Cycle Decompositions of Complete Digraphs

In this paper, we consider the problem of decomposing the complete directed graph $K_n^*$ into cycles of given lengths. We consider general necessary conditions for a directed cycle decomposition of $K_n^*$ into $t$ cycles of lengths $m_1, m_2, \ldots, m_t$ to exist and and provide a powerful construction for creating such decompositions in the case where there is one 'large' cycle. Finally, we give a complete solution in the case when there are exactly three cycles of lengths $\alpha, \beta, \gamma \neq 2$. Somewhat surprisingly, the general necessary conditions turn out not to be sufficient in this case. In particular, when $\gamma=n$, $\alpha+\beta > n+2$ and $\alpha+\beta \equiv n$ (mod 4), $K_n^*$ is not decomposable.

Study on the "Phase Hopping" AC-AC Frequency Conversion Method with Approximately Continuous Frequencies

Background: The output voltage frequency for the previously proposed "phase hopping" AC-AC frequency conversion technology is determined by the law that the number of output voltage cycles is reduced by one relative to the power frequency in a large cycle containing six jumps. According to the law, only a limited number of output frequencies, such as 37.5 Hz, 42.86 Hz and 45 Hz are found. Due to the large spacing between the output frequencies, the "phase hopping" frequency conversion technology is difficult to put into practical use. Methods: In this paper, the law of the output frequency control is generalized so that the number of output cycles in a large cycle is reduced by n relative to the power frequency. The analysis shows that the appropriate selection of large cycles, including the number of power frequency cycles and the value of n, can find more frequencies to be used. Reducing the interval between the output frequencies within 1Hz. Results: The analysis results were verified in simulation by MATLAB, and the harmonics and the feasibility of the actual application were analyzed. Conclusion: Finally, an experimental platform was built and an experimental analysis was carried out. The experimental results show that the theoretical and simulation analyses are correct.

Development of Rotating Stall Cell Under Coexisting Phenomena of Surge and Rotating Stall in an Axial-Flow Compressor

The relationship between the growth of the stall cell and variation in the surge behavior was experimentally investigated. The aim of this study was to reveal the effect of the stall cell on the surge behavior from the viewpoint of the inner flow structure. In the experiment, the unsteady compressor characteristics during the surge and rotating stall were obtained by using a precision pressure transducer and a one-dimensional single hotwire anemometer. Under the coexisting states of surge and rotating stall, various surge behaviors were observed by throttling the mass flow rate. When the flow rate was set such that the surge behavior switched, an irregular surge was observed. During the irregular cycle, two different cycles were selected randomly corresponding to the stall behavior. When the amplitude of the plenum pressure is relatively large among the measurement results, the absolute value of the time-change rate in the flow coefficient and the static pressure-rise coefficient tend to be high. This shows that the flow field during stable operation near the peak point of the unsteady characteristics changes rapidly. In this case, an auto-correlation function of the wall-pressure fluctuation data showed that the stall inception of the compressor was induced earlier in the large cycle compared with the case of the top cycle. When studying the growth of the stall cell during the stalling process of the large cycle, the wall-pressure fluctuation data showed that the stall cell rapidly grew by gathering more than one spike-type disturbance into one stall cell. In this case, the stall cell fully expanded along the circumferential direction and developed into a deep stall. Therefore, the key factors that determine the surge behavior are the sudden change in the flow field near the peak point of the unsteady characteristics and the rapid growth in the stall cell during the stalling process.

On periodic solutions associated with the nonlinear feedback loop in the non-dissipative Lorenz model

In this study, we discuss the role of the linear heating term and nonlinear terms associated with a nonlinear feedback loop in the energy cycle of the three-dimensional (X–Y–Z) non-dissipative Lorenz model (3D-NLM), where (X, Y, Z) represent the solutions in the phase space. Using trigonometric functions, we first present the closed-form solution of the nonlinear equation d2X/dτ2 + (X2/2)X = 0 without the heating term (i.e., rX), (where τ is a non-dimensional time and r is the normalized Rayleigh number), a solution that has not been previously documented. Since the solution of the simplified 3D-NLM is oscillatory (wave-like) and since the nonlinear term (X3) is associated with the nonlinear feedback loop, here, we suggest that the nonlinear feedback loop may act as a restoring force. When the heating term is considered, the system yields three critical points. A linear analysis suggests that the origin (i.e., the trivial critical point) is a saddle point and that the other two non-trivial critical points are stable. Here, we provide an analysis for three types of solutions that are associated with these critical points. Two of the solutions represent closed curves that travel around one non-trivial critical point or all three critical points. The third type of solution, appearing to connect the stable and unstable manifolds of the saddle point, is called the homoclinic orbit. Using the solution that contains one non-trivial critical point, here, we show that the competing impact of the nonlinear restoring force and the linear (heating) force determines the partitions of the averaged available potential energy from the Y and Z modes. Based on the energy analysis, an energy cycle with four different regimes is identified. The cycle is only half of a "large" cycle, displaying the wing pattern of a glasswinged butterfly. The other half cycle is antisymmetric with respect to the origin. The two types of oscillatory solutions with either a small cycle or a large cycle are orbitally stable. As compared to the oscillatory solutions, the homoclinic orbit is not periodic because it "takes forever" to reach the origin. Two trajectories with starting points near the homoclinic orbit may be diverged because one moves with a small cycle and the other moves with a large cycle. Therefore, the homoclinic orbit is not orbitally stable. In a future study, dissipation and/or additional nonlinear terms will be included in order to determine how their interactions with the original nonlinear feedback loop and the heating term may change the periodic orbits (as well as homoclinic orbits) to become quasi-periodic orbits and chaotic solutions.

Parametric Design of Gear Hob Cutter

The gears are some of the most used mechanical transmission due to the advantages its provide: constant transmission ratio, high power transmitted, high efficiency, small size, silent operation, precision kinematic, etc. This causes the wheel to be one of the most valuable machine elements both in terms of accuracy and processing technology as well. From the technological point of view due to high precision and productivity, one of the most methods used is processing with gear hob cutter. Although the tool profile does not depend on the number of teeth on the wheel, but only its module, large diversity of gear requires the design of gear hob cutters tailored to existing needs. As it is known the gear hob cutters are complicated and expensive gear cutting tools, both due to the manufacturing process and in terms of design, requiring a significant amount of calculations, high effort from the designer and consequently results in a large cycle of design. For this reason, the literature and in various websites, there are many studies and research information to optimize the design process and increase accuracy of gear hob cutters.