Indirect Inverse Substructuring Theory for Coupling Dynamic Stiffness Identification of Complex Interface Between Packaged Product and Vehicle Transport System

Inverse substructuring method has been recently proposed and applied for inverse analysis of the dynamical response of product transport system. The component‐level frequency response functions (FRFs) and the coupling dynamic stiffness for facilitating the cushioning packaging design are all predicted from only the system‐level FRFs. However, the system‐level FRFs from coupling degree of freedoms may not be measured accurately because of the difficulties of vibration excitation and response measurement for the coupled interface between packaged product and vehicle within the limited accessible space. The aim of this paper is to develop a new FRF‐based indirect inverse substructuring method for the analysis of the dynamic characteristics of a three‐substructure coupled product transport system without measuring system‐level FRFs at the coupling degree of freedoms. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed‐form analytical solution to inverse sub‐structuring analysis of multi‐substructure coupled product transport system is derived based on the relationship of easy‐to‐monitor component‐level FRFs and the system‐level FRFs at the coupling coordinates.. The proposed method is validated by a lumped mass‐spring‐damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. Then, the FRF tests of a physical prototype of multi‐substructure coupled product transport system are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The proposed method may help to obtain the main controlling factors and contributions from the various structure‐borne paths for product transport system. Copyright © 2014 John Wiley & Sons, Ltd.     

Investigation of Frequency Response Function of Product‐Transport System Based on Multi‐coordinate Coupled Inverse Substructure Method

The product and vehicle, through a packaging, constitute a complex product‐transport system in logistics. It is very important to obtain the frequency response functions of the product‐transport system and its substructures for the design of product packaging. In this paper, the product‐transport system is treated as a two‐substructures multi‐coordinate coupled system. It is composed of a product substructure and a vehicle substructure, which are connected by a packaging structure consisting of many packaging units. The multi‐coordinate coupled inverse substructure method is developed and used to analyse the dynamic characteristics of the product‐transport system. To verify the validity of this method for the product‐transport system, the experiment of a physical prototype is conducted. The results show that the predicted substructure‐level frequency response functions are in accordance with the measured. Copyright © 2013 John Wiley & Sons, Ltd.     

Inverse Sub‐Structuring Theory for Coupled Product Transport System Based On the Dummy Masses Method

It is of high importance to predict the components frequency response functions (FRFs) for obtaining the coupled product transport system's response. However, the components behaves much differently when coupled with another components compared with that in free state. Inverse sub‐structuring method has been recently proposed and applied for inverse analysis of the dynamical response of coupled product transport system. The component‐level FRFs and the coupling dynamic stiffness are all predicted from only the system‐level FRFs, facilitating the engineering design for product transport system. However, in most engineering application practices, the system‐level FRFs from coupling degrees of freedom may not be measured accurately because of the difficulties of vibration excitation and/or response measurement for the coupled interface between components within the limited accessible space. The aim of this paper is to develop a new FRF‐based indirect inverse sub‐structuring method for the analysis of the dynamic characteristics of a two‐component coupled product transport system without measuring system‐level FRFs at the coupling degrees of freedom. A so‐called dummy masses method is developed and applied for predicting the unmeasured FRFs at the coupling degrees of freedom, and the inverse sub‐structuring approach based on the dummy mass method is derived for inverse analysis of coupled product transport system, which is further verified by a lumped‐mass model, showing exact agreement. Finally, the experiment on a physical prototype of two‐substructure coupled product transport system is performed to further check the accuracy of the suggested method. The new method shows its great application prospect in coupled product transport system.     

Dynamic Characteristic Analysis of Refrigerator‐Truck Transport System by Using Inverse Substructure Method

