Third-order Function Research Articles

Application of bispectral analysis of vibroacoustic signals for diagnosis blades cracks in aircraft engines

Background . The work is devoted to vibroacoustical diagnosis of the crack-like damages of the gas-turbine engines (GTE) blades at the steady-state modes of GTE. Objective . The effectiveness of the usage bispectrum analysis is investigated for signal processing at the vibroacoustical diagnostic of the blades cracks of aviation gas-turbine engines. Methods . Bispectral analysis is used for diagnostic information processing. Bispectrum can be found using Fourier transform third-order autocorrelation functions: \[S_{3x}(f_{1},f_{2})=\sum_{k,l=-\infty }^{\infty}C_{3x}(k,l)\textrm{exp}(-j2\pi f_{1}k)\textrm{exp}(-j2\pi f_{2}l).\] Results . Simulation bispectrum analysis vibroacoustical signals, radiated by models blades of working wheels without crack and with crack at the stationary vibrating excitations are carried out and investigated. It demonstrates that appearence and evolution of the fatigue crack in a blade increases intensity of bispectral modules. Conclusions . Application of a modern signal processing methods allows to detect fault features, which are sensitive to small crack-like damages. The receive results can be used to create a vibroacoustical monitoring system for aircraft engine rotor components.

Inverse dynamic analysis type of MPPT control strategy in a thermoelectric-solar hybrid energy harvesting system

This study presents the development of a novel inverse dynamic analysis-maximum power point tracking (IDA-MPPT) scheme in a hybrid energy harvesting system between thermoelectric module (TEM) and solar array (SA). The proposed method initially changes the harvested voltage response from both sources to be the third-order exponential function. This input function selection is based on the capability of this function to stabilize the initial response system and maintain its final position despite a prolonged response time. The mV voltage value from TEM is easily boosted up to nearly 5 V using this method. With this hybridization, the total obtained voltage is doubled to become 9.7 V, which results in a total power of 0.722 W. Furthermore, the method also allows for a fast tracking system, which enables faster voltage boosting and supercapacitor charging. The supercapacitor only requires less than 5 min to complete charging and boost the voltage to almost 5 V. Thus, a satisfactory value is obtained as compared with that of the TEM system with a chosen MPPC board.

On [formula omitted] in higher-order QCD calculations and the NNLO evolution of the polarized valence distribution

We discuss the prescription for the Dirac matrix γ5 in dimensional regularization used in most second- and third-order QCD calculations of collider cross sections. We provide an alternative implementation of this approach that avoids the use of an explicit form of γ5 and of its (anti-)commutation relations in the most important case of no more than one γ5 in each fermion trace. This treatment is checked by computing the third-order corrections to the structure functions F2 and g1 in charged-current deep-inelastic scattering with axial-vector couplings to the W-bosons. We derive the so far unknown third-order helicity-difference splitting function ΔPns(2)s that contributes to the next-to-next-to-leading order (NNLO) evolution of the polarized valence quark distribution of the nucleon. This function is negligible at momentum fractions x≳0.3 but relevant at x≪1.

Response characteristics of esophageal balloon catheters handmade using latex and nonlatex materials.

The measurement of esophageal pressure allows for the calculation of several important and clinically useful parameters of respiratory mechanics. Esophageal pressure is often measured with balloon-tipped catheters. These catheters may be handmade from natural latex condoms and polyethylene tubing. Given the potential of natural latex to cause allergic reaction, it is important to determine whether esophageal catheter balloons can be fabricated, by hand, using nonlatex condoms as construction materials. To determine the static and dynamic response characteristics of esophageal balloon catheters handmade from latex and nonlatex materials, six esophageal catheter balloons were constructed from each of the following condom materials: natural latex, synthetic polyisoprene, and polyurethane (18 total). Static compliance and working volume range of each balloon catheter was obtained from their pressure-volume characteristics in water. The dynamic response of balloon catheters were measured via a pressure “step” test, from which a third-order underdamped transfer function was modeled. The dynamic ranges of balloon catheters were characterized by the frequencies corresponding to ±5% amplitude- and phase-distortion (fA5% and fφ5%). Balloon catheters handmade from polyurethane condoms displayed the smallest working volume range and lowest static balloon compliance. Despite this lower compliance, fA5% and fφ5% were remarkably similar between all balloon materials. Our findings suggest that polyisoprene condoms are an ideal nonlatex construction material to use when fabricating esophageal catheter balloons by hand.

