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挖掘机用多路阀性能检测试验台设计【说明书+CAD】

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湘潭大学兴湘学院毕业论文(设计)工作中期检查表系 机械工程 专业 机械设计制造及自动化 班级 四班 姓 名 于啸天学 号2010963134指导教师刘金刚指导教师职称教授题目名称挖掘机用多路阀性能检测试验台设计 题目来源 科研 企业 其它课题名称题目性质 工程设计 理论研究 科学实验 软件开发 综合应用 其它资料情况1、选题是否有变化 有 否2、设计任务书 有 否3、文献综述是否完成 完成 未完成4、外文翻译 完成 未完成由学生填写随着工程机械的迅速发展,挖掘机的销量越来越大。多路阀作为挖掘机液压系统重要元件,其性能对挖掘机整机的质量和技术水平有很大影响。由于一旦装机后多路阀液压系统出现故障就很难检测到故障源,因此在其装机前对其进行全面的性能检验显得很有必要。但是目前国内现有的多路阀测试试验系统功能不齐全,且系统操作复杂,测试效率低。因此本文以多路阀测试试验相关标准和相关设计技术要求为依据,结合现代挖掘机多路阀中常见的功能特性,利用电液比例技术设计了一套高效率,操作方便的全电液控制式液压挖掘机多路阀综合测试系统。结合多路阀测试系统中阀芯位移难于测量且其对测量结果的影响,依据测试结果对阀芯位移进行了补偿研究。由老师填写工作进度预测(按照任务书中时间计划) 提前完成 按计划完成 拖后完成 无法完成工作态度(学生对毕业论文的认真程度、纪律及出勤情况): 认真 较认真 一般 不认真质量评价(学生前期已完成的工作的质量情况) 优 良 中 差存在的问题与建议: 指导教师(签名): 年 月 日建议检查结果: 通过 限期整改 缓答辩系意见: 签名: 年 月 日注:1、该表由指导教师和学生填写。2、此表作为附件装入毕业设计(论文)资料袋存档。湘潭大学兴湘学院 毕业论文(设计)任务书论文(设计)题目:挖掘机用多路阀性能检测试验台设计 学号:2010963134 姓名:于啸天 专业:机械设计制造及其自动化 指导教师:刘金刚 系主任:签名 一、主要内容及基本要求 1、以多路阀测试试验相关标准和设计技术要求为依据确定测试系统的基本功能,明确各测试试验项目及其实现方法,采用全电液控制方式确定挖掘机多路阀综合测试系统整体技术方案。并完成系统各功能模块单元的方案设计在满足功能要求的前提下通过性能对比以及综合考虑整个测试系统的设计成本,确定各功能单元的最终方案。 2、基于系统整体布局美观、合理,试验台实用、安全、可靠并操作维护简单的理念,对测试系统液压系统进行设计。并根据系统参数,对相关元器件进行相应的计算及选型。结合挖掘机多路阀测试试验的原理和要求,完成综合测试系统的测控系统软硬件设计。3、在研制的全电液控制式挖掘机多路阀综合试验系统上对合作商开发的挖掘机多路阀进行相应的测试试验研究。结合该多路阀具体功能原理,对相关测试试验项目的实现方法进行了概述,并对试验结果进行简要分析。从而验证该全电液控制式挖掘机多路阀综合测试系统功能的全面性和操作的便捷性。二、重点研究的问题以多路阀测试试验相关标准和相关设计技术要求为依据,结合现代挖掘机多路阀中常见的功能特性,利用电液比例技术设计了一套高效率,操作方便的全电液控制式液压挖掘机多路阀综合测试系统。结合多路阀测试系统中阀芯位移难于测量且其对测量结果的影响,依据测试结果对阀芯位移进行了补偿研究。三、进度安排序号各阶段完成的内容 完成时间1查阅资料、调研2014.1.202开题报告、制订设计方案2014.1.253实验(设计)2014.3.104分析、调试等2014.3.205写出初稿2014.4.106修改,写出第二稿2014.5.107写出正式稿2014.5.208答辩2010.5.30四、应收集的资料及主要参考文献1 吴根茂,邱敏秀,王庆丰等. 新编实用电液比例技术M. 杭州:浙江大学出版社,2006.2 路甬祥主编. 液压气动技术手册M. 北京:机械工业出版社,2002.3 杨华勇,曹剑,徐兵等. 多路换向阀的发展历程与研究展望J. 机械工程学报,2005(10):15.4 JB/T 8729.1一1998液压多路换向阀技术条件.5 JB/T 8729.2一1998液压多路换向阀试验方法.6 JG/T 5116一1999液压挖掘机用整体多路阀技术条件.湘潭大学兴湘学院毕业论文(设计)评阅表学号 2010963134 姓名 于啸天 专业 机械设计制造及其自动化 毕业论文(设计)题目:挖掘机用多路阀性能检测试验台设计评价项目评 价 内 容选题1.是否符合培养目标,体现学科、专业特点和教学计划的基本要求,达到综合训练的目的;2.难度、份量是否适当;3.是否与生产、科研、社会等实际相结合。能力1.是否有查阅文献、综合归纳资料的能力;2.是否有综合运用知识的能力;3.是否具备研究方案的设计能力、研究方法和手段的运用能力;4.是否具备一定的外文与计算机应用能力;5.工科是否有经济分析能力。论文(设计)质量1.立论是否正确,论述是否充分,结构是否严谨合理;实验是否正确,设计、计算、分析处理是否科学;技术用语是否准确,符号是否统一,图表图纸是否完备、整洁、正确,引文是否规范;2.文字是否通顺,有无观点提炼,综合概括能力如何;3.有无理论价值或实际应用价值,有无创新之处。综合评 价评阅人: 2010年5月 日 湘潭大学兴湘学院 毕业论文(设计)鉴定意见 学号:2010963134 姓名: 于啸天 专业:机械设计制造及其自动化 毕业论文(设计说明书) 33 页 图 表 2 张论文(设计)题目: 挖掘机用多路阀性能检测试验台设计 内容提要:1、以多路阀测试试验相关标准和设计技术要求为依据确定测试系统的基本功能,明确各测试试验项目及其实现方法,采用全电液控制方式确定挖掘机多路阀综合测试系统整体技术方案。并完成系统各功能模块单元的方案设计在满足功能要求的前提下通过性能对比以及综合考虑整个测试系统的设计成本,确定各功能单元的最终方案。 2、基于系统整体布局美观、合理,试验台实用、安全、可靠并操作维护简单的理念,对测试系统液压系统进行设计。并根据系统参数,对相关元器件进行相应的计算及选型。结合挖掘机多路阀测试试验的原理和要求,完成综合测试系统的测控系统软硬件设计。 3、在研制的全电液控制式挖掘机多路阀综合试验系统上对合作商开发的挖掘机多路阀进行相应的测试试验研究。结合该多路阀具体功能原理,对相关测试试验项目的实现方法进行了概述,并对试验结果进行简要分析。从而验证该全电液控制式挖掘机多路阀综合测试系统功能的全面性和操作的便捷性。指导教师评语指导教师: 年 月 日答辩简要情况及评语答辩小组组长: 年 月 日答辩委员会意见答辩委员会主任: 年 月 日Load-independent control of a hydraulic excavatorEugeniusz Budny*, Miroslaw Chlosta, Witold GutkowskiInstitute of Mechanized Construction and Rock Mining, ul. Racjonalizacji 6/8, 02-673 Warsaw, PolandAccepted 23 August 2002AbstractThe primary focus of this study is to investigate the control of excavation processes by applying load-independent hydraulicvalves. This approach allows avoiding closed loop control system with sensors and transducers mounted on the excavatorattachment. There are, then, no sensor cells mounted on the machine attachment. The considered system is composed of twosubsystems: a microcomputer and a hydraulic unit (a pump and load-independent valves). In the microcomputer unit, the bucketvelocity vector is related to the oil flow into three cylinders through the application of inverse kinematics. Then, flows aretransferred into the electric signals