformed plate (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 0 mm) and the plate pre-deformed according to the 1st mode shape considering different maximum pre-deformations <math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math>.</p> "> Figure 12
<p>Three dimensional view of the investigated structures pre-deformed according to the 1st mode shape, including (<b>a</b>) hollow cuboid (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 90 mm), (<b>b</b>) thick hexagonal prism (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 15 mm), (<b>c</b>) cube (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 15 mm), (<b>d</b>) truncated pyramid (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 15 mm), (<b>e</b>) thin hexagonal prism (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 15 mm), (<b>f</b>) curved rectangular duct (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 15 mm), and (<b>g</b>) connector square (<math display="inline"><semantics> <msub> <mi>δ</mi> <mi>max</mi> </msub> </semantics></math> = 15 mm). For visualization purposes, blue curves are imbedded in the pre-deformed surfaces to illustrate the deformations.</p> "> Figure 13
<p>Eigenfrequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math> (<b>a</b>) and mode shape frequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>1</mn> </mrow> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>4</mn> </mrow> </msub> </semantics></math> (<b>b</b>) of the hollow cuboid pre-deformed according to mode 1, considering different maximum relative pre-deformations. For two pre-deformations, the frequency deviation compared to the undeformed structure of the mode shape adapted to the structure is given in red. In addition, the maximum obtained frequency increase is also noted.</p> "> Figure 14
<p>Eigenfrequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math> (<b>a</b>) and mode shape frequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>1</mn> </mrow> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>4</mn> </mrow> </msub> </semantics></math> (<b>b</b>) of the thick hexagonal prism pre-deformed according to mode 1, considering different maximum relative pre-deformations. For two pre-deformations, the frequency deviation compared to the undeformed structure of the mode shape adapted to the structure is given in red. In addition, the maximum obtained frequency increase is also noted.</p> "> Figure 15
<p>Eigenfrequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math> (<b>a</b>) and mode shape frequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>1</mn> </mrow> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>4</mn> </mrow> </msub> </semantics></math> (<b>b</b>) of the cube pre-deformed according to mode 1, considering different maximum relative pre-deformations. For two pre-deformations, the frequency deviation compared to the undeformed structure of the mode shape adapted to the structure is given in red. In addition, the maximum obtained frequency increase is also noted.</p> "> Figure 16
<p>Eigenfrequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math> (<b>a</b>) and mode shape frequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>1</mn> </mrow> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>4</mn> </mrow> </msub> </semantics></math> (<b>b</b>) of the truncated pyramid pre-deformed according to mode 1, considering different maximum relative pre-deformations. For two pre-deformations, the frequency deviation compared to the undeformed structure of the mode shape adapted to the structure is given in red. In addition, the maximum obtained frequency increase is also noted.</p> "> Figure 17
<p>Eigenfrequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math> (<b>a</b>) and mode shape frequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>1</mn> </mrow> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>4</mn> </mrow> </msub> </semantics></math> (<b>b</b>) of the thin hexagonal prism pre-deformed according to mode 1, considering different maximum relative pre-deformations. For two pre-deformations, the frequency deviation compared to the undeformed structure of the mode shape adapted to the structure is given in red. In addition, the maximum obtained frequency increase is also noted.</p> "> Figure 18
<p>Eigenfrequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math> (<b>a</b>) and mode shape frequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>1</mn> </mrow> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>4</mn> </mrow> </msub> </semantics></math> (<b>b</b>) of the curved rectangular duct pre-deformed according to mode 1, considering different maximum relative pre-deformations. For two pre-deformations, the frequency deviation compared to the undeformed structure of the mode shape adapted to the structure is given in red. In addition, the maximum obtained frequency increase is also noted.</p> "> Figure 19
<p>Eigenfrequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math> (<b>a</b>) and mode shape frequencies from <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>1</mn> </mrow> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>f</mi> <mrow> <mi>M</mi> <mn>4</mn> </mrow> </msub> </semantics></math> (<b>b</b>) of the connector square pre-deformed according to mode 1, considering different maximum relative pre-deformations. For two pre-deformations, the frequency deviation compared to the undeformed structure of the mode shape adapted to the structure is given in red. In addition, the maximum obtained frequency increase is also noted.</p> "> Figure 20
<p>Analytically and numerically (i.e., using the created software component) obtained frequencies of the first and third mode shape of the beam pre-deformed according the first mode shape. Some data points are almost identical, which is why some markers are printed above others.</p> "> Figure 21
<p>The numerical result of the first six eigenfrequencies of the square plate pre-deformed according to the first mode shape using the created shell component. Some data points are almost identical, which is why some markers are printed above others.</p> ">
Abstract
:1. Introduction
2. Materials and Methods
2.1. Bio-Inspired Mode Shape Adaptation of 3D Structures
- Hollow cuboid: 0 mm to 150 mm in steps of 15 mm;
- Thick hexagonal prism: 0 mm to 5 mm in steps of 1 mm, and 5 mm to 25 mm in steps of 5 mm;
- Cube: 0 mm to 5 mm in steps of 1 mm, and 5 mm to 30 mm in steps of 5 mm.