A CMOS readout circuit for microstrip detectors

In this work, we present the design and the results of a CMOS analog channel for silicon microstrips detectors. The readout circuit was initially conceived for the outer layers of the SuperB silicon vertex tracker (SVT), but can serve more generally other microstrip-based detection systems. The strip detectors considered show a very high stray capacitance and high series resistance. Therefore, the noise optimization was the first priority design concern. A necessary compromise on the best peaking time to achieve an acceptable noise level together with efficiency and timing accuracy has been investigated. The ASIC is composed by a preamplifier, shaping amplifier and a Time over Threshold (T.o.T) block for the digitalization of the signals. The chosen shaping function is the third-order semi-Gaussian function implemented with complex poles. An inverter stage is employed in the analog channel in order to operate with signals delivered from both p and n strips. The circuit includes the possibility to select the peaking time of the shaper output from four values: 250 ns, 375 ns, 500 ns and 750 ns. In this way, the noise performances and the signal occupancy can be optimized according to the real background during the experiment. The ASIC prototype has been fabricated in the 130 nm IBM technology which is considered intrinsically radiation hard. The results of the experimental characterization of a produced prototype are satisfactorily matched with simulation.

Universal mock theta functions and two-variable Hecke–Rogers identities

We obtain two-variable Hecke–Rogers identities for three universal mock theta functions. This implies that many of Ramanujan’s mock theta functions, including all the third-order functions, have a Hecke–Rogers-type double sum representation. We find new generating function identities for the Dyson rank function, the overpartition rank function, the \(M2\)-rank function and related spt-crank functions. Results are proved using the theory of basic hypergeometric functions.

Identification of embedded horizontal cracks in beams using measured mode shapes

Mode shapes (MSs) have been extensively used to detect structural damage. This paper presents two new non-model-based methods that use measured MSs to identify embedded horizontal cracks in beams. The proposed methods do not require any a priori information of associated undamaged beams, if the beams are geometrically smooth and made of materials that have no stiffness discontinuities. Curvatures and continuous wavelet transforms (CWTs) of differences between a measured MS of a damaged beam and that from a polynomial that fits the MS of the damaged beam are processed to yield a curvature damage index (CDI) and a CWT damage index (CWTDI), respectively, at each measurement point. It is shown that the MS from the polynomial fit can well approximate the measured MS and associated curvature MS of the undamaged beam, provided that the measured MS of the damaged beam is extended beyond boundaries of the beam and the order of the polynomial is properly chosen. The proposed CDIs of a measured MS are presented with multiple resolutions to alleviate adverse effects caused by measurement noise, and cracks can be identified by locating their tips near regions with high values of the CDIs. It is shown that the CWT of a measured MS with the n-th-order Gaussian wavelet function has a shape resembling that of the n-th-order derivative of the MS. The crack tips can also be located using the CWTs of the aforementioned MS differences with second- and third-order Gaussian wavelet functions near peaks and valleys of the resulting CWTDIs, respectively, which are presented with multiple scales. A uniform acrylonitrile butadiene styrene (ABS) cantilever beam with an embedded horizontal crack was constructed to experimentally validate the proposed methods. The elastic modulus of the ABS was determined using experimental modal analysis and model updating. Non-contact operational modal analysis using acoustic excitations and measurements by two laser vibrometers was performed to measure the natural frequencies and MSs of the ABS cantilever beam, and the results compare well with those from the finite element method. Numerical and experimental crack identification can successfully identify the crack by locating its tips.

PMSG Based WECS with PR Control Strategy for Grid Control

In this paper, a current feedback control scheme is proposed using Proportional Resonant (PR) for a grid-connected converter with an LCL Filter; PR controller with harmonic compensators has high gain at the desired frequencies for decrease harmonic distortion in the alternative voltage or current. Harmonic compensators of PR controller are limited to several low-order harmonics due to the system instability when the compensated frequency is out of system control loop’s bandwidth. The average value of current flow through inductors of LCL filter is feedback to the PR current regulator. Zero D-axis Control (ZDC) is to control the generator side converter to reduce the DC-link ripples, multi-loop harmonic current controllers are applied to regulate the higher-order harmonics by PR regulators. Consequently control strategy with LCL filter is to reduce third-order function to first-order function. By this way large control loop gain can be chosen to obtain control loop for the required harmonic components of the PR controller. The simulation and experimental results have been illustrated to validate the effectiveness of the proposed method in MATLAB Simulink.

