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

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 Mindlin–Hermann theories. First, the general solutions governing differential equations four rod theories developed. Then substituted into generalized displacement force boundary conditions (BCs), leading to elemental DS matrices utilising symbolic computation. As accurate modal solution technique, Wittrick-Williams (WW) algorithm is applied. The J0 count WW has been resolved all types elements explicit analytical expressions. method verified against some existing results subjected specific BCs. Comparisons natural frequencies mode shapes different slenderness ratios also made. Finally, benchmark provided individual subject BCs, a stepped truss. This research provides highly analysis tool within whole frequency range, which suitable parametric studies, optimization design, inverse problem analysis, important statistical energy analysis.