Conditional Weibull Control Charts Using Multiple Linear Regression

Because in Weibull analysis, the key variable to be monitored is the lower reliability index (R(t)), and because the R(t) index is completely determined by both the lower scale parameter (η) and the lower shape parameter (β), then based on the direct relationships between η and β with the log‐mean (μ_x) and the log‐standard deviation (σ_x) of the analyzed lifetime data, a pair of control charts to monitor a Weibull process is proposed. Moreover, because the fact that in Weibull analysis, right censored data is common, and because it gives uncertainty to the estimated Weibull parameters, then in the proposed charts, μ_x and σ_x are estimated of the conditional expected times of the related Weibull family. After that both, μ_x and σ_x are used to monitor the Weibull process. In particular, μ_x was set as the lower control limit to monitor η, and σ_x was set as the upper control limit to monitor β. Numerical applications are used to show how the charts work. Copyright © 2016 John Wiley & Sons, Ltd.     

Weibull analysis for normal/accelerated and fatigue random vibration test

In this paper, the formula to estimate the sample size n to perform a random vibration test is derived only from the desired reliability (R(t)). Then, the addressed n value is used to design the ISO16750‐3 random vibration test IV for both normal and accelerated conditions. For the normal case, the applied random vibration stress (S) is modeled by using the Weibull stress distribution [W(s)]. Similarly, for the testing time (t), the Weibull time distribution [W(t)] is used to model its random behavior. For the accelerated case, by using the over‐stress factor fitted from the W(t) and W(s) distributions, four accelerated scenarios are formulated with their corresponding testing's profiles. Additionally, from the W(s) analysis, the stress formulation to perform the fatigue and Mohr stress analysis is given. Since the given Weibull/fatigue formulation is general, then the formulas to determine the W(s) parameters, which correspond to any principal stresses values and/or vice versa, are given. Although the application is performed to demonstrate R(t) = 0.97 by testing only n₂ = 6 parts, the guidelines to use the values given in columns n, S, and t of the Weibull analysis table to generate several accelerated testing plans are given.     

Asymptotic analytical solutions of the two-degree-of-freedom strongly nonlinear van der Pol oscillators with cubic couple terms using extended homotopy analysis method

This paper adopts the extended homotopy analysis method (EHAM) to obtain the asymptotic analytical series solutions of the strongly nonlinear two-degree-of-freedom (2DOF) van der Pol oscillators with cubic couple terms. It turns out that the oscillators occur essentially in only two variations: If the system has periodic solutions, then it either has only one solution which is out-of-phase (i.e., x ₁(t) = −x ₂(t)) or has two solutions that are not only in-phase (i.e., x ₁(t) = x ₂(t)) but also out-of-phase. Two examples, as two types of the problem have been raised, correspondingly. Employing the EHAM for those two problems, the explicit analytical solutions of frequency ω and displacements x ₁(t) and x ₂(t) are well formulated, but the conventional homotopy analysis method (HAM) can hardly do it if the cubic couple terms are complex. Thus, the EHAM is rather general. Moreover, the fifth-order analytical solutions are then compared with those derived from the established Runge–Kutta method in order to verify the accuracy and validity of this approach. It is shown that there is excellent agreement between the two sets of results, even if the time variable t progresses to a comparatively large domain in the time-history responses. Finally, the convergence theorem for the present method is also presented and discussed. All these results confirm that the EHAM can solve the presented problem successfully and completely, and that the EHAM will be a powerful and efficient tool for solving other multi-degree-of-freedom (MDOF) dynamical systems in engineering and physical sciences.     

