Modeling Rotating Shafts Using Axisymmetric Solid Finite Elements with Matrix Reduction

An axisymmetric harmonic finite element representation is used to calculate shaft lateral critical speeds and perform stability analysis. Unlike a beam element model, an axisymmetric solid element representation allows the actual rotor geometry to be modeled. A Fourier series representation allows the three-dimensional shaft geometry to be modeled in two dimensions by only considering the radial and axial coordinates. Thus, the degrees of freedom of this element type are different from the usual two translations and two rotations at each node associated with bending of a three-dimensional beam element. A required gyroscopic matrix is also presented for completeness in analysis of rotating shafts. A matrix reduction technique is used to reduce the size of the shaft mass, gyroscopic, and stiffness matrices by condensing out slave degrees of freedom in terms of the retained master degrees of freedom. The formulation is applied to various examples for verification and to investigate the effect of selection of different master degrees of freedom for this element type on the results.