A Nonlinear Model of a Slack Cable With Bending Stiffness and Moving Ends With Application

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negative eigenvalues

Understanding negative eigenvaluesSometimes, negative eigenvalues are reported in an eigenvalue buckling analysis. In most cases such negative eigenvalues indicate that the structure would buckle if the load were applied in the opposite direction. A classical example is a plate under shear loading; the plate will buckle at the same value for positive and negative applied shear load. Buckling under reverse loading can also occur in situations where it may not be expected. For example, a pressure vessel under external pressure may exhibit a negative eigenvalue (buckling under internal pressure) due to local buckling of a stiffener. Such “physical” negative buckling modes are usually readily understood once they are displayed and can usually be avoided by applying a preload before the buckling analysis.Negative eigenvalues sometimes correspond to buckling modes that cannot be understood readily in terms of physical behavior, particularly if a preload is applied that causes significant geometric nonlinearity. In this case a geometrically nonlinear load-displacement analysis should be performed (“Unstable collapse and postbuckling analysis,”Section 6.2.4).Correcting an overconstrained modelA node set containing all the nodes in the chains of constraints associated with a particular zero pivot is generated automatically and can be displayed in the Visualization module of Abaqus/CAE.There is no unique way to remove the overconstraints in this model. For example, if one JOIN and REVOLUTE (five constraints) combination is replaced with a SLOT connector element, which enforces only the two translation constraints in the plane of the mechanism, there are no redundancies. Alternatively, you could remove the REVOLUTE from one of the connector elements and also use a SLOT connection instead of a JOIN in one of the other connector elements.Another alternative is to relax some of the constraints. In the example outlined here, an elastic body could replace one or more of the rigid bodies. You could also relax the Lagrange multiplier-based constraints (e.g., JOIN or REVOLUTE) by using CARTESIAN and CARDAN connection types with appropriate elastic stiffnesses (see “Connector behavior,”Section 28.2.1).After analyzing the chains of constraints, you have to decide which constraints have to be removed to render the model properly constrained and also best fit the modeling goals. For this example the three constraints cannot be removed randomly. Removing any three combinations of the six boundary conditions, for example, would make the problem worse: the model is still overconstrained, and three rigid body modes have been added to the model. Moreover, you should remove the constraints that do not affect the kinematics of the model. For example, you cannot completely remove a JOIN connection from any ofthe connector elements since the model would be different from that originally intended.。

红层泥质胶结物干缩裂缝深度与间距预测模型

岩土力学Vol.43 No. 4R o c k and Soil M e c h a n i c sApr.2022DOI ： 10.16285/j.rsm.2021.1084第43卷第4期 2022年4月红层泥质胶结物干缩裂缝深度与间距预测模型陈晓斌喻昭晟h 2，周雨晴1>2,张家生h 2，钱于3(1.中南大学土木工程学院，湖南长沙410083; 2.中南大学重载铁路工程结构教育部重点实验室，湖南长沙410075;3.南卡罗来纳大学土木与环境工程系，美国哥伦比亚29208)摘要：红层泥质胶结物在干燥条件下会因含水率变化产生开裂，裂缝进一步发展贯通最终导致红层泥岩崩解破坏。

目前对红层泥岩崩解机制的研究多为定性描述，缺乏定量分析方法。

因此，定量研究红层泥质胶结物开裂的产生及扩展对深化红层泥岩崩解理论研宄具有重要意义。

基于线弹性断裂力学及非饱和土力学，建立了考虑含水率变化的红层泥质胶结物裂缝深度及间距计算公式，并对公式参数敏感性进行了分析。

应用C T 试验数据对P F C 2D 数值模型进行了校准，并基于干缩开裂数值试验结果，对裂缝深度及间距计算公式进行了对比验证。

结果表明：提出的计算公式能较好地预测红层泥质胶结物的开裂深度及间距，裂缝深度随含水率减小呈先快速增加再缓慢增长的特点。

研究成果有助于加深对红层泥岩崩解的量化理论分析，并为预测工程中红层泥岩的开裂程度提供参考。

关键词：红层泥质胶结物；干缩开裂；断裂力学；裂缝深度；裂缝间距中图分类号：T U 411文献标识码：A文章编号：1000—7598 (2022) 04 — 0868—11Prediction model of desiccation shrinkage fracture depth and spacing ofargillaceous cement in red bed mudstoneCHEN Xiao -bin 1 2, YU Zhao -sheng 1,2, ZHOU Yu -qing 1，2, ZHANG Jia -sheng 12, QIAN Yu 3(1. School of Civil Engineering, Central South University, Changsha, H u n a n 410083, China;2. M O E K e y Laboratory of Engineering Structures of H e a v y Haul Railway, Central South University, Changsha, H u n a n 410075, China;3. Department of Civil and Environmental Engineering, T h e University of South Carolina, Columbia 29208, America)Abstract : U n d e r desiccation conditions, the argillaceous cement of red bed mudstone will crack due to the change of water content,and the cracks will develop further and finally lead to the disintegration of red mudstone. At present, the research on the disintegration m e c h a n i s m of red mudstone is mostly qualitative description, lack of quantitative analysis method. Therefore, the quantitative study of the occurrence and propagation of the red mudstone argillaceous cement cracking i s of great significance to the further theoretical study of the red mudstone disintegration. B ased on linear elastic fracture mechanics and unsaturated soil mechanics, a formula that considering water content change w a s developed to calculate crack depth and spacing of argillaceous cement in red mudstone, and the sensitivity of the fonnula parameters w a s analyzed. C T test data were used to calibrate the P F C 2D numerical model, and based on the numerical test results of desiccation cracking, the formulas of fracture depth and spacing were c o m p a r e d and verified. T h e proposed formula can well predict the crack depth and spacing of the red layer argillaceous cement, and the crack depth increases rapidly at first and then slowly with the decrease of water content. T h e research results are helpful to deepen the quantitative theoretical analysis of red mudstone disintegration and provide reference for predicting the degree of red mudstone cracking in engineering.Keywords : argillaceous cement of red bed mudstone; desiccation cracking; fracture mechanics; crack depth; crack spacingl 引言红层泥岩在我国分布广泛，易产生因湿度变化引发的崩解性地陷、滑坡、塌方等工程灾害。

WELLPLAN2003_1_TorqueDragAnalysis

The application determines how string loads, surface loads, frictional and contact forces vary with mud properties, wellbore trajectory, and other operational parameters.
WELLPLAN
Torque Drag Analysis
Force Equilibrium
Normal Force Weight of Component
Friction Force = Normal Force x Friction Factor
WELLPLAN
Torque Drag Analysis
Why Do I need to model Sheave Friction?
• Martin Decker-type deadline weight-indicators do not account for this problem, and actual pick-up loads are therefore always greater than indicated, while slack-off loads are always less than indicated. • When using pick-up or slack-off hook-load measurements as the basis for friction factor determinations, this error source is manifest in pick-up friction factors that are too low, and slack-off friction factors that are too high.

钢丝绳精密传动机构的分析与实验研究

钢丝绳精密传动机构的分析与实验研究分类号ＵＤＣ工学硕士学位论文学号钢丝绳精密传动机构的分析与实验研究硕士生姓名金思庆学科领域机械工程研究方向精密工程与计算机控制指导教师范大鹏教授国防科学技术大学研究生院二ｏｏ六年十一月国防科学技术大学研究生院学位论文摘要高性能光电稳定跟踪平台要求平台机械系统精度高、动态性能好，采用钢丝绳精密传动可以满足这种需求。