actuating the load-independent valves. Their motion is presented by applying transferfunction. The performance of the system is verified by assuming an abrupt change of the oil flow into cylinders. The last part ofthe paper is devoted to the obtained experimental results. The first result deals with vertical drilling. The second result dealswith an excavation along a horizontal trajectory.D 2002 Elsevier Science B.V. All rights reserved.Keywords: Excavator; Hydraulic systems; Control; Trajectory execution1. IntroductionDue to encouraging results of recent research,there are increasing possibilities for enhancement ofa large spectrum human efforts in excavation pro-cesses. This may occur mainly through control ofrepetitive work tasks, such as trenching and drilling,requiring constant attention of machine operatorsduring the performance of each task. Particularattention, in research, is paid to excavation alongprescribed trajectories subjected to varying soil envi-ronment.Fundamentals dealing with controlled excavationprocesses are discussed by Vaha and Skibniewski 1,Hemami 2, and Hiller and Schnider 3. An inter-esting approach to piling processes by a directangular sensing method is proposed by Keskinen etal. 8. Budny and Gutkowski 4,6 proposed asystem, applying kinematically induced motion ofan excavator bucket. In this approach, influence ofa small variation of hydraulic oil flow into cylinders,applying sensitivity analysis, is discussed by Gut-kowski and Chlgosta 5. Huang et al. 7 presentedan impedance control study for a robotic excavator.They applied two neural networks: first, as a feed-forward controller and the second as a feedbacktarget impedance. Another impedance system, apply-ing a hybrid position/force control, is proposed by Haet al. 9.The first generation of robots was conceived asopen loop positioning devices. This implied thatall parts had to be manufactured with a very high0926-5805/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.PII: S0926-5805(02)00088-2* Corresponding author.E-mail address: mch.pl (E. Budny).URL: .pl./locate/autconAutomation in Construction 12 (2003) 245254and costly accuracy. Next, positioning robots, withsensors, reduced this accuracy requirement consider-ably. Here were several approaches, mentioned inabove references, to extend the industrial robotscapabilities to robotic excavator. Systems of forcecells, longitudinal and angular sensors have beenapplied. However, two main differences betweenrequirements for manufacturing robots and roboticexcavators should be noted. The first difference isthat manufacturing robots are working in almostperfect conditions, free of vibrations, protectedagainst shocks, humidity, and other possible damag-ing conditions. The second difference is the require-ments for high accuracy of manufacturing robots,often within microns. On the contrary, robotic exca-vators are working in very difficult construction siteconditions, and required accuracy of the executedtrajectories, comparing with industrial robots, islimited, say within centimetres. With difficult con-ditions of excavations works, all sensors attached tothe boom, arm, and bucket have to be very wellprotected.Bearing in mind the above differences, it would beof interest to investigate the possibilities of controllingexcavation trajectory by a hydraulic module com-posed of a pump and load-independent valves. Inother words, to investigate a system free of sensorcells mounted at the excavator attachment, combinedwith a feedback controller, included in the hydraulicunit of the machine. The main objective of the presentpaper is to extend the discussion, initiated by theauthors 10, on the possibilities of applying load-independent valves installed inside of operator cabinonly. Under