Abstract
Pure analytical solution for variable cantilever beams subjected to free bending oscillations and inertial forces is presented in this paper. Following the Euler-Bernoulli scheme for light slender cross section beams and Henri Bouasses’s analytical procedure the in-plane linear motion differential equation is deduced. Taking into account the beam’s shape a generalized non-dimensional differential equation is derived and characterized by only four geometric parameters. Beam’s shape whose cross section area and moment of inertia variation obey a power law with respect to the longitudinal distance to the tip section of the bar and truncation are considered. The differential equation is analytically solved using the symbolic Wolfram Mathematica module for a number of typical shapes from uniform to zero area tip section with linear to cubic root variation and both rectangular and circular cross sections. The natural frequencies are determined and compared. Applications of results to high-rise building design and variable beams found in nature like tree stem are suggested.
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