Axial Fundamental Vibration Frequency of a Tapered Rod with a Linear Cross-Sectional Area Variation

The axial vibration frequency of a tapered rod is investigated. The method of Rayleigh is used for this purpose. The taper type is relative to the linear variation of the cross section. The objective of the present investigation is to express the fundamental axial angular vibration frequency ω₁ by a closed-form equation that takes account of the taper degree. For this purpose, the first mode shape function of the axial vibration of a uniform rod is adopted in the present investigation for a simplification aim. The necessary validation is made relatively to uniform and conical rods given in the scientific literature. Two formulas, depending on which end (large or small base) is fixed, are proposed for the fundamental angular frequency vibration of the tapered rod with a linear cross-sectional area variation. They are expressed as a function of the taper degree, Young’s modulus E, material density ρ, and the rod length L.