Reliability performance of safety instrumented systems: A common approach for both low- and high-demand mode of operation

Safety instrumented systems (SISs) are usually divided into two modes of operation, low-demand and high-demand. Unfortunately, this classification is not easy to justify and the available formulas that are used to quantify the reliability performance in these two modes of operation are unable to capture combined effects of functional testing, spurious activations, and successful responses to demands. This article discusses some important modeling issues for SIS reliability performance quantification, and demonstrates their implementation in a Markov model. The accuracy of the Markov model for a simple case study of a pressure transmitter is verified through comparison with a scenario-based formula, and it is shown that the Markov approach gives a sufficiently accurate result for all demand rates, covering both low- and high-demand modes of operation.     

Smart production systems: automating decision-making in manufacturing environment

Smart production systems (SPS) are manufacturing systems capable of autonomously diagnosing their health and autonomously designing continuous improvement projects, leading to the desired productivity improvement. The main component of SPS, developed in this paper, is the Programmable Manufacturing Advisor (PMA), which evaluates the system's health and calculates optimal steps for continuous improvement. The analytics of PMA are based on the theory of Production Systems Engineering (PSE); the numerics of PMA are based on PSE Toolbox, which implements the PSE methods. In this paper, the PMA-based SPS architecture with manager-in-the-loop is described, theoretical/analytical foundations of PMA are outlined, its software/hardware implementations are commented upon, and demonstrations of PMA-based SPS operation are provided using two production systems: automotive underbody assembly (large volume manufacturing) and hot-dip galvanisation plant (small manufacturing organisation).     

Suppression of spurious intermediate frequency modes in under‐integrated elements by combined stiffness/viscous stabilization

Solutions employing perturbation stiffness or viscous hourglass control with one‐point quadrature finite elements often exhibit spurious modes in the intermediate frequency range. These spurious frequencies are demonstrated in several examples and their origin is explained. Then it is shown that by critically damping the hourglass modes, these spurious mid‐range frequency modes can be suppressed. Estimates of the hourglass frequency and damping coefficients are provided for the plane 4‐node quadrilateral and a 4‐node shell element. Results are presented that show almost complete annihilation of spurious intermediate frequency modes for both linear and non‐linear problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real‐part dual BEM

It has been found recently that the multiple reciprocity method (MRM) (Chen and Wong. Engng. Anal. Boundary Elements 1997; 20 (1):25–33; Chen. Processings of the Fourth World Congress on Computational Mechanics, Onate E, Idelsohn SR (eds). Argentina, 1998; 106; Chen and Wong. J. Sound Vibration 1998; 217 (1): 75–95.) or real‐part BEM (Liou, Chen and Chen. J. Chinese Inst. Civil Hydraulics 1999; 11 (2):299–310 (in Chinese)). results in spurious eigenvalues for eigenproblems if only the singular (UT) or hypersingular (LM) integral equation is used. In this paper, a circular cavity is considered as a demonstrative example for an analytical study. Based on the framework of the real‐part dual BEM, the true and spurious eigenvalues can be separated by using singular value decomposition (SVD). To understand why spurious eigenvalues occur, analytical derivation by discretizing the circular boundary into a finite degree‐of‐freedom system is employed, resulting in circulants for influence matrices. Based on the properties of the circulants, we find that the singular integral equation of the real‐part BEM for a circular domain results in spurious eigenvalues which are the zeros of the Bessel functions of the second kind, Y (kρ), while the hypersingular integral equation of the real‐part BEM results in spurious eigenvalues which are the zeros of the derivative of the Bessel functions of the second kind, Y_n′(kρ). It is found that spurious eigenvalues exist in the real‐part BEM, and that they depend on the integral representation one uses (singular or hypersingular; single layer or double layer) no matter what the given types of boundary conditions for the interior problem are. Furthermore, spurious modes are proved to be trivial in the circular cavity through analytical derivations. Numerically, they appear to have the same nodal lines of the true modes after normalization with respect to a very small nonzero value. Two examples with a circular domain, including the Neumann and Dirichlet problems, are presented. The numerical results for true and spurious eigensolutions match very well with the theoretical prediction. Copyright © 2000 John Wiley & Sons, Ltd.     

