外文翻译--曲柄滑块机构基于连杆-滑块间隙流体润滑的动态分析

曲柄滑块机构基于连杆 -滑块间隙流体润滑的动态分析 雷戈瑞 .丹尼尔 ，凯迪 .凯文 机械设计部，坎皮纳斯大学，机械工程学院，邮 件 信箱 6122， 13083-970 坎皮纳斯， SP ，巴西 摘要 传统的曲柄滑块连杆机构广泛的应用于机械系统中。轴承在机械中的使用特别作用在于减小摩擦，主要用于特殊条件的润滑，如曲柄滑块的铰链连接。这类轴承属于一类交替旋转运动的轴承。考虑到曲柄滑块系统的动力学与轴承润滑现象是相互作用的，本文为这种特殊问题提供了一种数学模型。两种数学模型被使用与动态系统的分析。第一种模型（ Eksergian 的运动方程），假设，连杆一端与轴承表面接触时代表系统，在这种情况下，连杆的行为与刚性轴承相同（无间隙）。第二种模型（拉格朗日方法），当连杆一端在以在滑块的间隙中为润滑方式时，则代表系统， 在这种情况下，在关系到连杆滑块移动， 则 呈现出多自由度问题 。流体动力润滑的数学模型介绍了获得系统动态行为中更为现实的结果。 2011 Elsevier Ltd. 保留所有权利 . 1，介绍 在众多功能中 润滑系统 是评价机械最佳性能的一个至关重要的因素 ，例如，润滑和保护部件，减少摩擦，清洗和冷却的内部机制。 本系统的操 作需要适当的校准。因此，过多或者润滑不足会影响机器的动态行为，会严重损害所设计的机构。许多机器的机构，包含了大量的流体动压轴承，而轴承连杆 -滑块联合尤为重要，因为这是一个新的流体动力轴承，被称为流体动力轴承交替运动。不同于传统的动压轴承，这种轴承不是一个完整的旋转。因此，随着近来来日益增长的需要，研究这一特定类型的流体动力轴承和机器的动态行为的影响。目前为止，大多数关于滑块连接的铰链的流体动力轴承正在被研究所所研究着。事实上，该研究所已开发和建造的设备来研究摩擦润滑滑块销轴承超过 15 年。比如， Takiguchi 1 等人研究一种浮式活塞在汽油动力汽车引擎上采用旋转运动。三年后，他 2 开发一个测量的设备， 确定在连杆 滑块联合的基础上测得的摩擦力的流体动力轴承的润滑状态。 在另一项研究中，在武藏技术研究所进行， 有 Suhara等人 3 检查润滑条件下轴承的滑销的汽油发动的汽车发动机，作为考虑的参数分析的长度，内径和活塞销的材料。最近，张等人 4 开发 的 工具，探讨在活塞销轴承磨 损。 2005，利吉尔、拉戈分析了活塞销轴承的行为。一年后，这些作者 7 表达饿了 连杆 活塞接头在四冲程发动机轴承运行的 总的看法，强调在轴承供 油。 如前所述，流体动压轴承的往复运动的滑块曲柄机构 一部分 。因此，这一机制可靠的的数学模型必须考虑连杆 滑块联合轴承的行为。由于 曲柄滑块 机构的广泛应用，许多研究都集中在设计的数学模型，分析了机械系统的动力学行为。施瓦布，梅嘉德，梅耶尔 8 比较 了 滑块曲柄连杆机构的动力学行为，考虑到不同的连杆 滑块联合模型如耗散赫兹接触模型，影响模型和动压轴承模型在上述工作， 在 考虑连杆弹性与刚性构件 的同时， 作者还比较了动态行为的机制。那些作者所获得的结果表明，连杆的假设为弹性元件，以及在关节的润滑条 件，往往显着降低振动机构的动态响应。弗洛雷斯等人 9 分析了 曲柄滑块 连杆机构的动态行为建模，连杆 滑块联合作为一个干接点无摩擦，干摩擦接触和混合模型考虑了小偏心率和高偏心率的摩擦接触的流体动力润滑。他们的研究结果表明，由于在动态响应的振荡幅度 ， 干接触摩擦模型比干接触模型无摩擦更现实的。对于混合模型，他们获得的结果显示最小的振荡的动态响应，但在文献中 没有结果能 支持这一观察。此后不久，弗洛雷斯等人 10 分析了滑块曲柄机构考虑 干接触影响的动力学，在连杆 滑块摩擦和流体动力润滑 铰链 。得到的结果与干接触模 型无摩擦表现出较高的振荡机构的动态响应，使该模型比摩擦干接触模型不现实。当轴承模型被认为是，结果是那些与理想的滑块曲柄机构 铰链 获得非相似。 erkaya， su ,uzmay 11 进行了运动学和动力学的一个改进的滑块曲柄机构和一个额外的 偏心连杆之间的连杆和曲柄销。