Using a least squares support vector machine to estimate a local geometric geoid model

In this study, test-region global positioning system (GPS) control points exhibiting known first-order orthometric heights were employed to obtain the points of plane coordinates and ellipsoidal heights by using the real-time GPS kinematic measurement method. Plane-fitting, second-order curve-surface fitting, back-propagation (BP) neural networks, and least-squares support vector machine (LS-SVM) calculation methods were employed. The study includes a discussion on data integrity and localization, changing reference-point quantities and distributions to obtain an optimal solution. Furthermore, the LS-SVM was combined with local geoidal-undulation models that were established by researching and analyzing3 kernel functions. The results indicated that the overall precision of the local geometric geoidal-undulation values calculated using the radial basis function (RBF) and third-order polynomial kernel function was optimal and the root mean square error (RMSE) was approximately ± 1.5 cm. These findings demonstrated that the LS-SVM provides a rapid and practical method for determining orthometric heights and should serve as a valuable academic reference regarding local geoid models.

Implementation of Elman Neural Networks for Enhancing Reliability of Integrated Bridge Deterioration Model

Probabilistic modelling is one of the most prominent techniques in bridge deterioration forecast. It can be classified into two types, namely, state- and time-based models. Reliability of both modelling techniques in forecasting long-term performance rely heavily on sufficient amount of bridge condition rating data being available together with well-distributed deterioration pattern over the age of bridge. However, it can be problematic when the available condition rating records are insufficient. In order to overcome this problem, an integrated deterioration method incorporating both the state- and time-based models has recently been developed. Despite such development and advancement, certain issues still remain with some cases of given condition data that cannot be used to produce reliable long-term performance curve. Aiming to achieve enhanced prediction performance, an Elman neural networks (ENN) technique is incorporated in the integrated method to replace the third-order polynomial regression function, the latter being the core component for long-term prediction in the state-based model. In the present study, the ENN are able to generate more reliable deterioration patterns than a typical deterministic method. The results demonstrate that the integrated method incorporating ENN are more effective in handling various situations of condition data quantities and distributions for generating long-term performance curves.

Implementation of Elman neural networks for enhancing reliability of integrated bridge deterioration model

Probabilistic modelling is one of the most prominent techniques in bridge deterioration forecast. It can be classified into two types, namely, state- and time-based models. Reliability of both modelling techniques in forecasting long-term performance rely heavily on sufficient amount of bridge condition rating data being available together with well-distributed deterioration pattern over the age of bridge. However, it can be problematic when the available condition rating records are insufficient. In order to overcome this problem, an integrated deterioration method incorporating both the state- and time-based models has recently been developed. Despite such development and advancement, certain issues still remain with some cases of given condition data that cannot be used to produce reliable long-term performance curve. Aiming to achieve enhanced prediction performance, an Elman neural networks (ENN) technique is incorporated in the integrated method to replace the third-order polynomial regression function, the latter being the core component for long-term prediction in the state-based model. In the present study, the ENN are able to generate more reliable deterioration patterns than a typical deterministic method. The results demonstrate that the integrated method incorporating ENN are more effective in handling various situations of condition data quantities and distributions for generating long-term performance curves.

Using Gibbs Energies to Calculate the Pt(111) Hupd Cyclic Voltammogram

The cyclic voltammogram for hydrogen on Pt(111) has been calculated using potential-dependent Gibbs reduction energies obtained by the Interface 1.0 code. The reversible potentials, Urev, are predicted by the equilibrium condition where the Gibbs energy of the oxidized reactant plus an electron and the Gibbs energy of the reduced product, when graphed as functions of electrode potential, cross and are equal at the reversible potential. Reversible potentials are calculated for 12 different coverages of H(ads), and a third-order analytic function is fit to the results. Using the derivative of this function, the experimental voltage scan rate, and the experimentally observed maximum H(ads) coverage, the cyclic voltammogram can be calculated. With the Langmuir isotherm contribution −TΔS added to the Gibbs energies, the width of the predicted voltammogram and its maximum current density compare favorably with measurements from the literature. In detailed shape, the predicted current densities are curved more t...