Elimination of spurious static modes in meshes of hybrid compatible triangular and tetrahedral finite elements

In the assumed displacement, or primal, hybrid finite element method, the requirements of continuity of displacements across the sides are regarded as constraints, imposed using Lagrange multipliers. In this paper, such a formulation for linear elasticity, in which the polynomial approximation functions are not associated with nodes, is presented. Elements with any number of sides may be easily used to create meshes with irregular vertices, when performing a non‐uniform h‐refinement. Meshes of non‐uniform degree may be easily created, when performing an hp‐refinement. The occurrence of spurious static modes in meshes of triangular elements, when compatibility is strongly enforced, is discussed. An algorithm for the automatic selection, based on the topology of a mesh of triangular elements, of the sides in which to decrease the degree of the approximation functions, in order to eliminate all these spurious modes and preserve compatibility, is presented. A similar discussion is presented for the occurrence of spurious static modes in meshes of tetrahedral elements. An algorithm, based on heuristic criteria, that succeeded in eliminating these spurious modes and preserving compatibility in all the meshes of tetrahedral elements of uniform degree that were tested, is also presented. Copyright © 2001 John Wiley & Sons, Ltd.     

A SIL quantification approach based on an operating situation model for safety evaluation in complex guided transportation systems

Safety analysis in guided transportation systems is essential to avoid rare but potentially catastrophic accidents. This article presents a quantitative probabilistic model that integrates Safety Integrity Levels (SIL) for evaluating the safety of such systems. The standardized SIL indicator allows the safety requirements of each safety subsystem, function and/or piece of equipment to be specified, making SILs pivotal parameters in safety evaluation. However, different interpretations of SIL exist, and faced with the complexity of guided transportation systems, the current SIL allocation methods are inadequate for the task of safety assessment. To remedy these problems, the model developed in this paper seeks to verify, during the design phase of guided transportation system, whether or not the safety specifications established by the transport authorities allow the overall safety target to be attained (i.e., if the SIL allocated to the different safety functions are sufficient to ensure the required level of safety). To meet this objective, the model is based both on the operating situation concept and on Monte Carlo simulation. The former allows safety systems to be formalized and their dynamics to be analyzed in order to show the evolution of the system in time and space, and the latter make it possible to perform probabilistic calculations based on the scenario structure obtained.     

Reliability and uncertainty analysis of hardware failures of a programmable electronic system

This paper presents a methodology for evaluating the performance of Programmable Electronic Systems (PES) used for safety applications. The most common PESs used in the industry are identified. Markov modeling techniques are used to develop the reliability model. The major aspects of the analysis address the random hardware failures, the uncertainty associated with these failures, a methodology to propagate these uncertainties in the Markov model, and modeling of common cause failures. The elements of this methodology are applied to an example using a Triple Redundant PES without inter-processor communication. The performance of the PES is quantified in terms of its reliability, probability to fail safe, and probability to fail dangerous within a mission time. The effect of model input parameters (component failure rates, diagnostic coverage), their uncertainties and common cause failures on the performance of the PES is evaluated.     

Finite element formulations of strain gradient theory for microstructures and the C0–1 patch test

Based on finite element formulations for the strain gradient theory of microstructures, a convergence criterion for the C^0–1 patch test is introduced, and a new approach to devise strain gradient finite elements that can pass the C^0–1 patch test is proposed. The displacement functions of several plane triangular elements, which satisfy the C⁰ continuity and weak C¹ continuity conditions are evaluated by the C^0–1 patch test. The difference between the proposed C^0–1 patch test and the C⁰ constant stress and C¹ constant curvature patch tests is elucidated. An 18-DOF plane strain gradient triangular element (RCT9+RT9), which passes the C^0–1 patch test and has no spurious zero energy modes, is proposed. Numerical examples are employed to examine the performance of the proposed element by carrying out the C^0–1 patch test and eigenvalue test. The proposed element is found to be without spurious zero energy modes, and it possesses higher accuracy compared with other strain gradient elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Quantitative safety assessment of the ventilation recirculation system in an undersea mine

This paper describes a quantitative safety study carried out on a system which monitors the environmental conditions in an undersea mine. Of particular importance are the concentrations of methane and carbon monoxide present in the mine. Although the presence of these gases is of concern in all mines it is of particular concern in this undersea mine since up to 37 per cent of the return air of the ventilation system is recirculated. When high methane or carbon monoxide levels are detected recirculation is halted. Fault trees were constructed to represent two system failure modes for each of the trip conditions; failure to trip on demand and spurious trip. These logic diagrams were then analysed to produce the minimal cut sets and the probabilities for the system failure events. From this prediction of system performance the degree of improvement attainable by changing the system design, component repair times or test frequencies was investigated.     

