Dynamic Stability of Thin-Walled Member with Variable Stiffness Considering Damping

The dynamic stability is studied for thin-walled structural elements with variable stiffnesssubjected to periodically alternating axial force in this paper. Here, the variation stiffnessmeans that it changes with periodically alternating axial force as for nonlinear geometrystiffness matrix of thin-walled member. Damping is considered and the governing equations are expressed in terms of a system of two second-order differential equations of the Mathieu type, with periodic coefficients. MATLAB package is used to determine the stability boundary. Numerical example is presented for the dynamic stability boundary of a simply supported beam with I-shaped cross section. Comparison is made with finite element analysis. Considered damping, some conclusions are drawn out: Excited zone of thin-walled member is continuous, the dynamic instability is highly dominant in the first region while the second and third instability regions are of much less practical importance; The larger the ratio of damp, the less the dynamic instability region; The larger the ratio of damp, the more time dependent components of the load wanted, absorption of damping is commonly of no effect to prevent parametrically excited vibration from dynamic instability; Parametrically excited vibration considering damping is much more different from damped forced vibration in nature.