Machine tool spindle units

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

The static stiffnesses for torsional and flexural deformation of a tapered beam under axial loading are determined analytically, extending the Bernoulli-Euler/Bessel-function approach of Banerjee and Williams (1985). The derivation of the expressions is explained in detail, and numerical results for sample problems are presented in tables and shown to be in excellent agreement with those obtained using stepped beams of 400 or 500 elements.

Exact Bernoulli‐Euler static stiffness matrix for a range of tapered beam‐columns

Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods

C.W.S. To

Papers

A linearly tapered beam finite element incorporating shear deformation and rotary inertia for vibration analysis

Higher order tapered beam finite elements for vibration analysis