钢丝绳精密传动具有低噪声、无空回、高刚度、小惯量、传动平稳和维护费用较低等优点，它在工业、商业、航空、军事、医疗和研究的应用，ｌ：都具有较高的性价比。

本文以此为背景，研究钢丝绳精密传动的关键技术问题。

论文研究了钢丝绳精密传动不同的结构形式，提出了钢丝绳精密传动的设计方案，分析了钢丝绳在不同绳槽上的受力、运动情况和绳槽之间的匹配关系，得出有效传动的条件。

对钢丝绳精密传动精度进行了理论分析，求解出实际传动比、滑移角、传动系统各部分的刚度和总的扭转刚度。

运用计算机仿真分析出系统各部分刚度和负载之间的关系、系统总刚度相对于系统中各参数改变的规律和灵敏度。

设计了测试钢丝绳精密传动特性的试验台，验证了钢丝绳精密传动的精度，测试了传动系统的动态性能。

本文为开发基于钢丝绳精密传动的高性能光电稳定平台奠定了初步的基础。

关键词：精密传动，钢丝绳传动，绳槽匹配，刚度，灵敏度第ｉ页ＡＢＳＴＲＡＣＴＨｉｇｈｐｅｒｆｏｒｍａｎｃｅｐｈｏｔｏｅｌｅｃｔｒｉｃｓｔａｂｉｌｉｚａｔｉｏｎａｎｄａｕｔｏｍａｔｉｃｐｏｉｎｔｉｎｇａｎｄｔｒａｃｋｉｎｇｐｌａｔｆｏｒｍｒｅｑｕｉｒｅｓｍｏｌｌａｎｄｍｏｌｅｈｉｇｈｐｒｅｃｉｓｉｏｎａｎｄｄｙｎａｍｉｃｒｅｓｐｏｎｓｅｔｏｉｔｓｍｅｃｈａｎｉｔａＩｓｙｓｔｅｍ．Ｓｔｅｅｌｃａｂｌｅｄｒｉｖｅｒｃａｎｍｅｅｔｔｈｉｓｒｅｑｕｉｒｅｍｅｎｔ．Ｐｒｅｃｉｓｅｓｔｅｅｌｃａｂｌｅｄｒｉｖｅｒｈａｓｔｈｅａｄｖａｎｔａｇｅｓｏｆｂｏｔｈｈｉｇｈａｃｃｕｒａｃｙａｎｄｓｔｉｆｆｎｅｓｓ，ａｓｗｅｌｌａｓｃｏｍｐａｃｔｃｏｍｐｏｓｉｔｉｏｎ．Ｉｔｐｒｏｖｉｄｅｓｂｏｔｈｃｏｓｔａｎｄｐｅｒｆｏｒｍａｎｃｅａｄｖａｎｔａｇｅｓｉｎａｌｌｔｙｐｅｓｏｆｉｎｄｕｓｔｒｉａｌ，ｃｏｍｍｅｒｃｉａｌ，ａｅｒｏｓｐａｃｅ，ｍｉｌｉｔａｒｙ，ｍｅｄｉｃａｌａｎｄｒｅｓｅａｒｃｈａｐｐｌｉｃａｔｉｏｎｓ．Ｏｎｔｈｉｓｂａｃｋｇｒｏｕｎｄ，ｔｈｉｓｐａｐｅｒｓｔｕｄｉｅｓｍａｉｎｌｙｏｎｔｈｅ研ｍａｒｙｔｅｃｈｎｏｌｏｇｉｅｓｏｆｐｒｅｃｉｓｅｃａｂｌｅｄｒｉｖｅｒ．１１１ｉｓｐａｐｅｒｓｔｕｄｉｅｓｄｉｆｆｅｒｅｎｔｓｔｒｕｃｔｕｒａｌｆｏｒｍｓｏｆｐｒｅｃｉｓｅｓｔｅｅｌｃａｂｌｅｄｒｉｖｅｒ，ｔｈｅｄｅｓｉｇｎｐｒｏｊｅｃｔｏｆｐｒｅｃｉｓｅｓｔｅｅｌｃａｂｌｅｄｒｉｖｅｒ．ＩｔａｎａｌｙｓｅｓｔｈｅｍｏｖｅｍｅｎｔｏｆｓｔｅｅｌｃａｂｌｅａｎｄｔｈｅｆｏｒｃｅｌｏａｄｅｄｏｎｓｔｅｅｌｃａｂｌｅｉｎｄｉｆｆｅｌＩｎｔｇｒｏｏｖｅｓ．ａｎｄａｎａｌｙｓｅｓｔｈｅｍａｔｃｈｉｎｇｂｅｔｗｅｅｎｄｉｆｆｅｒｅｎｔｇｒｏｏｖｅｓ．Ａｎｄｉｔｅｄｕｃｅｓｔｈｅｑｕａｌｉｆｉｃａｔｉｏｎｏｆｔｒａｎｓｍｉｔｔｉｎｇａｖａｉｌａｂｌｙ．Ｉｔａｎａｌｙｓｅｓｔｈｅｐｒｅｃｉｓｉｏｎｏｆｔｈｅｐｒｅｃｉｓｅｓｔｅｅｌｃａｂｌｅｄｒｉｖｅｒｔｈｅｏｒｅｔｉｃａｌｌｙ．ａｎｄｅｄｕｃｅｓａｃｔｕａＩｒａｔｉｏ，ｓｌｉｐｐａｇｅａｎｇｅｌ，ｔｈｅｓｔｉｆｆｎｅｓｓｏｆｄｉｆｆｅｒｅｎｔｐａｒｔｓｏｆｔｈｅｐｒｅｃｉｓｅｓｔｅｅｌｃａｂｌｅｄｒｉｖｅｒａｎｄｔｈｅｔｏｔａｌｔｏｒｓｉｏｎｓｔｉｆｆｎｅｓｓｏｆｔｈｅｔｒａｎｓｍｉｓｓｉｏｎｓｙｓｔｅｍ．Ｔｈｅｃｈａｒａｃｔｅｒｉｓｔｉｃｓｏｆｂｏｔｈｔｏｒｓｉｏｎｓｔｉｆｆｎｅｓｓａｎｄｓｔｉｆｆｉｌｅｓｓｓｅｎｓｉｔｉｖｉｔｙａｒｅｔｈｅｏｒｅｔｉｃａｌｌｙａｎａｌｙｓｅｄａｎｄｓｉｍｕｌａｔｅｄ．Ｔｈｅｅｘｐｅｒｉｍｅｎｔｓｔｒａｔｅｇｙｉｓｓｅｔｕｐｉｎｏｒｄｅｒｔｏｔｅｓｔｔｈｅｐｅｒｆｏｒｍａｎｃｅｏｆｔｈｅｓｔｅｅｌｃａｂｌｅｄｒｉｖｅｒ．ＴｈｅｌｌＳＵＩｔｓｓｈｏｗｔｈａｔｔｈｅｐｒｅｃｉｓｅｓｔｅｅＩｃａｂｌｅｄｒｉｖｅｒｃｏｕｌｄｒｅａｃｈｒｅａｓｏｎａｂｌｅｇｏｏｄａｃｃｕｒａｃｙ．Ｔｈｉｓｐａｐｅｒｌａｙｓａｐｒｅｐａｒａｔｏｒｙｐｈｏｔｏｅｌｅｃｔｒｉｃｓｔａｂｉｌｉｚａｔｉｏｎａｎｄａｕｔｏｍａｔｉｃｐｒｅｃｉｓｅｓｔｅｅｌｃａｂｌｅｄｒｉｖｅｒ．ｆｏｕｎｄａｔｉｏｎｆｏｒｄｅｖｅｌｏｐｉｎｇｈｉｇ｝ｌ—ｐｒｅｃｉｓｉｏｎｐｏｉｎｔｉｎｇａｎｄｔｒａｃｋｉｎｇｐｌａｔｆｏｒｍｂａｓｅｄｏｎＫｅｙＷｏｒｄｓ：ｐｒｅｃｉｓｅｄｒｉｖｅｒ，ｓｔｅｅｌｃａｂｌｅｄｒｉｖｅ，ｃａｂｌｅｇｒｏｏｖｅｍａｔｃｈｉｎｇ，ｓｔｉｆｆｎｅｓｓ．ｓｅｎｓｉｔｉｖｉｔｙ第ｉｉ页表目录表格２．１表格２．２表格５．１表格５．２表格５．３表格５．４表格５．５表格５．６表格Ａｌ表格Ａ２钢丝绳精密传动结构形式综合比较……………………………………。

修正的Giuffre-Menegotto-Pinto钢筋滞回本构模型

LEI Yan-yun, XIE Xu
(College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China)
Abstract: The Giuffre-Menegotto-Pinto constitutive model of reinforcement (MP model) will develop an unrealistic sudden change of the strain and stiffness, and overestimate the stress under a circumstance of reloading subsequent to a small-amplitude unloading history. The definition of the elastic range radius and the reassignment of reversal points were proposed to overcome the defect of MP model. The criterion of small strain vibration was determined and divided into two different cases based on the radius of elastic range. A nonlinear analysis program of the reinforced concrete structure was developed based on the fiber beam element, in which the modified MP model and the Mander concrete model were included. The accuracy and applicability of the improved method were verified by both the analysis of steel bar’s hysteresis characteristic under small strain vibration and the simulation of the loaddisplacement curve for a pier through pseudo-static test. Results show that the stress-strain curve predicted by the modified MP model is more rational, and the overestimation of the horizontal bearing capacity and the stiffness of structures can be successfully avoided. The accuracy of hysteretic curve of reinforcement was significantly improved by the modified model in the reinforced concrete structural elasto-plastic analysis. Key words: steel bar; constitutive model; Giuffre-Menegotto-Pinto model; small strain vibration; fiber element model; elasto-plastic seismic response