Mechanical wave interaction at a diffuse interface

An investigation is made of the reflection coefficientR of an ultrasonic wave impinging on a boundary at which there is a gradual transition of material properties from one material to another. The study is confined to the one-dimensional case of compression waves incident at 90° on the boundary. Both analytical and numerical solutions forR are presented for various interfacial profiles, and are seen in general to be strongly dependent on wave frequency. Measurements are made on diffusion bonded samples of nickel to copper, and the results compared with theory. Significant scatter is found in the experimental results due primarily to the migration of oxygen into the diffusion zone. A second set of measurements on diffusion bonded samples of nickel to carbon steel show general agreement with theory.Nomenclature A _i ,B _i ,C _i ,i=1, 2 coefficients in the expression forX (x) - c _i concentration of speciesi - c _L compression wave velocity - D _i diffusion coefficient for speciesi - diffusion coefficient for a binary mixture - E(x) Young's modulus - stiffness for the case of one-dimensional strain, =(1–v)E(x)/(1+v)(1–2v) - one-dimensional stiffness within bluk materiali - J _i ,i=1, 2 flux of speciesi - k(x) wave number at spatial co-ordinatex, equal to 2/ - k _i ,i=1, 2 wave number within bulk materiali - K(x) pseudo-wavenumber, defined by Eq. (8). - k _B Boltzmann's constant - L thickness of interfacial zone - f wave frequency - m, N material profile parameters, used to specifyK(x) - Q activation energy for diffusion - R reflection coefficient - t time - T transmission coefficient - u(x, t) displacement of molecules due to wave action,u(x, t)=X(x) (t) - X(x) spatial portion of functionu(x, t) - x spatial variable - Y(x, t) (x, t)/ - Z _i ,i=1, 2 mechanical impedance of bulk materiali - strain - wavelength of sound - Poisson's ratio - material density - (x, t) stress - (t) temporal portion of functionu(x, t) - _i (x),i=1, 2 functional form ofX(x) within interfacial region - angular frequency of wave     

Blended infinite elements for paraboblic boundary value problems

A common method for numerically approximating two-point parabolic boundary value problems of the form u_t = L[u]+f(u) defined of the semi-infinite strip S = [0, 1]×[0, ∞] is to first discretize the spatial operator in the differential equation and then solve for the time evolution. Such an approach typically involves solving a system of algebriaic equations at a sequence of time steps. In this paper we take a different approach and subdivide S into a collection of semi-infinite substrips S_i = [x_i, x_i+1]×[0, ∞], and use blending function techniques to derive finite parameter functions e_i(x, t) defined on S_i. Spectral matching methods are used in deriving e_i to ensure that (u ? e_i) can be made small on S_i. Galerkin's method, with associated integration sover the entire space-time domain S, is then used to generate approximations to u(x, t) based upon the so defined infinite element (e_i, S_i). Approximations are hence found for all (x, t) in S by solving one well structed system of algebraic equations. We apply the method to several linear and non-linear problms.     

Global existence and blow-up analysis for a nonlinear diffusion equation with inner absorption and boundary flux

This article deals with a nonlinear diffusion equation with inner absorption u _t ?=?(log^σ(1?+?u)u _x ) _x ???λ(1?+?u)log ^p (1?+?u), in ?₊?×?(0,?+∞), subject to a logarithmic boundary flux ? log^σ(1?+?u)u _x (0,?t)?=?(1?+?u)log ^q (1?+?u)(0,?t), t?∈?(0,?+∞). We establish the critical global existence curve and give the asymptotic behaviour close to the blow-up time.     

Sampling Theory

A setup involving p + 1 normal populations Π i with unknown population means μ p and known population variances σ2 i (0 ≤ i ≤ p) is considered; Π0 is the “control” population and Π i (1 ≤ i ≤ p) are the “test” populations. Based on independent observations zii (j = 1, 2, …, Ni ) from Π i (0 ≤ i ≤ p) it is desired to make an exact joint confidence statement of the form {x0 – x i – d < μO – μ i < x 0 x i + rd (1 ≤ i ≤ p)). Here x i = Σ N i i = 1 x ij /N i (0 ≤ i ≤ p) is the observed sample mean from II i , and the constants N = Σ p i , r > ?1, and d > 0 are specified prior to experimentation.

The problem of choosing the N_i (0 ≤ i ≤ p) to maximize the confidence coefficient is considered. A procedure is described which is globally optimal if σ² ₁ = σ² ₂ = … = σ² _p and is optimal in a more restricted sense if not all of the σ² _i (1 ≤ i ≤ p) are equal.