Modeling safety instrumented systems with MooN voting architectures addressing system reconfiguration for testing

This paper addresses the modeling of probability of dangerous failure on demand and spurious trip rate of safety instrumented systems that include MooN voting redundancies in their architecture. MooN systems are a special case of k-out-of-n systems. The first part of the article is devoted to the development of a time-dependent probability of dangerous failure on demand model with capability of handling MooN systems. The model is able to model explicitly common cause failure and diagnostic coverage, as well as different test frequencies and strategies. It includes quantification of both detected and undetected failures, and puts emphasis on the quantification of common cause failure to the system probability of dangerous failure on demand as an additional component. In order to be able to accommodate changes in testing strategies, special treatment is devoted to the analysis of system reconfiguration (including common cause failure) during test of one of its components, what is then included in the model. Another model for spurious trip rate is also analyzed and extended under the same methodology in order to empower it with similar capabilities. These two models are powerful enough, but at the same time simple, to be suitable for handling of dependability measures in multi-objective optimization of both system design and test strategies for safety instrumented systems. The level of modeling detail considered permits compliance with the requirements of the standard IEC 61508. The two models are applied to brief case studies to demonstrate their effectiveness. The results obtained demonstrated that the first model is adequate to quantify time-dependent PFD of MooN systems during different system states (i.e. full operation, test and repair) and different MooN configurations, which values are averaged to obtain the PFD_avg. Also, it was demonstrated that the second model is adequate to quantify STR including spurious trips induced by internal component failure and by test itself. Both models were tested for different architectures with 1≤N≤5 and 2≤M≤5 subject to uniform staggered test. The results obtained also showed the effects that modifying M and N has on both PFD_avg and STR, and also demonstrated the conflicting nature of these two measures with respect to one another.     

Vibration analysis of fluid–solid systems using a finite element displacement formulation

This report presents a finite element solution for the vibration interaction between an inviscid fluid and a solid. The equation of motion governing the inviscid fluid is expressed in terms of the displacements. This ensures that compatibility and equilibrium will be satisfied automatically along the interface of the coupled systems. To suppress circulation modes with non-zero energy, reduced integration is used when computing the element stiffness matrix contributed by the fluid. In addition, a projection is used on the element mass matrix in order to remove the spurious modes which result from the use of reduced integration. Numerical examples for both fluid and coupled fluid–solid systems are performed and the results are shown.     

Determination of the natural frequencies and natural modes of a rod using the dual BEM in conjunction with the domain partition technique

In this paper, the dual BEM in conjunction with the domain partition technique is employed to solve both natural frequencies and natural modes of a rod. In this new approach, there exists no spurious eigenvalue using the complex-valued singular or hypersingular equation alone. In the derivation of the singular and hypersingular integral equations, if only the real parts of the kernel functions are chosen, the resulting eigenequations have spurious eigenvalues. Such spurious eigenvalues stem from adding the dummy links into the interior structures considered. Although the spurious eigenvalues exist in this approach which uses the real-valued kernel functions, the possible indeterminacy of eigenmodes using the conventional real-valued singular or real-valued hypersingular equations disappears when the domain partition technique is adopted. The conventional real-valued multiple reciprocity BEM results in spurious eigenvalues for the mixed boundary conditions and indeterminacy of eigenmodes owing to insufficient rank of the leading coefficient matrix for the Dirichlet and Neumann boundary conditions. Such problems can be solved by combining the singular and hypersingular equations together; however, they also can be treated by using the real-valued singular or hypersingular equation alone if the domain partition technique is adopted. Three examples including the Dirichlet, Neumann and mixed type boundary conditions are investigated to show the validity of current approach.     

Applications of dual MRM for determining the natural frequencies and natural modes of an Euler–Bernoulli beam using the singular value decomposition method

In this paper, a dual multiple reciprocity method (MRM) is employed to solve the natural frequencies and natural modes for an Euler–Bernoulli beam. It is found that the conventional MRM using an essential integral equation results in spurious eigenvalues and modes. By using the natural integral equation of dual MRM, the spurious eigendata can be filtered out. Four numerical examples are given to verify the validity of the present formulation. In one of these four examples, fixed–fixed supported beam, it is found that the boundary eigenvector cannot be determined by either the essential or natural integral equation alone since the rank of the corresponding leading coefficient matrix is insufficient. The singular value decomposition method is then used to solve the eigenproblem after combining the essential and natural integral equations. This method can avoid the spurious eigenvalue problem and possible indeterminancy of boundary eigenvectors at the same time.     

