Numerical Study of Duffing Nonlinearity in the Quantum Dot Embedded Nanomechanical Resonator

Geometry, electrostatics, and single-electron tunneling contribute to the nonlinearity in the quantum dot embedded nanomechanical resonator, while “Duffing term” is a kind of mathematics describing the third-order nonlinearity of the system as a whole. We study theoretically the influence of a variation of a mathematical parameter Fuffing term on the actual physical effect. The position probability distribution, the average current, and the displacement fluctuation spectrum with the different Duffing parameter and electromechanical coupling are obtained through numerically calculating the Fokker Planck equation. The mechanical bistability has been described by these quantities under different electromechanical coupling and Duffing parameters. We conclude that the nonlinearities of the nanotube resonator contribute to the mechanical bistability, which induces the asymmetry of the position probability distribution, compresses the current, and softens or stiffens the mechanical resonance frequency as the same as the electromechanical coupling to use it in mechanical engineering.

Predicting Equation of Transient Position-Probability Density for Nano-Tunnel Problem in SET

This analytic investigation intends to study the nano-tunnel problem of the single electron transistor (SET), which is the most important component in the nano-electronics industry. With a combined effort of quantum mechanics and similarity parameter, the PDE equation of transient position-probability density is attained and can be applied to predict the electron’s position inside the nano tunnel. Also, appropriate initial and the boundary conditions are set up in accordance to the actual electron behavior for solving this PDE of probability density function. Thereafter, a simple, closed-form solution for the probability density is obtained and expressed in terms of the error function for a new similarity variable η. In conclusions, this is an innovative approach by using the Schrödinger equation directly to solve the nano-tunnel problem. Moreover, with the aids of this analytic position-probability-density solution, it is illustrated that the free single electron in the SET’s tunnel can only appear at some specified regions, which are defined by a dimensionless parameter η within a range of 0≤η≤2. This result can be served as a valuable design reference for setting the practical manufacture requirement.

On the existence of solutions to the fractional derivative equations d(alpha)u/dt(alpha)+Au = f, of relevance to diffusion in complex systems

Fractional derivative equations account for relaxation and diffusion processes in a large variety of condensed matter systems. For instance, diffusion of position probability density displayed by a random walker in complex systems – such as glassy materials – is often modeled by fractional derivative partial differential equations. This paper deals with the existence of solutions to the general fractional derivative equation dαu/dtα+Au = f for 0 < α < 1, with A a self-adjoint operator. The results are proved using the von Neumann–Dixmier theorem.

Bi-Probability Fatigue Life Prediction for Bridge Crane Structures

Fatigue crack is very dangerous for safely operating of steel structures. To estimate precisely fatigue life of bridge cranes, the randomness of lifted load and trolley’s position should be considered. Therefore, bi-probability fatigue life prediction method, namely load and position probability, is put forward based on the miner linear cumulative damage theory. Stress cycle spectrum is constructed based on real-time monitoring data by rainflow counting method. This method can successfully explain the existence of girder cracks in a typical bridge crane RMG, so it would provide valuable reference for maintenance decision of in-service cranes.