Vibration of a variable cross-section beam

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.

Exact Bernoulli–Euler dynamic stiffness matrix for a range of tapered beams

Finite element vibration analysis of rotating timoshenko beams

Static, dynamic and stability analysis of structures composed of tapered beams

Finite element eigenvalue analysis of tapered and twisted Timoshenko beams

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.

C. B. Rogers

Papers

Transverse Vibrations of Double‐Tapered Cantilever Beams