This paper is a continuation of the previous research. The refrigerator‐truck system is treated as a two‐substructure multi‐coordinate coupled system, which is composed of a refrigerator substructure and a truck substructure coupled by numerous packaging units. Applying the two‐substructure multi‐coordinate coupled inverse substructure method to a product‐transport system, the paper makes an analysis of the dynamic characteristics for the refrigerator‐truck system. In order to validate the method in the refrigerator‐truck system, the measured system‐level frequency response functions (FRFs) were used to predict the substructure‐level FRFs that were compared with those directly measured and found to be in agreement in trend. To evaluate the vibration transmissibility of packaging units, the concepts of the vibration contribution, vibration contribution ratio, overall vibration contribution and overall vibration contribution ratio to product response through each packaging unit are introduced. The vibration contributions to the refrigerator through four coupling points almost coincide with each other in the whole frequency domain. However, the vibration contribution ratios are different at each frequency. The overall vibration contributions to the refrigerator through the frequency domain 10 to 50 Hz through four coupling points are respectively 9.4, 7.5, 9.8 and 11.2 mm/N, and the overall vibration contribution ratios are respectively 0.25, 0.20, 0.26 and 0.29. The vibration transmissibilities of four coupling points are similar. Copyright © 2014 John Wiley & Sons, Ltd.     

Inverse Sub‐structuring Method for Multi‐coordinate Rigidly Coupled Product Transport System based on a Novel Shearing Probe Technique

The inverse sub‐structuring method can predict the component‐level frequency response functions (FRFs) of product (critical component) for product transport system from only measured system‐level FRFs, facilitating the cushioning packaging design. However, the FRFs of the coupling interface between product and vehicle are usually of extreme difficulty to be measured due to the limited accessible space. To overcome this difficulty, the authors suggested a so‐called FRF probe technique method in the previous study, which may be more suitable for the single‐coordinate coupled system. In practice, most of the product transport systems should be treated as multi‐coordinate coupled system. The aim of this paper is to derive a new FRF‐based inverse sub‐structuring method for multi‐coordinate rigidly coupled product transport system and develop a new shearing probe technique to obtain the difficult‐to‐monitor FRFs at the coupling interface, which will be validated by a lumped mass model and finite element models, respectively, showing perfect agreement. Finally, the experiment on a physical prototype of multi‐coordinate rigidly coupled product transport system is performed to further check the feasibility of the application prospect of the shearing probe technique for inverse analysis of product transport system. The method proposed in this study will provide the packaging designers an alternative method to monitor the integrity of product transport system. Copyright © 2017 John Wiley & Sons, Ltd.     

Inverse Sub‐structuring Method for Rigidly Coupled Product Transport System based on Frequency Response Function Testing Probe Technique

The inverse sub‐structuring method has been recently proposed and applied for inverse analysis of product transport system, to predict the component‐level frequency response functions (FRFs) and the coupling dynamic stiffness from only the system‐level FRFs. However, previous applications of this method were all developed based on the assumption that the components were coupled by flexible couplings. Actually, increasing more components are welded or bolted to construct a coupled system, which should be treated as rigidly coupled system. The aim of this paper is to derive a new FRF‐based inverse sub‐structuring method for the analysis of the dynamic characteristics of a two‐component coupled product transport system with rigid couplings. And then a so‐called FRF testing probe technique is proposed and applied to measure the difficult‐to‐monitor FRFs at the coupling interface. The developed method is verified by a lumped‐mass model, showing exact agreement. Finally, the experiment on a physical prototype of two‐substructure coupled product transport system is performed to further check the accuracy of the suggested method. The proposed method is an extension of previous inverse sub‐structuring method and may help to obtain the main controlling factors and contributions from the various structure‐borne paths for product transport system. Copyright © 2016 John Wiley & Sons, Ltd.     

Inverse Sub‐Structuring Method for Multi‐Coordinate Coupled Product Transport System

A new high‐accuracy transfer function is selected, and an inverse sub‐structuring method is developed for the analysis of the dynamic characteristics of a three‐sub‐structure coupled product transport system. The closed‐form analytical solution to inverse sub‐structuring analysis of multi‐coordinate coupled multi‐ sub‐structure product transport system is derived. The proposed method is validated by a lumped mass spring damper model; the predicted frequency response functions (FRFs) of sub‐structures and the coupling stiffness, in addition to the most concerned system‐level FRF, are compared with the direct computations, showing exact agreement. Then, FRF tests of a physical prototype of the multi‐coordinate coupled product transport system with three sub‐structures are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown sub‐structure‐level FRFs and coupling stiffness purely from system‐level FRFs. The suggested method may help obtain the main controlling factors and contributions from the various structure‐borne paths for the product transport system, which may certainly facilitate the cushioning packaging design. Copyright © 2013 John Wiley & Sons, Ltd.     

