Dynamic Performance of Stacked Packaging Units

Stacked packaging units is the main form of distribution packaging of products. Its dynamic performance is not fully understood. This paper investigated the influence of the constraint, input vibration, location and contact nonlinearity on the dynamic performance of three layers stacked packaging units. The dynamic contact force between surfaces and acceleration response of products were obtained. In sine sweep vibrations, the constraint to stacked packaging units has an obvious influence on the dynamic characteristics. The acceleration response of product is associated with the vibration mode. The force amplification factor is in general between 1.5 and 2, but it can close to 3 on top layer in the case of no fixed. In random vibrations, non‐Gaussian data of dynamic contact force appear when Gaussian data of input vibration pass through the stacked packaging units, resulting a Weibull distribution of force level‐crossing. The force level‐crossing diagram becomes more abrupt with the decreasing of input vibration level, smoother from the top contact surface to middle and bottom ones, and moves right and becomes smoother with the constraint strengthen. In the case of lower input level, Gaussian distribution of force level‐crossing appears. The force power spectral density (PSD) between bottom box and table is much larger than that between boxes, which is significantly influenced by the first resonance frequency. However, the acceleration PSD of product is significantly influenced by both the first and second resonance frequencies, and controlled by the vibration mode. It depends much on the value of input acceleration PSD around the resonance frequencies.     

Multivariate probability distribution of ordered peaks of vector Gaussian random processes

The problem of determining the joint probability distribution of ordered peaks of jointly stationary Gaussian random processes is considered. The solution is obtained by modeling the number of times a specified threshold is crossed by the component processes as a multivariate Poisson process. Based on this, the joint probability distribution of the time required for the nth crossing of a specified level with positive slope is derived. This formulation is further extended to derive the joint distribution of ordered peaks in a given time interval. An illustrative example on a bivariate Gaussian random process is presented and the analytical predictions are shown to compare reasonably well with corresponding results from Monte Carlo simulations. Also presented is an analysis of response of a randomly driven multi-degree of freedom system with emphasis on the sensitivity of ordered peak characteristics with respect to changes in system parameters. It is demonstrated that higher order statistics are generally more sensitive to changes in system characteristics—a property that has potential for application in structural model updating and damage detection.     

Entropy propagation analysis in stochastic structural dynamics: application to a beam with uncertain cross sectional area

This paper investigates the impact of different probabilistic models of uncertain parameters on the response of a dynamical structure. The probabilistic models of the uncertain parameters are constructed using the maximum entropy principle, where different information is considered, such as bounds, mean value, etc. Nested probabilistic models are constructed with increasing information; as the information given increases, the level of entropy of the input model decreases. The response of the linear dynamical model is given in the frequency domain, and the propagation of the input uncertainty throughout the computational model is analyzed in terms of Shannon’s entropy. Low and high frequencies are analyzed because uncertainties propagate differently depending on the frequency band. A beam discretized by means of the finite element method with random cross sectional area (random field) is the application analyzed.     

A new reliability measure based on specified minimum distances before the locations of random variables in a finite interval

A new reliability measure is proposed and equations are derived which determine the probability of existence of a specified set of minimum gaps between random variables following a homogeneous Poisson process in a finite interval. Using the derived equations, a method is proposed for specifying the upper bound of the random variables' number density which guarantees that the probability of clustering of two or more random variables in a finite interval remains below a maximum acceptable level. It is demonstrated that even for moderate number densities the probability of clustering is substantial and should not be neglected in reliability calculations.In the important special case where the random variables are failure times, models have been proposed for determining the upper bound of the hazard rate which guarantees a set of minimum failure-free operating intervals before the random failures, with a specified probability. A model has also been proposed for determining the upper bound of the hazard rate which guarantees a minimum availability target. Using the models proposed, a new strategy, models and reliability tools have been developed for setting quantitative reliability requirements which consist of determining the intersection of the hazard rate envelopes (hazard rate upper bounds) which deliver a minimum failure-free operating period before random failures, a risk of premature failure below a maximum acceptable level and a minimum required availability. It is demonstrated that setting reliability requirements solely based on an availability target does not necessarily mean a low risk of premature failure. Even at a high availability level, the probability of premature failure can be substantial. For industries characterised by a high cost of failure, the reliability requirements should involve a hazard rate envelope limiting the risk of failure below a maximum acceptable level.     