他们 将 滑块曲柄机构开发得到的结果进行了评价 ，同时与 常规滑块曲柄机构比较。这种比较表明，虽然传统的和改进的滑块曲柄机构的行程和缸内气体压力的改性机理相同，具有比常规的机制更高的扭矩输出。 khemili 和 Romdhane 12 分析了平面柔性 滑块曲柄含间隙机构的动力学行为，进行仿真和实验测试 ， 比较利用亚当斯软件模拟的数值实验结果。他们发现，间隙的存在影响了系统的动态响应，和耦合器的灵活性作为机制的悬挂系 统。 estupi 和 santos 13 建立一个线性往复压缩机的数学型。他们检查了机械系统的动态行为，考虑到基于多体动力学的机械部件的动力学（刚性部件）和有限元法（弹性元件）。他们还评估了流体动力轴承的影响，描述在 铰链 处用雷诺兹方程的流体动力。他们的研究结果显示，得到当活塞接近上死点，由于倾斜振荡的曲柄的轨道的非线性行为的增加是最大力和最小 油膜厚度，这是由曲柄销的长度的影响。 这项工作涉及的滑块曲柄机构和轴承的流体动力润滑条件 在 滑块连杆 铰链处 的动态行为分析。为此，分析是基于两个不同的模型，其应用程序依赖于润滑条件（偏心值）的轴承。第一个模型（ 1DOF）认为，连杆 滑块联合作为一种理想的联合应用时，连杆的一端与轴承表面接触 14 。第二个模型（ 3DOF）时，连杆一端的滑块润滑孔的间隙。在这种情况下，在关系到连杆滑块，表征问题的多个自由度，通过流体的流体动力润滑连杆和滑块之间发生的相互作用。因此，解决的办法是从一个由滑块曲柄机构 理想 铰链 与连接杆 滑块联合轴承滑块曲柄机构的模型的混合模型得到的。 因此，在这种分析的一个重要参数是考虑最小油膜厚度在流体动力润滑。根据弗洛雷斯 9 ，高偏心率（低最小油膜厚度），压力使表面的弹性变形，这可以是相同的顺序作为润滑油膜厚度。这些情况与那些在流体动力润滑获得一个更现实的分析，可以基于弹流润滑理论。因为这个原因，这项工作被认为是最小油膜厚度超过 10%的径向间隙的流体动力润滑状态，这代表 0.9 的偏心率。 从混合模型得出的曲柄滑块机构的动态响应与从常规的模型相比（理想滑块曲柄机构 铰链 ）。 此外，在轴承的压力分布一个周期期间获得的润滑条件。它强调的是，通过与 Bannwart 15 合作以前开发 相比， 作者在目前的工作中使用的流体动力学模型 更为 重要 。 2 方法 在这项工作中，一个平面曲柄滑块 机构 模型， 被用来 确定其动态行为。然而，滑 液压动压 轴承 被认为是 在连杆滑块 的一个铰链， 代替传统的曲柄滑块机构 。这种假设认为这样当轴承不考虑间隙 时 ，销不局限于只有一个方向的运动。本节描述的混合模型（自由度）的平面曲柄滑块机构 ， 考虑连杆 滑块联合滑动轴承，这是用来分析在润滑条件下的动态行为。该数学 模型用于分析接触状况的动态行为是传统 的 滑块曲柄机构（ 1自由度） ，是 Doughty 14 提出的。 2.1 平面曲柄滑块机构的运动 学 分析 平面曲柄滑块机构的运动学 研 发考虑 了 图 1 中的方案。图 1a显示了平面滑块曲柄机构和图 1b描绘了自由体图的连杆销和滑块孔。 R是曲柄的长度， L 是连接杆的长度， Q是曲轴的角位移，一个是连杆的角位移， XP 和 YP是连杆销线性位移， XPT 和 YPT 是滑块的直线位移， FXP 和 FYP 是 分别在 x和 y方向上的流体动力。图 1c显示 的是 连接的平面滑块曲柄机构杆 液压动压轴承 轴承。 OH 是中心的轴承， OP 是连杆销中心， RH 是轴承的半径， RP 是连杆销半径， E是偏心的， hmix是最小油膜厚度， hmax 是最大油膜厚度。此外，偏心率（ ）之间的偏心率（ E）和径向间隙（ CR），在径向游隙的轴承半径和连杆销半径之间的差异 根据这个数字，滑块的销的位置可以被表示为公式 1所示： Xp = R:cos（ q ） + L:cos（ A） (1) Yp = R:sin（ q） L:sin（ A） : 公式 1对连杆销，速度和加速度 的派生公式 ，得到： 图 1。 