一种针对含非线性子结构多点耦合运输包装系统的解耦方法

目的 考虑到运输包装系统耦合形式复杂,包装材料及包装结构具有非线性特性,不容易测量局部物理参数,需要对传统逆向子结构方法进行优化,使之能够求解非线性多点耦合系统中子结构的动态响应特性。方法 使用描述函数法将非线性的运输包装系统线性化,测量其在若干特定振动幅值下的频率响应函数;之后,应用逆向子结构方法和参数识别方法,计算包装件的模态参数;最后,拟合包装件模态参数与振动幅值之间的关系,构建函数来描述包装件的动态响应特性。结果 在集总参数模型中,解耦预测值与实际值吻合;在有限元模型中,对响应峰值的预测误差小于5%,对响应跳跃现象所在频率的预测误差小于3%。结论 该研究将传统逆向子结构方法的应用范围拓展到了非线性多点耦合系统,对复杂运输包装系统动力学模型的构建和防振包装的设计具有指导意义。     

基于逆子理论的洗衣机运输系统动态特性分析

目的获取产品运输包装系统各部件的动态特性。方法采用多级系统分布解耦法,结合二级刚性耦合系统逆子结构理论,推导由产品、车辆部件水平和系统水平频响传函预测关键部件频响传函的理论公式。搭建电机-洗衣机-车辆三级刚柔耦合运输系统,对理论方法进行验证。结果通过在线实验验证,基于理论预测得到的关键部件频响传函和测试值相吻合。结论研究结果为产品运输包装设计和优化提供了参考依据。     

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.     

Multilevel dynamic substructures

The dynamic substructure method is extended to multilevel (recursive) substructures. The obvious distinction of the two approaches is that the stiffness and mass matrices before condensation are no longer frequency independent. The dynamic stiffness matrix at any substructure level is proved to be a function of the vibrating frequency in terms of some constant matrices which are derivable from the dynamic stiffness matrix at one lower substructure level. The method can accurately predict more modes than the number of degrees of freedom retained. The computational procedure, the generalized inverse iteration, the stationary principle of the system natural frequency and the generalized Rayleigh's quotient are derived for the frequency dependent matrices. Numerical examples are given to illustrate some engineering applications. A transcendental dynamic stiffness matrix can be transformed to a more convenient algebraic form by the present method.     

车内低频噪声与悬架特性参数的定量关系

基于车身乘坐室声振耦合的动态子结构修改方法，将汽车悬架系统视为附加于车身上的子结构（子系统），并结合悬架系统对路面不平度位移激励的振动传递效应，揭示出车内低频噪声的声压值与悬架系统刚度、阻尼、非悬挂质量以及轮胎径向刚度、径向阻尼问的直接定量关系。然后，通过算例及相应的实验验证了其正确性.     

离散化包装耦合体动刚度的逆子结构计算方法

提供离散化包装耦合体的动刚度一种间接的逆子结构计算方法,依据矩阵理论建立复杂包装耦合体等效离散化后动态刚度逆子结构分析计算方法及其普适性公式。应用典型运输包装系统的集总参数模型验证公式的完备有效性,并计算源函数（频率响应函数）误差的影响。     

考虑易损件的包装系统在半正弦波激励下的冲击响应

把运输包装系统简化为二自由度线性系统,运用数值求解法对具有线弹性材料的缓冲包装系统进行研究,得到了在半正弦激励下,易损件及主体的冲击响应谱,并讨论了频率比、质量比、阻尼比及冲击波幅值的影响;该方法和结论对缓冲包装设计具有重要意义.     

