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Using the total probability theorem, we propose a method to calculate the failure rate of a linear vibratory system with random parameters excited by stationary Gaussian processes. The response of such a system is non-stationary because of the randomness of the input parameters. A space-filling design, such as optimal symmetric Latin hypercube sampling or maximin, is first used to sample the input parameter space. For each design point, the output process is stationary and Gaussian. We present two approaches to calculate the corresponding conditional probability of failure. A Kriging metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure and the failure rate of the dynamic system. The proposed method is demonstrated using a vibratory system.

A twist beam rear suspension system is modeled, analyzed and optimized in this paper. An ADAMS model is established based on the REC (Rigid-Elastic Coupling) Theory, which is verified by FEM (Finite Element Method) approach, the effects of the geometric parameters on the twist beam suspension performance are investigated. In order to increase the calculation efficiency and improve the simulation accuracy, a neural network model and NSGA II (Non-dominated Sorting Genetic Algorithm II) are adopted to conduct a multi-objective optimization on a twist beam rear suspension system.

As vacuum suction cups are widely used in stamping plants, it becomes urgent and important to understand their performance and failure mode. Vacuum suction cups are employed to lift, move, and place sheet metal instead of human hands. Occasionally the vacuum cups would fail and drop parts, even it would cause expensive delays in the production line. In this research, several types of vacuum cups have been studies and compared experimentally. A new tensile device and test method was developed to measure the pulling force and deformation of vacuum cups. The digital image correlation technique has been adopted to capture and analyze the contour, deformation and strain of the cups under different working conditions. The experimental results revealed that the relevant influential parameters include cup type, pulling force angles, vacuum levels, sheet metal curvatures, etc.

Self-piercing riveting (SPR) has been used in production to join sheet materials since the early 1990s. A large amount of experimental trial work was required in order to determine an appropriate combination of rivet and anvil design to fulfill the required joint parameters. The presented study is describing the methodology of SPR joint design based on numerical simulation and experimental methods of defining required simulation input parameters. The required inputs are the stress-strain curves of sheet materials and rivets for the range of strains taking place in the SPR joining process, parameters required for a fracture model for all involved materials, and friction parameters for all interfaces of SPR process. In the current study, the normalized Cockroft-Latham fracture criterion was used for predicting fracture. Custom hole and tube expansion tests were used for predicting fracture of the riveted materials and the rivet, respectively.

It is challenging to perform probabilistic analysis and design of large-scale structures because probabilistic analysis requires repeated finite element analyses of large models and each analysis is expensive. This paper presents a methodology for probabilistic analysis and reliability based design optimization of large scale structures that consists of two re-analysis methods; one for estimating the deterministic vibratory response and another for estimating the probability of the response exceeding a certain level. The deterministic re-analysis method can analyze efficiently large-scale finite element models consisting of tens or hundreds of thousand degrees of freedom and large numbers of design variables that vary in a wide range. The probabilistic re-analysis method calculates very efficiently the system reliability for many probability distributions of the design variables by performing a single Monte Carlo simulation.

A Reliability-Based Design Optimization (RBDO) method for multiple failure regions is presented. The method uses a Probabilistic Re-Analysis (PRRA) approach in conjunction with an approximate global metamodel with local refinements. The latter serves as an indicator to determine the failure and safe regions. PRRA calculates very efficiently the system reliability of a design by performing a single Monte Carlo (MC) simulation. Although PRRA is based on MC simulation, it calculates “smooth” sensitivity derivatives, allowing therefore, the use of a gradient-based optimizer. An “accurate-on-demand” metamodel is used in the PRRA that allows us to handle problems with multiple disjoint failure regions and potentially multiple most-probable points (MPP). The multiple failure regions are identified by using a clustering technique. A maximin “space-filling” sampling technique is used to construct the metamodel. A vibration absorber example highlights the potential of the proposed method.

An efficient reliability estimation method is presented for engineering systems with multiple failure regions and potentially multiple most probable points. The method can handle implicit, nonlinear limit-state functions, with correlated or non-correlated random variables, which can be described by any probabilistic distribution. It uses a combination of approximate or “accurate-on-demand,” global and local metamodels which serve as indicators to determine the failure and safe regions. Samples close to limit states define transition regions between safe and failure domains. A clustering technique identifies all transition regions which can be in general disjoint, and local metamodels of the actual limit states are generated for each transition region.

