Monitoring Bernoulli processes considering measurement errors and learning effect

This paper presents the effect of measurement errors and learning on monitoring processes with individual Bernoulli observations. A cumulative sum control chart is considered to evaluate the possible impacts of measurement errors and learning. We propose a time‐dependent learning effect model along with measurement errors and incorporate them into the Bernoulli CUSUM control chart statistic. The performance of the Bernoulli CUSUM control chart is then merely assessed by comparing the average number of observations to signal (ANOS) under two proposed conditions with the condition of no possible errors. Thus, the ANOS values are obtained under different proportions of non‐conforming items, once considering errors due to measurement by inspectors, and once considering both errors and learning effect together. The experimental results show that the efficiency of the control chart to detect assignable causes deteriorates in the presence of measurement errors and enhances when learning affects operators' performance. The proposed approach has a potential to be used in monitoring high‐quality Bernoulli processes as well as disease diagnosis, and other health care applications with Bernoulli observations.     

Reduction of the Effect of Estimation Error on In‐control Performance for Risk‐adjusted Bernoulli CUSUM Chart with Dynamic Probability Control Limits

The in‐control performance of any control chart is highly associated with the accuracy of estimation for the in‐control parameter(s). For the risk‐adjusted Bernoulli cumulative sum (CUSUM) chart with a constant control limit, it had been shown that the estimation error could have a substantial effect on the in‐control performance. In our study, we examine the effect of estimation error on the in‐control performance of the risk‐adjusted Bernoulli CUSUM chart with dynamic probability control limits (DPCLs). Our simulation results show that the in‐control performance of risk‐adjusted Bernoulli CUSUM chart with DPCLs is also affected by the estimation error. The most important factors affecting estimation error are the specified desired in‐control average run length, the Phase I sample size, and the adverse event rate. However, the effect of estimation error is uniformly smaller for the risk‐adjusted Bernoulli CUSUM chart with DPCLs than for the corresponding chart with a constant control limit under various realistic scenarios. In addition, we found a substantial reduction in the mean and variation of the standard deviation of the in‐control run length when DPCLs are used. Therefore, use of DPCLs has yet another advantage when designing a risk‐adjusted Bernoulli CUSUM chart. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic Probability Control Limits for Lower and Two‐Sided Risk‐Adjusted Bernoulli CUSUM Charts

Because of its advantages of design, performance, and effectiveness in reducing the effect of patients' prior risks, the risk‐adjusted Bernoulli cumulative sum (CUSUM) chart is widely applied to monitor clinical and surgical outcome performance. In practice, it is beneficial to obtain evidence of improved surgical performance using the lower risk‐adjusted Bernoulli CUSUM charts. However, it had been shown that the in‐control performance of the charts with constant control limits varies considerably for different patient populations. In our study, we apply the dynamic probability control limits (DPCLs) developed for the upper risk‐adjusted Bernoulli CUSUM charts to the lower and two‐sided charts and examine their in‐control performance. The simulation results demonstrate that the in‐control performance of the lower risk‐adjusted Bernoulli CUSUM charts with DPCLs can be controlled for different patient populations, because these limits are determined for each specific sequence of patients. In addition, practitioners could also run upper and lower risk‐adjusted Bernoulli CUSUM charts with DPCLs side by side simultaneously and obtain desired in‐control performance for the two‐sided chart for any particular sequence of patients for a surgeon or hospital.     

The Bernoulli CUSUM Chart for Detecting Decreases in a Proportion

This article considers the Bernoulli CUSUM chart for detecting a decrease in the proportion p of nonconforming items when a continuous stream of Bernoulli observations from the process is available. The properties of the Bernoulli CUSUM chart can be obtained using a Markov chain model. However, in some cases, the number of transient states in the Markov chain may be so large that it is not feasible to work directly with the transition matrix. This article provides a solution to the equations used in defining the Markov chain so that properties of the chart can be obtained without explicitly using the transition matrix. This solution allows the practitioner to determine the control limit required to give a specified in‐control performance. A simple example is used to show that using the Bernoulli CUSUM chart is a better option than the standard practice of artificially grouping the observations into samples of size n > 1 and using a Shewhart chart. Copyright © 2012 John Wiley & Sons, Ltd.     