this assumption, the system is free ofsensors located on the excavator attachment. Afterdiscussing mathematical model of the system, pre-liminary experimental results are presented at the endof the paper.2. Statement of the problemThe paper deals with a controlled, stable motionof an excavator bucket along a prescribed path. Theproblem is based on previous authors theoreticalinvestigations 4 of quasi-static, kinematically in-duced excavation processes for assumed trajectories.In this study, the following assumptions are made.The excavator attachment is a planar mechanism,composed of a boom, an arm, and a bucket. Three,independently driven, hydraulic cylinders operate thesystem. They are assuring a unique representation ofthe three degrees of the planar bucket motion, twodisplacements and a rotation.The excavation process, in the experiments per-formed,isassumedtobeslowenoughtoconsideritasaquasi-static one. Inertia terms in motion equations ofattachment can be then neglected. Only spool of theservomechanism is assumed to move with accelera-tions, which cannot be neglected.The force (pressure) disturbances are assumed tohave sinusoidal form. The acceptable parameters of thesinusoid are defined from stability conditions of thesystem.The soil is assumed homogeneous. Some smallinclusions in the form of stones are acceptable.The proposed control system of excavation isoperator-assisted. It means that in a case of a largerobstacle, the operator has to intervene.If successful, the proposed control setup couldapply to standard excavators with the aim of enhance-ment of a large spectrum of human efforts in repetitiveprocesses such as trenching and drilling.The experiment is considered as a system com-posed of three subsystems, namely: microcomputerwith PLC; hydraulic arrangement (a pump, valves,cylinders); and the mechanism with three degrees offreedom of the bucket. Next, the subsystems areconsidered as sets of components. In the first sub-system, the following components are recognised:personal computer with appropriate software, trans-forming introduced equations and inequalities ofmotion and trajectory planers into electric signal.The latter is send to a PLC unit, which in turn causesan electrical actuation of solenoid valves. Pressuresfrom the solenoid valves are causing changes in spoolpositions, assuring assumed flow of the hydraulic oilinto cylinders. The spool position, in turn, is con-verted through a transducer to an electric feedbacksignal sent to the solenoid valves. Opened spools areletting the hydraulic oil to flow into the third sub-system, namely cylinders of the excavator mechanism.Finally, the last subsystem is composed of threecomponents: the hydraulic cylinders, the boom, thearm, and the bucket. With the motion of the excavator,arms and the bucket itself, the pressures in cylindersE. Budny et al. / Automation in Construction 12 (2003) 245254246are changing. Information about these changes is sentto the second, hydraulic subsystem, where the feed-back signal corrects position of spools assuring the oilflow according to the designed trajectory.In the paper, transfer functions of all systemcomponents are investigated from the point of viewof stability. The functions are defined theoretically, ornumerically from diagrams presented in catalogues ofhydraulic equipment. Joining all transfer function ofparticular component, the transfer function of thewhole system is discussed from the point of view ofperformance under abrupt unit signal.Several experiments were performed, showing thatit is possible to assure stable, assumed motion of thebucket. Among experiments, one was devoted to drill-ing. In other words, the kinematically induced trajec-tory was a straight, vertical line. Experimentallyobtained