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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.

Piston design is a challenging engineering problem which involves complex physics and requires satisfying multiple performance objectives. Uncertainty in piston operating conditions and variability in piston design variables are inevitable and must be accounted for. The piston assembly can be a major source of engine mechanical friction and cold start noise, if not designed properly. In this paper, an analytical piston model is used in a deterministic and probabilistic (reliability-based) multi-objective design optimization process to obtain an optimal piston design. The model predicts piston performance in terms of scuffing, friction and noise, In order to keep the computational cost low, efficient and accurate metamodels of the piston performance metrics are used. The Pareto set of all optimal solutions is calculated allowing the designer to choose the “best” solution according to trade-offs among the multiple objectives.

The system response of many engineering systems depends on time. A random process approach is therefore, needed to quantify variation or uncertainty. The system input may consist of a combination of random variables and random processes. In this case, a time-dependent reliability analysis must be performed to calculate the probability of failure within a specified time interval. This is known as cumulative probability of failure which is in general, different from the instantaneous probability of failure. Failure occurs if the limit state function becomes negative at least at one instance within a specified time interval. Time-dependent reliability problems appear if for example, the material properties deteriorate in time or if random loading is involved which is modeled by a random process. Existing methods to calculate the cumulative probability of failure provide an upper bound which may grossly overestimate the true value.

Styrene-Butadiene Rubber (SBR) is a copolymer of butadiene and styrene. It has a wide range of applications in the automotive industry due to its high durability, resistance to abrasion, oils and oxidation. SBR applications vary from tires to vibration isolators and gaskets. SBR is also used in tuned dampers which aim to reduce and control the angular vibrations of crankshafts, acting as an isolator and energy absorber between the tune damper's hub and the inertia ring. The dynamic properties of this polymer are therefore, very important in developing an appropriate analytical model. This paper presents the results of a series of experiments performed to determine the dynamic stiffness and damping properties of SBR. The frequency, temperature and displacement dependent properties are determined in a low frequency range from 0.4 to 150 Hz, and in a mid frequency range from 150 to 550 Hz. The most interesting property of SBR is its frequency dependent behavior.

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.

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.

The cathode gas diffusion layer (GDL) is an important component of polymer electrolyte membrane (PEM) fuel cell. Its design parameters, including thickness, porosity and permeability, significantly affect the reactant transport and water management, thus impacting the fuel cell performance. This paper presents an optimization study of the GDL design parameters with the objective of maximizing the current density under a given voltage. A two-dimensional single-phase PEM fuel cell model is used. A multivariable optimization problem is formed to maximize the current density at the cathode under a given electrode voltage with respect to the GDL parameters. In order to reduce the computational effort and find the global optimum among the potential multiple optima, a global metamodel of the actual CFD-based fuel cell simulation, is adaptively generated using radial basis function approximations.

Quality and performance are two important customer requirements in vehicle design. Driveline clunk negatively affects the perceived quality and must be therefore, minimized. This is usually achieved using engine torque management, which is part of engine calibration. During a tip-in event, the engine torque rate of rise is limited until all the driveline lash is taken up. However, the engine torque rise, and its rate can negatively affect the vehicle throttle response. Therefore, the engine torque management must be balanced against throttle response. In practice, the engine torque rate of rise is calibrated manually. This paper describes a methodology for calibrating the engine torque in order to minimize the clunk disturbance, while still meeting throttle response constraints. A set of predetermined engine torque profiles are calibrated in a vehicle and the transmission turbine speed is measured for each profile. The latter is used to quantify the clunk disturbance.

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.

This paper explores the trade-off between reliability-based design and robustness for an automotive under-hood thermal system using the iSIGHT-FD environment. The interaction between the engine cooling system and the heating, ventilating, and air-conditioning (HVAC) system is described. The engine cooling system performance is modeled using Flowmaster and a metamodel is developed in iSIGHT. The actual HVAC system performance is characterized using test bench data. A design of experiment procedure determines the dominant factors and the statistics of the HVAC performance is obtained using Monte Carlo simulation (MCS). The MCS results are used to build an overall system response metamodel in order to reduce the computational effort. A multi-objective optimization in iSIGHT maximizes the system mean performance and simultaneously minimizes its standard deviation subject to probabilistic constraints.