基于Pasternak双参数模型的滑坡段埋地管道受力分析方法

第51卷第5期2020年5月中南大学学报(自然科学版)Journal of Central South University (Science and Technology)V ol.51No.5May 2020基于Pasternak 双参数模型的滑坡段埋地管道受力分析方法张家铭，尚玉杰，王荣有，王殿龙(中国地质大学(武汉)工程学院，湖北武汉，430074)摘要：为改进基于Winkler 模型的弹性地基梁法固有缺陷，提高计算精度，引入Pasternak 双参数模型，考虑土弹簧间相互作用，提出一种考虑轴向载荷的滑坡段埋地管道受力分析方法，通过参数分析讨论轴向载荷、地基反力系数及地基剪切刚度对滑坡段管道受力变形性状的影响。

研究结果表明：基于Pasternak 双参数模型的滑坡段管道受力变形分析方法比基于Winkler 模型的弹性地基梁法具有更高的精度，更符合工程应用；轴向载荷对滑坡段管道受力变形影响显著，在管道强度设计和校核中不能忽略；地基反力系数较大时，荷载对临近单元体的传力性减弱，管道受力变形性状主要受地基反力系数影响；地基剪切刚度较大时，荷载对临近单元体的传力作用较强，减弱土体对管道的约束作用，影响管道受力变形性状。

关键词：Pasternak 双参数模型；参数分析；轴向载荷；地基反力系数；地基剪切刚度中图分类号：TB125文献标志码：A开放科学(资源服务)标识码(OSID)文章编号：1672-7207（2020）05-1328-09Force analysis method of buried pipeline in landslide sectionbased on Pasternak double-parameter modelZHANG Jiaming,SHANG Yujie,WANG Rongyou,WANG Dianlong(Faculty of Engineering,China University of Geosciences,Wuhan 430074,China)Abstract:In order to improve the inherent defects and improve the calculation accuracy of elastic foundation beam method based on Winkler model,a force analysis method based on Pasternak double-parameter model was introduced.Considering the shear action between soil spring,an analysis method of buried pipeline in landslide section which considers axial load was presented.The effects of axial load,foundation reaction coefficient and foundation shear stiffness on the deformation behavior of pipelines,caused by force,in landslide section were discussed through parameter analysis.The results show that the force analysis of buried pipeline in landslide based on Pasternak double-parameter model is more accurate and suitable than the elastic foundation beam method based on Winkler model.The effect of axial load on the deformation of pipeline which is aroused by force in landslide section is significant.It can not be neglected during the verification and design of buried pipeline in landslide section.When the coefficient of subgrade reaction is greater,the deformation behavior of the pipeline is mainly affected by the coefficient of subgrade reaction because force transmission of the load to the adjacent element is weakened.When the shear stiffness of foundation is greater,force transmission of the load to the adjacent unitDOI:10.11817/j.issn.1672-7207.2020.05.017收稿日期：2019−06−12；修回日期：2019−09−06基金项目(Foundation item)：湖北省安全生产科技专项资金资助项目(XQDSJC18032)(Project(XQDSJC18032)supported by theSpecial Funds for Safe Production and Technology Program of Hubei Province)通信作者：张家铭，博士，副教授，从事滑坡灾害治理研究；E-mail ：*************第5期张家铭，等：基于Pasternak双参数模型的滑坡段埋地管道受力分析方法body is strengthened,which affects the deformation behavior of the pipeline.Key words:Pasternak double-parameter model;parametric analysis;axial load;coefficient of subgrade reaction;shear stiffness of foundation近年来，长输管道工程迅猛发展，已成为能源输送的重要手段之一，其中很大一部分管道需要穿越地质环境复杂、滑坡灾害频发的地段，严重威胁着管道的安全运营。

