Theoretical Considerations about the Influence of the Work Environment and of Mixer Hydrodynamic Mass on the Deformations and Implicitly on the Shaft Own Pulsation

Loading shaft of the mixing device is complex. Because the operating regime is characterized by a random variation of the regime parameters, it is difficult to determine an exact theoretical approach, from the point of view of calculation of the shaft. The stress shaft of a mixing device takes into account only partally of the real state of loading. At present, there is no unitary methodology for calculating of shaft for mixing devices. Although the effect of the mixer weight on the own frequency, has been partially taken into account when the shaft-mixer system rotates in the air, it must be reconsidered if the shaft-mixer system rotates in a liquid. The calculation presented in this paper will take into account "hydrodynamic mass" mam of liquid, corresponding to the mixer that actually vibrates with it. This contributes to an increase of its inertia, and to reducing of the own frequency (respectively of the critical speed). In this paper, the influence of the working environment on the deflection of vertical cantilever shaft will be considered, and calculation relations for own pulsation of the shaft equipped with a mixer will be set, with consideration of mixer hydrodynamic mass. For this purpose, it is calculated hydrodynamic mass mh attached to the mixer, and the reduction factor of pulsation due to hydrodynamic mass.

A Semi-Analytic Model for Free Vibration Analysis of Longitudinally Loaded Cantilever Shaft-Disk Systems

A semi-analytic model is proposed to study the critical speeds and critical loads of the cantilever shaft-disk systems subjected to longitudinal loads. In the present work, both the exact and approximate stiffness influence coefficients of the longitudinally loaded cantilever shafts are derived. In order to obtain deeper insight into the dynamic behavior of such a shaft-disk system, the present shaft mathematical model has taken account of shear deformation effect. Because the exact stiffness influence coefficients of the shaft are transcendental and difficult to get their characteristics on longitudinal load intuitively, each stiffness influence coefficient is expanded in a Taylor’s series about the longitudinal load. Based on the present approximate stiffness influence coefficients, the critical speeds and critical loads of the cantilever shaft-disk systems can be easily and quickly obtained. Numerical simulations show that the present results are seen to be quite in agreement with the exact solutions.