It is shown that the globally optimal proportion of observations _i to be taken from II; (0 ≤ i ≤ p) depends only on the θ _i = σ² _i /σ² ₀ (1 ≤ i ≤ p) and λ = d √N/σ₀. The behavior of the _i as functions of r and λ is studied analytically; formulae are derived for determining the _i . Detailed results are given for p = 2 including tables of the _i (0 ≤ i ≤ 2) as a function of λ for selected r when σ² ₀ = σ² ₁ = σ² ₂. Examples of the use of the tables are provided. These results generalize earlier results for one-sided comparisons (r. = + ∞) given by Bechhofer.     

The influence of surface kinetics in modelling chemical vapour deposition processes in porous preforms

The isothermal chemical vapour infiltration (ICVI) process is a well known technique for the production of composites and the surface modification of porous preforms. Mathematical modelling of the process can provide a better understanding of the influence of individual process parameters on the deposition characteristics such as final porosity or deposition profiles in the pore network. The influence of different rate expressions for several binary compounds on the ICVI process is discussed. Experimental work is used to validate the importance of correct kinetic expressions in a continuous ICVI model for cylindrical pores. The predicted infiltration characteristics are compared with experimental results. The final densification and Thiele modulus, i.e. a number which is a measure for the diffusion limitations in a pore, are used for the evaluation of the presented model, and conditions are given for an optimal densification of a porous preform by the ICVI process for several binary compounds. The deposition profiles as predicted by the model calculations are in agreement with the experimentally determined deposition profiles of TiN and TiC in small tubes. Moreover, it can be concluded that the shape of the deposition profiles is determined by the heterogeneous reaction kinetics. There is only a qualitative agreement between the predicted densification and measured densification for the synthesis of TiN and TiB₂ in sintered porous alumina. This mismatch can be explained in terms of a complexity of the pore network and differences in reaction kinetics. Model calculations reveal that there is a scattering for the predicted residual porosity as a function of the Thiele modulus for TiN. Moreover, this Thiele modulus can not fully account for the changes in densification at different temperatures. Given these uncertainties it is likely that a residual porosity of less than one percent can be obtained if the Thiele modulus is smaller than 1 × 10^–4. However, a CVI process with such a small Thiele modulus will not be practical, because of the concomitant long process times. Therefore, more precise conditions for the individual process parameters, i.e. concentration, reactor pressure, and temperature are deduced from the model calculations.Nomenclature a, b, c reaction order constants - C _i(x, t) concentration of species i at axial position x and time t (mole m^–3) - C _i ^o bulk concentration of species i (mole m^–3) - C _i ^* (x, t) dimensionless concentration of species i at axial position x and time t - D _e(x, t) effective diffusion coefficient at axial position x and time t (m²s^–1) - D _ij(x, t) binary diffusion coefficient (m²s^–1) - D _K(x, t) Knudsen diffusion coefficient at position x and time t (m²s^–1) - F correction factor for effective diffusion coefficient - k growth rate constant (ms^–1(m³mole^–1)^a+b-1) - K _i adsorption-desorption equilibrium constant (m³mole^–1) - L length of a pore (m) - M _i molecular weight of species i (g mole^–1) - M _ij harmonic mean of the molecular weights of species i andj (g mole^–1) - M _s molecular weight of deposit (g mole^–1) - m _t measured mass increase (g) - n _i stoichiometric number - P reactor pressure (Pa) - R(C _i) growth rate (mole(m^–2s^–1)) - r(x, t) pore radius at position x and time t (m) - r ^o initial pore radius (m) - r ^* dimensionless pore radius - S geometrical surface area (m²) - s ^t fraction of free titanium sites at the surface of TiN - s ⁿ fraction of free nitrogen sites at the surface of TiN - T temperature (K) - t time (s) - t _p process time (s) - U K _HCl/(K _{H 2} C _{H 2})^1/2 (m³ mole^–1) - V volume of alumina substrate (m³) - W K _TiCl3(m³ mole^–1) - X volume of infiltrated deposit relative to initial pore volume - x axial distance (m) - x ^* dimensionless axial distance - z number of time steps - dummy variable for integration - porosity of sintered porous alumina substrate - ratio of the volume over the surface area perpendicular to the flux (m) - density deposit (kg m^–3) - _ij a characteristic length (Å) - tortuosity factor of substrate - Thiele modulus - _D collision integral     