tip in tip out控制

/lib-publicationsOriginal citation:Bin Raja Ahsan Shah, Raja Mazuir, Cheng, Caizhen, Jones, R. Peter and Pawar, Jasjit (2011) Modelling of 4WD vehicle driveability during tip-in/tip-out events. In: 22nd International Symposium on Dynamics of Vehicle on Road and Tracks (IAVSD), Manchester, UK, 14-19 Aug 2011. Published in: IAVSD 2011 : 22nd International Symposium on Dynamics of Vehicles on Roads and Tracks : 14-19 August 2011 (No.0085). pp. 1-6.Permanent WRAP url:/51652Copyright and reuse:The Warwick Research Archive Portal (WRAP) makes the work of researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher’s version . Please see the ‘permanent WRAP url’ above for details on accessing the published version and note that access may require a subscription.For more information, please contact the WRAP Team at: wrap@MODELLING OF4WD VEHICLE DRIVEABILITY DURING TIP-IN/TIP-OUT EVENTSRaja Mazuir Raja Ahsan Shah1,Caizhen Cheng1,R.Peter Jones1,Jasjit Pawar21School of Engineering,University of Warwick,Coventry,United Kingdom2Jaguar Land Rover,United Kingdome-mail:esriau@AbstractThis paper describes a modelling method to investigate the dynamic behaviour of4WD vehicle under a severe driving condition,where the driver applies a rapid tip-in on the accelerator pedal in2nd gear to achieve maximum engine torque.This is followed by a tip-out event by releasing the accelerator quickly.The Tip-In/Tip-Out events are one of important elements to assess the vehicle driveability.During these test events,the vehicle is expected to generate low frequency vibration between2Hz and10Hz and gives discomfort feelings induced by resonance effects on sensitive human organs.The aim of this paper is to develop a4WD vehicle model in a modern object-oriented multi-body simulation tool and study its driveability.1.INTRODUCTIONNot many evidences have been seen on the research carried out to examine the eminence of the vibration response in4WD vehicle system predominantly in the Tip-In/Tip-Out events.The basic test method is to study the interaction between the driver and vehicle in terms of driveability at low speed events.The results can be analysed to identify the root cause of the low frequency responses.From the past investigation[1],the exposure of low frequencies to the driver leads to a sense of discomfort induced by resonance effects on different human organs such as upper torso,lower abdomen and shoulders.The previous effort[2]that influenced this work has resulted in the development of a single Torsional Model, a single Fore-Aft Model using the ADAMS modelling environment to study the above behaviours.Both models have been correlated with the ADAMS full nonlinear model as a reference model and the vehicle testing data.The development of these single models was due to the long computation time of the ADAMS full nonlinear model. Furthermore,it was not possible to use the full nonlinear model for a real time application due to the complexity of its architecture,which consisted of more than550Degree of Freedom(DoF).One aim of the work was to accelerate the process of tuning the properties of the system components to improve the vehicle driveability.However,these two models were not connected to represent the actual physical behavior of the vehicle system due to the lack of tyre model.Hence,it did not demonstrate the actual interaction between these two sub systems.In addition,there were limitations to the development of control strategies in real time using ADAMS.This is due to long computation time and the ability to host the model in a real time platform.Considering the above factors,Dymola modelling platform[3]has been selected to develop the model of vehicle system based on the open-source Modelica language.For this research,the vehicle model was constructed from standard and extended Modelica libraries by setting up the vehicle parameters.The model was also integrated in a MATLAB/Simulink environment and implemented in the real time platform,which allows the development of control strategies.From this experimental result,the vehicle model characteristics at low frequencies have been derived to give the prediction of the vehicle driveability behavior.In the remaining of this paper,the modelling methods and modelling details are presented in the next topics,which explained the results of simulation in Dymola,real time simulation and conclusion of this research.2.MODELLINGThe architecture of the vehicle model was based on a4x4platform with2.2L Diesel engine mounted in East-West orientation.It consists of all drive-train subsystems and the chassis components acting on longitudinal direction.The physical characteristics of the components were represented by combining the torsional and fore-aft elements with the correct mechanical properties.The interaction of the drive-train and chassis systems was captured by a nonlinear tyre model in longitudinal direction,which was derived from simulation data to give a true behavior of the tractive force on different road surfaces.In order to build the complete reduced order architecture of the vehicle system,the Dymola single models were correlated with ADAMS single models by using the same inputs,which were extracted from vehicle measurement.The models then were coupled after the results have satisfied the specific output characteristics i.e.crank speed and seat acceleration.This Coupled model then has been correlated with the ADAMS full nonlinear model.Figure 1shows the architecture of the single Torsional Model and single Fore-Aft Model.a)Torsional Model b)Fore-Aft Model Figure 1:4WD Single Dymola Vehicle System Architecture2.1Single Torsional ModelWith the adopted approach,the torsional system of engine and drive subsystem components had been simulated to determine the transient response when subjected to the torque signal from the driver.The number of DoF has been reduced to less than 20to reduce the computation time [2,4].All sub systems were linear except the Dual Mass Flywheel (DMF),where two stages of stiffness properties (Figure 2)were applied to capture shuffle frequency as a function of angular displacement and angular speed.The vehicle inertias were attached to each of wheel inertias to imitate the friction load acting on the system.The torsional system of a 4WD vehicle (Figure 1a )comprises of two drive system,namely a FWD system and a RWD system.4WD is engaged by connecting the FWD system through the power take-off unit,and linked to the rear differential unit with propeller shaft and traction control unit.Figure 2:DMF Characteristic2.2Single Fore-Aft ModelThe chassis sub-system was represented by the Fore-Aft model as shown in Figure 1(b).It consists of high level components,which have the dominant effect on the system i.e.vehicle body,front and rear wheels,power-unit,front and rear sub-frames.Each of the components was connected through nonlinear compliances,and its displacement has been fixed on longitudinal direction except the mass carrier compliances.Both of the mass carrier compliances (right hand and left hand side)have 3DoF to perceive the effect of thepower-unit’s1st Stage2nd StageAngular DisplacementT o r q u eAngular SpeedDriveReversepitch and yaw to the vehicle system dynamics.Similarly to Torsional Model,no tyre model has been included to represent dynamics characteristics of the wheels,i.e.tractive force,slip ratio.The characteristics have been replaced by the forces taken from vehicle test data and used as the inputs to the front and rear wheels to excite the system.2.3Coupled ModelThe architecture of the Coupled Model consists of 40DoF from 4sub-systems and driver environment,.Each component of the sub-systems has the combination of torsional elements,masses,inertias and mechanical properties.From Figure 3,the engine is mounted to the transmission bell housing and the energy from the combustion is transferred to input shaft via DMF.The integrated front differential inside the transmission splits the torque to the front half-shafts and Power Take Off (PTO)unit.The rear differential then divides the torque to right and left side of the rear half-shafts.Both of the front and rear half-shafts are connected to the tyre model to provide the longitudinal force as a function of vertical load and wheel slip ratio.The longitudinal force drives the chassis sub-system where each of the chassis components is inter-connected by the nonlinear compliances.Figure 3:Coupled Model of 4WD Vehicle SystemA generic tyre model has been developed based on the standard tyre parameters fitted on this particular vehicle.Car Maker software [5]was used to generate the 3dimensional data ,on tarmac surface with surface coefficient, =0.85.Fundamentally,the road surface coefficient is reduced as a function of vehicle speed;hence reduce the tractive force further.But this effect has been neglected in this analytical model.The input torque is controlled by the driver environment block to represent the combustion process to induce the crank train system.The gear ratio is fixed at 2nd Gear with the assumption that no mechanical losses in the gearing system.By coupling the two single models as discussed earlier,the vehicle model was expected to be more robust and produce true physical representation of the sub-system behaviour.The common root causes of vehicle driveability were always associated with the rise rate of the torque demand as well as the overshoot of the oscillation,which can be sensed by most drivers [6,7].The explanation for this occurrence was due to the rapid excitation of torque demand to the driveline system and its compliances.With this Coupled Model,it has allowed more accurate study on driveability and to understand the energy flow from throttle demand to the seat rail via contact patch of the wheels.To validate the accuracy and robustness of the Dymola Coupled Model,a correlation was carried out with an ADAMS full nonlinear model and had been verified with vehicle measurement data.The simulation parameters were set-up to be the same as the ADAMS full nonlinear model.3.SIMULATIONDrivelinePower UnitDriver EnvironmentTransmissionChassis3.1Off-line SimulationThe standard practices of the Tip-In/Tip-Out tests are to accelerate the vehicle up to 45kilometer per hour (kph)and coast down to 30kph.An abrupt torque then applied within 0.12seconds to excite the torsional system until it reached maximum torque.And finally the torque will be removed instantaneously from the crank train system.For the simulation set-up,the model was run up to 17seconds to replicate the test condition above,with fixed step integrator of 0.001seconds.The Euler solver was used together with Implicit Euler as the inline integration for real time solver.Based on the single Dymola sub-systems simulation,the correlation results show a good agreement with the single ADAMS models in terms of crank speed and seat acceleration as presented in Figure 4and Figure 5.These results verified that the single Dymola Torsional Model and Fore-Aft Model can be used to form the Coupled Model.For the Coupled Model,the computation time against the ADAMS full nonlinear system has improved significantly from 9hours 30minutes to only 10.5seconds.In Figure 6,the crank speed is having the same behavior as full nonlinear model but the oscillation is seen been phased in 2nd to 4th orders.However,these discrepancies can be neglected as the occurrence was due to the effect of the generic tyre model properties.The correlation result demonstrates that the Coupled Model has performed the same manner as the ADAMS full nonlinear model.Figure 4:Correlation results against single ADAMS Torsional Model without tyre modelFigure 5:Correlation results against single ADAMS Fore-Aft Model without tyre modelTime [sec]S e a t A c c e l e r a t i o n [m /s 2]Time [sec]C r a n k S p e e d [r p m ]The next stage of the simulation for the Coupled Model was to conduct the Tip-In/Tip-Out events and observed the response of the system based on 70%throttle position at 2nd Gear.Figure 7shows that the system generates shuffle frequencies of 3.42Hz during Tip-In and 3.14Hz during Tip-Out with exposure time of 2seconds measuring at the seat rail.As discussed earlier,these frequencies range (2Hz -10Hz)will stimulate the human organs that can cause the motion sickness.These shuffle frequencies were dominated by the half-shafts [4]and can be tuned by changing its mechanical properties.The overshoots of the shuffle frequencies,which attributes to the customer perception of vehicle driveability,exceed more than 85%for both events as shown in Table 1.From the graph,the rise rates have also been calculated during Tip-In and Tip-Out to characterize the vehicle driveability.The Tip-In event has generated higher value of rise rate at 3.62g/s as the result of the sudden excitation of the driveline system from steady state condition .When torque is removed instantaneously from the system,the rise rate on the seat acceleration is reduced by 30%compare to Tip-In event due to the nonlinearity and asymmetric of the system compliances.With these two outputs,it is possible to develop the driveability matrix in the next phase of the research to assess the vehicle behaviour acting on the transient torque response as well as developing the control strategies to damp the shuffle frequencies.Figure 6:Correlation between Dymola Coupled Model and ADAMS full nonlinear model with Throttle Input100%at 2nd Gear,surface coefficient,μ=0.85Figure 7:Tip-In/Tip-Out with Throttle Position 70%at 2nd Gear,μ=0.85T h r o t t l e P o s i t i o n [%]S e a t A c c e l e r a t i o n [m /s 2]Time [sec]T h r o t t l e P o s i t i o n [%]C r a n k S p e e d [r p m ]Time [sec]S e a t A c c e l e r a t i o n [m /s 2]T h r o t t l e P o s i t i o n [%]S e a t A c c e l e r a t i o n [m /s 2]Time [sec]Tip-In Event Tip-Out EventDamping Ratio,ζ0.03660.0393Natural Frequency,Hz 3.42 3.14Overshoot,%89.188.4Rise rate,g/s 3.62-2.36Table1:Overshoot and Rise Rate during Tip-In/Tip-Out with Throttle Position70%at2nd Gear,μ=0.85 3.2Real Time SimulationFor the future control strategies development,it is essential to see how the vehicle reacts in real time and to validate the response of the vehicle sub-system as close as the actual condition.From the real time simulation,the model has performed the same behaviour as the Dymola Coupled Model.The computation time for this model is 0.5milliseconds to process every1millisecond of sampling time with no overrun.It indicates that the coupled vehicle model can be used in real time simulation for the control development.4.CONCLUSIONSThe Coupled Model of4WD vehicle system built in Dymola has performed equally as good as the ADAMS full nonlinear system.The model has produced frequencies that can affect the human organs during Tip-In/Tip-Out events.The oscillation is mainly dominated by the half shafts,where these frequencies can be shifted by tuning the properties of stiffness and damping.The simulation time has also reduced from9hours30minutes with an ADAMS full nonlinear model to10.5seconds with the Coupled Model with equally valid outputs.The Coupled Model also performs well in a real time test environment as seen by the measurement of the computation time to give a prediction of the vehicle behaviour during torque transient mode.ACKNOWLEGEMENTThe authors would like to thank Jaguar Land Rover and Warwick Manufacturing Group(WMG)for providing us the support and information in this research project.REFERENCES[1]Atsumi,B.,Tokunaga,H.,Kanamori,H.,Sugawara,T.,Yasuda,E.,and Inagaki,H.,Evaluation of vehiclemotion sickness due to vehicle vibration,JSAE Review,Vol.23(3),2002,pp.341-346.[2]Pawar,J.,Low Frequency Powertrain and Vehicle System Dynamic,Eng.D Theses,University ofWarwick,Coventry,2009.[3]Dymola,“Multi-Engineering Modeling and Simulation”/products/catia/portfolio/dymola,[retrieved in July2011][4]Shah,R.M.R.A.,Dhadyalla,G.,Jones,P.,Pawar,J.,Biggs,S.,and Cheng,C.,A Low Fidelity Non LinearModel of4WD Torsional Stiffness at Tip-In/Tip-out Events,FISITA2010World Automotive Congress Proceedings,2010,F2010C137[5]CarMaker3.0-Shift the test drive into simulation!http://www.ipg.de/CarMaker.609.0.html,[retrieved inJuly2011][6]List,H.O.and Schoeggl,P.,Objective Evaluation of Vehicle Driveability,International Congress&Exposition.1998,SAE Paper No.980204[7]Dorey,R.E.and Holmes,C.B.,Vehicle Driveability-Its Characterisation and Measurement,InternationalCongress&Exposition.1999,SAE Paper No.2004-01-1338。

