Shock spectra and damage boundary curves for non‐linear package cushioning systems

This paper suggests a general approach for obtaining shock spectra and damage boundary curves for cushioning packaging systems. This approach can treat both linear and non‐linear cushioning systems such as cubic, tangent and hyperbolic tangent systems. Corresponding software has been developed for analysing different cushioning systems, and the shock spectra and the damage boundary curves are given for a tangent non‐linear cushioning system under the action of a rectangular, half‐sine, terminal‐peak saw‐tooth and initial‐peak saw‐tooth pulse, respectively. It is worth noting that the shock spectrum is affected not only by the damping parameter but also by the dimensionless pulse peak, and that both the damping parameter and the dimensionless fragility also influence the damage boundary curve. These are important features of non‐linear cushioning system. Copyright © 1999 John Wiley & Sons, Ltd.     

Shock spectra and damage boundary curves for hyperbolic tangent cushioning system and their important features

By applying the method suggested in the author's previous paper, shock spectra and damage boundary curves are investigated for a hyperbolic tangent cushioning system under the action of rectangular, half‐sine, terminal‐peak saw‐tooth and initial‐peak saw‐tooth acceleration pulses, respectively. The shock spectrum is affected not only by the damping parameter but also by the dimensionless pulse peak, and both the damping parameter and the dimensionless fragility influence the damage boundary curve for this cushioning system. Some important features of a hyperbolic tangent cushioning system that differs from a tangent cushioning system are discussed in detail. Copyright © 2001 John Wiley & Sons, Ltd.     

On evaluation of product dropping damage

As an extension of the classical damage boundary curve in the case of product dropping shock, the concept of a dropping damage boundary curve for linear and non‐linear packaging system to evaluate the dropping damage of product is developed in this paper. The dropping damage boundary curves are given for linear and hyperbolic tangent packaging systems with different damping. For a linear packaging system the dropping damage of a product is determined only by the natural frequency of the corresponding packaging system without damping and the dropping shock velocity of package except the system damping, and they compose the basic evaluation quantities of product dropping damage. For a non‐linear hyperbolic tangent packaging system, the system parameter and the dimensionless dropping shock velocity are two basic quantities in the evaluation of product dropping damage. It should be emphasized that the dimensionless dropping shock velocity is related not only to the dropping height of package box but also to the system parameter integration. This is the important feature differentiating a non‐linear packaging system from a linear one. The influence of system damping on the dropping damage boundary curves is also discussed. This concept and the results have important value in the design of cushioning packaging. Copyright © 2002 John Wiley & Sons, Ltd.     

Evaluation of product dropping damage based on key component

This paper analyses the packaging system of a product as a two‐degrees‐of‐freedom system, one degree for the key component and the other for the main part of the product. The dropping damage boundary curve was developed based on the key component for linear and non‐linear packaging systems to predict product damage as a result of drop impacts. The dynamic models of two‐degrees‐of‐freedom dropping shock were obtained. For a linear packaging system, the dropping response of the key component was determined by the dimensionless dropping shock velocity, the frequency parameter ratio, the mass ratio and the damping parameters; for a non‐linear system, the system parameter was also used. The frequency parameter ratio of the packaging system and the dimensionless dropping shock velocity were selected as the basic evaluation quantities for the dropping damage of the key component. As an example, the dropping damage boundary curves based on the key component were given for linear and tangent packaging systems. The influence of related parameters such as the mass ratio, the system parameter and the damping parameters on the dropping damage boundary curve was investigated. To verify the theory, experiments were designed and completed. Experiment results for both linear and tangent packaging systems were consistent with the theory suggested in this paper. These results have important value not only for the design of cushioning packaging but also for the improvement of products. Copyright © 2010 John Wiley & Sons, Ltd.     