line is presented in Refs. 6 and 10. It isinteresting to note that the variation of experimentalline does not exceed 10 cm.3. Three subsystems of the experimental setupThe discussed system is divided in three sub-systems, namely: microcomputer, hydraulic valves,and excavator arms with a bucket. Below, they arediscussed separately and then a joint control prob-lem is defined.3.1. Microcomputer as a subsystemWe start with defining a model of the end-effector (bucket, drill, hammer) motion. The end-effector, in its plane motion, has three degrees offreedom aj(j=1,2,3) (Fig. 1). They are rotations ofthe boom, of the arm, and of the effector.Denoting by x1p, x2pposition of the end-effectortip, and by x3its rotation, the kinematics of theconsidered mechanism is represented by vectorrelation:x1px2px3p266664377775c1c2c30s1s2s30000a3266664377775?l1l2l3266664377775;1where cjand sjdenote cos ajand sin aj, respec-tively. In further considerations, the sub index p isomitted as the position of only one point is con-sidered.Velocity of the point P, v=v1, v2, v3T=x 1, x 2,x 3Tis obtained by taking time derivative of Eq.(1), and by reducing 3?4 matrix to a 3?3 matrix: x v A a a a Aw;2whereA ?l1s1?l2s2l3s3l1c1l2c2l3c3001266664377775:3Taking inverse of A matrix equal to:A?1l2c2l1c10?l2s2?l1s10l2l3f23l1l3f13l1l2f12266664377775?1l1l2c1s2? s1c24with fij=sicj?cisj, we find the inverse kinematics,relating angular velocities of mechanism elementsto the tip displacement vectorw A?1v:5Angular velocities xj, in turn, are dependent on theelongation velocities hiof hydraulic cylinders. Thisdependence has to be determined from geometricalrelations between cylinder lengths, constant param-eters of attachment, and aj.We start with the first cylinder. From Fig. 2 we findcoordinates of two cylinders hinges, A1and B1.They are:x1A1 a0;x2A1 b0;x1B1 b1c1 a1s1;x2B1 b1s1? a1c1:Takingh21 x1B1? x1A12 x2B1? x2A22;E. Budny et al. / Automation in Construction 12 (2003) 245254247after transformation we obtainh21 p01 q01c1 r01s1;6wherep01 a20 a21 b20 b21;q01 2a1b0? a0b1;r01 ?2a0a1 b0b1:Taking time derivative of Eq. (6) we find:h1?q01s1 r01c12h1? x1G1112h1? x1:7Repeating the same consideration for the secondcylinder length (Fig. 3) we obtainh22 p02 q02f12 r02g128wherep02 a22 a23 b22 b23;q02 ?2a2a3 b2b3;r02 2a2b3? b2a3;Fig. 1. The mini-excavator considered.E. Budny et al. / Automation in Construction 12 (2003) 245254248andfij cicj? sisj;gij sicj cisj:9Taking again time derivatives of Eqs. (8) and (9), wearrive ath2?q02g12 r02f122h2? x1 x2G2122h2? x1 x2:10An expression representing the length h3of thethird cylinder is more complex, and requires intro-duction of an auxiliary variable a4(Fig. 4). With anew variable, there is a need to introduce an addi-tional relation. In this case, the relation joins varia-bles a2, a3, and a4, through the condition thatdistance between B3and D3is constant and equalto b7. After some lengthy transformation, theserelations take the following form:h23 p03 q03f24 r03g24;11b27 p04 q04f23 r04g23 q05f24 r05g24;12wherep03 a24 a25 a27 b24 b25 b4b5;q03 ?2a7a4? a5;r03 2a7b4? b5;p04 a25 a26 a27 b25 b26;q04 2b5b6? a5a6? a6a7;r04 ?2a5b6 a6b5;q05 2a5a7;r05 ?2a6a7:Fig. 4. The length h3of the third cylinder.Fig. 2. The length h1of the first cylinder.Fig. 3. The length h2of the second cylinder.E. Budny et al. / Automation in Construction 12 (2003) 245254249Taking time derivative of Eq. (11) and recallingthat aj=xj, the velocity h3can be presented as:h3?q03g24 r03f242h3? x2 x4G3242h3? x2 x413The mentioned condition for b7in the form of Eq. (12)allows to find a4, and eliminates it from the otherequations. Taking now time derivative of Eq. (12), wecan express x4in terms of x2and x3x4 ?G423G524 1? x2?G423G524? x3;14whereG423 ?q04g23 r04f23;G524 ?q05g24 r05f24:Combining, now, together Eqs. (7), (10), (13), and(14) in a vector notation, we can write:h H ? w15withH12 H13 H23 H31 0;H11G1112h1;H12 H22G2122h2;H32 H33 ?G324G4232h3G524:The flow of the hydraulic fluid into jth cylinder,denoted by qj, is equal to hjSj, where Sjis the cross-section area of the cylinder. With above notations,we can write the final relation between assumedvelocity vector of the end-effector and flow vectorq asq S ? H ? A?1? v16where S is diagonal matrix with components Sj(j=1, 2, 3). The flow (Eq. (16) is a calculatedflow, which in our model is needed to move the endeffector according to its assumed motion. In a realsystem, this amount of oil has to be supplied to realcylinders through valves. The latter must be thenactuated by an electrical signal vector u. The relationof qj=qj(uj) between this signal and oil flow is givenby valve characteristic, which in general has theform presented in Fig. 5. The positive values of qjare related to the elongation of the cylinder. Thenegative ones are related to its shortening.The curve representing graphically qj(uj) can beassumed to be represented by the following function:qj a1u ? b a3u ? b3 a5u ? b5;17with constraints d imposed on maximum openings ofthe valve. Coefficients a1, a2, and a3can be deter-mined by fitting the function (17) at three points of thecharacteristic curve. In order to find electrical signal ujin terms of qj, we have to take the inverse of Eq. (17).In general, this can be achieved only through anumerical solution method.3.2. Hydraulic valve subsystem (HVS)The calculated in microcomputer, reference elec-trical signal is now converted into real electricalsignal, actuating the valve. In the problem discussedhere, this is a load-independent, proportional valvePVG 32 by DanfossR. The discussed subsystem ispresented in Fig. 6. Below, all of its parts and theirtransfer functions are discussed.Fig. 5. The oil flow q leaving the valve, as a function of uj.E. Budny et al. / Automation in Construction 12 (2003) 245254250The difference between reference signal ujand ud,and a signal coming from the feedback, is actuatingthe controller. The controller in turn, is adjusting thepump pressure ppto a pressure pcneeded for anadequate position of the spool. This adjustment isdone by four solenoid valves. Denoting by capitalletters the Laplace transforms, we find:Ucs Ujs ? Uds Ujs ? Huds ? Ds;18where D(s) is Laplace transform of spool displace-ment d; Hudis a transfer function between the spooldisplacement and feedback signal ud.The latter is obtained by a transducer, with constantmultiplier, giving:Huds UdsDs Kd:19The relation between UCentering the controller and pcleaving it is also constant:Gpus PcsUcs Kc:20The pressure acting on the spool is causing its motion,defined by an equation for one degree of freedom,with a spring constant ks, spool mass m, dampingcoefficient c, and cross-section area on which thepressure is acting As:md cd ksd pcAs:21The transfer function between spool displacement dand pressure pcis then as followss2m sc ks ? Ds Pcs ? As:22Considering now Eqs. (18)(21), we obtain the rela-tion between the transformed output of spool displace-ment and transformed reference input of electricalsignal:Djs As? KcAsKcKd ks sc s2m? Ujs;23or considering feedback electrical signal Ujd, we haveUjds AsKcKds2m sc AsKcKd ks;24With a constant nominator and denominator, in theform of a second order polynomial, we can verify theperformance of our control setup by assuming elec-trical signal equal to a unit step functionujt ujut25which implies an abrupt change in the cylinder length.Considering now Eqs. (24) and (25), carrying apartial fraction expansion, and taking inverse Laplacetransforms, we find the error e(t) as a function of time:et