Free Vibration Analysis of Thin-Walled Open Cross-Section Beams with Initial Axial Loads

• Main Authors: Peng, Shih-Chuen, 彭詩淳
• Other Authors: 蕭國模
• Format: Others
• Language: zh-TW
• Published: 2016
• Online Access: http://ndltd.ncl.edu.tw/handle/yq3e8j

碩士 === 國立交通大學 === 機械工程系所 === 105 === The geometrical nonlinear static behavior and infinitesimal free vibration around the static equilibrium position are studied using total Lagrangian finite element method for three dimensional thin-walled beams with point-symmetric open section subjected to axial load with its resultant passing through the centroid of beam cross section. The bimoment induced by axial load is considered in this study. The element employed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear center of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. The kinematics of the beam element is described in the current element coordinate system constructed at the current configuration of the element. The current element coordinate system is regarded as an inertial local coordinate system. Thus, the first and the second time derivative of the position vector defined in the element coordinates are the absolute velocity and absolute acceleration. The element deformation nodal forces and inertia nodal forces are systematically derived by the d'Alembert principle, the virtual work principle and consistent second order linearization in the current element coordinates. The equilibrium equations may be obtained by dropping the terms of the inertia forces in the equation of motion. The governing equations for linear vibration around the static equilibrium position are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of the nonlinear equilibrium equations. The subspace iterative method is used for the solution of natural frequencies and vibration modes for the free vibration. Numerical examples are studied to investigate the effects of the axial load and bimoment induced by axial load on the critical state, critical load and the natural frequencies of z cross section beams with different lengths and boundary conditions under axial loading. The objective of the paper is to analyze the influence of bimoment induced by constant axial loads on the free motion of thin-walled beams with point-symmetric open cross- section. For various boundary conditions, a closed-form solution for natural frequencies of free harmonic vibrations was derived by using a general solution of governing differential equations of motion based on Vlasov’s theory. In order to investigate the effect of the bimoment on natural frequencies, the numerical examples with symmetric Z cross-section are given. The obtained results, verified using an ANSYS finite element model, demonstrate that the influence of the bimoment is important in the assessment of torsional natural frequencies. A force with its resultant passing through the centroid of a particular section and being perpendicular to the plane of the section. A force in a direction parallel to the long axis of the structure