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.

Low vibration and noise level in internal combustion engines has become an essential part of the design process. It is well known that the piston assembly can be a major source of engine mechanical friction and cold start noise, if not designed properly. The piston secondary motion and piston-bore contact pattern are critical in piston design because they affect the skirt-to-bore impact force and therefore, how the piston impact excitation energy is damped, transmitted and eventually radiated from the engine structure as noise. An analytical method is presented in this paper for simulating piston secondary dynamics and piston-bore contact for an asymmetric half piston model. The method includes several important physical attributes such as bore distortion effects due to mechanical and thermal deformation, inertia loading, piston barrelity and ovality, piston flexibility and skirt-to-bore clearance. The method accounts for piston kinematics, rigid-body dynamics and flexibility.

An analytical method is presented in this paper for simulating piston secondary dynamics and piston-bore contact for an asymmetric half piston model including elastohydrodynamic (EHD) lubrication at the bore-skirt interface. A piston EHD analysis is used based on a finite-difference formulation. The oil film is discretized using a two-dimensional mesh. For improved computational efficiency without loss of accuracy, the Reynolds’ equation is solved using a perturbation approach which utilizes an “influence zone” concept, and a successive over-relaxation solver. The analysis includes several important physical attributes such as bore distortion effects due to mechanical and thermal deformation, inertia loading and piston barrelity and ovality. A Newmark-Beta time integration scheme combined with a Newton-Raphson linearization, calculates the piston secondary motion.

The objective of this paper is to assess the effect of snow density on tire-snow interaction in the presence of uncertainty. The snow-depth dependent finite element analysis (FEA) and semi-analytical models we have developed recently can predict tire-snow interfacial forces at a given density under combined slip conditions. One drawback of the models is that they are only applicable for fresh, low-density snow due to the unavailability of a density-dependent snow model. In reality, the snow density on the ground can vary between that of fresh snow to heavily compacted snow that is similar to ice. Even for fresh snow on the ground, as a vehicle moves forward, the rear wheels experience higher snow densities than the front wheels. In addition, being a natural material, snow's physical properties vary significantly even for the same density.

The overstress probe fatigue testing method, although codified to characterize fatigue strength, is poorly understood. While it yields data confirming whether minimum fatigue strength may be met, it does not directly reveal the mean fatigue strength. Procedures for conducting the test are somewhat arbitrary and rely on fitting a 3-parameter Weibull model. In this paper, a Monte Carlo procedure is developed to simulate the overstress probe test. The effect of various parameters used in the test is also discussed. A comparison is made between Weibull and Gaussian models. Suggestions for conducting the overstress probe test are provided.

Existing methods for design optimization under uncertainty assume that a high level of information is available, typically in the form of data. In reality, however, insufficient data prevents correct inference of probability distributions, membership functions, or interval ranges. In this article we use an engine design example to show that optimal design decisions and reliability estimations depend strongly on uncertainty characterization. We contrast the reliability-based optimal designs to the ones obtained using worst-case optimization, and ask the question of how to obtain non-conservative designs with incomplete uncertainty information. We propose an answer to this question through the use of Bayesian statistics. We estimate the truck's engine reliability based only on available samples, and demonstrate that the accuracy of our estimates increases as more samples become available.

Fatigue strength mean and standard deviation may be estimated by the Probit and 2-Point test methods. In this paper, methodologies for conducting the tests are developed and results from Monte Carlo simulation are presented. The results are compared with those from concurrent testing with the staircase method. While the Probit and 2-Point methods are intuitively attractive, their results are significantly different from those from the staircase method. The latter remains the best of the three.

The staircase fatigue test method is a well-established, but poorly understood probe for determining fatigue strength mean and standard deviation. The sensitivity of results to underlying distributions was studied using Monte Carlo simulation by repeatedly sampling known distributions of hypothetical fatigue strength data with the staircase test method. In this paper, the effects of the underlying distribution on staircase test results are presented with emphasis on original normal, lognormal, Weibull and bimodal data. The results indicate that the mean fatigue strength determined by the staircase testing protocol is largely unaffected by the underlying distribution, but the standard deviation is not. Suggestions for conducting staircase tests are provided.