Abaqus仿真分析-CONT-LA03-DynamicImp

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E.g.: 100,000 increments for a 0.1 second event
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A3.4
Time Integration Issues (2/4)
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A2.7
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Overall general contact surface
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Contact Logic in Abaqus/Standard
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5
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损伤力学资料

Effect of manufacturing defects on mechanical properties and failure features of 3D orthogonal woven C/CcompositesAi Shigang a ,Fang Daining b ,⇑,He Rujie b ,1,Pei Yongmao ba Institute of Engineering Mechanics,Beijing Jiaotong University,Beijing 100044,PR ChinabState Key Laboratory of Turbulence and Complex System,College of Engineering,Peking University,Beijing 100871,PR Chinaa r t i c l e i n f o Article history:Received 8September 2014Received in revised form 1November 2014Accepted 3November 2014Available online 10November 2014Keywords:A.Carbon-carbon composites (CCCs)B.DefectsC.Damage mechanics C.Numerical analysisD.Non-destructive testinga b s t r a c tFor high performance 3D orthogonal textile Carbon/Carbon (C/C)composites,a key issue is the manufac-turing defects,such as micro-cracks and voids.Defects can be substantial perturbations of the ideal archi-tecture of the materials which trigger the failure mechanisms and compromise strength.This study presents comprehensive investigations,including experimental mechanical tests,micron-resolution computed tomography (l CT)detection and ﬁnite element modeling of the defects in the C/C composite.Virtual C/C specimens with void defects were constructed based on l CT data and a new progressive dam-age model for the composite was proposed.According to the numerical approach,effects of voids on mechanical performance of the C/C composite were investigated.Failure predictions of the C/C virtual specimens under different void fraction and location were presented.Numerical simulation results showed that voids in ﬁber yarns had the greatest inﬂuences on performance of the C/C composite,espe-cially on tensile strength.Ó2014Elsevier Ltd.All rights reserved.1.IntroductionCarbon ﬁber reinforced carbon composites (C/C)have high ther-mal stability,thermal shock resistance,strength and stiffness in non-oxidizing atmosphere.Due to its superior speciﬁc strength and toughness,C/C composites can be considered as favourite materials for highly demanding thermostructural lightweight applications e.g.in aerospace and nuclear industry [1–6].Nowa-days C/C components are leading candidates for applications under extreme conditions.C/C composites are produced by chemical vapor inﬁltration (CVI)of a textile ﬁber preform.After the CVI pro-cess and high temperature heart-treatments,generally,manufac-turing defects exist inner the materials.In particular,porosity/voids and micro-cracks are typical defects in C/C composites,and seriously affect the performance of the composites [7–9].So,it is mandatory to account for the effects of defects and their evolution,even in the early stages of the design process.With the increasing use of C/C composites as advanced structural materials,the deter-mination of damage criticality and structural reliability of compos-ites has become an important issue in recent years.Defects–mechanical property relationships of ﬁber reinforced composites have always been of interest to scientists addressing the composite performance.In Gowayed et al.’s work [10],defects in an as-manufactured oxide/oxide and two non-oxide (SiC/SiNC and MI SiC/SiC)ceramic matrix composites were categorized and their volume fraction quantiﬁed using optical imaging and image analysis.Aslan and Sahin [11]investigated the effects of delamin-ations size on the critical buckling load and compressive failure load of E-glass/epoxy composite laminates with multiple large del-aminations by experiments and numerical simulations.In Masoud et al.’s work [12]effects of manufacturing and installation defects on mechanical performance of polymer matrix composites appear-ing in civil infrastructure and aerospace applications were studied.Damage onset and propagation were studied used time-dependent nonlinear regression of the strain ﬁeld.In Refs.[13–17],the ﬁnite element method (FEM)was followed by various authors to study the delamination problems.FEM is preferred than analytical solu-tions because it can handle various laminate conﬁgurations and boundary conditions.In recent decades,high-ﬁdelity X-ray micro-computed tomog-raphy (l CT)technology has been used to characterize defects and reconstruct meso-structure of textile composites [18].In Cox et al.’s work [19–21],three-dimensional images of textile com-posites were captured by X-ray l CT on a synchrotron beamline.Based on a modiﬁed Markov Chain algorithm and the l CT data,/10.1016/positesb.2014.11.0031359-8368/Ó2014Elsevier Ltd.All rights reserved.⇑Corresponding author.E-mail addresses:sgai@ (F.Daining),rujh@ (H.Rujie).1Co-corresponding author.a computationally efﬁcient method has been demonstrated for generating virtual textile specimens.In Fard et al.’s work [22],manufacturing defects in stitch-bonded biaxial carbon/epoxy composites were studied through nondestructive testing (NDT)and the mechanical performance of the composite structures was investigated using strain mapping technique.In Desplentere et al.’s work [23],X-ray l CT was used to characterize the micro-structural variation of four different 3D warp-interlaced fabrics.And the inﬂuence of the variability of the fabric internal geometry on the mechanical properties of the composites was estimated.In Guillaume et al.’s work [24]effects of porosity defects on the interlaminar tensile (ILT)fatigue behavior of car-bon/epoxy tape composites were studied.In that work,CT mea-surements of porosity defects present in specimens were integrated into ﬁnite element stress analysis to capture the effects of defects on the ILT fatigue behavior.In Thomas et al.’s work [25]X-ray microtomography technology was adopted to measure the dimensions and orientation of the critical defects in short-ﬁber reinforced composites.Generally,geometry reconstruction based on l CT data is a huge and complex work,sometimes,virtual specimens explored through this approach are difﬁcult to use for numerical analysis.For 3D fabric composites,because of the 2.Material and experimentsMaterial studied in this article is C/C 3-D orthogonal woven ceramic composite (fabricated by National Key Laboratory of Ther-mostructure Composite Materials,Northwestern Polytechnical University,China)in which T300carbon ﬁber (Nippon Toray,Japan)tows rigidiﬁed by carbon matrix.The C/C composite was prepared using chemical vapor inﬁltration (CVI)method.T300car-bon ﬁber was used as reinforcement of the C/C composites with the ﬁber volume fraction was 56.5%.The ﬁber preforms,as shown in Fig.1a,were inﬁltrated with carbon matrix using multiple cycles of inﬁltration and heat treatment at 1373K,0.03MPa (the thick-ness of the ﬁber preforms is about 5mm).With increasing cycles,a matrix with near full density can be asymptotically approached,generally,it was about 10cycles (1200h).The C/C specimens are illustrated in Fig.1b (the thickness of the tensile specimen is 5.0mm).However,from the l CT images of the C/C materials,it was found that manufacturing defects such as voids and micro-cracks appeared inner the composites.It is because of the special material preparation process.The manufacturing defects are illus-trated in Fig.1c.Uniaxial tensile experiments were carried out under a Shima-Fig.1.C/C 3-D orthogonal woven composite.Fig.2.Stress–strain curve of the C/C composite under uniaxial tension.114 A.Shigang et al./Composites:Part B 71(2015)113–121In the tensile experiments,ﬁve specimens in total were tested and the tensile strengths were217.3,185.1,219.8,176.5and 187.3MPa correspondingly.The average value of the tensile strength was197.2MPa and the dispersion of the experimental results was less than11.5%.Other more,the fracture behaviors of theﬁve specimens were similar with the failure locations almost all located in the middle of the specimens.From the experiments, deformation of the C/C3-D orthogonal composite under uniaxial tension comprises with three stages:linear elastic stage,damage initiation/evolution stage and the material fracture stage.In the ﬁrst stage the stress–strain curve increased linearly and in the sec-ond stage the stress–strain curve increased nonlinearly.In the frac-ture stage the stress–strain curve rapidly declined.3.Numerical programmer3.1.3Dﬁnite element modelFiber tows in the3-D orthogonal architecturesﬁt together snugly in the woven pattern by a system of periodic motions, and approximately in the same cross-sectional geometry.In this study,cross-sections of the warpﬁber yarns and weftﬁber yarns wereﬁtted as rectangle.The cross-sections of the z-binder tows wereﬁtted as circular.Geometric parameters of theﬁber yarn cross-sections