Unbiased Weibull capabilities indices using multiple linear regression

Although the recently proposed Weibull process capability indices (PCIs) actually measure the times that the standard deviation (σ_x) is within the tolerance specifications, because they not accurately estimate neither the log‐mean (μ_x) nor the σ_x values, then the actual PCIs are biased. This actually because μ_x and σ_x are both estimated without considering the effect that the sample size (n) has over their values. Hence, μ_x is subestimated and σ_x is overestimated. As a response to this issue, in this paper, μ_x and σ_x are estimated in function of n. In particular, the PCIs' efficiency is based on the following facts: (1) the derived n value is unique and it completely determines η, (2) the μ_x value completely determines the η value, and (3) the σ_x value completely determines the β value. Thus, now, since μ_x and σ_x are in function of n and they completely determine β and η, then the proposed PCIs are unbiased, and they completely represent the analyzed process also. Finally, a step by step numerical application is given.     

On the location of roots of non-reciprocal integer polynomials

Let P be a polynomial of degree d with integer coefficients such that P(0) ≠ 0. Assuming that P has no reciprocal factors we obtain a lower bound on the modulus of the smallest root of P in terms of its degree d, its Mahler measure M(P) and the number of roots of P lying outside the unit circle, say, k. We derive from this that all d roots of P must lie in the annulus R ₀ < |z| < R ₁, where R ₀ = R ₀(d, k, M(P)) and R ₁ = R ₁(d, k, M(P)) are given explicitly. As an application, for non-reciprocal conjugate algebraic numbers α, α′ of degree d ≥ 2 and of Mahler’s measure M(α), we prove the inequality ${|\alpha\alpha'-1|\,{ > }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}${|\alpha\alpha'-1|\,{ > }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}. Some lower bounds on the moduli of the conjugates of a Pisot number are also given. In particular, it is shown that if α is a cubic Pisot number, then the disc |z| ≤ α ⁻¹ + 0.1999α ⁻² contains no conjugates of α. Here the constant 0.1999 cannot be replaced by the constant 0.2. We also show that if α is a Pisot number of degree at least 4 and α′ is its conjugate, then |α α′ − 1| > (19α ²)⁻¹.     

A methyl parathion electrochemical sensor based on Nano-TiO2, graphene composite film modified electrode

A methyl parathion electrochemical sensor based on nano-TiO₂ and graphene composite film modified glassy carbon electrode has been developed. The electrochemical behavior of MP at the sensor was investigated by cyclic voltammetry and linear sweep voltammetry. Scanning electron microscopy was used to characterize the surface morphology of nano-TiO₂ and graphene composite film. Compared with a bare glassy carbon electrode or a mono-film modified electrode, the redox peak currents of methyl parathion on the composite film modified electrode improved greatly, indicating that the sensor showed good catalytic effects. The optimal experimental condition was obtained. The results indicated that the linear sweep voltammetry responses of methyl parathion, in pH 5.2 acetate buffer solution with open-circuit accumulation for 2 min, were linear with concentrations of methyl parathion in two ranges of 0.002～5 μM and 5～100 μM. The linear equations were i_Pc(μA) = 0.0136 + 4.965c_MP(μM) (R₁² = 0.9911) and i_Pc(μA) = 21.87 + 0.8206c_MP(μM) (R₂² = 0.9914), respectively. The detection limit was 1.0 nM (S/N = 3). The sensor exhibited high sensitivity and good reproducibility as well as certain anti-interference ability. It was applied to the determination of residual MP in fresh apple sample with the recovery of 92%～102%. The result was satisfactory.     

