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In the field of engineering, the fractional moments of random variables play a crucial role and are widely utilized. They are applied in various areas such as structural reliability assessment and analysis, studying the response characteristics of random vibration systems and optimizing signal processing and control systems. This study focuses on calculating the fractional moments of positive random variables encountered in engineering. This study focuses on calculating the fractional moments of positive random variables encountered in engineering.

By integrating Laplace transforms with fractional derivatives, both analytical and practical numerical solutions are derived. Furthermore, specific practical application methods are provided.

This approach allows for the stable and highly accurate calculation of fractional moments based on the integer moments of random variables. Data experiments included in this study demonstrate the effectiveness of this method in solving fractional moment calculations in engineering. Compared to traditional methods, the proposed method offers significant advantages in stability and accuracy, which can further advance research in the engineering field that employs fractional moments.

(1) Accuracy: Although the proposed method does involve some error, its error level is significantly lower than traditional methods, such as the Taylor expansion method. (2) Stability: The computational error of the proposed method is not only minimal but also remains stable within a narrow range as the fractional order varies. (3) Efficiency: Compared to the widely used Taylor expansion method, the proposed method requires only a minimal number of integer-order moments to achieve the desired results. Additionally, it avoids convergence issues during computation, greatly reducing computational resource requirements. (4) Simplicity: The application steps of the proposed method are very straightforward, offering significant advantages in practical applications.

The behaviours of four indeterminate frame-support combinations namely Type I (with fixed supports), Type II (with pinned supports), Type III (with fixed-pinned supports) and Type IV (with fixed-roller supports) frames under the exposure conditions and loads as existing on site were simulated. Two categories of these combinations (I and II) were studied namely single storey-single bay and multiple storey-single bay frames, as illustrated in the case studies treated. A procedure for determining the probability of failure at different sections along the frame types, the range between the probability of failure bounds and the reliability ratings of the frame types were developed based on the kinetic method of plastic moment analysis, minimum weight design method, piecewise method of moment analysis and first order-second moment (FOSM) methods. The analysis results of the Category I frames showed that the Type I frame was most reliable (with the lowest probability of failure range of 0.3269), while the Type II frame was least reliable (with the highest probability of failure range of 0.4918). These results were consistent with those of the Category II frames. The paper aims to discuss these issues.

Collapse mechanisms were generated for four frame-support types and the corresponding plastic moments were determined using both the kinematic plastic analyses and minimum weight design methods. The members were designed and the plastic moments were distributed at sections of constant interval along the frame length to generate corresponding envelopes. A similar process was carried out to determine the elastic moment variables due to the loads. The reliability index and the corresponding probability of failure at each frame section were determined. Then, the probabilities of failure bounds for the frames were then compared to determine the most reliable.

It was observed that there existed a wide margin between the elastic and plastic moments indicating that design of steel structures at the elastic limit does not take full advantage of its strength. Hence, the design can be carried out beyond the elastic limit and within the safety margin given in equation (3). However, the safety of the entire frame is assessed on the basis of range of values between the highest and the lowest probability of failure bounds. The lower this range is (not exceeding 0.5 or 50 per cent), the more reliable the frame is.

The equations developed in this study can only be directly applied to multi storey-single bay frames. However, the reliability-based analysis and design procedure developed can be extended to other types of frames.

A practical approach for analysing steel frames with different supports with the overall goal of producing safe and economical designs has been developed and presented in this paper.

The procedure adopted is very original and can be backed up by existing literature. The piecewise method for analysing moments at various sections along a frame is also innovative. The whole concept can be adopted to determine the reliability of other types of frames such as multiple bay-multistorey frames with different support types.

The purpose of this paper is to develop new types of direct expansion method of moments (DEMM) by using the n/3th moments for simulating nanoparticle Brownian coagulation in the free molecule regime. The feasibilities of new proposed DEMMs with n/3th moments are investigated to describe the evolution of aerosol size distribution, and some of the models will be applied to further simulation of physical processes.

The accuracy and efficiency of some kinds of methods of moments are mainly compared including the quadrature method of moments (QMOM), Taylor-expansion method of moments (TEMOM), the log-normal preserving method of moments proposed by Lee (LMM) and the derived DEMM in this paper. QMOM with 12 quadrature approximation points is taken as a reference to evaluate other methods.

The newly derived models, namely DEMM(4/3,4) and DEMM(2,6), as well as the previous DEMM(2,4), are considered to be qualified models due to their high accuracy and efficiency. They are confirmed to be valid and alternative models to describe the evolution of aerosol size distribution for particle dynamical process involving the n/3th moments.

The n/3th moments, which have clear physical interpretations when n stands for first several integers, are first introduced in the DEMM method for simulating nanoparticle Brownian coagulation in the free molecule regime.

THE method which is to be outlined is not a new way of approaching continuous beam problems, it was first introduced in America by Professor Hardy Cross, and appears to be little known in this country. It is felt that the method is much easier to handle than the normal approach to such problems by the Theorem of Three Moments. Engineers dealing with structural design will find it particularly useful, as a very good approximation to the bending moments at the supports can be obtained during the early stages of the work. The method will be helpful to the engineering student, in removing any doubts he may have in determining the directions of bending moments at the supports.

The main purpose of this paper is to consider the role of discretization of random variables in analyzing statistical tolerancing, and to propose a new discretizing method along with a study on its usefulness.

