In the metal strip-multiroll leveling process, the action behavior of each roll is different. However, modeling each roll individually will result in redundancy which is not conducive to the modeling of the entire leveling process. To overcome this problem, a roll-strip unit (RSU) model is proposed to uniformly describe the behavior of each roll during the leveling process. The RSU and its equivalent model are defined on the basis of analyzing the force relationship between the roll and the strip. According to the linear distribution of the bending moment in the longitudinal direction of the strip, the position of the zero bending moment, that is, the virtual fulcrum, is obtained to determine the interval of the RSU. A plastic deformation function is established to describe the influence of plastic extension on the tension and velocity of the strip. The fitting of the deformation curve of the strip is optimized by the tension influence factor and the zero curvature moment. The static friction condition between the roll and the strip which ensures the normal operation of the RSU is given. The AMESim model of the RSU is established to lay the foundation for the dynamic modeling of the multiroll leveler.
Gene transcription and protein synthesis can be widely found in
biological research. Based on this viewpoint, this article focuses on a
two-dimensional stochastic delay differential equation (SDDE) to
investigate the gene expression. A wealth of dynamic behaviors from a
mathematical perspective are observed. An optimized method to handle the
stochastic delayed system is proposed and we applied it to the model in
this paper. Then an averaging Ito equation is obtained. Based on this,
we studied the stochastic bifurcation of the system and the
corresponding probability distribution with time delay, degradation
rate, the intensity of the stochastic process, and the randomization
degree as parameters. The numerical results revealed the transition of
the probability density function of the system from a Dirac δ function
to a nontrivial stationary state. In addition, stochastic P-bifurcation
occurs when the presentation of the stationary probability density
change from singular peak to crater.
In this paper, Myc/E2F/miR-17-92 network under Gaussian white noise is studied. Taking the time delay as the parameter, the Hopf bifurcation of the system is obtained, which causes the protein concentration to oscillate periodically. Under the influence of time delay and noise, the stochastic D-bifurcation of the system is obtained. It is worth noting that the occurrence of stochastic P-bifurcation is successfully captured. Thus a pattern of coexistence of high and low protein concentrations is founded in the network. The specific research methods of this paper are as follows: firstly, the system is reduced to a finite dimensional system by using stochastic center manifold and normal form theory. Then, using the stochastic averaging method, the Fokker–Planck–Kolmogorov equation of the system is constructed in which the statistical response in the stationary state is the probability density. Finally, the stochastic bifurcation analysis and numerical simulation are carried out. The agreements between the analytical method and those obtained numerically validate the effectiveness of the analytical investigations.
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