Scaling impacted structures when the prototype and the model are made of different materials

By properly applying scaling laws, it is possible to infer the behaviour of a structure from the response of a similar model whose dimensions are scaled by a factor β. In some cases, however, e.g. in the case of strain rate sensitive structures under severe dynamic loads, these laws become distorted, severely limiting this approach. In this article, a methodology for the correction of this distortion is explored for the case when the structure and the model are made of different materials. It is shown that the behaviour of a structure, say, made of mild steel, can be forecast from the response of a model, say, made of aluminium. The technique here detailed is shown to be valid for simple structures subject to axial and transverse impact loads.     

High-Order Vibrations’ Dynamic Scaling Laws of Distorted Scaled Models of Thin-Walled Short Cylindrical Shells

In this study, we investigate the dynamic scaling laws of geometrically distorted models for predicting dynamic characteristics of thin-walled short cylindrical shells. Approximate and accurate analysis methods for obtaining the scaling laws are introduced. Two coefficient functions are established in deriving high-order scaling laws for a narrow range distorted model. Then a modified function is obtained by using numerical analysis, in order to modify the errors of wide range distorted models. The general form of the high-order scaling laws of thin-walled short cylindrical shells is also developed. To be practical, a process of detecting scaling laws by experimental operation is summarized, and the applicability of scaling laws is validated by using experimental data. Although there are some limitations in practice, the scaling laws of the thin-walled short cylindrical shell still have the ability to predict the prototype with good accuracy.     

受轴向冲击薄壁圆管的几何畸变相似律研究

对于受轴向冲击载荷作用的薄壁圆管动态响应的相似律问题,由于圆管的薄壁特性导致厚度无法与高度和半径按相同的比例进行结构缩放,从而产生模型的几何畸变,此时传统的相似律已无法描述原型与畸变模型之间的动态响应规律。基于薄壁圆管轴向冲击问题的控制方程,通过能量守恒和量纲分析,推导了考虑几何畸变条件下轴向冲击载荷作用的理想弹塑性薄壁圆管动态响应的相似律。通过在给定应变与应变率区间上建立比例模型预测的流动屈服应力与原型流动屈服应力的最佳逼近关系,将几何畸变相似律进一步推广至包含应变率和应变硬化的材料。通过数值方法验证了提出的几何畸变模型相似律的适用性。分析结果表明,提出的考虑厚度畸变的受轴向冲击薄壁圆管的相似律可用于预测原型结构的冲击动态响应,并显著降低比例模型与原型结构平均载荷和能量的偏差。     

轮胎破片冲击机身的非等比例相似模型研究

为降低机身结构抗冲击性能的实验成本,利用相似理论建立机身的非等比例缩放模型,开展模型实验是行之有效的方法。基于量纲分析的方法,建立Johnson-Cook线性应变率函数的修正关系;鉴于生产制造技术的限制,考虑扭曲厚度的非等比例机身模型对相似性行为的影响,采用指数函数法建立了非等比例模型的相似修正关系。通过对比实验中破片冲击过程的变形形态、靶板的应变时间历程曲线和最终变形轮廓,验证了数值模型的有效性。此外,分析了破片偏航姿态、机身材料、厚度和质量等因素对机身结构抗冲击性能的影响。结果表明:(1) 150 m/s的冲击速度下,破片冲击角度90o和着靶角度180o是最严苛的冲击条件。综合多种因素,分析认为3.5 mm厚的钛合金为机身结构的最佳选择,并以此作为全尺寸原型验证相似模型;另外,提出了一种可以快速获取缩比模型的设计方法。(2)应变率效应对轮胎破片冲击机身结构的影响并不显著,等比例缩放模型与原型结果吻合较好。(3)厚度扭曲的非等比例模型能够有效地预测原型结构的变形行为;虽然,在时间尺度上,模型与原型存在一定的偏差;但是,在空间尺度上,非等比例相似模型能够有效地修正扭曲厚度造成中心最大挠度的预测误差,修正后的最大误差不超过5.1%,这表明该方法能够有效地指导机身结构的相似模型设计。     

Scaling of structures subject to impact loads when using a power law constitutive equation

It is well known that structures subjected to dynamic loads do not follow the usual similarity laws when the material is strain rate sensitive. As a consequence, it is not possible to use a scaled model to predict the prototype behaviour. In the present study, this problem is overcome by changing the impact velocity so that the model behaves exactly as the prototype. This exact solution is generated thanks to the use of an exponential constitutive law to infer the dynamic flow stress. Furthermore, it is shown that the adopted procedure does not rely on any previous knowledge of the structure response. Three analytical models are used to analyze the performance of the technique. It is shown that perfect similarity is achieved, regardless of the magnitude of the scaling factor. For the class of material used, the solution outlined has long been sought, inasmuch as it allows perfect similarity for strain rate sensitive structures subject to impact loads.     

