Parametric resonance of shaft with variable slenderness ratio

Parametric instability of a shaft occurs within a certain range of speed when the shaft parameters such as stiffness and mass are disturbed periodically. At high speed a shaft may carry high torque and as such the effect of torsion on the parametric instability need to be considered. The use of the Timoshenko’s beam theory in the parametric instability formulation of shaft in the past has seen this torsional effect being neglected. In this paper the torsional degree of freedom is added to the formulation for the parametric instability of shaft that is based on the mentioned Timoshenko’s theory. The focus is on the effect of slenderness ratio of shaft on the occurrence of parametric instability on the shaft. The finite element method is used to produce the Mathieu-Hill equation and upon applying the Bolotin’s method the parametric insta-bility equations of shaft can be determined. The formulation and its corresponding MATLAB source codes were successfully validated based on past results. It was found that the increase in the slenderness ratio has shifted the instability chart to the right by 20 %. Furthermore the additional torsional degree of freedom has significantly changed the instability chart of the shaft at high speed.

Bars in Cusps

In order to investigate the stability properties of galaxy models with central density cusps, TV-body simulations of oblate models with density ρ ∝ m–1 (m+a)–3 where m2=R2+[z/q]2 and distribution functions f(E, Lz) (computed as in Dehnen, 1995) have been performed with the following results. 1.An E7 model with identical amounts of stars of either sense of rotation was stable over 30 tdyn(r=a). This is interesting for the bending instability has been argued to set in at about this flattening and be responsible for the absence of flatter elliptical galaxies (Merritt & Sellwood, 1994).2.Rapidly rotating E≳E5 models quickly form weak bars inside the cusp, which are stronger for the more flattened, faster rotating initial configurations. The bars grow in a self similar fashion from inside out: the pattern speed decreases with increasing bar length and time. This process is initiated at the origin, where, because of finite AT, the actual density no longer follows the power law, and stops when the edge of the cusp is reached. A typical example is given in the figure showing the z-y-coordinates of particles with |z|<0.1a after ≃20tdyn(r=a) for an initially rapidly rotating E7-model. The bar has axis ratios of about 5:3:1, and extends almost to corotation. However, it has no sharp edge, but an inhomogenous density with a cusp steeper than the initial model. No sign of a buckling insta-bility has been observerd.