The Free Vibration Characteristics of Variable Cross-Section Beams under Different Profiles

Based on Euler beam theory and Hamilton’s principle, the free vibration control equation for the variable cross-section beam was established. Galerkin discretization and eigenvalue methods were used to solve the natural frequencies of beams under simply supported-simply supported (SS) and simply-clamped supported (SC) boundary conditions. The vibration characteristics of beams with annular, rectangular frame and rectangular cross-section at different cross-sectional parameters (taper, thickness, and aspect ratio) are explored. The calculation example shows that the natural frequency of the beam is closely related to changes in taper. Especially for rectangular cross-section, the change in taper in the height direction of the beam has a large effect on the frequency, while the change in taper in the width direction has a smaller effect. Increasing the outer thickness of the annular cross-section beam has a different effect on the beam’s natural frequency than the inner thickness, while increasing the thickness of the rectangular frame cross-section beams will result in a decrease in the natural frequency of the beams. Increasing the aspect ratio of a rectangular variable cross-section section beam increases the natural frequency of the beam. Under the same cross-sectional area, the rectangular frame cross-section beam frequencies are higher than other cross-section beams.