结构冲击畸变问题的直接相似方法研究

结构受冲击时由不同材料引起比例模型与全尺寸原型的畸变, 通常通过修正比例模型的速度或密度进行补偿. 然而传统的修正方法, 存在需预先测试结构响应、依赖特殊本构方程、不能反映动态过程的缺陷, 因而限制了相似理论在比例模型试验中的直接应用. 本文提出了一种不同材料畸变问题的直接相似方法. 它通过建立应变率区间上比例模型预测的流动屈服应力与原型流动屈服应力的最佳逼近关系, 直接获得了修正速度或修正密度的比例因子, 完成了比例模型与原型的动态相似关系. 基于Norton-Hoff, Cowper-Symonds, Johnson-Cook三种经典的本构模型, 研究了材料应变率敏感性特征参数、参考应变率、屈服应力、密度在动态相似关系中的作用. 并通过受质量冲击的折板结构算例, 验证了直接相似方法的有效性. 分析表明, 本文提出的直接相似方法不需要预先测试结构的响应, 不依赖特殊的本构方程、强调动态相似特性, 具有直接、高效、通用的特点. 此外, 动态相似关系的最佳逼近效果, 受材料应变率敏感性特征参数控制, 屈服应力、密度和参考应变率影响不大; 当比例模型应变率敏感性特征参数与原型相近, 可获得最佳逼近.      

Prediction of vibration characteristics of blisks using similitude models

This study investigates the determination approach of distorted scaling laws for predicting the dynamic characteristic of an aero engine’s blisk. Based on the dynamic scaling laws of typical thin-walled structures, an assumption of geometrically complete scaling laws is firstly proposed and numerically validated. For distorted models of disk thickness, in order to simplify the design procedure, a simplification condition is proposed and applied to the first 10 orders’ distorted scaling laws (blade-dominated vibrations) by combining sensitivity analysis. Next, the 11th–14th orders’ distorted scaling laws are determined for disk-dominated vibrations. Numerical validation demonstrates that distorted scaling laws possess a good accuracy. Finally, the applicability of these new scaling laws is validated by the experimental data. The results indicate that, by using the new scaling laws, the simple models can predict vibration characteristics of blisk by employing similitude models.     

薄板在冲击载荷下线弹性理想塑性响应的相似性研究

对于比例模型和原型使用不同弹塑性材料的冲击相似性问题,由于弹性和塑性阶段材料特性的差别及其在不同变形阶段的弹性塑性共存现象,将导致原有的结构冲击相似性理论失效。基于薄板冲击问题的理论模型,采用方程分析方法重新推导了材料线弹性以及理想刚塑性共存时的冲击响应的相似律。提出了一种能够同时考虑弹性变形和塑性变形的结构缩放响应相似性分析的厚度补偿方法,并利用数值分析验证了提出方法的适用性。分析结果表明,使用厚度补偿方法得到的比例模型结构响应能够准确地预测原型结构的冲击响应。     

弹体侵彻混凝土靶体的尺寸效应分析

由于混凝土靶体抗刚性弹侵彻实验大多基于缩比弹体展开,侵彻深度相似律是否成立显得尤为重要。在侵彻相似模型基础上,综合分析已有侵彻实验数据及经验公式,发现侵彻深度在通常情况下存在尺寸效应,且无量纲侵彻深度随弹体尺寸变大而增大。但如果模型以及原型实验中弹体与混凝土靶体（包括粗骨料）严格等比例设计,侵彻深度相似律是成立的。不变的骨料特征（粗骨料未随弹体尺寸缩放）是引起侵彻实验以及侵彻经验公式中尺寸效应的主要原因。为研究由粗骨料引起的侵彻尺寸效应,开发了混凝土二维细观有限元建模程序,细观数值实验成功地反映出了侵彻尺寸效应,考虑模拟尺寸效应后的侵彻公式能较好地预测不同尺寸侵彻实验。     

模糊桁架结构在模糊激励下的动力响应分析

研究了模糊参数桁架结构在模糊荷载激励下的动力响应分析问题. 不仅模糊桁架结构的物 理参数、几何尺寸具有模糊性,而且外载荷幅值也具有模糊性时,从Duhamel积分式出发,利用模糊因子法、区间运算和振型叠加法求出了结构动力响应模糊变量的表达式. 最后通过算例考察了结构参数和作用荷载的模糊性对结构动力响应的影响,表明所提出的计算模型和 分析方法是正确与可行的.     

