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Journal of Computational and Nonlinear Dynamics | Volume 3 | Issue 1 | Research Paper
Research Papers

Efficient and Robust Approaches to the Stability Analysis of Large Multibody Systems

[+] Author and Article Information
Olivier A. Bauchau

Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332olivier.bauchau@ae.gatech.edu

Jielong Wang

Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332gtg136j@prism.gatech.edu

J. Comput. Nonlinear Dynam 3(1), 011001 (Oct 24, 2007) (12 pages) doi:10.1115/1.2397690 History: Received March 31, 2006; Revised October 05, 2006; Published October 24, 2007
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Linearized stability analysis methodologies that are applicable to large scale, multiphysics problems are presented in this paper. Two classes of closely related algorithms based on a partial Floquet and on an autoregressive approach, respectively, are presented in common framework that underlines their similarity and their relationship to other methods. The robustness of the proposed approach is improved by using optimized signals that are derived from the proper orthogonal modes of the system. Finally, a signal synthesis procedure based on the identified frequencies and damping rates is shown to be an important tool for assessing the accuracy of the identified parameters; furthermore, it provides a means of resolving the frequency indeterminacy associated with the eigenvalues of the transition matrix for periodic systems. The proposed approaches are computationally inexpensive and consist of purely post processing steps that can be used with any multiphysics computational tool or with experimental data. Unlike classical stability analysis methodologies, it does not require the linearization of the equations of motion of the system.
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References

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Figures

Grahic Jump Location
+Strutt’s diagram. Stability boundaries predicted by Hill’s infinite determinant, solid lines. Present predictions for μ=0.15 and 0.25: stable solution, (엯), unstable solution, (×).
View Large  |  Save Figure  |  Download Slide (.ppt)
Strutt’s diagram. Stability boundaries predicted by Hill’s infinite determinant, solid lines. Present predictions for μ=0.15 and 0.25: stable solution, (엯), unstable solution, (×).
Figure 1

Strutt’s diagram. Stability boundaries predicted by Hill’s infinite determinant, solid lines. Present predictions for μ=0.15 and 0.25: stable solution, (엯), unstable solution, (×).

Grahic Jump Location
+Frequency (top figure), damping (middle figure), and norm (bottom figure) of the maximum eigenvalue of the system versus excitation frequency, for μ=0.15. Floquet’s classical analysis: solid line; present approach: (엯).
View Large  |  Save Figure  |  Download Slide (.ppt)
Frequency (top figure), damping (middle figure), and norm (bottom figure) of the maximum eigenvalue of the system versus excitation frequency, for μ=0.15. Floquet’s classical analysis: solid line; present approach: (엯).
Figure 2

Frequency (top figure), damping (middle figure), and norm (bottom figure) of the maximum eigenvalue of the system versus excitation frequency, for μ=0.15. Floquet’s classical analysis: solid line; present approach: (엯).

Grahic Jump Location
+Strutt’s diagram. Stability boundaries predicted by Hill’s infinite determinant, solid lines. Present predictions for μ=0.15 and 0.25: stable solution, (엯), unstable solution, (×).
View Large  |  Save Figure  |  Download Slide (.ppt)
Strutt’s diagram. Stability boundaries predicted by Hill’s infinite determinant, solid lines. Present predictions for μ=0.15 and 0.25: stable solution, (엯), unstable solution, (×).
Figure 3

Strutt’s diagram. Stability boundaries predicted by Hill’s infinite determinant, solid lines. Present predictions for μ=0.15 and 0.25: stable solution, (엯), unstable solution, (×).

Grahic Jump Location
+Frequency (top figure), damping (middle figure), and norm (bottom figure), of the maximum eigenvalue of the system versus excitation frequency, for μ=0.15. Floquet’s classical analysis: solid line; present approach: (엯).
View Large  |  Save Figure  |  Download Slide (.ppt)
Frequency (top figure), damping (middle figure), and norm (bottom figure), of the maximum eigenvalue of the system versus excitation frequency, for μ=0.15. Floquet’s classical analysis: solid line; present approach: (엯).
Figure 4

Frequency (top figure), damping (middle figure), and norm (bottom figure), of the maximum eigenvalue of the system versus excitation frequency, for μ=0.15. Floquet’s classical analysis: solid line; present approach: (엯).

Grahic Jump Location
+Flexible shaft with end flexible couplings. Configuration (a): the shaft is supported by compliant bearings. Configuration (b): the right support consists of spatial clearance joint.
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Flexible shaft with end flexible couplings. Configuration (a): the shaft is supported by compliant bearings. Configuration (b): the right support consists of spatial clearance joint.
Figure 5

Flexible shaft with end flexible couplings. Configuration (a): the shaft is supported by compliant bearings. Configuration (b): the right support consists of spatial clearance joint.

Grahic Jump Location
+Frequency and damping of the two least-damped modes versus shaft angular velocity. No damping. Simplified modal solution: solid line; proposed approach: first mode, (엯), second mode (◻).
View Large  |  Save Figure  |  Download Slide (.ppt)
Frequency and damping of the two least-damped modes versus shaft angular velocity. No damping. Simplified modal solution: solid line; proposed approach: first mode, (엯), second mode (◻).
Figure 6

Frequency and damping of the two least-damped modes versus shaft angular velocity. No damping. Simplified modal solution: solid line; proposed approach: first mode, (엯), second mode (◻).

Grahic Jump Location
+Original (dashed line) and reconstructed (solid line) generalized signals corresponding to the three proper orthogonal modes at Ω=24rad∕s
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Original (dashed line) and reconstructed (solid line) generalized signals corresponding to the three proper orthogonal modes at Ω=24rad∕s
Figure 7

Original (dashed line) and reconstructed (solid line) generalized signals corresponding to the three proper orthogonal modes at Ω=24rad∕s

Grahic Jump Location
+Frequency and damping of the two least-damped modes versus shaft angular velocity in the presence of damping. Simplified modal solution: solid line; proposed approach: first mode, (엯), second mode (◻).
View Large  |  Save Figure  |  Download Slide (.ppt)
Frequency and damping of the two least-damped modes versus shaft angular velocity in the presence of damping. Simplified modal solution: solid line; proposed approach: first mode, (엯), second mode (◻).
Figure 8

Frequency and damping of the two least-damped modes versus shaft angular velocity in the presence of damping. Simplified modal solution: solid line; proposed approach: first mode, (엯), second mode (◻).

Grahic Jump Location
+Frequencies and damping of the least-damped modes versus shaft angular velocity. Simplified modal solution for compliant bearings, configuration (a): dashed line. Predictions for configuration (b): symbols.
View Large  |  Save Figure  |  Download Slide (.ppt)
Frequencies and damping of the least-damped modes versus shaft angular velocity. Simplified modal solution for compliant bearings, configuration (a): dashed line. Predictions for configuration (b): symbols.
Figure 9

Frequencies and damping of the least-damped modes versus shaft angular velocity. Simplified modal solution for compliant bearings, configuration (a): dashed line. Predictions for configuration (b): symbols.

Grahic Jump Location
+Frequencies and damping of the cantilevered wing. Case 1: dashed line (◇): Case 2: solid line (◻).
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Frequencies and damping of the cantilevered wing. Case 1: dashed line (◇): Case 2: solid line (◻).
Figure 10

Frequencies and damping of the cantilevered wing. Case 1: dashed line (◇): Case 2: solid line (◻).

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