Correcting hot-wire spatial resolution effects in third- and fourth-order velocity moments in wall-bounded turbulence

Spatial averaging, resulting from the finite size of a hot-wire probe, significantly affects the accuracy of velocity measurements in turbulent flows close to walls. Here, we extend the theoretical model, introduced in Segalini et al. (Meas Sci Technol 22:104508, 2011) quantifying the effect of a linear spatial filter of hot-wire probes on the mean and the variance of the streamwise velocity in turbulent wall-bounded flows, to describe the effect of the spatial filtering on the third- and fourth-order moments of the same velocity component. The model, based on the three-(four) point velocity-correlation function for the third-(fourth-) order moment, shows that the filtering can be related to a characteristic length scale which is an equivalent of the Taylor transverse microscale for the second-order moment. The capacity of the model to accurately describe the attenuation is validated against direct numerical simulation (DNS) data of a zero pressure-gradient turbulent boundary layer. The DNS data allow the filtering effect to be appraised for different wire lengths and for the different moments. The model shows good accuracy except for the third-order moment in the region where a zero-crossing of the third-order function is observed and where the equations become ill-conditioned. An “a posteriori” correction procedure, based on the developed model, to correct the measured third- and fourth-order velocity moments is also presented. This procedure, based on combining the measured data by two single hot-wire sensors with different wire lengths, is a natural extension of the one introduced by Segalini et al. (Exp Fluids 51:693–700, 2011) to evaluate both the turbulence intensity and the transverse Taylor microscale in turbulent flows. The technique is validated against spatially averaged simulation data showing a good capacity to correct the actual profiles over the entire height of the boundary layer except, as expected, for the third-order moment in the region where the latter exhibits a zero-crossing. Moreover, the proposed method has been tested on experimental data from turbulent pipe flow experiments.

Optimization Design of Heterogeneous Metal Sandwich Plate Using Response Surface Method

Heterogeneous metal sandwich plate is composed of high strength material face-sheets and low-density material core. In this paper, the crashworthiness property of the heterogeneous metal sandwich plate was investigated with combining the response surface method and finite element simulation. The specific absorption energy of sandwich plate is made as the crashworthiness performance index, and approximation written as a third-order function of the section parameters by means of the response surface method. According to the BB experiment design method, 29 groups of sandwich plate with different section structure parameters were selected and finite element numerical simulations were conducted to study on deformation behavior of sandwich plates during the collision process. Then, the coefficients of approximate specific absorption energy function were solved with regressive analysis. Consequently, the relationship between the crashworthiness performances of sandwich plate and the section parameters α, β, D, d was obtained.

Mathematical expression for the relationship between internode number and internode length for bamboo, Phyllostachys pubescens

We analyzed the relationship between internode number and internode length for one of the largest bamboo, Phyllostachys pubescens Mazel ex Houz. For 50 sample culms with various sizes felled in a pure stand of P. pubescens, the internode number was assigned from base to tip and the length for each internode was directly measured. The result indicated that the internode length should be cumulated from base to tip, and then the cumulated internode length should be relativized by the total culm length. It was inappropriate to relativize the internode length by the maximum intenode length. In addition, the relationship between the relative internode number (the internode number relativized by the total number of internodes) and the relative cumulated internode length should be described not by a power function but by a sigmoid function such as the third-order function. The determined function enabled us to estimate the actual internode length, with the root mean squared error being 4 cm. In conclusion, the mathematical expression presented here, i.e., the relativization of the cumulated internode length by the total culm length and the application of the sigmoid function, will be useful in describing the relationship between internode number and internode length for P. pubescens.