were recorded.For the warp yarns and weft yarns the side lengths of the cross-section rectangle were0.786mm and0.340mm.For the z-binderﬁber yarns the diameter of the cross-section circular was0.790mm.The smallest repeatable rep-resentative volume element(RVE)of the textile architecture was constructed and shown in Fig.3.The lengths of the RVE model in X and Y direction both were1.96mm and the height of the RVE model in Z direction was0.76mm.To reveal the internal defects in theﬁnite element model,l CT technology was used to investigate the meso-structures of the fore,three local coordinates were constructed to identify the mate-rial directions.Then,an interface zone with a constant thickness 0.01mm was generated based on the geometrical model of the ﬁber yarns,as shown in Fig.3d.Finally,a solid block model with the same size of the composite specimen was constructed.Boolean operation were carried out among the solid block,interfaces and theﬁber yarns to generate the geometrical model of the carbon matrix,which is shown in Fig.3b.A whole RVE model of the com-posite is illustrated in Fig.3a.A Monte Carlo algorithm was adopted to choose elements one-by-one randomly as‘‘void defects’’until the volume fraction of the voids satisﬁed the threshold values in the three zones respectively. For the C/C composite studied in this paper,the void fractions of theﬁber yarns,matrix and the interfaces are0.51%,0.47%and 1.94%respectively.It must be noted out that those elements which identiﬁed as‘‘void defects’’were not moved away from the FE model,but the stiffness was degenerated by10eÀ6times in the simulation process.The void defects in the three zones are high-light as‘‘red’’,as shown in Fig.3.3.2.Progressive damage modelThe failure criterion proposed here is a strain-based continuum damage formulation with different failure criteria applied for matrix andﬁber yarns.A gradual degradation of the material prop-erties is assumed.This gradual degradation is controlled by the individual fracture energies of matrix andﬁber yarns,respectively. Theﬁber yarn is in the X(1)–Y(2)–Z(3)Cartesian coordinate sys-tem,and the X direction corresponds to theﬁber longitudinal direction.For theﬁber yarns,two different modes of failure are considered:ﬁber failure in longitudinal direction and matrix fail-ure in transverse direction.The damage mechanism consists of two ingredients:the damage initiation criteria and the damage evolution law.orthogonal textile C/C composite,(a)RVE model,(b)carbon matrix,(c)ﬁber yarns and(d)ﬁber yarns-matrixinterpretation of the references to colour in thisﬁgure legend,the reader is referred to the web version of thisA.Shigang et al./Composites:Part B71(2015)113–121115failure strains inﬁber direction in tension and compression,F f;tX and F f;cXare the tensile and compressive strength of theﬁberyarns in X direction,respectively.Once the above criterion is sat-isﬁed,theﬁber damage variable,f Xf,evolves according to the fol-lowing equation law:d X f ¼1Àe f;t11f XfeÀC11e f;t11f X fÀe f;t11ðÞL c=G fðÞð2Þwhere L c is the characteristic length associated with the material point.For matrix failure the following failure criterion is used:f Y m ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃe f;t22e f;c22ðe22Þ2þe f;t22Àe f;t222e f;c22B@1C A e22þe f;t22e f;s12!2ðe12Þ2>e f;t22v uu uu tð3Þf Z m ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃe f;t3333ðe33Þ2þe f;t33Àe f;t33233B@1C A e33þe f;t3313!2ðe13Þ2>e f;t33v uu uu tð4Þwhere e f;t22;e f;t33;e f;c22and e f;c33are the failure strains perpendicular to the ﬁber direction in tension and compression,respectively.The failure strain for shear are e f;s13and e f;s12.Failure occurs when f Y m exceeds its threshold value e f;t22or f Z m exceeds its threshold value e f;t33.The evolu-tion law of the matrix damage variable,d m,is:d Ym¼1Àe f;t22f YmeÀC22e f;t22f Y mÀe f;t22ðÞL c=G mðÞð5Þd Zm¼1Àe f;t33fmeÀC33e f;t33f Z mÀe f;t33ðÞL c=G mðÞð6ÞAs damage progressing,the effective elasticity matrix isreduced as functions of the three damage variables f Xf,d Ymand d Zm, as follows:3.2.2.Failure criterion for matrixDamage in theﬁber is initiated when the following criterion is reached:where e f;t and e f;c are the failure strains in tension and compression respectively and e f,t=r f,t/C11,e f,c=r f,c/C11.Once the above criterionis satisﬁed,theﬁber damage variable,f XðY=ZÞm,evolves according to the equation:d XðY=ZÞm¼1Àef;tfmeÀC11e f;t f XðY=ZÞmÀe f;tL c=G mð9ÞThe modulus matrix of the matrix will be reduced according to:In user subroutine UMAT the stresses are updated according to the following equation:C f d ¼1Àd XfC111Àd Xf1Àd YmC121Àd Xf1Àd ZmC130001Àd YmC221Àd Ym1Àd ZmC230001Àd ZmC330001Àd Xf1Àd YmC4400Symmetric1Àd Xf1Àd ZmC5501Àd Ym1Àd ZmC66ð7Þf XðY=ZÞm ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃef;tef;cðe11ð22=33ÞÞ2þe f;tÀðe f;tÞ2ef;c!e11ð22=33Þþef;tef;s2ðe12ð23=13ÞÞ2þef;tef;s2ðe13ð12=23ÞÞ2v uu t>e f;tð8ÞC m d ¼1Àd XmC111Àd Xm1Àd YmC121Àd Xm1Àd ZmC130001Àd YmC221Àd Ym1Àd ZmC230001Àd ZmC330001Àd Xm1Àd YmC4400Symmetric1Àd Xm1Àd ZmC5501Àd Ym1Àd ZmC66ð10Þ116 A.Shigang et al./Composites:Part B71(2015)113–121r ¼C d :eTo improve convergence,a technique based ization (Duvaut–Lions regularization [27])of is implemented in the user subroutine.In this age variables are ‘‘normalized’’via the _d t ;X ðY =Z Þf ðm Þ¼1gd X ðY =Z Þf ðm ÞÀd t ;X ðY =Z Þf ðm Þwhere d X f and d X ðY =Z Þm are the ﬁber and matrix culated according to the damage evolution d t ;X f and d t ;X ðY =Z Þm are the ‘‘normalized’’the real calculations of the damaged elasticity bian matrix,and g is the viscosity parameter.and d t ;X ðY =Z Þm can be calculated according to the d t ;X ðY =Z Þf ðm Þt 0þD t¼D t t 0þt d X ðY =Z Þf ðm Þ t 0þD t þg g þt dt ;X ðY =f ðm ÞTherefore,for the ﬁber yarns and matrix,can be further formulated as Eqs.(14)and (15)correspondingly@r e ðf Þ¼C f d þ@C f d@d f:e !@d X f @f f Á@f Xfe!þ@C f d @d Y m :e !@d Y m @f Y m Á@f Y m @e !þ@C f d @d Z m :e !@d Z m @f Zm Á@f Z m@e !ð14Þ@r @e ðm Þ¼C m d þ@C md @d m :e !@d X m @f m Á@f X m@e!þ@C md @d m :e !@d Y m @f m Á@f Y m @e !þ@C m d @d m :e !@d Z m @f m Á@f Z m @e!ð15Þ3.3.Material parameters3D orthogonal C/C composites are composed by T300ﬁber yarns and carbon matrix.The ﬁber yarns can be regarded as unidi-rectional ﬁber-reinforced C/C composites and are assumed to be one transversely isotropic entity in each local material coordinate system.The mechanical properties of the ﬁber yarns can be calcu-lated using the properties of the component materials (ﬁbers and matrix):E 1¼e E f 11þð1Àe ÞE mE 2¼E 3¼E m1Àﬃﬃe p 1ÀE m =E f 22ðÞG 12¼G 13¼G m 1Àﬃp 1ÀG m =G f 12ðÞG 23¼G m1Àﬃp 1ÀG m =G f23ðÞl 12¼l 13¼e l f 12þð1Àe Þl m l 23¼E 222G 23À19>>>>>>>>>>>>=>>>>>>>>>>>>;ð16Þwhere e is the yarn pack factor,for the C/C composite studied in this paper,e =0.81.E f 11,E f 22are the Young’s elastic modulus of the ﬁberin the principal axis 1and 2,respectively.Axis 1is the longitudinal direction of the ﬁber yarns.G f 12,G f 23are the shear modulus of the ﬁber in the 1–2and 2–3plane,respectively.l f 12is the primary Pois-son’s ratio of the ﬁber,E m ,l m and G m represent the Young’s elastic modulus,Poisson’s ratio and shear modulus of the matrix,respec-tively.Materials parameters are listed in Table 1.It should be noted that the mechanical parameters of the carbon matrix and the T300ﬁbers changed after the CVI process.In particular,strength of the ﬁber will had a greater decline.The tensile and com-pressive strength of the T300ﬁber yarns were tested with the values listed in Table 1.The elasticity modular of the carbon matrix was tested by a nanoindentor system,which developed by Fang’s research team from Peking University [28].In the carbon matrix modular tests,the experiments repeated 20times for statistical averaging.The val-ues in the 20measurements were 7.18,9.77,8.58,10.01,11.92,5.30,9.98,8.14,8.63,7.31,6.19,10.69,11.10,13.45,9.15,9.13,11.27,9.20,10.14and 10.06GPa;average value was 9.36GPa.Mate-rial parameters of the ﬁber/matrix interface are not very clear so far,in this study the Young’s elastic modulus and Poisson’s ratio of the inter-faces were assumed as same as the carbon matrix.G f is one of the key parameters which control the failure pro-gress of the ﬁber yarns,however,different values were recom-mended in reported articles.In this study,inﬂuences of G f on the mechanical properties of the C/C composite were investigated ﬁrstly.Based on the values reported in Refs.