The Laplace Transform: Motivating the Definition

Most undergraduate texts in ordinary differential equations (ODE) contain a chapter covering the Laplace transform which begins with the definition of the transform, followed by a sequence of theorems which establish the properties of the transform, followed by a number of examples. Many students accept the transform as a Gift From The Gods, but the better students will wonder how anyone could possibly have discovered/developed it. This article outlines a presentation, which offers a plausible (hopefully) progression of thoughts, which leads to integral transforms in general, and the Laplace transform in particular

Integrating Factors and Repeated Roots of the Characteristic Equation

Most texts on elementary differential equations solve homogeneous constant coefficient linear equations by introducing the characteristic equation; once the roots of the characteristic equation are known the solutions to the differential equation follow immediately, unless there is a repeated root. In this paper we show how an integrating factor can be used to find all of the solutions in the case of a repeated root without depending on an assumption about the form that these solutions will take. We also show how an integrating factor can be used to explain the extra power of t which appears in the trial form of the solution when using the method of undetermined coefficients on a nonhomogeneous equation in the case where the right hand side is a polynomial multiple of the corresponding homogeneous solution

Direct load measurement of a wind turbine high speed shaft bearing in the field

Small piezoelectric transducers bonded to the outer raceway of a rolling element bearing can be used to send ultrasonic pulses to the element-raceway contact and receive reflections. Interpretation of the magnitude and time of flight (ToF) of the reflections can be used to deduce amount of load imparted between roller and raceway. The approach is most successful when the sensor is smaller than the area of contact. Then the sound field falls directly on the contact region. It is thus suited to large bearings, such as those found in wind turbines. In this work, we have used the approach on the rotor side high speed shaft bearing in a Vestas V42 600 kW wind turbine operating in the Barnesmore windfarm in Northern Ireland. The bearing is a SKF 32222 tapered roller bearing. The nature of the contact geometry meant that careful location of the transducer was required to ensure direct pulseecho reflection. The load directly imparted by the roller onto the raceway can be deduced from the change in time of flight of the reflected signal. This measurement is challenging because the change in time of flight is only a few nanoseconds. Nevertheless loads on each roller were determined and results were consistent with expectation. Functional over a period of 2 years, this validated method allows confident application in other sites such as pitch and main bearings to measure key inputs to prognostic algorithms derived from rig-based sensitivity studies, rather than optimistic ISO 281 calculations

Measurement of roller load, load variation, and lubrication in a wind turbine gearbox high speed shaft bearing in the field

The variability of applied load and the integrity of lubrication are crucial factors dictating the durability of wind turbine gearbox bearings. In this work, new ultrasonic reflectometry techniques have been implemented to measure both load and lubrication in the high-speed shaft bearing of a field operational Vestas V42 wind turbine gearbox. Miniature piezoelectric transducers were bonded onto the bearing outer raceway. The reflected pulses were used to infer bearing load and lubrication. Roller load measurements were seen to vary by 33% across the roller complement, with similar trend observed for reflection coefficient. Inspection of the reflection coefficient patterns were able to show whether the roller inlet region was fully flooded or partially starved, capturing the stochastic behaviour of bearing lubrication

Long-duration gamma-ray emissions from 2007 and 2008 winter thunderstorms

The Gamma-Ray Observation of Winter THunderclouds (GROWTH) experiment, consisting of two radiation-detection subsystems, has been operating since 2006 on the premises of Kashiwazaki-Kariwa nuclear power plant located at the coastal area of Japan Sea. By 2010 February, GROWTH detected 7 long-duration $\gamma$-rays emissions associated with winter thunderstorms. Of them, two events, obtained on 2007 December 13 and 2008 December 25, are reported.On both occasions, all inorganic scintillators (NaI, CsI, and BGO) of the two subsystems detected significant gamma-ray signals lasting for >1 minute. Neither of these two events were associated with any lightning. In both cases, the gamma-ray energy spectra extend to 10 MeV, suggesting that the detected gamma-rays are produced by relativistic electrons via bremsstrahlung. Assuming that the initial photon spectrum at the source is expressed by a power-law function,the observed photons can be interpreted as being radiated from a source located at a distance of 290-560 m for the 2007 event and 110-690 m for the 2008 one, both at 90% confidence level.Employing these photon spectra, the number of relativistic electrons is estimated as 10^9 - 10^{11}. The estimation generally agrees with those calculated based on the relativistic runaway electron avalanche model. A GROWTH photon spectrum, summed over 3 individual events including the present two events and another reported previously, has similar features including a cut-off energy, to an averaged spectrum of terrestrial gamma-ray flashes.Comment: 46 pages, 13 figures, accepted for publication in JGR-Atmospher

Observations on acoustic emissions from a line contact compressed into the plastic region

Some observations from acoustic emissions recorded during a yield test of a bearing raceway compressed into plasticity using a rolling element are presented. The general objective of the study is to establish whether there is enough evidence of the onset of sub-surface plasticity in the acoustic emissions signature. It is discussed here how acoustic emissions monitoring during compression could indicate the onset of subsurface plasticity as a precursor to damage propagation to the surface. Some comparisons are drawn between the acoustic emissions activity levels and time-frequency response during elastic deformation and at yield loads

Overview of the NASA Entry, Descent and Landing Systems Analysis Study

NASA senior management commissioned the Entry, Descent and Landing Systems Analysis (EDL-SA) Study in 2008 to identify and roadmap the Entry, Descent and Landing (EDL) technology investments that the agency needed to make in order to successfully land large payloads at Mars for both robotic and human-scale missions. This paper summarizes the approach and top-level results from Year 1 of the Study, which focused on landing 10-50 mt on Mars, but also included a trade study of the best advanced parachute design for increasing the landed payloads within the EDL architecture of the Mars Science Laboratory (MSL) mission

Heavy Drinking Is Associated with Poor Blood Pressure Control in the REasons for Geographic and Racial Differences in Stroke (REGARDS) Study

Alcohol intake has been shown to have a J-shaped association with blood pressure (BP). However, this association has not been examined in mixed race populations or in people with diabetes where tighter blood pressure control is recommended. Participants in the REGARDS study who were 45 years or older (n = 30,239) were included. Medical history (including self-reported alcohol intake) was collected by telephone while blood collection and clinical measurements were done during an in-home visit. We defined diabetes as use of medications and/or fasting glucose ≥ 126 mg/dL and hypertension as use of blood pressure lowering medications and/or BP ≥ 140/90 mmHg or BP ≥ 130/80 mmHg in people with diabetes. After adjustment for confounders, heavy drinking was associated with an increased odds of hypertension (OR = 1.59; 95% CI = 1.37, 1.87). Diabetes and gender significantly modified (interaction P < 0.05 for both) the association between alcohol use and hypertension, although heavy drinking remained associated with increased odds of hypertension in sub-group analyses. We did not observe the previously described J-shaped relationship in any sub-group except white females. These data suggest heavy alcohol consumption is associated with poor BP control and that heavy drinkers may want to consider limiting alcohol intake in order to manage hypertension