Bounds on effective magnetic permeability of three-phase composites with coated spherical inclusions

Abstract An effective model is developed to bound the effective magnetic permeability of three-phase composites with coated spherical inclusions. In the present model, the trial magnetic potential for the upper bound and the trial magnetic induction field for the lower bound are constructed to satisfy continuity interface conditions. According to the variational principle, the upper and lower bounds on the effective magnetic permeability of three-phase composites with coated spherical inclusions are derived. In this paper, trial magnetic potentials with different function forms are taken and the optimal upper bound is obtained for the trail magnetic potential corresponding to the third-order function. When the three-phase model degenerates into the composite spheres assemblage model [1] , it is interesting that the optimal upper and lower bounds are the same. The effects of the volume fraction of coated spherical inclusions and the thickness and magnetic permeability of coated layers between the matrix and spherical inclusions on the effective magnetic permeability of composites are analyzed. The upper and lower bounds are finite non-zero values when the magnetic permeability of spherical inclusions tends to ∞ and 0, respectively.

Prediction of Global Solar Radiation in Abu Dhabi

Mathematical expressions have been employed to estimate global solar radiation on horizontal from relative sunshine duration for two weather stations in the United Arab Emirates (UAE), which are Abu Dhabi and Al Ain. These expressions include the original Angstrom-Prescott regression function (linear), quadratic function, third-order function, single-term exponential function, power function, logarithmic, and linear-logarithmic function. The predicted values were compared to the measured values using number of statistical methods to validate the goodness of the fits, such as residual analysis and goodness of fit statistics. All the used mathematical models performed generally well in both cities of Abu Dhabi and Al Ain, with all values of the coefficient of determination (R2) higher than 75%. Specifically, the linear Angstrom-Prescott model estimated the average monthly global radiation on horizontal best for the city of Abu Dhabi, providing the second lowest mean absolute percentage error (MAPE) of 1.89% and the highest value of R2, which is approximately 94%, while the third-order model proved to be the best estimator for the city of Al Ain, providing the lowest MAPE value (3.06%) and a corresponding R2 of 83%.

Free vibration analysis of sandwich plates with functionally graded face sheets and temperature-dependent material properties: A new approach

Abstract Improved high-order sandwich plate theory is used to analyze the free vibration of sandwich plates with functionally graded (FG) face sheets in various thermal environments. The material properties of FG face sheets are assumed to be temperature-dependent by a third-order function of temperature and vary continuously through the thickness according to a power-law distribution in terms of the volume fractions of the constituents. Also, the material properties of the core are assumed to be temperature-dependent by a third-order function of temperature. The governing equations of motion in free natural vibration are derived using Hamilton's principle. A new approach is used to reduce the equations of motion and then solved them for both un-symmetric and symmetric sandwich plates. In-plane stresses of the core that usually are ignored in the vibration characteristics of the sandwich structures are considered in this formulation. The results show that the fundamental frequency parameter increases and decreases with increasing the volume fraction index for soft core and hard core sandwich plates, respectively. The results indicate that as the side-to-thickness ratio, the core-to-face sheet thickness ratio and the temperature are changed, a significant effect on the fundamental frequency parameter is observed. Good agreement is found between the theoretical predictions of the fundamental frequency parameters and the results obtained from other references for simply supported sandwich plates with functionally graded face sheets in the literature.

On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews’ partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order f(q), ϕ(q) and ψ(q). An identity of Ramanujan is proved combinatorially. Several new identities are also established.

Perceptually informed synthesis of bandlimited classical waveforms using integrated polynomial interpolation

Digital subtractive synthesis is a popular music synthesis method, which requires oscillators that are aliasing-free in a perceptual sense. It is a research challenge to find computationally efficient waveform generation algorithms that produce similar-sounding signals to analog music synthesizers but which are free from audible aliasing. A technique for approximately bandlimited waveform generation is considered that is based on a polynomial correction function, which is defined as the difference of a non-bandlimited step function and a polynomial approximation of the ideal bandlimited step function. It is shown that the ideal bandlimited step function is equivalent to the sine integral, and that integrated polynomial interpolation methods can successfully approximate it. Integrated Lagrange interpolation and B-spline basis functions are considered for polynomial approximation. The polynomial correction function can be added onto samples around each discontinuity in a non-bandlimited waveform to suppress aliasing. Comparison against previously known methods shows that the proposed technique yields the best tradeoff between computational cost and sound quality. The superior method amongst those considered in this study is the integrated third-order B-spline correction function, which offers perceptually aliasing-free sawtooth emulation up to the fundamental frequency of 7.8 kHz at the sample rate of 44.1 kHz.