Analysis of whispering-gallery superconducting dielectric resonator modes

The whispering-gallery (WG) modes of a superconducting dielectric resonator (SDR) based on a sapphire cylindrical dielectric resonator and a YBa₂Cu₃O_{7 –} shielding cylinder have been studied. A method for the determination of the resonant frequencies and the maximum quality factor of such modes is presented. Calculations have shown that most of the mode energy could be confined between the caustic surface of the WG modes provided the dimensions of the SDR are properly selected, and a magnitude of 10⁹ forQ of the SDR could be estimated. A phenomenal explanation is given to account for such outstanding microwave behavior.     

Nonlinear oscillations of laminated plates using an accurate four-node rectangular shear flexible material finite element

The objective of the present paper is to investigate the large amplitude vibratory behaviour of unsymmetrically laminated plates. For this purpose, an efficient and accurate four-node shear flexible rectangular material finite element (MFE) with six degrees of freedom per node (three displacements (u, v, w) along thex, y andz axes, two rotations (θ _x and θ _y ) abouty andx axes and twist (θ _xy )) is developed. The element assumes bi-cubic polynomial distribution with sixteen generalized undetermined coefficients for the transverse displacement. The fields for section rotations θ _x and θ _y , and in-plane displacementsu andv are derived using moment-shear equilibrium and in-plane equilibrium equations of composite strips along thex- andy-axes. The displacement field so derived not only depends on the element coordinates but is a function of extensional, bending-extensional coupling, bending and transverse shear stiffness as well. The element stiffness and mass matrices are computed numerically by employing 3×3 Gauss-Legendre product rules. The element is found to be free ofshear locking and does not exhibit any spurious modes. In order to compute the nonlinear frequencies, linear mode shape corresponding to the fundamental frequency is assumed as the spatial distribution and nonlinear finite element equations are reduced to a single nonlinear second-order differential equation. This equation is solved by employing thedirect numerical integration method. A series of numerical examples are solved to demonstrate the efficacy of the proposed element.     

Sound Modes of Superfluid 3He in Aerogel

We present the results of experiments on sound propagation at audio frequencies in ³ He-filled aerogel. Sound modes were observed at temperatures of 0.8–100 mK in an aerogel sample of 98% porosity. We find that below T _c for superfluid ³ He in the aerogel matrix the speed of sound in the composite system increases by as much as 1.5%. Also below the aerogel T _c new modes appear which correspond to propagation speeds of up to 10 m/s.     

A triangular hybrid equilibrium plate element of general degree

Hybrid stress‐based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

Role of electron-phonon interaction in lattice dynamics and superconductivity of oxide spinel LiTi2O4

The lattice dynamics of an oxide spinel LiTi₂O₄ is studied by taking account of the electron-phonon (EP) interaction derived on the basis of the realistic tight-binding bands fitted to the first-principles bands. Due to the characteristic dependences of the EP interaction on wavevectors and vibrational modes, a remarkable frequency renormalization of the O vibrational modes, which hybridize with the Ti vibrational modes, is obtained over a wide region of the Brillouin zone. The overall features of the calculated phonon density of states are in agreement with those observed by the inelastic neutron scattering measurements. By using the EP interaction and the renormalized phonon frequencies we have calculated the EP spectral function ² F(). The superconducting transition temperature, gap function, and tunneling spectra are calculated by solving the Éliashberg equation. The results agree well with the observations.     

Two-dimensional analysis of spurious modes in aluminum nitride film resonators

In this paper, a hybrid method, which combines the traditional concept of guided waves and the finite element method (FEM), is proposed to analyze the spurious modes of aluminum nitride (AIN) film with electrodes. First, the guided wave modes in the plated area are obtained by 1-D FEM. Second, a mode-match method is used to satisfy the boundary conditions. The vibration of the film resonator is a superposition of all of the guided modes. With respect to an A1N film resonator, which is a thickness-stretch mode resonator, we have identified three families of spurious modes: extension, thickness-stretch, and thickness-shear. The spectrum of spurious modes is calculated and the influence of the spurious modes is discussed.     

Hierarchical theories for the free vibration analysis of functionally graded beams

This work addresses a free vibration analysis of functionally graded beams via several axiomatic refined theories. The material properties of the beam are assumed to vary continuously on the cross-section according to a power law distribution in terms of the volume fraction of the material constituents. Young’s modulus, Poisson’s ratio and density can vary along one or two dimensions all together or independently. The three-dimensional kinematic field is derived in a compact form as a generic N-order polynomial approximation. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the Principle of Virtual Displacements. They are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted. Higher-order displacements-based theories that account for non-classical effects are formulated. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Bending, torsion and axial modes are investigated. Slender as well as short beams are considered. Numerical results highlight the effect of different material distributions on natural frequencies and mode shapes and the accuracy of the proposed models.