Frequency effect on the electrical and dielectric properties of (TlGaS2) 1 ? x(TlInSe2)x (x = 0.005, 0.02) single crystals

The electrical properties (loss tangent (tanδ), real (ɛ) and imaginary (ɛ″) parts of complex dielectric permittivity, and ac conductivity across the layers (σ_ac)) of (TlGaS₂)_{1 − x} (TlInSe₂) _x (x = 0.005, 0.02) layered single crystals have been studied in the frequency range f = 5 × 10⁴ to 3.5 × 10⁷ Hz. The results demonstrate that the dielectric dispersion in the crystals has a relaxation nature. Almost throughout the frequency range studied, their ac conductivity follows the relation σ_ac ∼ f ^0.8, characteristic of hopping conduction through localized states near the Fermi level. The Fermi-level density of states (N _F ), the spread of their energies, the mean hop time τ and distance R, and the concentration of deep traps determining the ac conductivity of the crystals (N _t ) have been estimated. With increasing x in (TlGaS₂)_{1 − x} (TlInSe₂) _x , N _F and N _t increase, while τ and R decrease.     

Accelerated Life Testing for a Class of Linear Hazard Rate Type Distributions

Accelerated life testing for distributions with hazard rate functions of the form r(t) = Ag(t) + Bh(t) are considered. Let V ₁, …, V _k be stress levels larger than V ₀—the stress level under normal conditions [V ₀ > 0]—and let a(v) be a nondecreasing function on (0, ∞). We discuss a generalization of the common accelerated models (the power rule model and the Arrhenius model) by assuming that the hazard rate under the stress level V, is of the form (a(V _t )) ^P (Ag(t) + Bh(t)). The maximum likelihood estimators of A, B and P for complete and censored samples are studied. The estimation procedure reduces to a solution of one equation with one unknown parameter. The estimation procedure under the assumption of aging is also described. The asymptotic variance-covariance matrix is given.     

Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration

For R a Galois ring and m ₁, . . . , m _l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} is an R[x]-submodule of R[x]/(x^m₁-1)×?×R[x]/(x^m_l-1){R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}. Suppose m ₁, . . . , m _l are all coprime to the characteristic of R and let {g ₁, . . . , g _t } be the set of all monic basic irreducible polynomials in the factorizations of x^m_i-1{x^{m_i}-1} (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} are identified with G₁×?×G_t{{\mathcal G}_1\times\cdots\times {\mathcal G}_t}, where G_j{{\mathcal G}_j} is an R[x]/(g _j )-submodule of (R[x]/(g_j))^n_j{(R[x]/(g_j))^{n_j}}, where n _j is the number of i for which g _j appears in the factorization of x^m_i-1{x^{m_i}-1} into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.     

Manipulating the Interfacial Band Bending For Enhancing the Thermoelectric Properties of 1T′-MoTe2/Bi2Te3 Superlattice Films

Interfacial charge effects, such as band bending, modulation doping, and energy filtering, are critical for improving electronic transport properties of superlattice films. However, effectively manipulating interfacial band bending has proven challenging in previous studies. In this study, (1T′-MoTe₂)_x(Bi₂Te₃)_y superlattice films with symmetry-mismatch were successfully fabricated via the molecular beam epitaxy. This enables to manipulate the interfacial band bending, thereby optimizing the corresponding thermoelectric performance. These results demonstrate that the increase of Te/Bi flux ratio (R) effectively tailored interfacial band bending, resulting in a reduction of the interfacial electric potential from ≈127 meV at R = 16 to ≈73 meV at R = 8. It is further verified that a smaller interfacial electric potential is more beneficial for optimizing the electronic transport properties of (1T′-MoTe₂)_x(Bi₂Te₃)_y. Especially, the (1T′-MoTe₂)₁(Bi₂Te₃)₁₂ superlattice film displays the highest thermoelectric power factor of 2.72 mW m⁻¹ K⁻² among all films, due to the synergy of modulation doping, energy filtering, and the manipulation of band bending. Moreover, the lattice thermal conductivity of the superlattice films is significantly reduced. This work provides valuable guidance to manipulate the interfacial band bending and further enhance the thermoelectric performances of superlattice films.     