Uncertainty propagation in inverse reliability-based design of composite structures

An approach for the analysis of uncertainty propagation in reliability-based design optimization of composite laminate structures is presented. Using the Uniform Design Method (UDM), a set of design points is generated over a domain centered on the mean reference values of the random variables. A methodology based on inverse optimal design of composite structures to achieve a specified reliability level is proposed, and the corresponding maximum load is outlined as a function of ply angle. Using the generated UDM design points as input/output patterns, an Artificial Neural Network (ANN) is developed based on an evolutionary learning process. Then, a Monte Carlo simulation using ANN development is performed to simulate the behavior of the critical Tsai number, structural reliability index, and their relative sensitivities as a function of the ply angle of laminates. The results are generated for uniformly distributed random variables on a domain centered on mean values. The statistical analysis of the results enables the study of the variability of the reliability index and its sensitivity relative to the ply angle. Numerical examples showing the utility of the approach for robust design of angle-ply laminates are presented.     

Performance optimization of irreversible refrigerators based on a new thermo-ecological criterion

A performance analysis and optimization based on a new thermo-ecological optimization criterion has been carried out for refrigerators. The ecological objective function is defined as the ratio of the cooling load to the loss rate of availability (or entropy generation rate). The maximum of the ecological performance criterion and the corresponding optimal conditions have been derived analytically. The optimum performance parameters which maximize the objective function have been investigated and the effects of irreversibility parameters on the general and optimal performances are discussed detailed. The obtained results may provide a general theoretical tool for the ecological design of refrigerators.     

Comparison study on simulation effect of improved simulation methods for packaging random vibration test

In general, existing testing methods for packaging design employ Gaussian distributions to represent vibration induced by road profiles. However, the actual road vehicle vibration is non‐Gaussian. The key limitation is that the simulated Gaussian vibration cannot reconstruct the shock events buried in the vibration, which will result in inaccurate reliability evaluation of cushion packaging. A new simulation technique called shock extraction has been proposed and validated in our earlier study. This article is a further study of our previous research. The shock extraction method has been compared with other three representative simulation methods including the single‐level PSD, three‐way split spectral, and wavelet decomposition in terms of simulation effect. Signal simulated by the shock extraction method possesses the same vibration intensity, duration, approximate statistical characteristics, and PSD plots with the original signal. The results show that the simulation effect of the shock extraction method is the best, and the worst is the single‐level PSD. The wavelet decomposition and three‐way split spectral are somewhere in the middle.     

Inverse reliability based structural design for system dependent critical earthquake loads

The problem of reliability based design of structures subjected to partially specified random earthquake loads is considered. A procedure for the determination of structure–excitation pair that maximizes a specified response variable and, at the same time, achieves a target reliability is outlined. The procedure combines concepts from inverse first order reliability methods and methods for determining random critical earthquake loads. The formulation is shown to lead to a problem in constrained non-linear optimization. Issues related to spatial variability of earthquake loads are also addressed. Illustrative examples on a singly supported multi-degree of freedom system and a doubly supported single degree of freedom system are included.     

A new model for non-Gaussian random excitations

Very often one is called upon to model time series data which are clearly non-Gaussian, but which retain some aspects of a Gaussian process. In the present paper, a novel methodology which helps in modelling such data is presented. The method is essentially to express the process as a series with finite number of terms, wherein the first term is a Gaussian process with zero mean and unit standard deviation. Non-Gaussian higher order correction terms are added to this such that each succeeding term is orthogonal or uncorrelated with all the previous terms. The unknown coefficients in the series representation can be expressed in terms of the estimated moments of the data. Further the autocorrelation or PSD of the data can be exactly reproduced by the non-Gaussian model. The use of the proposed model is illustrated by considering the unevenness data of railway tracks. Application to response of systems under non-Gaussian excitation is also briefly discussed.     

基于离散小波变换的滑移隔震结构非平稳随机响应计算

将地震地面运动模拟成非平稳随机过程,基于离散小波变换得到地面运动离散小波系数的统计值,并以此作为输入,使用等价线性方法获得了滑移隔震结构在非平稳地震作用下的均方滑动位移和速度反应。通过计算实例与Monte-Carlo法的计算结果进行对比,验证了本文方法的正确性。最后,给出了这种结构的最大滑动位移响应随滑移面摩擦系数的变化曲线,可供设计时参考。     

Crossing rate statistics of quadratic transformations of Gaussian processes

The main focus of this paper is the development of a numerical procedure for calculating the average crossing rates of a stochastic process that can be expressed as a sum of a linear and a nonlinear, quadratic transformation of a Gaussian process. Such a representation applies for instance to the motion response of a linear structure subjected to wind loading, when the loading model is proportional to the square of a Gaussian wind velocity process. It is also the standard model for expressing the total wave forces or horizontal excursion responses of a moored floating offshore platform in a random sea way. Knowledge of the crossing rate is a key to many important quantities related to response statistics and reliability applications. It is demonstrated how the proposed numerical procedure can be used for calculating the average crossing rate of the type of response processes considered.     

