On the Leibniz rule and Laplace transform for fractional derivatives

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results

Feedback cancellation with probe shaping compensation

Adaptive feedback cancellation methods may integrate the use of probe signals to assist with the biased optimal solution in acoustic systems working in closed-loop. However, injecting a probe noise in the loudspeaker decreases the signal quality perceived by users of assistive listening devices. To counter this, probe signals are usually shaped to provide some level of perceptual masking. In this letter we show the impact of using a shaping filter on the system behavior in terms of convergence rate and steady state error. From this study, it can be concluded that shaping the probe signal may have detrimental influence in terms of system performance. Accordingly, we propose to use the unshaped probe signal combined with an inverse filter of the shaping filter to identify the feedback channel. This restructure of the problem restores convergence rate of LMS type algorithms. Furthermore, we also show that an adequate forward path delay is required to obtain an unbiased solution and that the suggested scheme reduces this delay

Open CNC Machine Tool's State Data Acquisition and Application Based on OPC Specification

AbstractFor open CNC machine tool, user can develop third-party software based on OPC specification. Besides, CNC machine tool produce massive state data in machining process, which real time reflect the states of the machine. In this paper, the structure of open numerical control system is introduced, then data acquisition and compression based on OPC specification is implemented, finally data mining of the status data and its application in the intelligent fault warning and diagnostic is proposed. This research is critical to realize the digit and smart factory

Source Mechanism and Rupture Directivity of the 18 May 2009 M_W 4.6 Inglewood, California, Earthquake

On 18 May 2009, an M_w 4.6 earthquake occurred beneath Inglewood, California, and was widely felt. Though source mechanism and its location suggest that the Newport–Inglewood fault (NIF) may be involved in generating the earthquake, rupture directivity must be modeled to establish the connection between the fault and the earthquake. We first invert for the event’s source mechanism and depth with the cut-and-paste method in the long-period band (>5 s). Because of the low velocity shallow sediments in the Los Angeles (LA) basin, we use two velocity models in the inversion for stations inside and outside the LA basin. However, little difference is observed in the resolved source mechanism (M_w 4.6, strike 246°/145°, dip 50°/77°, rake 17°/138°) and depth (7 to ~9 km), compared to an inversion using the standard southern Calfornia model. With the resolved source parameters, we calibrate the amplitude anomaly of the short-period (0.5–2 Hz) P waves with amplitude adjustment factors (AAF). These AAFs are used as corrections when retrieving source mechanisms of the smaller aftershocks using short-period P waves alone. Most of the aftershocks show similar source mechanisms as that of the mainshock, providing ideal empirical Green’s functions (EGFs) for studying its rupture process. We use a forward modeling approach to retrieve rupture directivity of the mainshock, consistent with movement on the NIF with rupture toward the southeast. Although we focus on P waves for analyzing rupture directivity, the resolved unilateral pattern is also confirmed with the azimuthal variation of the duration of SH waves observed in the basin. The high rupture velocity near the shear velocity and relatively low stress drop are consistent with the hypothesis of rupture on a mature fault