Integral equation formulation and analysis of the dynamic stability of damped beams subjected to subtangential follower forces

Abstract

This paper presents a mathematical model based on integral equations for numerical investigations of stability analyses of damped beams subjected to subtangential follower forces. A mathematical formulation based on Euler–Bernoulli beam theory is presented for beams with variable cross sections on a viscoelastic foundation and subjected to lateral excitation, conservative and non-conservative axial loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebro-differential system related to internal and boundary unknowns. Generalized formulations for the deflection, the slope, the moment and the shear force are presented. The free vibration of loaded beams is formulated in a compact matrix form and all necessary matrices are explicitly given. The load–frequency dependence is extensively investigated for various parameters of non-conservative loads, of internal and viscous dampings and for various positions of the concentrated foundation. For an undamped beam, a dynamic stability analysis is illustrated numerically based on the coalescence criterion. The flutter load and instability regions with respect to various parameters are identified. The effects of internal and viscous dampings on the critical flutter load are examined separately and relative effects are evaluated. The dynamic responses, before, near and after the flutter are investigated. A simple and quite general methodological approach is presented. Comprehensive numerical tests for flutter analysis are reported and discussed.