Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes
Abstract
:1. Introduction
2. Dynamic Stiffness Formulation for 1D and 2D Elements and the Classical Wittrick-Williams Algorithm
2.1. General Procedure for the DS Formulation for 1D Element, 2D Element
2.2. The DS Formulation for 2D Plate Element
2.3. The DS Formulation of 1D Rod and Beam Element
2.4. Massless Spring
2.5. Classical Wittrick-Williams Algorithm in Original Form
3. The DS Formulation and Extended Wittrick-Williams Algorithm for Built-Up Structures with Line Nodes Connected to Point Nodes
3.1. The Direct Constraint Method
3.2. The Global DS Formulation and Mode Count for Line Nodes Subjected to Point Supports/Connections
3.2.1. Rigid Point Supports/Connections
3.2.2. Elastic Point Supports/Connections without Slave Degrees of Freedom
- (1)
- (2)
- (3)
3.2.3. Elastic Point Supports/Point Connections to Substructures with Slave Degrees of Freedom
4. Results and Applications
4.1. Example 1: Plate with Rigid Point Supports
4.2. Example 2: Plates Supported at Points by Massless Springs
4.3. Example 3: A Plate Connected by a Rod and/or a Beam
4.4. Example 4: Two Plates Coupled by a Spring or a Beam
5. Conclusions
- The theory is systematic and general with a wider scope for applications. The W-W algorithm is extended to cover a wide range of cases, including plates with point rigid constraint, point elastic support, rod support, rigid support, beam constraint, point elastic coupling constraint, plate-rod coupling, and plate-beam coupling.
- The modeling procedure is more direct and convenient with much better numerical stability. The work by Williams and Anderson [84] are based on dynamic flexibility formulation, which involved matrix inversions that might become numerically unstable for larger systems with a larger number of DOFs.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Appendix A. DS Matrix for Classical Plate Theory
Appendix B. DS Matrix for the Classical Rod Thory
Appendix C. DS Matrix for the Euler-Bernoulli Beam Theory
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Mode | DSM (Hz) | FEM (Hz) | |||||
---|---|---|---|---|---|---|---|
m = 5 | m = 25 | m = 50 | m = 100 | 100 × 100 | 200 × 200 | 500 × 500 | |
1 | 12.39 | 12.26 | 12.26 | 12.26 | 12.26 | 12.25 | 12.25 |
2 | 16.58 | 16.41 | 16.40 | 16.40 | 16.40 | 16.40 | 16.39 |
3 | 23.37 | 23.21 | 23.20 | 23.20 | 23.21 | 23.20 | 23.19 |
4 | 29.12 | 28.69 | 28.67 | 28.67 | 28.68 | 28.66 | 28.65 |
5 | 36.71 | 36.04 | 36.02 | 36.02 | 36.04 | 36.01 | 35.99 |
6 | 39.74 | 39.51 | 39.50 | 39.50 | 39.52 | 39.49 | 39.48 |
7 | 45.36 | 45.20 | 45.20 | 45.19 | 45.23 | 45.19 | 45.17 |
8 | 52.61 | 52.11 | 52.09 | 52.08 | 52.11 | 52.07 | 52.04 |
9 | 63.11 | 61.68 | 61.64 | 61.64 | 61.68 | 61.61 | 61.58 |
10 | 64.42 | 64.28 | 64.27 | 64.27 | 64.33 | 64.27 | 64.24 |