Optimization of Welding Process Parameters for 9Cr-1Mo Steel Using RSM and GA

Optimization of A-TIG welding process parameters for 9Cr-1Mo steel has been carried out using response surface methodology (RSM) and genetic algorithm (GA). RSM has been used to obtain the design matrix for generating data on the influence of process parameters on the response variables. A second-order response surface model was developed for predicting the response for the set of given input variables. Then, numerical and graphical optimization was performed using RSM to obtain the target depth of penetration (DOP) and heat-affected zone (HAZ) width using desirability approach. Multiple regression models were developed based on the generated data, and then the models were used in GA to determine the optimum process parameters for achieving the target DOP and HAZ width. GA-based models employed two different selection processes. Both the RSM- and GA-based models suggested a number of solutions in terms of process parameters, and the identified solutions were validated by experiments. GA-based model employing tournament selection has been found to be a more accurate method for determining the optimum A-TIG welding process parameters.     

Sensitivity analysis and optimal design of non-linear structures

Sensitivity analysis for non-linear elastic structures in regular and critical states is first discussed including design parameters and initial imperfections. Next, the optimal design problem is formulated by considering imperfect structures and setting constraints on deflections and stresses. For structures with unstable post-critical response the limit load constraint is introduced in the optimization procedure. Several examples of truss optimization are provided. The level of initial imperfections can be regarded as design parameter and specified from the optimal solution. © 1998 John Wiley & Sons, Ltd.     

Spectral color constancy using a maximum entropy approach

This paper proposes a solution to the spectral color constancy problem. The method is based on a statistical model for the surface reflectance spectrum and applies a maximum entropy constraint. Unlike prior methods based on linear models, the solution process does not require a set of basis functions to be defined, nor does it require a database of spectra to be specified in advance. Experiments on simulated and real data show that spectral estimation using the maximum entropy approach is feasible and performs similarly to existing spectral methods in spite of the lower level of a priori information required.     

Maximum roll angle estimation of a ship in confused sea waves via a quasi-deterministic approach

This paper considers the maximum roll motion of a ship in confused sea waves. The ship motion is described by a nonlinear differential equation including quadratic damping and cubic restoring force. The excitation of the ship is represented by a stationary mean-zero Gaussian process of a given power spectral density function. It is shown that a reliable estimate of the maximum roll motion is found considering the ship response to an approximate deterministic representation of an appropriately large and adequately rich (frequency-wise) load. Specifically, the time variation of the load is approximated by a normalized autocovariance function; the maximum amplitude of the load is taken as a certain multiple of the standard deviation of the stochastic load process. This approximation relates to the method of quasi-deterministic representation of extreme realizations of a stationary Gaussian process; the method is interpreted as a tool for generating deterministic time histories of the load which are compatible with a certain power spectral density function. The efficacy of this perspective is shown by comparison with the results from pertinent Monte Carlo simulations.Next, the paper addresses the ship stability problem in the space of initial conditions. In this context, it shows that the proposed approximation can be adequately utilized for a ship safety assessment.     

Reliability analysis of a microwave tower for fluctuating mean wind with directional effect

The reliability analysis of a 75 m tall steel lattice tower is presented for the annual variation of mean wind velocity taken as a function of both mean wind speed and direction. A parametric study is also conducted in order to investigate the effect of different important parameters on the component and system reliability of the tower. Variation of the mean wind velocity is determined from the continuous records of the hourly mean wind speed and the hourly mean values of the two components of the wind velocity. Annual probability of maximum response of a component of the tower exceeding a threshold level is determined as the mean number of crossing per day of the velocity vector out of the safe resistance boundary surface defined as a function of mean wind speed and direction. Using the above method of analysis, reliability against allowable stress failure of critical members of the tower and the consequent, system reliability are determined for a number of parametric variations. It is shown that the probability of failure of the tower increases with the increase in the standard deviation of mean wind speed, the zero crossing rate and the correlation coefficient between the velocities in the two directions. Further, the variation of the probability of failure is found to be sensitive to the variation of the above parameters in the lower range of the threshold levels of stresses.     