跌落损伤脆值及损伤边界

提出了跌落损伤脆值与跌落损伤边界的新概念,并提出了测定获取跌落冲击损伤脆值与跌落损伤边界曲线的技术过程与方法.以嘎拉苹果为研究对象,设计多组高度对嘎拉苹果进行自由跌落刚性冲击试验.在测试获得冲击力-时间数据的基础上,通过理论分析计算,得到加速度-时间、变形量-时间等动态关系曲线,获得了嘎拉苹果跌落损伤脆值;构造了嘎拉苹果跌落损伤边界曲线.结果表明,跌落损伤边界曲线与传统产品跌落破损边界曲线存在较大差异.对应果品类产品,即使速度很小,只要产品瞬时加速度达到一定值,也将产生冲击损伤;对应不同的损伤率,可构造相应的多条损伤边界曲线.提出的概念及研究结果为控制果品类产品的跌落损伤、合理地进行缓冲包装设计提供了新的理论基础.     

物品包装临界跌落高度与缓冲衬垫特性参数关系初探

运用经典的加速度损坏边界和新提出的位移损坏边界理论，研究了线性包装系统临界跌落高度与缓冲衬垫特性参数（缓冲初垫刚度、最大可压缩量）间的关系，并给出了数值解，为包装动力学理论在工程缓冲设计中的简化应用提供了一种途径和方法。     

缓冲包装系统跌落破损边界曲线研究

建立了线性和非线性缓冲包装系统的无量纲跌落冲击方程,得到了产品最大加速度响应,提出了评价线性和非线性包装系统产品安全与否的跌落破损边界曲线概念。对于线性包装系统,固有频率和跌落冲击速度是产品跌落破损的评价量;对于非线性包装系统,系统参数和无量纲跌落冲击速度是产品跌落破损的评价量。     

Damage boundary of a packaging system under rectangular pulse excitation

Newton's theory of an acceleration damage boundary (ADB) in packaging technology is based on an assumption that there is no restriction to the deformation of cushion materials. This assumption is in fact not true for a real packaging system since the compressible range of any cushion material is limited. In this paper, the concept of a displacement damage boundary (DDB) is introduced and combined with that of ADB to determine an actual damage boundary for a packaging system. The method is applied to linear and bilinear cushion materials, with damage boundaries obtained analytically. It is shown that the safe region enclosed by an actual damage boundary may be substantially smaller than that determined solely by ADB. © 1998 John Wiley & Sons, Ltd.     

Non‐local boundary integral formulation for softening damage

A strongly non‐local boundary element method (BEM) for structures with strain‐softening damage treated by an integral‐type operator is developed. A plasticity model with yield limit degradation is implemented in a boundary element program using the initial‐stress boundary element method with iterations in each load increment. Regularized integral representations and boundary integral equations are used to avoid the difficulties associated with numerical computation of singular integrals. A numerical example is solved to verify the physical correctness and efficiency of the proposed formulation. The example consists of a softening strip perforated by a circular hole, subjected to tension. The strain‐softening damage is described by a plasticity model with a negative hardening parameter. The local formulation is shown to exhibit spurious sensitivity to cell mesh refinements, localization of softening damage into a band of single‐cell width, and excessive dependence of energy dissipation on the cell size. By contrast, the results for the non‐local theory are shown to be free of these physically incorrect features. Compared to the classical non‐local finite element approach, an additional advantage is that the internal cells need to be introduced only within the small zone (or band) in which the strain‐softening damage tends to localize within the structure. Copyright © 2003 John Wiley & Sons, Ltd.     

Relaxation iterative algorithms for solving cathodic protection systems with non‐linear polarization curves

This paper discusses the calculation of potential distribution of impressed cathodic protection (CP) models with non‐linear polarization curves. We propose a relaxation iterative algorithm for the non‐linear problem and prove both theoretically and numerically that this iterative sequence is convergent for any physical polarization curves. This feature is of significant importance in developing a computer code for the design of CP systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Sensitivity and optimization for shape and non‐linear boundary conditions in thermal boundary elements