平面滑块曲柄机构，（一）普通视图，（ b）扩展视图，（ C）的流体动力轴承。 在 KP 是的滑块销速度系数矩阵，和 LqP和 LAP是滑块的销的速度系数的偏导数矩阵，定义为： 2.2. 对曲柄连杆系统 质心运动学分析 这种分析的目的是确定的组件的质量中心的动态行为，使 它 找到一个滑块曲柄机构 14 的 运动方程。 图 2a 和 b 显示 的 分别 是 曲柄连杆与曲柄质心杆子系统和连杆。 Upm 和 VPM 是曲轴的质量中心的坐标参考系统（ PM）在位于曲柄（ UM， VM）， UPb 和 VPb 是 质心坐标（ PB）的参考系统位于连 杆（ UB， VB）， XPM 和 YPM 的曲轴的质量中心直线位移（ PM）在惯性参考系统（ X， Y）和 XPB 和有排搏是连杆的质心线位移（ PB）在惯性参考系统（ X， Y）。 根据图 2a，曲柄的质量中心位置可以被描述为： 速度可写式的 派生 形式（ 4）为： 考虑在图 2b 的方案 ， 得到同样的连杆质心位移和速度。 图 2。曲柄连杆系统，（一）对曲轴质量中心子系统，（ b）在质量上的连杆中心子系统。 为了确定曲柄连杆系统的动态行为，线性和角的这个子系统的所有组件的速度必须分 开 14 。这些速度是需要确定的动能（旋转 和平移）的子系统，因此，采用拉格朗日方法得到的运动方程。因此，根据公式 7和公式 9，它可能是写： 在 KC代表曲柄连杆系统的广义速度系数矩阵。导出 KC相对独立的变量 Q 和，有： 在 LQ和 LA是广义速度系数的偏导数矩阵。 最后，分别考虑到 MM， MB， IM E IB 曲柄连杆质量，质量，转动惯量和曲柄连杆转动惯量，质量矩阵可以得到： 2.3 在曲柄连杆系统 的 广义力 通过虚功的概念，得到了广义的激振力，在这里提出的混合模型的情况下，由于流体动力润滑在连杆销 滑块间隙应用 于 耗散粘滞力。根据强 14 ，广义力可确定使用广义速度系数矩阵的连接点，在外力施加的 条件下 。因此，广义力是由公式 14给出了： 由于 液压 动力施加在滑动 铰链 （ FXP 和 FYP） 中，导致 广义力发生。在这项工作中， 液压 动力是由滑动轴承的润滑模型确定的振荡运动，通过邦瓦特 15 的 提议。 2.4 的曲柄连杆系统的动能和势能 曲柄连杆系统动能的质量矩阵和广义速度系数矩阵的确定，根据式 15： 图 3显示考虑到这些组件的质量中心 ， 子系统由曲柄、连杆 组成 。 根据图 3A，潜在的能量可以写成： 3.结果 本研究的目的是确定模型的适用性。计算机模拟在 FORTRAN执行，允许在流体动力轴承的润滑状态分析。 图 4中的图表明求解过程在动态分析中的应用。首先，在轴承的流体动力分析的初始条件。连杆销和滑块加速度的确定和 液压 动力计算。因此，对初始条件求解运动方程，并与加速度的水动力进行了评价。因此，连杆的销和滑块的位移和速度计算到下一个时间步长（ T +T ）。在确定了新的速度和位移，销轴承偏心必须验证。如果偏心率小于 0.9 时，该 销 是在润滑条件下。在这种情况下，新的液压 力计算，先后对销和滑块的加速度的重新确定 。 在 流体 动力润滑条件下，考虑在第 2.7 节润滑模型 ，对 系统的动态行为进行评估，。然而，当滑块销超过阈值（偏心 = 0.9 ），它被认为是与轴承的表面接触。在这种情况下，它是假定系统的动态行为是由传统的滑块曲柄机构的，在连杆 滑块联合没有间隙。因此，滑块的曲柄连杆机构 常规模型应用到一个反向运动发生在滑块。在滑块的反向运动后，滑块销叶片轴承表面，改变润滑条件 。 至于运动微分方程的分辨率，传统的模型并不复杂，因此可以解决比较快（在 2 GHz处理器， 3 GB RAM 的计算机系统 10 分钟）。 然而，由于高数值刚度的润滑条件下的运 动方程是高度复杂的。因此，一个特定的数值积分方法必须根据这些方程的特征选择。在这项工作中，对 3自由度的解决方案。混合模型是通过使用一个僵硬的初始值问题的积分多步法的发现，rowmap 方法 16 。然而，尽管这种方法的效率，对微分方程的解还需要计算成本高（约 48 小时，在 2 GHz 处理器， 3 GB RAM 的计算机系统）。 