De la Vallée Poussin inequality for impulsive differential equations

The de la Vallée Poussin inequality is a handy tool for the investigation of disconjugacy, and hence, for the oscillation/nonoscillation of differential equations. The results in this paper are extensions of former those of Hartman and Wintner [Quart. Appl. Math. 13 (1955), 330–332] to the impulsive differential equations. Although the inequality first appeared in such an early date for ordinary differential equations, its improved version for differential equations under impulse effect never has been occurred in the literature. In the present study, first, we state and prove a de la Vallée Poussin inequality for impulsive differential equations, then we give some corollaries on disconjugacy. We also mention some open problems and finally, present some examples that support our findings.

THE DEVELOPMENT OF THEORETICAL BACKGROUND OF CONTROLLING THE CAVITATION-IMPULSE EFFECT ON A BOTTOM-HOLE ON THE BASIS OF THE THEORY OF SPECTRA

The authors consider the technology of intensification of the rock failure during the drilling of the wells using the substantiation of physico-mechanical, cavitation and technological processes. Further development of the mechanism of rock failure due to the created cavitation processes, the manifestation of which is possible at the well bottom when drilling with modern types of drill bits, is an important scientific and technical problem. The solution of this problem will significantly increase the efficiency and reliability of drilling the wells. The development of the mechanism is of great practical importance for oil-and-gas industry. The authors have further developed the mechanisms of rock failure during drilling, which allow to take into account as constant actions both the mechanical effect of the drill bit cutting structure on the rock and the cavitation effect of the cooling flushing fluid on the bottom-hole surface. For the first time it has been proved that cavitation-impulse treatment of a bottom during drilling allows to evaluate the erosion effect of cavitation at various distances from the cavitator, taking into account dissipative losses, and to increase the proportion of energy directed to the rock. For the first time, the possibility of choosing the most optimal mode of cavitation-impulse load at the bottom of a well has been substantiated. To evaluate the effectiveness of the cavitation-pulsation washing technology, analytical dependencies have been proposed. Those dependencies allow to predict the frequency distribution of energy from the collapse of cavitation bubbles created by the cavitator at the bottom of the well. It allows to control actively the process of cavitation-impulse impact on rocks in course of their failure during drilling. The authors provide characteristics that show the cavitation-pulsation process fully. Thus, these characteristics allow to evaluate the effectiveness of the process in the rocks failure at the bottom-hole more accurately. When conducting cavitation-impulse treatment of the bottomhole, in order to create artificial cracking, the load mode, namely the distribution of the load energy over frequency ranges, is of importance. To expand the area of the cavitation-impulse treatment of ​​the rock mass, it is necessary to form such loads that the main part of the energy is concentrated in the low frequency range. With the increase of the distance from the perturbance source (cavitator) low frequencies attenuate less in comparison with high frequencies. In order to choose the most optimal mode of cavitation-impulse load on the bottom hole, the distribution of energy over various frequency ranges in the process of the spread of cavitation-impulse effect on a rock massif has been studied. The suggested analytical dependencies allow to predict the frequency distribution of energy which is released when the cavitation bubbles collapse at the bottom-hole. It gives a possibility to control the process of cavitation-impulse effect on rocks in the process of their failure during drilling.

Research on the Blow-Off Impulse Effect of a Composite Reinforced Panel Subjected to Lightning Strike

The blow-off impulse effect of a composite reinforced panel subjected to lightning strike is studied combing electric-thermal coupling with explicit dynamic methods. A finite element model of a composite reinforced panel is established under the action of 2.6/10.5 µs impulse current waveform with current peak 60 kA. Blow-off impulse elements are selected according to numerical results of electric-thermal coupling analysis. Elements failure, pressure, and von Mises stress distribution are discussed when blow-off impulse analysis is completed. The results show that the blow-off impulse effect can alter the damage forms of a composite reinforced panel and causes the damage distribution to deviate from the initial fiber direction in each layer. Elements failure modes around the blow-off impulse area are similar to that around the attachment area of the lightning strike. The blow-off impulse effect can well model the internal damage, concave pit, and bulge phenomenon around the attachment area. Additionally, pressure contours are not presented as an anisotropic characteristic but an isotropic characteristic under the blow-off impulse effect, which indicates that the mechanical behavior of composite materials presents as an anisotropic characteristic in low pressure while as an isotropic characteristic in high pressure. This method is suitable to evaluate shock damage of a composite reinforced panel induced by lightning strike.