Bessel Equation in the Semiunbounded Intervalx∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point

This study expresses the solution of the Bessel equation in the neighbourhood ofx=∞as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval[x0,∞]. In order to guarantee the series’ uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution atx→∞(not to mention a so-called formal solution), but a solution in the interval[x0,∞]with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal.

Effect of Fiber Fineness on Acoustic Absorption Property of Nylon 6/Low-Melting PET/Recycled PET Nonwoven

In recent years, as global technical progress grows, many mechanized machines are developed continuously. Mechanization has substituted for manpower in many factories for promotion of production efficiency, leading to mechanical noise happening more severely. If people are subjected to noise pollution for long term, they would happen dysphoria and absent mindedness, resulting in accident occurrence. Therefore, how to effectively reduce noise pollution becomes an urgent subject. Moreover, rapid population development and fast-growth economy raise consumption in every country, driving industry into high-production and high-consumption times and meanwhile generating much of wastes. In order to achieve sustainable development, these wastes should be recycled effectively. Therefore, 20 wt% recycled PET fibers were added in this study. This paper mainly used 3D and 15D Nylon fibers (70, 60, 50, 40, 30 wt%), low-melting PET fibers (10, 20, 30, 40, 50 wt%) and 20 wt% high-strength PET fibers, to fabricate Nylon 6/Low melting PET/ Recycled high-strength PET nonwoven via needle-punching process. After that, tensile strength, air permeability and sound absorption coefficient property of resulting nonwoven were tested and then evaluated respectively.

Cutting force monitoring in the endmilling operation for chatter detection

The chatter phenomenon shows the dynamic characteristics of tool vibration in the endmilling process and will change according to the irregular dynamic characteristics of that tool vibration. This chatter produces an adverse effect on tool life, machining integrity, surface quality of the workpiece, and other geometric accuracy. Chatter behaviour in endmilling is a complex, non-linear phenomenon, which is very difficult to detect and diagnose. It is therefore necessary to suggest a new method for analysing chatter mechanics. This paper presents a new method for the detection of chatter in the endmilling operation based on the wavelet transform. This wavelet transform method provides various ways to determine chatter characteristics. The fundamental coefficient property of the wavelet transform is reviewed. The reliability of the wavelet transform method is verified by comparing the spectra using the fast Fourier transform (FFT). The behaviour of the detail coefficients obtained by wavelet transform reveals the possibility to detect and analyse chatter and other malfunction states using tool dynamometer cutting force. Because wavelets are closely related to filter, the method presented in this paper can be applied to other real-time cutting force monitoring and analysis in a range of endmilling processes.