A Fast Calculation Methods And Application For Vibration Characteristic Of Beam With Variable Cross-Section

Beam, as the basic element for engineering structures, is widely used in many kinds of structures. Beams with variable section showed excellent mechanical properties, such as equal intensity, the optimization of mass distribution, and it has been widely used in aviation, machinery, construction and many other fields. However, besides the finite element method, there is no unified and rapid method to calculate vibration characteristics of the beam with variable section.The method to calculate vibration characteristics of the beam with variable section was discussed from some aspects as followings:1) Based on beam with variable section transverse oscillation equation which is vary in a manner of continuous or non-continuous in bending stiffness and mass per unit, from Euler-Bernoulli beam theory, the beam with variable section is modeled as a number of segments connected by continuity condition, and each segment of the continuous beam is assumed to obey characteristic of uniform cross section. The relation of modal functions between two adjacent contiguous segments is derived by using the conditions for continuity of displacement slope of deflection curve, moment, and shear force at the connecting point between contiguous segments. The characteristic equation of transverse vibration natural frequency of variable section beam and vibration mode function was proposed, and the natural frequencies can be obtained by using Newton-Raphson method. To verify the effect of this method, the natural frequencies and modal shape of beam are presented. Compared with the results from the finite element method, the precision and validity of the proposed method have been verified.2) By using Timoshenko beam model to replace the Euler-Bernoulli beam model, the calculation method of the beams with variable section vibration characteristics of this model is established. Compared to the results of the finite element method, the accuracy and validity of this method have been verified.3) Based on the Euler-Bernoulli beam model and Timoshenko beam model, respectively, two types of models of varying axial force beams with variable section natural frequencies and mode functions solution method were established. 4) Faced with the beam in varies boundary conditions (simply support, clamped, free-shear, free end with added mass, elastic support) when the vibration, the characteristic equation of the unified boundary condition shave been proposed.5) As an example of a wind turbine tower, the new calculation method for dynamic characteristics of it is proposed. According to the present method, mathematical model of the wind turbine tower has been established and the natural frequency and modal were calculated.6) Unilateral opening crack beam with variable cross-section seen as a variable cross-section Timoshenko beam. Using the function created by Ostachowicz and Krawczuk to establish the relationship between crack punishment segment at the left and right ends of the corner. Finally, the calculation method of natural frequency and modal of cracked beam with variable cross-section are obtained.