Stress finite element models for determining the frequencies of free oscillations

Structural mechanics

Development of calculation methods, allowing determining the lower limits frequencies of free oscillations is actual. Such methods, in combination with the decisions by the method of finite element in displacements, will let to assess the accuracy of the frequencies of free oscillations calculated values. The frequencies of free oscillations constant cross section rods with different supports of the ends are calculating by the stress finite element analysis. The proposed method is based on combination of the additional potential energy functional and the virtual displacements principle. The last is used to construct the equilibrium equations. The solution reduces to finding the minimum of the additional energy functional with constraints in form of the linear algebraic system equilibrium equations. The equilibrium equations, taking into account inertia forces, are writing for the finite element mesh nodes in the directions of coordinate axes. Using the Lagrange multipliers the equilibrium equations are included in the functional. The Lagrange multipliers are the nodes displacements values. Considered two variants of bending moment’s approximation on the finite element field: linear and piecewise constant. The free oscillations forms are represented as polygonal lines. According to the proposed method the first three frequencies of free oscillations were defined for constant cross-section rods with different supports of the ends. The calculated values of the frequencies were compared with the exact values. In comparison with the method of finite elements in displacements, it is shown that the proposed method allows to get the opposite bound values for the frequencies of free oscillations.