Estimation of Weibull parameters by omission of some data in a sample

Methods of estimating Weibull parameters by omission of some data in a sample are considered for various functions for survival probability in linear Weibull plots. Computer simulation is used for the consideration. A use of the survival probability function, (P(i) = 1 – (i – 0.5)N, in whichP(i) is survival probability,i is thei-th order of failure strength andN is the total number of data in the sample, and the omission of 2 to 4% of the data, which are of the smallest value, the largest one and their neighbours, are recommended to get the best value of the Weibull parameters.     

Physical properties of pulsed current copper electrodeposits

Square-wave unipolar pulse electroplating and bipolar pulse reverse electroplating of copper using an acid copper electrolyte of the type used for printed circuit board production has been studied. The effects of pulse-plated coatings on cathode efficiency and on the coatings properties, such as surface roughness, microhardness, stress-strain characteristics and elongation, are reported. The results show that microhardness, tensile strength and elongation are directly affected by the pulse technique used.Nomenclature A Cross-sectional area (m²) - CE Relative cathode efficiency - i Current density (A dm^–2) - i _m,i _p Mean, peak current density (A dm^–2) - M _a Actual deposit mass (g) - M _t Theoretical mass (g) - R _a Average roughness - R _tm Mean roughness - t Coating thickness (m) - t _off Off-time (ms) - t _on On-time (ms) - duty cycle (%) - density (g m^–3)     

A new interpretation of Legendre's transformation and consequences

The Legendre transformation has found widespread application in thermodynamics, Hamilton‐Lagrange‐mechanics and optics. It attributes the values of the coordinates (x, y (x)) representing the points of a monotonic piecewise smooth functional curve y (x) the slopes m_x (x) = dy (x)/dx and the intercepts y (m_x) of their tangents on the y‐axis. Thus, the initial curve y (x) is represented by the ordered set of all slopes m_x (x) = dy (x)/dx of its tangents together with their intercepts y (m_x) on the y‐axis. It is shown that the transformed or conjugated function must basically be supplemented by a homogeneous linear function of the relevant variable. This is usually neglected in the literature. In addition, a new interpretation of the Legendre transformation is presented and discussed: For this purpose the derivative m_x (x) is considered as the proper initial function and integrated between x₀ and x. This integral is complemented by the integral of x (m_x) ((the inverse function of m_x (x)) over m_x between m_x0 = m_x (x₀) and m_x (x), if m_x (x) and x (m_x) are strictly monotonic. The sum of both integrals yields the “area” (xm_x – x₀m_x0). Legendre's transformation is obtained by reordering the respective terms. The procedure of transformation corresponds to integration by parts. Some examples and consequences of the properties considered are demonstrated and discussed using the simple model of two‐state systems. The general results of the present work remove possible internal inconsistencies in thermodynamics.     

Mean stress and plasticity effect prediction on notch fatigue and crack growth threshold,combining the theory of critical distances and multiaxial fatigue criteria

In the present work, we propose a robust calibration of some bi‐parametric multiaxial fatigue criteria applied in conjunction with the theory of critical distances (TCD). This is based on least‐square fitting fatigue data generated using plain and sharp‐notched specimens tested at two different load ratios and allows for the estimation of the critical distance according to the point and line method formulation of TCD. It is shown that this combination permits to incorporate the mean stress effect into the fatigue strength calculation, which is not accounted for in the classical formulation of TCD based on the range of the maximum principal stress. It is also shown that for those materials exhibiting a low fatigue‐strength‐to‐yield‐stress ratio σ_fl,R = ?1/σ_YS, such as 7075‐T6 (σ_fl,R = ?1/σ_YS = 0.30), satisfactorily accurate predictions are obtained assuming a linear‐elastic stress distribution, even at the tip of sharp notches and cracks. Conversely, for any materials characterized by higher values of this ratio, as quenched and tempered 42CrMo4 (σ_fl,R = ?1/σ_YS = 0.54), it is recommended to consider the stabilized elastic‐plastic stress/strain distribution, also for plain and blunt‐notched samples and even in the high cycle fatigue regime still with the application of the TCD.