Bending Analysis of Curved Sandwich Beams with Functionally Graded Core

In this article, bending analysis of curved sandwich beams with transversely and functionally graded (FG) core is studied. The Euler–Bernoulli beam theory is used to model the thin face-sheets and high-order shear theory is used to analyze the core. Equilibrium/field equations, compatibility and boundary conditions are used to derive the set of governing equations. The numerical solution of the governing nonlinear differential equations is based on the series Fourier–Galerkin method. Finally, the effect of geometric properties on radial deflection of core and the effect of core radius and Young's modulus on radial deflection, circumferential displacement, and stresses are investigated.     

A Generalized Likelihood Ratio Chart for Monitoring Bernoulli Processes

This paper considers the problem of monitoring the proportion p of nonconforming items when a continuous stream of Bernoulli observations is available and the objective is to effectively detect a wide range of increases in p. The proposed control chart is based on a generalized likelihood ratio (GLR) statistic obtained from a moving window of past Bernoulli observations. The Phase II performance of this chart in detecting sustained increases in p is evaluated using the steady state average number of observations to signal. Comparisons of the Bernoulli GLR chart to the Shewhart CCC‐r chart, the Bernoulli cumulative sum chart, and the Bernoulli exponentially weighted moving average chart show that the overall performance of the Bernoulli GLR chart is better than its competitors. In addition, methods are provided for designing the Bernoulli GLR chart so that this chart can be easily applied in practice. Copyright © 2012 John Wiley & Sons, Ltd.     

A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads

A mixed method is presented to study the dynamic behavior of functionally graded (FG) beams subjected to moving loads. The theoretical formulations are based on Euler–Bernoulli beam theory, and the governing equations of motion of the system are derived using the Lagrange equations. The Rayleigh–Ritz method is employed to discretize the spatial partial derivatives and a step-by-step differential quadrature method (DQM) is used for the discretization of temporal derivatives. It is shown that the proposed mixed method is very efficient and reliable. Also, compared to the single-step methods such as the Newmark and Wilson methods, the DQM gives better accuracy using larger time step sizes for the cases considered. Moreover, effects of material properties of the FG beam and inertia of the moving load on the dynamic behavior of the system are investigated and analyzed.     

Flow of Prandtl-Mayer type for a one-velocity model of dispersion medium

A one-velocity model of disperse medium finds application in the investigations of wave phenomena in two-phase media of foamy structure, bubble liquids, foamed polymers (such as polyurethane), and aerosols. Treated in [1,2] are numerical methods for the calculation of a complete set of model equations [3], which is associated with a fairly laborious procedure of integration of partial differential equations. For homogeneous isentropic flows, the solution in a number of cases may be derived by simpler means. For example, in the case of the problem of expansion flow in the vicinity of the exterior obtuse angle treated in this paper, the Bernoulli integral may be used to reduce the problem to a set of ordinary differential equations. Note that the modification of the Bernoulli integral in application to a one-velocity model of multicomponent mixture has not been described in the literature; therefore, related problems are also treated below such as the calculation of critical parameters, maximum possible velocities, and stagnation parameters.     

Free vibration of magneto-electro-elastic nanobeams based on modified couple stress theory in thermal environment

In this paper, the size-dependent free vibration of magneto-electro-elastic (MEE) nanobeams in thermal environment is investigated. Size effects are taken into account using the modified couple stress theory, which is capable of accounting for higher-order electromechanical coupling, and the equations are developed on the basis of Euler–Bernoulli beam model and using von Karman nonlinear strain. The vibration of hinged–hinged nanobeams is investigated by way of example. Effects of various parameters such as temperature, thickness, and length on natural frequencies are demonstrated, and it is indicated that increased length and decreased thickness lead to decreased nanobeam natural frequencies.     

Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory

Accurate modeling and analysis of micro-/nanoelectromechanical systems (MEMS/NEMS) has an immense contribution in identification and improvement of the performance of such systems. This article investigates a nonclassical formulation for dynamicmodeling and vibration analysis of a piezo-actuatedmicrocantilever considering the Euler–Bernoulli beam model. Regarding the size effects in micro- to nanoscales, the size-dependent nonlocal continuum theory is employed to derive dynamic equations of the nonclassical microbeam taking into account the beam discontinuities. The nonlocal formulation of the beam and piezoelectric actuator is taken into consideration. Furthermore, the size effects on the resonant vibration characteristics of the beam are studied and some results are obtained. The results illustrate the size-dependent behavior of the beam particularly at higher resonant modes of vibrations. Also, it is indicated that the nonlocality and piezoelectric characteristics have a significant influence on the resonance behavior of the beam. However, this effect is more significant at higher resonant modes of vibrations.     

A power series for vibration of a rotating nanobeam with considering thermal effect

In this article, the equation of motion for a rotating nanocantilever has been developed based on the Euler–Bernoulli beam model, which includes the effect of temperature, small scale effect, and centrifugal force. A power series method has been employed to obtain the exact solution of the natural frequencies. The results also compared with other solutions of exact and approximate differential quadrature method. The effects of temperature, angular velocity, and small scale in the vibration characteristics of a rotating nanocantilever beam are investigated. It is shown that the effect of temperature plays a significant role in the behavior of the vibration of a rotating nanocantilever. Nondimensional frequency increases in the first mode with increasing the nonlocal parameter while it is inverse for the second and third modes of vibration.     

Semi-analytical solution for free transverse vibrations of Euler–Bernoulli nanobeams with manifold concentrated masses

In the present study, transverse vibrations of nanobeams with manifold concentrated masses, resting on Winkler elastic foundations, are investigated. The model is based on the theory of nonlocal elasticity in the presence of concentrated masses applied to Euler–Bernoulli beams. A closed-form expression for the transverse vibration modes of Euler–Bernoulli beams is presented. The proposed expressions are provided explicitly as the function of two integrated constants, which are determined by the standard boundary conditions. The utilization of the boundary conditions leads to definite terms of natural frequency equations. The natural frequencies and vibration modes of the concerned nanobeams with different numbers of concentrated masses in different positions under some typical boundary conditions (simply supported, cantilevered, and clamped–clamped) have been analyzed by means of the proposed closed–form expressions in order to show their efficiency. It is worth mentioning that the effect of various nonlocal length parameters and Winkler modulus on natural frequencies and vibration modes are also discussed. Finally, the results are compared with those corresponding to a classical local model.     

Stability analysis of an embedded single-walled carbon nanotube with small initial curvature based on nonlocal theory

Many experimental observations have shown that most nanostructures, such as carbon nanotubes, are often characterized by a certain degree of waviness along their axial direction. This geometrical imperfection has profound effects on the mechanical behavior of carbon nanotubes. In the present work, stability of a slightly curved carbon nanotube under lateral loading is investigated based on Eringen's nonlocal elasticity theory. Euler Bernoulli and Timoshenko beam theories are employed to obtain equilibrium equations. Winkler-Pasternak elastic foundation is used to approximate the effect of matrix. Effects of initial curvature, nonlocal parameter, beam length, and elastic foundation parameters on initiation of critical conditions are investigated.     

Postbuckling characteristics of nanobeams based on the surface elasticity theory

In the present paper, an attempt is made to numerically investigate the postbuckling response of nanobeams with the consideration of the surface stress effect. To accomplish this, the Gurtin–Murdoch elasticity theory is exploited to incorporate surface stress effect into the classical Euler–Bernoulli beam theory. The size-dependent governing differential equations are derived and discretized along with various end supports by employing the principle of virtual work and the generalized differential quadrature (GDQ) method. Newton’s method is applied to solve the discretized nonlinear equations with the aid of an auxiliary normalizing equation. After solving the governing equations linearly, to obtain each eigenpair in the nonlinear model, the linear response is used as the initial value in Newton’s method. Selected numerical results are given to show the surface stress effect on the postbuckling characteristics of nanobeams. It is found that by increasing the thickness of nanobeams, the postbuckling equilibrium path obtained by the developed non-classical beam model tends to the one predicted by the classical beam theory and this anticipation is the same for all selected boundary conditions.     

A higher-order micro-beam model with application to free vibration

The free vibration of micro-beams is analyzed employing three different beam models. The previously obtained equations of motion for Bernoulli–Euler and Timoshenko models are solved analytically. A higher-order model is devised, which satisfies the lateral boundary conditions of micro-beams. The equations of motion with associated boundary conditions are derived by means of Hamilton's principle. The generalized differential quadrature method is used for the solution of the equations. The first five natural frequencies are obtained for micro-beams with three length-scale parameter/height ratios and five different boundary conditions.     

