Recently,scrambling index and competition index are widely applied to stochastic matrices and food webs. By analyzing the relationship of scrambling index and 2-competition index,n-「d/2」+ 1 was proved to be an upper bound of the 2-competition2 index of a primitive digraph with exact d loops in this article.Moreover,the maximum index problem and the index set problem for the 2-competition index of primitive digraphs with minimally strong digraphs were settled.
In order to better application of brush seal in rotating machinery,the leakage flow characteristics of the brush seal considering geometry effects are numerically analyzed using Reynolds-Averaged Navier-Stokes( RANS) model coupling with a non-Darcian porous medium model. The reliability of the present numerical method is proved,which is in agreement with the experimental and numerical results from literatures. Three different bristle pack thicknesses,fence heights and initial clearances under different pressure ratios,rotational velocities and other operating conditions are utilized to investigate the effects of geometry modification on the brush seal leakage flow behaviors. It discusses the effectiveness of various geometry configurations outlining important flow features. The results indicate that the increase of fence height and clearance would lead to the increase of leakage rate. But the leakage is not linearly with respect to the bristle pack thickness,and the effect of rotational velocity is not obvious. Moreover,the detailed leakage flow fields and pressure distributions along the rotor surface,free bristle height,and fence height of the brush seals are also presented. The static pressure drop amplitude through the bristle pack and the pressure rise amplitude through the cavity would increase while the pressure differential increases. And the axial pressure is the main reason of bristle blow down. The results provide theoretical support for the brush seal structure optimal design.
According to the dimensional tolerances on hydrodynamic journal bearing system, a nonlinear oil film force model was established,and the Reynolds' equation was solved by adopting finite difference method. In order to fulfill different dimensional tolerances in the system,adopting 2kfactor design and using the eccentricity ratio corresponding to the stability critical curve,the effects of the friction power loss brought by the dimensional tolerances of the dynamic viscosity,bearing width,bearing diameter and journal diameter were analyzed. The effect on dynamic characteristics of the hydrodynamic journal bearing system was quantitatively analyzed,and the nonlinear dynamic analysis, modeling and calculation methods were studied while considering the manufacturing tolerances. The results show that in contrast to the impacts of the tolerances in journal diameter,dynamic viscosity and bearing width,the bearing diameter tolerance would lead to the rise in the power loss, and the dimensional tolerances have different degrees of impacts on the journal bearing system. The friction power loss decreased as the eccentricity ratio increased, and when the eccentricity ratio was 0. 695 the power loss came to the minimum.The investigation would find the best solution and reduce energy consumption,then control varieties of nonlinear dynamical behavior effectively,and provide a theoretical basis for hydrodynamic journal bearing system in parameter design.
