Stability Analysis of Fractional-Order Mathieu Equation with Forced Excitation

The advantage of fractional-order derivative has attracted extensive attention in the field of dynamics. In this paper, we investigated the stability of the fractional-order Mathieu equation under forced excitation, which is based on a model of the pantograph–catenary system. First, we obtained the approximate analytical expressions and periodic solutions of the stability boundaries by the multi-scale method and the perturbation method, and the correctness of these results were verified through numerical analysis by Matlab. In addition, by analyzing the stability of the k’T-periodic solutions in the system, we verified the existence of the unstable k’T-resonance lines through numerical simulation, and visually investigated the effect of the system parameters. The results show that forced excitation with a finite period does not change the position of the stability boundaries, but it can affect the expressions of the periodic solutions. Moreover, by analyzing the properties of the resonant lines, we found that when the points with k’T-periodic solutions were perturbed by the same frequency of forced excitation, these points became unstable due to resonance. Finally, we found that both the damping coefficient and the fractional-order parameters in the system have important influences on the stability boundaries and the resonance lines

Synergistic Effect of Tungsten Carbide and Palladium on Graphene for Promoted Ethanol Electrooxidation

The synergistic effect of WC and Pd has large benefit for ethanol electrooxidation. The small-sized Pd nanoparticles (NPs) decorated tungsten carbide on graphene (Pd-WC/GN) will be a promising anode catalyst for the direct ethanol fuel cells. The density functional theory (DFT) calculations reveal that the strong interaction exists at the interface between Pd and WC, which induces the electron transfer from WC to Pd. Fortunately, the nanoscale architecture of Pd-WC/GN has been successfully fabricated in our experiments. X-ray photoelectron spectrum further confirms the existence of electron transfer from WC to Pd in a Pd-WC/GN nanohybrid. Notably, electrochemical tests show that the Pd-WC/GN catalyst exhibits low onset potential, a large electrochemical surface area, high activity, and stability for ethanol electrooxidation in alkaline solution compared with Pd/graphene and Pd/commercial Vulcan 72R carbon catalysts. The enhancement can be attributed to the synergistic effect of Pd and WC on graphene. At the interface between Pd and WC, the electron transfer from WC to Pd leads to the increased electron densities of surface Pd, which is available for weakening adsorption of intermediate oxygen-containing species such as CO and activating catalyst. Meanwhile, the increased tungsten oxide induced by electron transfer can facilitate the effective removal of intermediate species adsorbed on the Pd surface through a bifunctional mechanism or hydrogen spillover effect