The approach for discretization of a continuous distribution is based on the concept of moment equalization with the original random variable, conditionally given a set of points of realization. For the purpose of demonstration the normal distribution has been discretized into seven points. Application of the discretization method in approximating the distribution/survival function of a complex system has also been studied. Numerical analysis on two important engineering items has been undertaken and the closeness between the values of the distribution/survival functions obtained by simulation and the proposed method has been examined to indicate the advantage of the proposed approach.

A comparative study with the earlier reported discretizing methods indicates that the proposed method, which is easy to implement, provides better results for most of the cases studied in this work.

Using the proposed approach one can approximate the probability distribution of a complex system with random component values, which cannot be analytically expressed.

This paper is able to provide a new direction in reliability management research, because it can be used for product design of many important engineering items such as solid‐shaft, hollow cylinder, torsion bar, I‐beam etc.

This research gives a new linear method of discretization. It gives better results than the existing discretization methods of Experimental design, Moment equalization, and Discrete Concentration for reliability (survival probability) determination of solid‐shaft and power resistor.

Statistical inference (estimation and testing) for the stochastic volatility (SV) model Taylor (1982, 1986) is challenging, especially likelihood-based methods which are difficult to apply due to the presence of latent variables. The existing methods are either computationally costly and/or inefficient. In this paper, we propose computationally simple estimators for the SV model, which are at the same time highly efficient. The proposed class of estimators uses a small number of moment equations derived from an ARMA representation associated with the SV model, along with the possibility of using “winsorization” to improve stability and efficiency. We call these ARMA-SV estimators. Closed-form expressions for ARMA-SV estimators are obtained, and no numerical optimization procedure or choice of initial parameter values is required. The asymptotic distributional theory of the proposed estimators is studied. Due to their computational simplicity, the ARMA-SV estimators allow one to make reliable – even exact – simulation-based inference, through the application of Monte Carlo (MC) test or bootstrap methods. We compare them in a simulation experiment with a wide array of alternative estimation methods, in terms of bias, root mean square error and computation time. In addition to confirming the enormous computational advantage of the proposed estimators, the results show that ARMA-SV estimators match (or exceed) alternative estimators in terms of precision, including the widely used Bayesian estimator. The proposed methods are applied to daily observations on the returns for three major stock prices (Coca-Cola, Walmart, Ford) and the S&P Composite Price Index (2000–2017). The results confirm the presence of stochastic volatility with strong persistence.

PROBLEMS relating to built‐in or rigid‐end members under transverse loading are frequently encountered by the aircraft engineer. In the following paper discussion of relevant theorems leads to the development of Clapeyron's Theorem of Three Moments. The latter is particularly valuable in, for example, estimating the crankshaft bearing loads in a non‐radial engine. Attention is also drawn to Wilson's method of solving continuous beam problems. This simple method produces results identical with those given by the Theorem of Three Moments and deserves wider recognition.

In this work, an efficient and easy statistical method to find an equivalent discrete distribution for a continuous random variable (r.v.) is proposed. The proposed method is illustrated by applying it to the treatment of the anthropometrical noise factors in the context of Robust Ergonomic Design (RED; Lanzotti 2006; Barone S. and Lanzotti A., 2007).

Bearing friction moments are important factors that affect the vibrations of rotor systems. The bearing friction moments are related to the dimension parameters, lubrication conditions and manufacturing errors of support bearings. This work studies the effects of the bearing friction moments on the vibrations of rotor systems.

The rotor is separated into several shaft elements for formulating a flexible rotor. The time-varying friction moment (TFM) is affected by the time-varying contact loads. The vibrations of FRS from the TFM and Palmgren's friction moment (PFM) calculation methods are compared. Moreover, the effects of the rotor offset and radial clearance on the frequency-amplitude characteristics of FRS are studied.

The TFM method is more consistent with the actual operation mechanisms. The rotor offset and radial clearance can significantly affect the nonlinear vibrations of FRS. This work provides a new reference and research method for the vibration analysis of rotor systems considering the friction effects.

The elastohydrodynamic lubrication (EHL), elastic hysteresis and differential sliding are considered. A flexible rotor system (FRS) dynamic model considering the TFM is proposed. The vibrations of FRS from the TFM calculation method and empirical calculation formula are compared. The effects of the rotor offset and radial clearance on the frequency–amplitude characteristics of FRS are studied.

In this paper, the authors applied the empirical likelihood method, which was originally proposed by Owen, to the copula moment based estimation methods to take advantage of its properties, effectiveness, flexibility and reliability of the nonparametric methods, which have limiting chi-square distributions and may be used to obtain tests or confidence intervals. The authors derive an asymptotically normal estimator of the empirical likelihood based on copula moment estimation methods (ELCM). Finally numerical performance with a simulation experiment of ELCM estimator is studied and compared to the CM estimator, with a good result.

In this paper we applied the empirical likelihood method which originally proposed by Owen, to the copula moment based estimation methods.

We derive an asymptotically normal estimator of the empirical likelihood based on copula moment estimation methods (ELCM). Finally numerical performance with a simulation experiment of ELCM estimator is studied and compared to the CM estimator, with a good result.

In this paper we applied the empirical likelihood method which originally proposed by Owen 1988, to the copula moment based estimation methods given by Brahimi and Necir 2012. We derive an new estimator of copula parameters and the asymptotic normality of the empirical likelihood based on copula moment estimation methods.