Free Vibration Analysis of Elastically Restrained Tapered Beams with Concentrated Mass and Axial Force

This study proposes a new numerical method for the free vibration analysis of elastically restrained tapered Rayleigh beams with concentrated mass and axial force. The beam model had elastic support, concentrated mass at both ends, and axial force at the right end. The elastic supports were modeled as translational and rotational springs. The shear force and bending moment were determined under the assumption that the sum of the forces at arbitrary positions and the joint between the beam and elastic supports always becomes zero. Therefore, a frequency determinant is established considering the free-free end condition at both ends, but various boundary conditions were constructed by adjusting the values of the elastic springs in the frequency equation. This assumption simplified the deduction procedure, and the method’s efficiency was demonstrated through various comparisons. In particular, the value of compressive loading at which the first natural frequency vanished was investigated by considering the taper ratio based on the relationship between the elastic support and compressive loading. The analyzed results can be adopted as benchmark solutions for other approaches. The frequency determinant employs the transfer matrix method; however, numerical methods can easily be utilized in other approaches.