2.2. Integration of the Bio-Inspired Adaptation Method into a Low-Code Software
2.2.1. Automatization of the Mode Shape Adaptation Method
2.2.2. Validation of the Created Software Components
- (i)
- Mode Shape Adaptation of Beam Components
- (ii)
- Mode Shape Adaptation of Shell Components
- (iii)
- Mode Shape Adaptation of Solid Components
3. Results
3.1. Bio-Inspired Mode Shape Adaptation of 3D Structures
3.2. Integration of the Bio-Inspired Adaptation Method into a Low-Code Software
- (i)
- Mode Shape Adaptation of Beam Components
- (ii)
- Mode Shape Adaptation of Shell Components
- (iii)
- Mode Shape Adaptation of Solid Components
4. Discussion
4.1. Bio-Inspired Mode Shape Adaptation of 3D Structures
4.2. Integration of the Bio-Inspired Adaptation Method into a Low-Code Software
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Hollow Cuboid | Thick Hexagonal Prism | Cube | Truncated Pyramid | Thin Hexagonal Prism | Curved Rectangular Duct | Connector Square | |
---|---|---|---|---|---|---|---|
Mesh element type | CTETRA (volume) | CTETRA (volume) | CTETRA (volume) | CQUAD1 (shell) | CQUAD1 (shell) | CQUAD1 (shell) | CQUAD1 (shell) |
Element edge length | 9 mm | 3 mm | 5 mm | 4 mm | 3 mm | 1 mm | 1 mm |
Number of elements | 1,303,473 | 26,040 | 2402 | 3104 | 13,464 | 2996 | 2370 |
Material properties | Structural steel (Young’s modulus: 210,000 MPa, density: 7850 kg m3, Poisson’s ratio: 0.3) | ||||||
Studied max. pre-deformation | 0–150 mm ( = 0–15) | 0–25 mm ( = 0.0–2.5) | 0–30 mm ( = 0.0–3.0) | 0–25 mm ( = 0.0–13.9) | 0–25 mm ( = 0.0–13.5) | 0–25 mm ( = 0.0–13.0) | 0–25 mm ( = 0.0–13.3) |
Wall thickness | 10 mm (constant) | 10 mm (constant) | - | 1.80–2.00 mm (varied) | 1.85–2.00 mm (varied) | 1.92–2.00 mm (varied) | 1.88–2.00 mm (varied) |
Mass | 0.65–0.71 t (varied up to 9%) | 8.88–9.49 kg (varied up to 7%) | 7.85–8.35 kg (varied up to 6%) | 0.78 t (constant) | 1.88 kg (constant) | 1.14 t (constant) | 0.565 t (constant) |
(mm) | 0 | 1 | 2 | 3 | 4 | 5 | 10 | 15 | 20 | 25 | |
---|---|---|---|---|---|---|---|---|---|---|---|
Adaptation according to the 1st mode shape | Truncated pyramid | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.96 | 1.92 | 1.86 | 1.80 |
Thin hexagonal prism | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.98 | 1.95 | 1.90 | 1.85 | |
Curved rectangular duct | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.98 | 1.94 | 1.92 | |
Connector square | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.98 | 1.96 | 1.92 | 1.88 | |
Adaptation according to the 2nd mode shape | Truncated pyramid | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.96 | 1.92 | 1.86 | 1.80 |
Thin hexagonal prism | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.98 | 1.97 | 1.95 | 1.92 | |
Curved rectangular duct | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.98 | 1.94 | 1.92 | |
Connector square | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.98 | 1.96 | 1.92 | 1.88 |
Property | Value |
---|---|
Young’s modulus E (MPa) | 69,000 |
Material density (kg m3) | 2688 |
Poisson’s ratio (-) | 0.34 |
Beam length l (m) | 0.6 |
Beam cross-section radius r (m) | 0.002 |
Average Eigenfrequency Deviation | |
---|---|
1.0 | 0.0% |
2.0 | 0.0% |
3.0 | 0.1% |
4.1 | 0.2% |
5.1 | 0.4% |
10.7 | 2.8% |
17.4 | 4.7% |
25.3 | 8.7% |
Average Eigenfrequency Deviation | |
---|---|
1.5 | 0.4% |
3.0 | 0.3% |
4.5 | 0.2% |
6.0 | 0.2% |
7.5 | 0.2% |
9.0 | 1.4% |
10.5 | 0.3% |
12.0 | 1.0% |
13.5 | 0.8% |
15.0 | 0.7% |
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Andresen, S.; Ahmad Basri, A.B. Diatom-Inspired Structural Adaptation According to Mode Shapes: A Study on 3D Structures and Software Tools. Biomimetics 2024, 9, 241. https://doi.org/10.3390/biomimetics9040241
Andresen S, Ahmad Basri AB. Diatom-Inspired Structural Adaptation According to Mode Shapes: A Study on 3D Structures and Software Tools. Biomimetics. 2024; 9(4):241. https://doi.org/10.3390/biomimetics9040241
Chicago/Turabian StyleAndresen, Simone, and Ahmad Burhani Ahmad Basri. 2024. "Diatom-Inspired Structural Adaptation According to Mode Shapes: A Study on 3D Structures and Software Tools" Biomimetics 9, no. 4: 241. https://doi.org/10.3390/biomimetics9040241