e?fxntcosxdt fxnxdsinxdt?ujdt;26where2fxncm;x2nAsKcKd ksm;xd 1 ? f21=2xn;f 1 weak damping:Fig. 6. Hydraulic valve subsystem.E. Budny et al. / Automation in Construction 12 (2003) 245254251The relation (26) shows that the error asymptoti-cally tends to zero with the increase of time.4. Experimental realization4.1. The mini-excavator used for experimentsThe mini-excavator K-111 is used for experiments.It was assumed to minimize part replacements, in aserial machine, needed to perform the consideredcontrol. The main components in the hydraulic systemto be replaced were valves. Moreover, the hydrauliccylinders are supplied with additional valves assuringrequired pressure. This ensures that unpredictedmotion of the attachment is not taking place. Thehydraulic load-independent valves used in this experi-ment were supplied by DanfossR. The transfer ofinformation from a microcomputer to load-independ-ent hydraulic valves is conducted by a ControlledArea Network (CAN).After modification of the hydraulic system, theexcavator can be controlled in two different ways.The first method consists in using joysticks mountedin the operator cab. This way, using a joystick, theoperator can move the mechanism in an arbitraryposition, and with desirable velocity. The secondmethod consists of programming the bucket motionin a microcomputer. The information from it is thentransformed in elongation rates of cylinders, and inthe flow of the oil moving them. The latter isconverted in an electrical signal send through CANto load-independent valves. The organization of theelectrical system for load-independent valves isshown in Fig. 7.The control algorithm is written in Borland Pas-cal and executed under MS Window 98 operatingsystem. The CAN communication rate is assumed tobe 250 kbit/s with sampling time between 0.5 and2.0 s.Fig. 7. Hardware of the control subsystem.Fig. 8. Experimental results of vertical drilling.E. Budny et al. / Automation in Construction 12 (2003) 2452542524.2. Experimental resultsTo examine the performance of the proposed con-trol system, the above-mentioned mini excavator withthree hydraulic cylinders and three degrees of freedomis used. An electro-hydraulic, load-independent, pro-portional valve controls separately each cylinder.Experiments were done in order to control themotion of the bucket tip along straight lines, a verticalone, and a horizontal one. As mentioned in Statementof the problem, the motion was controlled both in freespace and soil box, filled with homogenous mildlyhumid sand.The experiments were performed with relativelysmall velocity of about 2 m/min. Trajectories obtainedfor drilling along a vertical line and the movement ofthe bucket tip along a horizontal line are presented inFigs. 8 and 9.5. ConclusionsA relatively simple control system for an excavatoris proposed. The system is free of sensor mounted onthe attachment of the machine. The whole controlhardware is within load-independent valves, located inthe operator cabin. After some additional study andimprovements, the system could be applied on mass-manufactured excavators, helping in works with repet-itive processes, like digging trenches or drilling.The experimental results are showing good qual-itative performance. Also, quantitative results, in thecase of horizontal line, are promising. Here, theobtained trajectory varies from straight line by nomore than 4% of the traveled distance. Less accurateresults are in the second case, namely for vertical line.Here, errors, for some experimental runs, are even upto 15%. This might be caused, among others, by thefact that elongation of one of the cylinders is changingsign during the motion. It means that for a moment, itsspool is closing oi
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