THE AXIAL LOAD INFLUENCE COEFFICIENT OF NATURAL FREQUENCIES FOR LATERAL VIBRATION OF A BERNOULLI-EULER BEAM

A high precision formula is proposed in order to consider the influence coefficient of natural frequencies for the lateral vibration of a Bernoulli-Euler beam subjected to axial loads. It is different from the dynamic stiffness matrix obtained by dynamic stiffness method for the free vibration of the uniform beam. According to the governing differential equation for the lateral vibration of a Bernoulli-Euler beam subjected to a constant axial load, the general solution is obtained. The exact shape function can be obtained by eliminating the undetermined constants of displacement boundary conditions. With the finite element method, the differential formulation for a dynamic stiffness matrix of a Bernoulli-Euler beam is proposed and expressed exactly in a dynamic shape function. The differential formulation’s stiffness matrix is the same with the stiffness matrix obtained by a dynamic stiffness method. To follow the Timoshenko method used in the axial load influence factor of bending beam’s static deflection formula, the formula to compute an axial load influence coefficient of natural frequencies for the lateral vibration of a Bernoulli-Euler beam is proposed. The Wittrick-Williams algorithm and dynamic stiffness matrix are used to prove that the maximum relative error of the proposed formula is less than 2%, when the axial load is between the positive and negative half of the first order Euler critical load. Furthermore, the higher the order of the natural frequencies is, the smaller the error is.