[29,30],ﬁve virtual specimens with different G f values (0.5,2.0,6.0,10.0,14.0)were constructed and numerical tested.Simulation results were com-pared with the experimental result,as illustrated in Fig.4.It was found that G f has inﬂuences on tensile strength and fracture strain of the C/C composite.When G f were 0.5,2.0,6.0,10.0and 14.0,ten-sile strengths of the specimens were 200.5,205.1,205.3,214.2and 214.8MPa.When G f were 0.5,2.0,6.0,the failure strains were 0.36%,0.43%and 0.57%correspondingly.When G f is bigger than 10.0,failure strains of the C/C specimens were bigger than 1.0%.So,by the simulation results,in the present study the value of G f was set to 6.0.Table 1Materials parameters.E 11(GPa)E 22(GPa)ˆ12G 12(GPa)G 23(GPa)F t (MPa)F c (MPa)S (MPa)G f (m )(N/mm)gT300ﬁber 230400.262414.389075650 6.00.001C matrix 9.360.338210050 1.00.001Interface9.360.3382100501.00.001Fig.4.Stress–strain curves of the C/C virtual specimens under different G f .4.Simulation results and discussionThe anisotropic damage model of theﬁber yarns and the isotro-damage model of the matrix and the interface were carriedmaterial constitutive equations by User subroutine UMAT ABAQUS nonlinearﬁnite element codes.Static uniaxial tensile sim-ulations were carried out.In order to keep forces continuity and displacements compatibility of the opposite faces of the unit cell, periodic boundary conditions were imposed in the simulation. Because the opposite faces of the unit cell have the same geomet-rical features,the nodes on the faces were controlled in the same position to form the corresponding nodes in the process of meshing.The periodic BCs were imposed on the corresponding nodes by FORTRAN pre-compiler code,detailed in Ref.[26].The RVE model subjected to a constant displacement load in Y direction and the loading strain is1%.4.1.Effects of the void defectsIn order to investigate the void defects on the mechanical prop-erties and failure behaviors of the C/C composite,two RVE models of the C/C composite were numerical simulated.In one RVE model (FE_D),theﬁber yarns,interface and matrix all had void defects with the void fractions are0.51%,1.94%and0.47%respectively. For the other RVE model(FE_Intact),no defect inside.The stress–strain curves of the C/C composite in the simulations and experi-ment are illustrated in Fig.5.By the experimental results,the elas-ticity modular of this C/C composite was58.4GPa.By the numerical simulations,for the intact model,the elasticity modular was56.6GPa;for the‘defected’model the modular was56.3GPa. In the view of modular,the simulation error of the two models were3.1%and3.6%compared with the experimental results.The difference between the two FE models was only0.53%,so,void defects have relatively limited effects on the elastic modular of the C/C composite.The uniaxial tensile strength of the C/C compos-ite was197.2MPa by the experiments.In the simulations,the ten-sile strengths were231.4MPa and205.3MPa corresponding to the intact model and the‘defected’model.It was about17.3%and4.1% difference compared with experimental results.It is clear that,theFig.5.Stress–strain curves of the C/C composite under uniaxial tension.Fig.6.Damage evolution inﬁber yarns,(a)RVE model with voids defects,(b)intact model.Part B71(2015)113–121yarns are corresponding to the three pictures‘o’,‘p’and‘q’in Fig.6. For the RVE model with defects,it was found that damages were ﬁrstly generated besides the defects.During the loading process, damages were growing in several sections in theﬁber yarns.How-ever,for the intact model,damages were almost generated in one section in theﬁber yarns.Damage evolution in carbon matrix and the interface zone are illustrated in Figs.7and8.From the simulation results,in all of the three zones,damages wereﬁrstly generated in the‘defected’RVE model.For the‘defected’model,when e=0.27%damages appeared in the interface zone,while for the intact model the strain was0.33%.In the matrix zone,the strains in the two models were0.31%and0.33%,respectively,when damages appeared.In ﬁber yarns,the strains when damages appear for the two models were0.37%and0.44%.So,because of the internal defects,in load-ing progress damages will generate early inner the material.Fail-ure strain of the materials which with defects is comparatively small when compared with the materials without defects.4.2.Inﬂuence of void locationBy the l CT images,it is clear that voids and micro cracks exist in ﬁber yarns,carbon matrix and the interface zones.By statistical analysis for those defects,fraction of the voids in those three zones was calculated.To study the inﬂuence of void location on the mechanical properties of the C/C materials,threeﬁnite elementFig.7.Damage evolution in carbon matrix,(a)model with voids defects,(b)intact model.Fig.8.Damage evolution in interface,(a)model with void defects,(b)intact model.models were constructed and numerically analyzed.In the three RVE models,one model has defects only in theﬁber yarns(FE_DF) and another model has the defects only in carbon matrix(FE_DM), while the other one has defects only in the interface zone(FE_DI). Simulation results were compared with the experimental results. Stress–strain curves in the numerical simulations and experiments are illustrated in Fig.9.Tensile strength calculated by the simulations were208.5MPa, 229.7MPa and230.1MPa,corresponding to the threeﬁnite ele-ment models:FE_DF,FE_DI and FE_DM.By the simulation results of theﬁnite element models FE_D and FE_Intact,as mentioned in above section,the tensile strengths were205.3MPa and 231.4MPa.It can give the conclusion that,defects inﬁber yarns has the biggest effects on the mechanical properties of the C/C composite.If theﬁber yarns are perfect and defects only exist in carbon matrix and interfaces,void defects have limited inﬂuences on the mechanical properties of the C/C composites under the cur-rent void volume fractions.4.3.Inﬂuence of void volume fractionBy the statistical analysis in Section3.1,volume fractions of voids in theﬁber yarns,matrix and the interface zones are0.51%, 0.47%and1.94%.Under this defect fraction,as calculated in Section,tensile strength of the C/C composite declined12.7%compared with the material which contains no defects.So,it is important and meaningful that if we can make sure about the mechanical behav-iors of C/C composites when we exactly know the void defect frac-tion.If so,it will be helpful for the performance evaluation of C/C composites and structures.To investigate the inﬂuence of the void defect fraction on the mechanical performance of the C/C composite,ﬁve RVE models were constructed and the defect fractions of theﬁber yarns were 0.25%,0.5%,1.0%,2.0%and4.0%.In this study,void defect was assumed only exist inﬁber yarns.Because,as calculated in Section 4.2,voids in carbon matrix and interfaces zone had very little effects on the mechanical properties of the C/C composite.Uniaxial tension simulations were carried out and the stress–strain curves of theﬁve C/C virtual specimens are illustrated in Fig.10.From the simulation results,it is clear that as the defect fraction increased tensile strength of the C/C composite decreased.For the intact FE model,the tensile strength was231.4MPa.For theﬁve FE models with voids defects,the tensile strengths were214.8MPa, 206.6MPa,197.1MPa,182.3MPa and152.8MPa.For the FE model under the defect density0.25%,tensile strength decreased7.2% compared with the intact model.So,if there exist defects inner the C/C materials,even if the volume fraction of the defects was small,it will has obvious effects on the mechanical performance of the composite,especially on the tensile strength.When the defect density was4.0%,tensile strength of the C/C virtual speci-men declined33.9%compared with the intact specimen.5.ConclusionUniaxial tensile properties and meso-structure of the3D orthogonal C/C composite were studied by experimental approaches.Manufacturing defects inner the C/C composite were investigated though a micron-resolution computed tomography (l CT)approach.From the l CT photos of the3-D orthogonal car-bon/carbon composite,it was found that voids and microcracks are two classic type of manufacture defects inner the C/C materials. Base on the statistical analysis of the l CT data,ﬁnite element mod-els of the C/C composite were constructed.According to a new pro-gressive damage model,failure behaviors and mechanical properties of the C/C composites were studied by ABAQUS code. Effects of the void defects on the mechanical performances of the C/C material were numerically investigated.From the numerical simulation results,manufacturing defects such as voids have great effects on the mechanical performance of the carbon/carbon com-posite,especially on the tensile strength.With0.51%void volume fraction,tensile strength of the carbon/carbon composite has 13.2%declines compared with the intact material.When void defects exist inﬁber yarns,even if the volume fraction of the defects is small it still will has great inﬂuence on tensile strength of the C/C composite.However,the defects which exist in carbon matrix and interface have limited effects on the mechanical prop-erties of the C/C materials.So,keep the continuity and improve the density of the carbonﬁber yarns in C/C composite manufacture process is the key to improve the mechanical properties of the C/ C composites.AcknowledgementsFinancial support from the National Natural Science Founda-tions of China(Nos.11202007,11232001,11402132)and the Foundation of Beijing Jiaotong University(KCRC14002536)are gratefully acknowledged.Fig.9.Stress–strain curves of the C/C composite in experiment and simulations.10.Stress–strain curves of the virtual specimens with different void defectfraction.Part B71(2015)113–121。