远距离爆炸荷载作用下钢框架几何相似律研究

根据Π定理推导了远距离爆炸荷载作用下钢框架原型结构与缩比模型的几何相似律表达式。基于已有的钢框架子结构爆炸实验,采用AUTODYN建立了钢框架子结构数值模型,验证了流固耦合方法在结构爆炸响应分析中的可靠性。在此基础上,对比了流固耦合方法和解析爆炸边界方法在钢框架远距离爆炸数值模拟中的准确性和计算效率,结果表明,解析爆炸边界方法可以合理地模拟远距离爆炸荷载作用下钢框架的动态响应,且计算效率较高。最后,采用该方法分析了具有不同相似比的两层三跨钢框架结构在远距离爆炸荷载作用下的动态响应及毁伤效应,结果表明:该结构的动态响应和毁伤效应符合几何相似规律。     

DYNAMIC CHARACTERISTIC ANALYSIS OF FUZZY- STOCHASTIC TRUSS STRUCTURES BASED ON FUZZY FACTOR METHOD AND RANDOM FACTOR METHOD

A new fuzzy stochastic finite element method based on the fuzzy factor method and random factor method is given and the analysis of structural dynamic characteristic for fuzzy stochastic truss structures is presented. Considering the fuzzy randomness of the structural physical parameters and geometric dimensions simultaneously, the structural stiffness and mass matrices are constructed based on the fuzzy factor method and random factor method; from the Rayleigh's quotient of structural vibration, the structural fuzzy random dynamic characteristic is obtained by means of the interval arithmetic; the fuzzy numeric characteristics of dynamic characteristic are then derived by using the random variable's moment function method and algebra synthesis method. Two examples are used to illustrate the validity and rationality of the method given. The advantage of this method is that the effect of the fuzzy randomness of one of the structural parameters on the fuzzy randomness of the dynamic characteristic can be reflected expediently and objectively.     

A general method for impact dynamic analysis of a planar multi-body system with a rolling ball bearing joint

Impact affects the dynamic characteristics of mechanical multi-body systems and damages those rotating parts, such as the joint rolling element bearings, which are high-precision, defect intolerant components. Based on multi-body dynamic theory, Hertzian contact theory, and a continuous contact model, this study proposed a modelling method that can describe the dynamic behaviour of planar mechanical multi-body systems containing a rolling ball bearing joint under impact. In this method, the rigid bodies and bearing joint were connected according to their joint force constraints; the impact constraint between the multi-body system and the target rigid body was constructed using a continuous contact force model. Based on this method, the reflection relationship between the external impacts of the mechanical multi-body system and the variation law governing the dynamic load on the rolling bearing joint were revealed. Subsequently, an impact multi-body system, which was composed of a sliding–crank mechanism containing a rolling ball bearing joint and the target rigid body with an elastic support, was analysed to explore the dynamic response of such a complex discontinuous dynamic system andthe relevant relationship governing the dynamic load on the rolling bearing joint. In addition, a multi-body dynamic simulation software was used to build a virtual prototype of the impact slider–crank system. Compared with the theoretical model, the prototype had an additional deep groove ball bearing. That is to say, the prototype model took account of the specific geometric structural characteristics and the complex contact relationship of the inner and outer races, rolling balls, and bearing cage. Finally, the effectiveness of the theoretical method proposed in this study was verified by comparative analysis of the results. The results suggested that the external impact of a mechanical multi-body system was prone to induce sudden changes in the equivalent reaction force on its bearing joint and the dynamic load carried on its rolling balls. This study provided an effective method for exploring the distribution characteristics of dynamic loads on rolling ball bearing joints under working impact load conditions. Moreover, it offered support for the parameter optimisation of geometric structure, performance evaluation, and dynamic design of the rolling ball bearings.     

平稳随机激励下随机桁架结构动力可靠性分析

研究了随机桁架结构在平稳随机激励下的动力可靠度求解方法。同时考虑结构物理参数、几何尺寸的随机性,利用求解随机变量数字特征的代数综合法和矩法,从结构随机响应的频域表达式出发,导出了随机桁架结构在平稳随机激励下的位移、应力及其导数响应均方值的数字特征,再由动力可靠性的Poisson公式导出了结构动力可靠度的计算公式。通过算例分析了各参数的随机性对结构动力可靠度的影响,验证了该方法的可行性和有效性。     

Dynamic limit analysis formulation for impact simulation of structural members

This paper introduces an extended concept of limit analysis to deal with the dynamic equilibrium condition considering the inertia and strain-rate effect for dynamic behavior of structures. The conventional limit analysis method has been applied to only static collapse analysis of structures without consideration of dynamic effects in the structural behavior. A dynamic formulation for the limit analysis has been derived for incremental analysis dealing with time integration, strain and stress evaluation, strain hardening, strain-rate hardening and thermal softening. The time dependent term in the governing equation is integrated with the WBZ-α method. The dynamic material behavior is described by the Johnson–Cook model in order to consider strain-rate hardening and thermal softening as well as strain hardening. Simulations have been carried out for impact analysis of a Taylor bar and an S-rail and their numerical results are compared with elasto-plastic explicit analysis results by LS-DYNA3D. Comparison demonstrates that the dynamic finite element limit analysis can predict the crashworthiness of structural members effectively with less effort and computing time than the commercial code compared. The crashworthiness of a structure with the rate-dependent constitutive model is also compared to that with the quasi-static constitutive relation in order to investigate the dynamic effect on deformation of structures.     

