Author
H. H. Mabie
Bio: H. H. Mabie is an academic researcher. The author has contributed to research in topics: Fundamental frequency & Vertical plane. The author has an hindex of 1, co-authored 1 publications receiving 84 citations.
Papers
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TL;DR: In this article, the Bernoulli-Euler equation for the free vibrations of a double-tapered cantilever beam is developed from a computer solution of this equation, and a table has been developed from which the fundamental frequency, second, third, fourth, and fifth harmonic can easily be obtained for various taper ratios.
Abstract: The differential equation is developed from the Bernoulli‐Euler equation for the free vibrations of a double‐tapered cantilever beam. The beam tapers linearly in the horizontal and in the vertical planes simultaneously. From a computer solution of this equation, a table has been developed from which the fundamental frequency, second, third, fourth, and fifth harmonic can easily be obtained for various taper ratios. Charts are plotted for selected taper ratios in the vertical plane to show the effect of taper ratios on frequency.
88 citations
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TL;DR: In this paper, the governing equation is reduced to an ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying width, and analytical solutions of the vibration of the beam are obtained for three different types of boundary conditions associated with simply supported, clamped and free ends.
167 citations
TL;DR: In this article, Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as yn and GJ and I both vary as y(n + 2), where y = (cx/L) + 1; c is a constant such that c > − 1; L is the length of the beam; and x is the distance from one end of a beam.
Abstract: Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as yn and GJ and I both vary as y(n + 2), where A, GJ and I have their usual meanings; y = (cx/L) + 1; c is a constant such that c > − 1; L is the length of the beam; and x is the distance from one end of the beam. Numerical checks give better than seven-figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements. A procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped. This enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members.
154 citations
TL;DR: In this article, the stiffness and mass matrices of a rotating twisted and tapered beam element were derived and the effects of shear deformation and rotary inertia were also considered in deriving the elemental matrices.
143 citations
109 citations
TL;DR: In this article, the stiffness and mass matrices of a twisted beam element with linearly varying breadth and depth are derived, where the angle of twist is assumed to vary linearly along the length of the beam.
93 citations