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Dynamic stiffness method for exact longitudinal free vibration of rods and trusses using simple and advanced theories

Liu, X., Zhao, Y., Zhou, W. & Banerjee, J. R. (2022). Dynamic stiffness method for exact longitudinal free vibration of rods and trusses using simple and advanced theories. Applied Mathematical Modelling, 104, pp. 401-420. doi: 10.1016/j.apm.2021.11.023


Closed-form dynamic stiffness (DS) formulations coupled with an efficient eigen-solution technique are proposed for exact longitudinal free vibration analyses of rods and trusses by using classical, Rayleigh-Love, Rayleigh-Bishop and Mindlin–Hermann theories. First, the exact general solutions of the governing differential equations of the four rod theories are developed. Then the solutions are substituted into the generalized displacement and force boundary conditions (BCs), leading to the elemental DS matrices utilising symbolic computation. As an accurate and efficient modal solution technique, the Wittrick-Williams (WW) algorithm is applied. The J0 count for the WW algorithm has been resolved for all four types of DS elements with explicit analytical expressions. The method is verified against some existing exact results for rods subjected to specific BCs. Comparisons of the natural frequencies and mode shapes for different theories and slenderness ratios are also made. Finally, benchmark solutions are provided for individual rods subject to different BCs, a stepped rod and a truss. This research provides an exact and highly efficient modal analysis tool for rods and trusses within the whole frequency range, which is suitable for parametric studies, optimization design, inverse problem analysis, and important for statistical energy analysis.

Publication Type: Article
Additional Information: © 2022. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Publisher Keywords: Dynamic stiffness method, Wittrick-Williams algorithm, Rayleigh-Bishop theory, Mindlin–Hermann theory, Broadband dynamics, Modal analysis
Subjects: Q Science > QA Mathematics
T Technology > TA Engineering (General). Civil engineering (General)
T Technology > TE Highway engineering. Roads and pavements
Departments: School of Science & Technology > Engineering
SWORD Depositor:
[thumbnail of Manuscript DSM of rods.pdf]
Text - Accepted Version
Available under License Creative Commons Attribution Non-commercial No Derivatives.

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