Scaling impacted structures

Summary The problem of non-scalability of structures under impact loads caused by strain-rate effects is solved in this article by properly correcting the impact velocity. The technique relies on the use of an alternative dimensionless basis, together with a mathematical model which allows the calculation of a correction factor for the impact velocity. This new velocity, when applied to the model, makes it to assure the satisfaction of the scaling laws. The indirect similitude method detailed here is applied to two strain-rate sensitive structures, a double plate under in-plane impact and a beam subjected to a blast load. The results show a very good agreement so that the model and a prototype made from strain rate sensitive materials behave the same.accepted for publication 3 March 2004     

区间桁架结构动力特性分析的区间因子法

对于具有区间参数的桁架结构的动力特性分析问题,提出了一种新的分析方法即区间因子法。利用区间因子法,结构材料物理参数、几何尺寸均可表达为其区间因子和确定性量的乘积,进而结构的固有频率和振型也可显式表达成区间因子们的函数。利用区间算法,推导出了结构固有频率和振型的上、下限与均值的计算表达式。通过算例,分析了结构参数的不确定性对结构动力特性的影响,并验证了本文模型和方法的合理性与可行性。本文方法的优点是能够反映结构某一参数的不确定性对结构动力特性的影响。     

Meshfree modeling of dynamic response of mechanical structures

Transient nature of the loading conditions applied to the structural components makes dynamic analysis one of the important components in the design-analysis cycle. Time-varying forces and accelerations can substantially change stress distributions and cause a premature failure of the mechanical structures. In addition, it is also important to determine dynamic response of the structural elements to the frequency of the applied loads. In this paper we describe the first application of the meshfree Solution Structure Method (SSM) to the structural dynamics problems. SSM is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure. In the proposed approach we will use SSM to enforce initial and boundary conditions, while transient behavior is enforced by time marching schemes used to solve systems of ordinary differential equations. In the paper we will explain the key points of the SSM as well as investigate the accuracy and convergence of the proposed approach by comparing our results with the ones obtained using traditional finite element analysis. Despite in this paper we are dealing with 2D in-plane vibrations, the proposed approach has a straightforward application to model vibrations of 3D structures.     

Nonlinear modeling of tuned liquid dampers (TLDs) in rotating wind turbine blades for damping edgewise vibrations

Tuned liquid dampers (TLDs) utilize the sloshing motion of the fluid to suppress structural vibrations and become a natural candidate for damping vibrations in rotating wind turbine blades. The centrifugal acceleration at the tip of a wind turbine blade can reach a magnitude of 7–8g. This facilitates the use of a TLD with a relatively small fluid mass and with feasible geometric dimensions to mitigate the lightly-damped edgewise vibrations effectively. In the present paper, modal expansions are carried out directly on the velocity field and the free surface of the sloshing liquid in the rotating coordinate system. A formulation has been proposed leading to coupled nonlinear ordinary differential equations, which have been obtained through the Galerkin variational approach together with the modal expansion technique. Two models, with one sloshing mode and three sloshing modes, have been studied in the numerical simulation. It is shown that the one-mode model is able to predict the sloshing force and the damped structural response accurately, since the primary damping effect on the structure is achieved by the first sloshing mode of the fluid. Although it is unable to predict the fluid free-surface elevation equally well, the one-mode model can still be utilized for the design of TLD. Parametric optimization of the TLD is carried out based on the one-mode model, and the optimized damper effectively improves the dynamic response of wind turbine blades.     

Application of the NSCD method to analyse the dynamic behaviour of stone arched structures

In this paper, a masonry arch is simulated in order to assess both its structural and seismic vulnerability. The non-smooth contact dynamics (NSCD) computational method is used to simulate this type of structure as a collection of bodies under the hypothesis of unilateral constraints and frictional contact, with or without cohesion. Sinusoidal oscillations in three dimensions and real earthquake data have been applied to the supporting base element of the arch model.The primary aim of this study is to better understand the dynamic behaviour of the masonry arch, a typical component of historic unreinforced masonry (URM) structures. This study also assesses the influences of the input parameters on the mechanical and dynamic behaviour of the arch structure. Its collapse mechanism is studied for both cohesive and non-cohesive contact.In addition, we examine the behaviour under seismic loading of the Arles aqueduct, a real historical arched structure located in the south-east of France. Significant information can be obtained from the comparison of the results of advanced numerical analysis, taking into account the precise geometry of the model, the mechanical characteristics of the materials and the observation of the in situ monuments after their collapse.