In this paper, a finite element methodology is given in which finite element models of a three-weld Al-Cu plate is created with support and loading conditions emulating those seen in an optical lab. Harmonic response is sought for the models under the presumption that various defective welds are present. The numerical results are carefully examined to determine the guideline frequency range so the actual optical experiment can be carried out more efficiently.

In today's light-weight vehicles, the strength of spot welds plays an important role in overall product integrity, reliability and customer satisfaction. Naturally, there is a need for a quick and reliable technique to inspect the quality of the welds. In the past, the primary quality control tests for detecting weld defects are the destructive chisel test and peel test [1]. The non-destructive evaluation (NDE) method currently used in industry is based on ultrasonic inspection [2, 3, 4]. The technique is not always successful in evaluating the nugget size, nor is it effective in detecting the so-called “cold” or “stick” welds. Therefore, it is necessary to develop a precise and reliable noncontact NDE method for spot welds. There have been numerous studies in predicting the weld nugget size by considering the spot-weld process [5, 6].

In this paper, the development of random vibration testing schedules for durability design verification of engine mounted products is presented, based on the equivalent fatigue damage concept and the 95th-percentile customer engine usage data for 150,000 miles. Development of the 95th-percentile customer usage profile is first discussed. Following that, the field engine excitation and engine duty cycle definition is introduced. By using a simplified transfer function of a single degree-of-freedom (SDOF) system subjected to a base excitation, the response acceleration and stress PSDs are related to the input excitation in PSD, which is the equivalent fatigue damage concept. Also, the narrow-band fatigue damage spectrum (FDS) is calculated in terms of the input excitation PSD based on the Miner linear damage rule, the Rayleigh statistical distribution for stress amplitude, a material's S-N curve, and the Miles approximate solution.

Design of sheet metal drawing processes requires accurate information about the distribution of restraining forces, which is usually accomplished by a set of drawbeads positioned along the perimeter of the die cavity. This study is targeting bringing together the results of finite element analysis and experimental data in order to understand the most critical factors influencing the restraining force. The experimental study of the restraining force was performed using drawbead simulator tool installed into a tensile testing machine. Based upon the experimental results, it was observed that the restraining force of the given drawbead configuration is dependent upon the depth of bead penetration, friction between the drawbead surfaces as well as the clearance between the flanges of the drawbead simulator. This clearance is often adjusted during stamping operations to increase or decrease material inflow into the die cavity without any modification in the die.

In this study, a unique multi-camera three-dimensional digital image correlation (3D-DIC) system was designed and applied to an engine dynamometer cell to measure the displacement and strain of the exhaust manifold while an engine was running in a durability test. In the engine dynamometer cell, the ambient temperature varies from 25°C to 80°C~100°C cyclically and the exhaust manifold experiences high temperatures up to 900°C with high frequency vibrations. In order to obtain reliable data under such conditions, two 3D-DIC systems were designed and set up in the engine dynamometer. One is a high-speed 3D-DIC system, consisting of cameras with a sampling rate of 1250 frames per second. It was used to measure the local displacement of the bolted joint in the exhaust manifold. The high-speed measurement system is able to record the behavior of the bolt during the thermal cycles.

Finite element analysis is a well-established methodology in structural dynamics. However, optimization and/or probabilistic studies can be prohibitively expensive because they require repeated FE analyses of large models. Various reanalysis methods have been proposed in order to calculate efficiently the dynamic response of a structure after a baseline design has been modified, without recalculating the new response. The parametric reduced-order modeling (PROM) and the combined approximation (CA) methods are two re-analysis methods, which can handle large model parameter changes in a relatively efficient manner. Although both methods are promising by themselves, they can not handle large FE models with large numbers of DOF (e.g. 100,000) with a large number of design parameters (e.g. 50), which are common in practice. In this paper, the advantages and disadvantages of the PROM and CA methods are first discussed in detail.

An efficient approach for series system reliability-based design optimization (RBDO) is presented. The key idea is to apportion optimally the system reliability among the failure modes by considering the target values of the failure probabilities of the modes as design variables. Critical failure modes that contribute the most to the overall system reliability are identified. This paper proposes a computationally efficient, system RBDO approach using a single-loop method where the searches for the optimum design and for the most probable failure points proceed simultaneously. Specifically, at each iteration the optimizer uses approximated most probable failure points from the previous iteration to search for the optimum. A second-order Ditlevsen upper bound is used for the joint failure probability of failure modes. Also, an easy to implement active strategy set is employed to improve algorithmic stability.