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W.D.Zhu1Professore-mail:wzhu@umbc.edu

H.RenGraduateResearchAssistante-mail:renhui1@umbc.edu

C.XiaoVisitingGraduateResearchAssistante-mail:chuangx@umbc.edu

DepartmentofMechanicalEngineering,UniversityofMaryland,BaltimoreCounty,1000HilltopCircle,Baltimore,MD21250

ANonlinearModelofaSlackCableWithBendingStiffnessandMovingEndsWithApplicationtoElevatorTravelingandCompensationCables

Anonlinear,planarmodelofaslackcablewithbendingstiffnessandarbitrarilymovingendsisdeveloped.Themodelusestheslopeangleofthecentroidlineofthecabletodescribethemotionofthecable,andtheresultingintegropartialdifferentialequationwithconstraintsisderivedusingHamilton’sprinciple.Anewmethodisdevelopedtoobtainthespatiallydiscretizedequations,andtheBaumgartestabilizationprocedureisusedtosolvetheresultingdifferential-algebraicequations.Themodelcanbeusedtocalculatetheequilibriaandcorrespondingfreevibrationcharacteristicsofthecable,aswellasthedynamicresponseofthecableunderarbitrarilymovingends.Theresultsforanequilibriumandfreevibrationcharacteristicsaroundtheequilibriumareexperimen-tallyvalidatedonalaboratorysteelband.Themethodologyisappliedtoelevatortrav-elingandcompensationcables.Itisfoundthataverticalmotionofthecarcanintroduceahorizontalvibrationofatravelingorcompensationcable.Theresultspresentedareveriﬁedbyacommercialﬁniteelementsoftware.Thecurrentmethodisshowntobemoreefﬁcientthantheﬁniteelementmethodasitusesamuchsmallernumberofelementstoreachthesameaccuracy.Someotherinterestingfeaturesincludetheconditionforatravelingorcompensationcableequilibriumtobeclosesttoanaturalloopandadirectproofthatthecatenarysolutionisunique.͓DOI:10.1115/1.4003348͔

Keywords:elevatortravelingcables,slopeangle,integropartialdifferentialequation,ﬂexiblemultibodydynamics,differential-algebraicequations

1IntroductionAtypicalelevatorsystem,asshowninFig.1,hasthreetypesofcables:suspensioncables,compensationcables,andtravelingcables.Atravelingcabletransmitspowerandcontrolsignalsbe-tweentheelevatorcarandcontroller.Oneendofthetravelingcableisusuallyﬁxedtothewallinthemiddleofthehoistway,andtheotherendisattachedtothecar,whichmovesupanddownalongthehoistway.Acompensationcablebetweenthecarandcounterweightcompensatesfortheweightofthesuspensioncables.Asthecarmovesupanddown,thetravelingandcompen-sationcablescanundergohorizontalvibrationsinthecableplanes.Whilethedynamicsofthesuspensioncableshasbeenstudiedbyseveralresearchers͓1,2͔,thatofthetravelingandcom-pensationcablesinFig.1hasnotbeenstudiedinliterature.Whenthebendingstiffnessofacableisnegligible,Cartesiancoordinatesareoftenusedtodescribethemotionofthecable.ThelinearandnonlineardynamicsofcableswithoutbendingstiffnesswerestudiedbyIrvineandCaughey͓3͔andIrvine͓4,5͔.Whenthesag-to-spanratioislessthan1/8,theequilibriumofthecable,whichcorrespondstothecatenarysolution͓4͔,canbeapproxi-matedbyaparabolicfunction.Animportantdimensionlesspa-rameterthatdeterminesthedynamicsofthecableis͓3͔

␭c2󰀁

ͩ␳gLTͪ2EA

T͑1͒

whereEAand␳aretheaxialstiffnessandmassperunitlengthof

thecable,respectively;Listhelengthofthecable;Tistheten-sion;andgisthegravitationalacceleration.When␭c

2Ͻ24,the

cablebehaveslikeatautstring,andwhen␭c

2Ͼ24,thecablebe-

haveslikeaseesaw͓5͔.ThelineardynamicsoftautinclinedcableswasstudiedbyTriantafyllou͓6͔andTriantafyllouandGrinfogel͓7͔usingasymptoticanalyses.Anonlinear,three-dimensionaltranslatingcablemodelwasderivedandanalyzedbyPerkinsandMote͓8͔.SincetherearenoexternaltensionsinthetravelingandcompensationcablesinFig.1besidesthoseduetogravity,theaxialdeformationsofthecablescanbeneglected.Duetosmalltensionsinthelowerloopsofthecables,thebendingstiffnessesofthecablesneedtobeconsidered.Whenthebendingstiffnesseffectisconsidered,bendingandtorsionalanglescanbeusedtodescribethemotionofthecable,andtheformulationisreferredtoastheKirchhoff–Loverodtheory͓9͔.Antman͓10͔studiedindetailthemathematicalprob-lemsassociatedwiththerodtheory.Forlinearelasticmaterials,thedynamicequationsfromtherodtheoryreduceto12ﬁrst-ordernonlinearpartialdifferentialequations͑PDEs͒,whichcanalsobederivedusingtheNewton–Eulermethod͓11͔.Goyal͓12͔andGoyaletal.͓13͔usedtheﬁnitedifferencemethodtosolvethe12ﬁrst-orderPDEsintheirstudiesofthedynamicsofcablesandDNA.KimandChirikjian͓14͔presentedagrouptheoryapproachtocalculatingtheequilibriaandfreevibrationcharacteristicsoftherod.Santillanetal.͓15͔theoreticallyandexperimentallystud-iedtheequilibriaandstabilityofaloopwithtwoendsclampedtogether.Therearetwomainapproachesinthenumericalsimula-