Natural frequency of bending vibration for stepped beam of different geometrical characters and materials. (January 2019) - Li, Shun-Cai; Liang, Li; Yu, Qiu
Based on the bending vibration theory of the beam with equal cross-sectional area, the free bending vibration equation of each subsection of stepped beam is given. The modal function of the stepped beam is assumed by the method of separating variable. First, substitute the modal function into free vibration equation so as to obtain the corresponding vibration shape function and then the modal solution with undetermined coefficients and natural frequency is developed. Next, substitute the modal solution into the boundary condition and the continuity condition to get the corresponding equations. Solving the equations by elimination method, the frequency equation of the step beam is obtained by the condition that the system of linear equations has non-zero solution. Taking the cantilever step beam and simplified supported step beam as examples, the first-order natural frequency of transverse bending vibration of stepped beams with different cross-sectional shapes, different materials, different areas, and different lengths for the two portions was obtained by solving the frequency equation with Mathcad software, and the corresponding curves of natural frequency for the circular and rectangular cross-section beam have been given.