Reliability-based design optimization accounts for variation. However, it assumes that statistical information is available in the form of fully defined probabilistic distributions. This is not true for a variety of engineering problems where uncertainty is usually given in terms of interval ranges. In this case, interval analysis or possibility theory can be used instead of probability theory. This paper shows how possibility theory can be used in design and presents a computationally efficient sequential optimization algorithm. The algorithm handles problems with only uncertain or a combination of random and uncertain design variables and parameters. It consists of a sequence of cycles composed of a deterministic design optimization followed by a set of worst-case reliability evaluation loops. A crank-slider mechanism example demonstrates the accuracy and efficiency of the proposed sequential algorithm.

Probabilistic studies can be prohibitively expensive because they require repeated finite element analyses of large models. Re-analysis methods have been proposed with the premise to estimate accurately the dynamic response of a structure after a baseline design has been modified, without recalculating the new response. Although these methods increase computational efficiency, they are still not efficient enough for probabilistic analysis of large-scale dynamic systems with low failure probabilities (less or equal to 10-3). This paper presents a methodology that uses deterministic and probabilistic re-analysis methods to generate sample points of the response. Subsequently, tail modeling is used to estimate the right tail of the response PDF and the probability of failure a highly reliable system. The methodology is demonstrated on probabilistic vibration analysis of a realistic vehicle FE model.

In reliability design, often, there is scarce data for constructing probabilistic models. It is particularly challenging to model uncertainty in variables when the type of their probability distribution is unknown. Moreover, it is expensive to estimate the upper and lower bounds of the reliability of a system involving such variables. A method for modeling uncertainty by using Polynomial Chaos Expansion is presented. The method requires specifying bounds for statistical summaries such as the first four moments and credible intervals. A constrained optimization problem, in which decision variables are the coefficients of the Polynomial Chaos Expansion approximation, is formulated and solved in order to estimate the minimum and maximum values of a system’s reliability. This problem is solved efficiently by employing a probabilistic re-analysis approach to approximate the system reliability as a function of the moments of the random variables.

The paper studies the distributions of residual stresses in auto components after induction hardening. Three prototype parts are analyzed in this paper. Firstly, the temperature fields of the analyzed parts are quantitatively simulated during quenching by simulating surface heating to the austenitization temperature of the material. Secondly, the formation and states of the residual stresses are predicted. Therefore the distribution of residual stress is simulated and shows compressive stresses on the surface of components so that the strength can be improved. The simulated results by computer are compared with experimental results. The good comparison indicates that the results obtained by the FEA analysis are reliable. Thus, it can be concluded that the FEA (Finite element analysis) program is effectively developed to simulate heating and quenching processes and residual stresses distribution.

An accurate and efficient Monte Carlo simulation (MCS) method is developed in this paper for limit state-based reliability analysis, especially at system levels, by using a response surface approximation of the failure indicator function. The Moving Least Squares (MLS) method is used to construct the response surface of the indicator function, along with an Optimum Symmetric Latin Hypercube (OSLH) as the sampling technique. Similar to MCS, the proposed method can easily handle implicit, highly nonlinear limit-state functions, with variables of any statistical distributions and correlations. However, the efficiency of MCS can be greatly improved. The method appears to be particularly efficient for multiple limit state and multiple design point problem. A mathematical example and a practical example are used to highlight the superior accuracy and efficiency of the proposed method over traditional reliability methods.

An efficient simulation-based method is proposed for the reliability analysis of a vehicle body-door subsystem with respect to an important quality issue -- wind noise. A nonlinear seal model is constructed for the automotive wind noise problem and the limit state function is evaluated using finite element analysis. Existing analytical as well as simulation-based methods are used to solve this problem. A multi-modal adaptive importance sampling method is then developed for reliability analysis at system level. It is demonstrated through this industrial application problem that the multi-modal adaptive importance sampling method is superior to existing methods in terms of efficiency and accuracy. The method can easily handle implicit limit-state functions, with variables of any statistical distributions.