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

• Published in: Applied Mathematical Modelling Vol. 104; pp. 401 - 420
• Main Authors: Liu, Xiang, Zhao, Yaxing, Zhou, Wei, Banerjee, J. Ranjan
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.04.2022; Elsevier BV
• Subjects: Algorithms; Boundary conditions; Broadband dynamics; Design optimization; Differential equations; Dynamic stiffness method; Exact solutions; Free vibration; Frequency ranges; Inverse problems; Mindlin plates; Mindlin–Hermann theory; Modal analysis; Rayleigh-Bishop theory; Resonant frequencies; Rods; Slenderness ratio; Statistical energy analysis; Stiffness; Trusses; Wittrick-Williams algorithm; Wittrick-Williams algorithm; Mindlin–Hermann theory; Modal analysis; Broadband dynamics; Dynamic stiffness method; Rayleigh-Bishop theory
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2021.11.023

•Closed-form dynamic stiffness are developed for simple and advanced rod theories.•Analytical J0 count expressions of different theories for W-W algorithm are given.•Comparisons on natural frequencies and modes for different rods and BCs are made.•Benchmark solutions are provided for bars and trusses based different theories.•High efficiency and accuracy within whole frequency range is demonstrated. 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.