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  1. Bearing fault detection plays a crucial role in ensuring machinery reliability and safety. However, the existing bearing-fault-detection sensors are commonly too large to be embedded in narrow areas of bearings and too vulnerable to work in complex environment. Here, we demonstrate an approach to distinguish the presence of race faults in bearings and their types by using an optomechanical micro-resonator. The principle of the amplitude-frequency modulation model mixing fault frequency with mechanical frequency is raised to explain the asymmetrical sideband phenomena detected by the optical microtoroidal sensor. Kurtosis estimation used in this work can distinguish normal and faulty bearings in the time domain with the maximum accuracy rate of 91.72% exceeding the industry standard rate of 90%, while the amplitude-frequency modulation of the fault signal and mechanical mode is introduced to identify the types of the bearing faults, including, e.g., outer race fault and inner race fault. The fault-detection methods have been applied to the bearing on a mimic unmanned aerial vehicle (UAV), and correctly confirmed the presence of fault and the type of outer or inner race fault. Our study gives new perspectives for precise measurements on early fault warning of bearings, and may find applications in other fields such as vibration sensing.

     
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  2. We construct new first- and second-order pressure correctionschemes using the scalar auxiliary variable approach for the Navier-Stokes equations. These schemes are linear, decoupled and only require solving a sequence of Poisson type equations at each time step. Furthermore, they are unconditionally energy stable. We also establish rigorous error estimates in the two dimensional case for the velocity and pressure approximation of the first-order scheme without any condition on the time step. 
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  3. null (Ed.)
    We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme. 
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  4. null (Ed.)
  5. The Cable equation is one of the most fundamental equations for modeling neuronal dynamics. In this article, we consider a high order compact finite difference numerical solution for the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. The resulting finite difference scheme is unconditionally stable and converges with the convergence order ofin maximum norm, 1‐norm and 2‐norm. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix‐vector multiplications arising from finite difference discretization. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work fromrequired by traditional methods towithout using any lossy compression, whereandτis the size of time step,andhis the size of space step. Moreover, we give a compact finite difference scheme and consider its stability analysis for two‐dimensional fractional Cable equation. The applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

     
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