Time (s) | 0.195 | 0.910 | 1.934 | 5.291 | 11.81 | 21.18 | 368.6 |
= 10 N/m | = 3 kN/m | = 20 KN/m | ||||||
---|---|---|---|---|---|---|---|---|
Mode | DSM (Hz) | FEM (Hz) | Mode | DSM (Hz) | FEM (Hz) | Mode | DSM (Hz) | FEM (Hz) |
1 | 4.801 | 4.812 | 1 | 6.930 | 6.929 | 1 | 10.49 | 10.48 |
2 | 8.035 | 8.040 | 2 | 9.879 | 9.876 | 2 | 12.97 | 12.96 |
3 | 18.27 | 18.27 | 3 | 19.53 | 19.52 | 3 | 21.46 | 21.45 |
4 | 19.38 | 19.38 | 4 | 20.24 | 20.24 | 4 | 22.51 | 22.51 |
5 | 23.25 | 23.26 | 5 | 25.08 | 25.08 | 5 | 31.01 | 31.00 |
6 | 35.20 | 35.18 | 6 | 36.06 | 36.04 | 6 | 37.06 | 37.05 |
7 | 37.45 | 37.45 | 7 | 38.66 | 38.65 | 7 | 43.21 | 43.20 |
8 | 43.77 | 43.78 | 8 | 44.14 | 44.15 | 8 | 45.86 | 45.87 |
9 | 47.78 | 47.78 | 9 | 48.29 | 48.29 | 9 | 52.66 | 52.65 |
10 | 55.23 | 55.21 | 10 | 56.08 | 56.06 | 10 | 58.64 | 58.62 |
20 | 117.5 | 117.5 | 20 | 117.5 | 117.5 | 20 | 117.5 | 117.5 |
30 | 175.8 | 176.0 | 30 | 176.0 | 176.1 | 30 | 177.2 | 177.4 |
40 | 237.0 | 237.0 | 40 | 237.0 | 236.9 | 40 | 237.0 | 236.9 |
50 | 295.5 | 295.5 | 50 | 295.5 | 295.5 | 50 | 295.5 | 295.5 |
A Plate Connected to a Rod with the Other End Free | A Plate Connected to a Beam with the Other End Rotation Constrained | ||||
---|---|---|---|---|---|
Mode | DSM (Hz) | FEM (Hz) | Mode | DSM (Hz) | FEM (Hz) |
1 | 6.811 | 6.815 | 1 | 1.427 | 1.426 |
2 | 11.39 | 11.40 | 2 | 7.709 | 7.700 |
3 | 25.91 | 25.94 | 3 | 14.37 | 14.36 |
4 | 27.51 | 27.54 | 4 | 19.03 | 19.01 |
5 | 32.93 | 32.99 | 5 | 24.08 | 24.01 |
6 | 49.78 | 49.89 | 6 | 35.40 | 35.36 |
7 | 53.13 | 53.18 | 7 | 54.70 | 54.45 |
8 | 62.20 | 62.25 | 8 | 56.81 | 56.74 |
9 | 67.81 | 67.88 | 9 | 58.18 | 58.21 |
10 | 78.07 | 78.25 | 10 | 69.72 | 69.58 |
Two Plates Coupled by a Spring (Hz) | Two Plates Coupled by a Beam (Hz) | ||||
---|---|---|---|---|---|
Mode | DSM | FEM | Mode | DSM | FEM |
1 | 11.98 | 11.98 | 1 | 14.37 | 14.36 |
2 | 12.09 | 12.18 | 2 | 14.42 | 14.49 |
3 | 20.06 | 20.00 | 3 | 24.05 | 23.94 |
4 | 20.23 | 20.33 | 4 | 24.14 | 24.26 |
5 | 45.67 | 45.48 | 5 | 54.71 | 54.33 |
6 | 45.75 | 45.69 | 6 | 54.81 | 54.83 |
7 | 48.43 | 48.42 | 7 | 58.05 | 57.92 |
8 | 48.49 | 48.55 | 8 | 58.12 | 58.12 |
9 | 58.12 | 58.03 | 9 | 69.55 | 69.07 |
10 | 58.23 | 58.26 | 10 | 69.72 | 69.66 |
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Liu, X.; Qiu, S.; Xie, S.; Banerjee, J.R. Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes. Mathematics 2022, 10, 57. https://doi.org/10.3390/math10010057
Liu X, Qiu S, Xie S, Banerjee JR. Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes. Mathematics. 2022; 10(1):57. https://doi.org/10.3390/math10010057
Chicago/Turabian StyleLiu, Xiang, Shaoqi Qiu, Suchao Xie, and Jnan Ranjan Banerjee. 2022. "Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes" Mathematics 10, no. 1: 57. https://doi.org/10.3390/math10010057