Fiber Bragg Gratings Strain Sensors for Railway Applications

Fiber optical sensors have played an important role in applications for monitoring the health of civil infrastructures, such as bridges, oil rigs, and railroads. Due to the reduction in cost of fiber-optic components and systems, fiber optical sensors have been studied extensively for their higher sensitivity, precision and immunity to electrical interference compared to their electrical counterparts. A fiber Bragg grating (FBG) strain sensor has been employed for this study to detect and distinguish normal and lateral loads on rail tracks. A theoretical analysis of the relationship between strain and displacement under vertical and horizontal strains on an aluminum beam has been performed, and the results are in excellent agreement with the measured strain data. Then a single FBG sensor system with erbium-doped fiber amplifier broadband source has been carried out. Force and temperature applied on the system have resulted in changes of 0.05 nm per 50 με and 0.094 nm per 10 oC at the center wavelength of the FBG. Furthermore, a low cost fiber-optic sensor system with a distributed feedback (DFB) laser as the light source has been implemented. We show that it has superior noise and sensitivity performances compared to strain gauge sensors. The design has been extended to accommodate multiple sensors with negligible cross talk. When two cascaded sensors on a rail track section are tested, strain readings of the sensor 20 inches away from the position of applied force decay to one seventh of the data of the sensor at the applied force location. The two FBG sensor systems can detect 1 ton of vertical load with a square wave pattern and 0.1 ton of lateral loads (3 tons and 0.5 ton, respectively, for strain gauges). Moreover, a single FBG sensor has been found capable of detecting and distinguishing lateral and normal strains applied at different frequencies. FBG sensors are promising alternatives to electrical sensors for their high sensitivity,ease of installation, and immunity to electromagnetic interferences.

Dispersive estimates for the Schrödinger equation

In this document we explore the issue of $L^1\to L^\infty$ estimates for the solution operator of the linear Schr\"{o}dinger equation, \begin{align*} iu_t-\Delta u+Vu&=0 &u(x,0)=f(x)\in \mathcal S(\R^n). \end{align*} We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto the absolutely continuous spectrum of $H=-\Delta+V$ satisfies the following estimate $\|e^{itH} P_{ac}(H)\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}$ under certain conditions on the potential $V$. Specifically, we prove the dispersive estimate is satisfied with optimal assumptions on smoothness, that is $V\in C^{\frac{n-3}{2}}(\R^n)$ for $n=5,7$ assuming that zero is regular, $|V(x)|\lesssim \langle x\rangle^{-\beta}$ and $|\nabla^j V(x)|\lesssim \langle x\rangle^{-\alpha}$, $1\leq j\leq \frac{n-3}{2}$ for some $\beta>\frac{3n+5}{2}$ and $\alpha>3,8$ in dimensions five and seven respectively. We also show that for the five dimensional result one only needs that $|V(x)|\lesssim \langle x\rangle^{-4-}$ in addition to the assumptions on the derivative and regularity of the potential. This more than cuts in half the required decay rate in the first chapter. Finally we consider a problem involving the non-linear Schr\"{o}dinger equation. In particular, we consider the following equation that arises in fiber optic communication systems, \begin{align*} iu_t+d(t) u_{xx}+|u|^2 u=0. \end{align*} We can reduce this to a non-linear, non-local eigenvalue equation that describes the so-called dispersion management solitons. We prove that the dispersion management solitons decay exponentially in $x$ and in the Fourier transform of $x$.