This paper is concerned with the minimization of functionals of the form ∫_Γ(b) f( h ,T( b, h )) dΓ( b ) where variation of the vector b modifies the shape of the domain Ω on which the potential problem, ?²T=0, is defined. The vector h is dependent on non‐linear boundary conditions that are defined on the boundary Γ. The method proposed is founded on the material derivative adjoint variable method traditionally used for shape optimization. Attention is restricted to problems where the shape of Γ is described by a boundary element mesh, where nodal co‐ordinates are used in the definition of b . Propositions are presented to show how design sensitivities for the modified functional ∫_Γ(b) f( h ,T ( b, h )) dΓ( b ) +∫_Ω(b) λ( b, h ) ?²T( b, h ) dΩ( b ) can be derived more readily with knowledge of the form of the adjoint function λ determined via non‐shape variations. The methods developed in the paper are applied to a problem in pressure die casting, where the objective is the determination of cooling channel shapes for optimum cooling. The results of the method are shown to be highly convergent. Copyright © 2002 John Wiley & Sons, Ltd.     

Test method for enhanced mechanical shock fragility statistics accuracy

An optimum cushioning package, which is neither excessive nor inadequate, must be designed to ensure cushioning performance that maintains an acceptable failure rate during transportation while also minimizing packaging costs. For this purpose, statistics pertaining to transport hazards and product shock strength must be engaged. The proposed study presents a test method to enhance the statistical accuracy of mechanical shock fragility of products. Sample statistics are invariably unknown; hence, optimum test‐setting values cannot be determined at the beginning. The proposed test method has been devised for determining optimum test‐setting values of the (n + 1)th sample using statistics of n samples being tested. An improvement in the estimation accuracy of the variation coefficient for the critical‐velocity‐change test was confirmed via simulations performed using the proposed method. Optimization of the test‐setting value has also been experimentally confirmed. A comparison of histograms and statistics obtained using experimental results has demonstrated that the proposed method can better estimate distribution shapes compared with the simple method. An example of the application of experimental results to stress‐strength models has also been described. The observed result has a considerable influence on the design of cushioning packages, thereby demonstrating effectiveness of the proposed method.     

Strategy for selecting representative points via tangent spheres in the probability density evolution method

A strategy of selecting efficient integration points via tangent spheres in the probability density evolution method (PDEM) for response analysis of non‐linear stochastic structures is studied. The PDEM is capable of capturing instantaneous probability density function of the stochastic dynamic responses. The strategy of selecting representative points is of importance to the accuracy and efficiency of the PDEM. In the present paper, the centers of equivalent non‐overlapping tangent spheres are used as the basis to construct a representative point set. An affine transformation is then conducted and a hypersphere sieving is imposed for spherically symmetric distributions. Construction procedures of centers of the tangent spheres are elaborated. The features of the point sets via tangent spheres, including the discrepancy and projection ratio, are observed and compared with some other typical point sets. The investigations show that the discrepancies of the point sets via tangent spheres are in the same order of magnitude as the point sets by the number theoretical method. In addition, it is observed that rotation transformation could greatly improve the projection ratios. Numerical examples show that the proposed method is accurate and efficient for situations involving up to four random variables. Copyright © 2007 John Wiley & Sons, Ltd.     

A sinh transformation for evaluating nearly singular boundary element integrals

An implementation of the boundary element method requires the accurate evaluation of many integrals. When the source point is far from the boundary element under consideration, a straightforward application of Gaussian quadrature suffices to evaluate such integrals. When the source point is on the element, the integrand becomes singular and accurate evaluation can be obtained using the same Gaussian points transformed under a polynomial transformation which has zero Jacobian at the singular point. A class of integrals which lies between these two extremes is that of ‘nearly singular’ integrals. Here, the source point is close to, but not on, the element and the integrand remains finite at all points. However, instead of remaining flat, the integrand develops a sharp peak as the source point moves closer to the element, thus rendering accurate evaluation of the integral difficult. This paper presents a transformation, based on the sinh function, which automatically takes into account the position of the projection of the source point onto the element, which we call the ‘nearly singular point’, and the distance from the source point to the element. The transformation again clusters the points towards the nearly singular point, but does not have a zero Jacobian. Implementation of the transformation is straightforward and could easily be included in existing boundary element method software. It is shown that, for the two‐dimensional boundary element method, several orders of magnitude improvement in relative error can be obtained using this transformation compared to a conventional implementation of Gaussian quadrature. Asymptotic estimates for the truncation errors are also quoted. Copyright © 2004 John Wiley & Sons, Ltd.     