Formulation of equations of motion of finite element form for vehicle–track–bridge interaction system with two types of vehicle model

Vehicle, track and bridge are considered as an entire system in this paper. Two types of vertical vehicle model are described. One is a one foot mass–spring–damper system having two‐degree‐of‐freedom, and the other is four‐wheelset mass–spring–damper system with two‐layer suspension systems possessing 10‐degree‐of‐freedom. For the latter vehicle model, the eccentric load of car body is taken into account. The rails and the bridge deck are modelled as an elastic Bernoulli–Euler upper beam with finite length and a simply supported Bernoulli–Euler lower beam, respectively, while the elasticity and damping properties of the rail bed are represented by continuous springs and dampers. The dynamic contact forces between the moving vehicle and rails are considered as internal forces, so it is not necessary to calculate the internal forces for setting up the equations of motion of the vehicle–track–bridge interaction system. The two types of equations of motion of finite element form for the entire system are derived by means of the principle of a stationary value of total potential energy of dynamic system. The proposed method can set up directly the equations of motion for sophisticated system, and these equations can be solved by step‐by‐step integration method, to obtain simultaneously the dynamic responses of vehicle, of track and of bridge. Illustration examples are given. Copyright 2004 © John Wiley & Sons, Ltd.     

Maximum pressure, temperature and resistance at supersonic flow of a body by condensed medium

Algebraic equations for determining the peak values of pressure and temperature inside a condensed medium at its supersonic flow around an absolutely rigid body are deduced based on the conservation laws for head shock wave, Mie-Grüneisen (or Tait) equation of state and Bernoulli integral. The results of calculations for two types of sands, clay and water are presented.     

Nonlocal theories for bending,buckling and vibration of beams

Various available beam theories, including the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories, are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived, and variational statements in terms of the generalized displacements are presented. Analytical solutions of bending, vibration and buckling are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies. The theoretical development as well as numerical solutions presented herein should serve as references for nonlocal theories of beams, plates, and shells.     

ACCURATE CALCULATION OF ELASTIC BUCKLING LOADS FOR SPACE FRAMES BUILT UP OF UNIFORM BEAMS WITH OPEN THIN-WALLED CROSS-SECTION

A so-called exact static stiffness matrix for a uniform beam element with open thin-walled cross-section carrying an axial compressive load is derived. This stiffness matrix is useful in an accurate calculation of bifurcation loads and corresponding buckling modes of space frames built up of such beam elements. One may also calculate displacements and sectional forces caused by external joint loads taking into account the second-order effect of the axial beam loads. The exact stiffness matrix is derived by use of the general solution to a set of three coupled differential equations. This means that no preselected shape functions need be introduced and that discretization errors are avoided. The differential equations model coupled Euler–Bernoulli bending in the two principal planes and Saint-Venant/Vlasov torsion and warping with respect to the shear centre axis. No cross-sectional symmetries are assumed. Numerical examples are given. One application will be to loaded pallet racks. The ‘effective length’ for a rack column is calculated.     

Dynamic analysis of a functionally graded simply supported Euler–Bernoulli beam subjected to a moving oscillator

The dynamic behavior of a functionally graded (FG) simply supported Euler–Bernoulli beam subjected to a moving oscillator has been investigated in this paper. The Young’s modulus and the mass density of the FG beam vary continuously in the thickness direction according to the power-law model. The system of equations of motion is derived by using Hamilton’s principle. By employing Petrov–Galerkin method, the system of fourth-order partial differential equations of motion has been reduced to a system of second-order ordinary differential equations. The resulting equations are solved using Runge–Kutta numerical scheme. In this study, the effect of the various parameters such as power-law exponent index and velocity of the moving oscillator on the dynamic responses of the FG beam is discussed in detail. To validate the present formulation, the mid-point displacement of the beam is compared with that of the existing literature, and also a comparison study is performed for free vibration of an FG beam. Good agreement is observed. The results indicated that the above-mentioned parameters have a significant role in the analysis.