Determination of particle size distribution by light extinction method using improved pattern search algorithm with Tikhonov smoothing functional

An inversion technique which combines the pattern search algorithm with the Tikhonov smoothing functional for retrieval of particle size distribution (PSD) by light extinction method is proposed. In the unparameterized shape-independent model, we first transform the PSD inversion problem into an optimization problem, with the Tikhonov smoothing functional employed to model the objective function. The optimization problem is then solved by the pattern search algorithm. To ensure good convergence rate and accuracy of the whole retrieval, a competitive strategy for determining the initial point of the pattern search algorithm is also designed. The accuracy and limitations of the proposed technique are tested by the inversion results of synthetic and real standard polystyrene particles immersed in water. In addition, the issues about the objective function and computation time are further discussed. Both simulation and experimental results show that the technique can be successfully applied to retrieve the PSD with high reliability and stability in the presence of random noise. Compared with the Phillips–Twomey method and genetic algorithm, the proposed technique has certain advantages in terms of reaching a more accurate and steady optimal solution with less computational effort, thus making this technique more suitable for quick and accurate measurement of PSD.     

The random barrier-crossing problem

This paper addresses the time-variant reliability analysis of structures with random resistance or random system parameters. It deals with the problem of a random load process crossing a random barrier level. The implications of approximating the arrival rate of the first overload by an ensemble-crossing rate are studied. The error involved in this so-called “ensemble-crossing rate” approximation is described in terms of load process and barrier distribution parameters, and in terms of the number of load cycles. Existing results are reviewed, and significant improvements involving load process bandwidth, mean-crossing frequency and time are presented. The paper shows that the ensemble-crossing rate approximation can be accurate enough for problems where load process variance is large in comparison to barrier variance, but especially when the number of load cycles is small. This includes important practical applications like random vibration due to impact loadings and earthquake loading. Two application examples are presented, one involving earthquake loading and one involving a frame structure subject to wind and snow loadings.     

Hybrid multi-response optimization of friction stir spot welds: failure load,effective bonded size and flash volume as responses

Friction stir spot welding (FSSW) is a multi-input multi-response process. Effective multi-response optimization of welds is desirable to create welds with a balance of quality responses. In order to eliminate the subjectivity (uncertainty and engineering judgment) with the existing multi-response Taguchi-based Grey relational analysis, principal component analysis (PCA) was integrated into it. The PCA helps in determining the effective optimal weighting values required for the estimation of Grey relational grade (GRG). As a result, tool rotational speed, plunge depth and dwell time were employed as input parameters while failure load (FL), expelled flash volume (EFV) and effective bonded size (EBS) of conical pin friction stir spot-welded joint of AA2219-O alloy were the chosen output responses. EFV was minimized while FL and EBS of the joints were maximized using this hybrid multi-response approach. From the analysis of variance of GRG and its response graphs, the significant parameters and their levels were obtained. Experimental results confirmed the effectiveness and robustness of this method. In addition, three critical zones were observed on the fracture surfaces of joints, namely, tool impelled unbonded zone, partially bonded zone and effective bonded/nugget zone. The weld nugget failed by circumferential nugget shear mode.     

Optimization of a Low-Voltage Load Switch for a Smart Meter Based on a Double Response Surface Model

The existing electric energy meter load switches have issues such as a low electrical life, poor resistance to short-circuit currents, high contact point resistivity and severe heating. Research has thus been conducted on the optimization of the electric energy meter built-in load switch mechanism to establish a double response surface model for the electromagnetic attractive force and reed reaction force. The model introduces a niche fitness sorting strategy and a Gaussian mutation mechanism that are based on the standard particle swarm algorithm and form an improved multi-objective particle swarm optimization algorithm in which the static attractive force is regarded as the optimization objective. The multi-objective optimization of the structural parameters of the electromagnetic system and the reed system was carried out, and the optimization results were applied in the devices’ design and implementation. The new structure improves the electromagnetic suction and reed force at the suction position. And the results from simulation show that the proposed method exceeds that of the back propagation neural network method in achieving the optimization goal and the new structure can effectively improve the performance of meter built-in load switch by avoiding the recoil phenomenon.