获取界面响应不可测数据的频响探针关键技术研究

目的针对传统逆子结构理论在求解过程中界面响应难以实测的问题,提出一种利用频响探针技术来获取该界面响应的关键技术。方法首先基于该频响探针的动力学微分方程,从理论上推导了该频响探针技术的理论公式,然后对建立的二级单点刚性耦合系统进行了有限元数值验证,将利用频响探针技术预测得到的难测原点频响函数与有限元计算值进行比较,并将该预测值代入逆子结构理论公式中,得到了部件频响函数的预测值,将该预测值与有限元计算值进行了对比验证。结果预测值与有限元计算值高度吻合,验证了该理论的准确性。结论该频响探针关键技术在获取界面响应不可测数据方面,有很好的应用价值。     

The dynamic behaviour of stacked shipping units during transport. Part 1: model validation

This paper deals with the dynamic behaviour of stacked packaging units when subjected to vertical vibrational inputs as experienced in transport vehicles. Although the vibrational performance of single‐unit packaging systems has been thoroughly studied, the behaviour of stacked packaging units is not fully understood. The complexity of the problem is compounded when the effects of vertical restraints are taken into account. The paper presents the development of a numerical computer model designed to predict the dynamic response of stacked package systems when subjected to vertical vibrational excitation. Provisions have been made to account for the effects of vertical restraint tension and stiffness. In addition, a physical model representative of a generic stacked packaging system has been developed to assist in validating the numerical model. The paper includes results from preliminary experiments in which the frequency response functions of the models were evaluated and compared. The validity of the numerical model in the time domain was tested using random burst excitation signals. These preliminary experiments reveal that, when the effects of frictional damping are taken into account, the numerical model can be used to generate reasonably accurate predications of the dynamic behaviour of the equivalent physical system. Copyright © 2004 John Wiley & Sons, Ltd.     

逆子结构理论在运输包装中的研究进展

微动态逆子结构方法在解决由产品、包装和运载体(如车辆)组成的产品运输包装系统的相关问题上效果显著。综述了运输包装中逆子结构理论的发展与完善及其在其他方面的应用成果、测量信息不完备下的逆子结构分析以及间接逆子结构分析。逆子结构理论的发展为了解运输过程中产品的动态特性提供了新的方法,也为今后运输包装的进一步发展提供了参考。     

地震作用下水-轴对称柱体相互作用的子结构分析方法

针对水-轴对称柱体动力相互作用问题,提出了一种地震作用下水-结构相互作用的时域子结构分析方法.基于三维不可压缩水体的波动方程和边界条件,利用分离变量法将其转换为环向解析、竖向和径向数值的二维模型;基于比例边界有限元推导了截断边界处无限域水体的动力刚度方程,并将水体内域有限元方程和人工边界处的动水压力进行耦合,从而得到结...     

具有弹性耦合结构振动系统的自由界面模态综合法

本文给出了以弹性连结件为耦合件的振动系统自由界面模态综合技术,把弹性连结件定义为“软子结构”单独处理,其余部件自然划分为若干子结构。利用有限单元法获取各子结构自由界面模态信息,用子结构的截断主模态及其高阶剩余柔度修正项作为结构 RayLeigh-Ritz 分析的假设模态,从而大幅度压缩了结构分析自由度。实例计算表明,该方法具有良好的综合精度,是分析具有弹性耦合结构振动问题的有效方法。     

Monitoring the Condition of Non‐linear Packaging Materials Subject to Varying Excitation Levels Using a Reverse Multiple Input/Single Output Based Approach

During transportation, protective packaging is subjected to random dynamic compressive loads that arise from random vibrations generated by the vehicle. The ability of the protective packaging to withstand these dynamic compressive loads depends on the environmental vibration levels, the nominal stresses and the material's characteristics. Previous research has shown that cumulative damage, in the packaging system under random dynamic compression, will result in a change in the overall stiffness of the system. This change is manifested as a shift in the system's fundamental resonant frequency. Natural frequency estimates are often extracted using a least squares regression curve fit applied to an estimate of the system's frequency response function. Frequency response function estimates are generally obtained using the Fourier transform with a single input/single output (SISO). This approach is suitable for many applications; however, it is not well suited to non‐linear systems subjected to non‐stationary excitation where the vibration level (overall root‐mean‐square value) can vary. This paper investigates the use of an optimised reverse multiple input/single output algorithm for reliably tracking variations in the condition of packaging elements subjected to excitation with varying magnitude (root‐mean‐square). Results are presented from the analysis of physical experiments performed on expanded polystyrene cushions as well as empty corrugated paperboard containers. The experiments performed using the polystyrene samples were designed to limit natural variation in the system's natural frequency; whereas the paperboard samples were allowed to naturally damage under dynamic loading. Copyright © 2015 John Wiley & Sons, Ltd.