A new hybrid method for solving linear–non‐linear systems of equations

This paper presents a new method for solving any combination of linear–non‐linear equations. The method is based on the separation of linear equations in terms of some selected variables from the non‐linear ones. The linear group is solved by means of any method suitable for the linear system. This operation needs no iteration. The non‐linear group, however, is solved by an iteration technique based on a new formula using the Taylor series expansion. The method has been described and demonstrated in several examples of analytical systems with very good results. The new method needs the initial approximations for non‐linear variables only. This requires far less computation than the Newton–Raphson method. The method also has a very good convergence rate. The proposed method is most beneficial for engineering systems that very often involve a large number of linear equations with limited number of non‐linear equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Predicting lower bound damage curves for high‐strength low‐alloy steels

Hindered by the distinctive toughness requirements of the current European standards, the high‐strength low‐alloy (HSLA) steels are rarely applied to the pressure vessels industry. The reason is that the design rules specified by the standards define local plastic deformation as limit state. This results in an over‐conservative application of materials. To achieve an effective, economical and energy‐efficient use of HSLA steels, a strain‐based criterion, the damage curve, which considers crack initiation instead of the beginning of plastic deformation as limit state, is proposed in this study for the improved design rules. In the view of the interaction of microstructure and mechanical properties of materials, the new design rule is derived on the basis of the correlation of microstructural features of HSLA steels with the micromechanical damage models. The experimental verification of the result is furthermore investigated with sufficient agreement so that the general applicability of the procedure can be expected. However, further studies for a reliable parameter calibration are necessary.     

A sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals

A new transformation technique is introduced for evaluating the two‐dimensional nearly singular integrals, which arise in the solution of Laplace's equation in three dimensions, using the boundary element method, when the source point is very close to the element of integration. The integrals are evaluated using (in a product fashion) a transformation which has recently been used to evaluate one‐dimensional near singular integrals. This sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element. The method is straightforward to implement and, when it is compared with a number of existing techniques for evaluating two‐dimensional near singular integrals, it is found that the sinh method is superior to the existing methods considered, both for potential integrals across the full range of b values considered (0<b?10), and for flux integrals where b>0.01. For smaller values of b, the use of the Lmethod is recommended for flux integrals. Copyright © 2006 John Wiley & Sons, Ltd.     

On non‐linear transformations for accurate numerical evaluation of weakly singular boundary integrals

This paper presents a study of the performance of the non‐linear co‐ordinate transformations in the numerical integration of weakly singular boundary integrals. A comparison of the smoothing property, numerical convergence and accuracy of the available non‐linear polynomial transformations is presented for two‐dimensional problems. Effectiveness of generalized transformations valid for any type and location of singularity has been investigated. It is found that weakly singular integrals are more efficiently handled with transformations valid for end‐point singularities by partitioning the element at the singular point. Further, transformations which are excellent for CPV integrals are not as accurate for weakly singular integrals. Connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated; cubic transformation is seen to be the optimum choice for single precision, and quartic or quintic one, for double precision computations. A new approach which combines the method of singularity subtraction with non‐linear transformation has been proposed. This composite approach is found to be more accurate, efficient and robust than the singularity subtraction method and the non‐linear transformation methods. Copyright © 2001 John Wiley & Sons, Ltd.     

A Generalised Conjugate Residual method for the solution of non‐symmetric systems of equations with multiple right‐hand sides

This paper presents an iterative algorithm for solving non‐symmetric systems of equations with multiple right‐hand sides. The algorithm is an extension of the Generalised Conjugate Residual method (GCR) and combines the advantages of a direct solver with those of an iterative solver: it does not have to restart from scratch for every right‐hand side, it tends to require less memory than a direct solver, and it can be implemented efficiently on a parallel computer. We will show that the extended GCR algorithm can be competitive with a direct solver when running on a single processor. We will also show that the algorithm performs well on a Cray T3E parallel computer. Copyright © 1999 John Wiley & Sons, Ltd.