列于表 1 和表 2参数分别表示轴承的几何形状和 滑块曲柄机构。从传统的内燃发动机，得到了这些参数。表 3 给出的初始条件，在计算机模拟中考虑并与文献一致，在操作过程中发现了一些实验值。 仿真 我们开始考虑在表 1和表 2给出的物理参数，初始条件列于表 3和 20M 轴承径向间隙。滑块的曲柄连杆机构的动力学与连杆轴承滑块 铰链的 分析 在这项工作中的计算机模拟，考虑流体动力润滑的阈值条件。因此，流体动力润滑条件下高达 0.9 的偏心率是极低的膜的厚度，这是流体动力润滑不充分的条件。因此，最大限度的模拟偏心率的值是 0.9，这表明 销在 假设是接触支承面 上不 是液压 动力条件 。 图 5a 和 b 分别显示在第一和第二圈获得的轨道。如图 5a表示，销的初始偏心率是 0.6。当运动开始，销向左移动直到超过偏心极限，之后它可能仍然在与轴承表面接触到滑块的运动倒置。在这种情况下，即，当销与轴承的接触面，动力学是基于传统的滑块曲柄机构的 14 的数学模型分析。当运动倒置，销从轴承表面的轴承上的区域。 表一 流体动力轴承的参数。 表二 曲柄滑块机构的参数 表三 采用计算机模拟的初始条件。 接近表面的轴承，销和滑块推向相反的方向上的流体动力，使它们相对于彼此移动。这使得 销 回到轴承的下部区域，再次超过偏心极限。销并保持与表面接触到底圈。在第二圈，活塞销的动态行为类似于第一圈。 图 6说明了曲轴的动态行为。可以看出， 曲柄的速度和加速度的两个系统，即是相似的，滑动轴承滑块曲柄机构（ 3自由度）和常规滑块曲柄机构（ 1自由度）。然而，请注意，速度和加速度都倾向于作为一个在 3自由度模型的时间函数的两个应用模型之间的过渡的增加。 图 5。轨道滑块的联合模拟 , (a)第一圈 ,(b)第二圈。 图 6曲柄分析模拟 (a)角速度 ,角加速度 (b)。 图 7.连杆分析模拟 (a)角位移、角速度 (b)。 当销超过偏心极限，最后重点分析了在水动力条件的 1 自由度的初始条件。传统的机制（接触）。因此，在这一点上的运动，是由上述 自由度模型 给定初始条件的动态行为描述。因此，在混合模型与传统的过渡，输入的速度不同，给出了两种模型在 t = 0 的初始条件。因此，主要作用是在模型的每个过渡曲柄的角速度的增大，这是相对于 1 自由度位移 的 传统的模型。 可以看到在图 6 b,曲轴的角加速度小断层与传统的模型。虽然曲柄不能直接连接到水动力轴承 ,这些不再生产的动态响应是由销在润滑条件。相同的行为由施瓦布报道 ,Meijaard 和 Meijers8。根据 Schwab8,与 液压 动力轴承的动态行为曲柄滑块机构在 曲柄滑块 联合常规机制类似 ,所代表的动态响应的平滑曲线。然 而 ,反应略有不同 ,只有当销进入轴承。这穿越涉及高速度和期刊中心 ,因此 ,它增加了高峰。 图 7 描述了连杆的位移和速度。注意连杆的位移、速度和轴承滑块曲柄机构（ 3自由度）和常规 滑块曲柄机构（ 1 自由度）是相似的。 图 8 所示，滑块的动态行为的两个系统是相似的（ 1DOF 和 3 自由度 ）。 图 9 显示了连杆的角加速度和滑块的直线加速。注意连杆和滑块的加速度与轴承滑块曲柄机构达到（ 3 自由度）类似于传统的滑块曲柄机构实现加速度（ 1 自由度）。然而，在与轴承表明振荡由于在流体动力润滑条件的 铰链 行为的滑块曲柄机构达到加速度。 图 8。 滑块分析模拟 (a)线性位移、 (b)线性速度。 图 9。 在模拟加速度分析 ,(a)连杆加速度 , (b)滑块加速度 如图 9显示， 在 润滑时更重要的是在滑块的加速度和连杆轴承的原因引起不连续，因为这些组件的交互。当销是不是在润滑条件下，它被认为是接触支承面。的数学模型，用来分析在接触条件的动态行为是传统的滑块曲柄机构模型。因此，机构的动态行为是传统。 4 结论 这项工作涉及的滑块曲柄机构和 在 润滑条件 下 滑块 铰链 的动态行为分析。为此，考虑连杆滑块 铰链 的流体动力轴承 ，建立 一个曲柄滑块连杆机构 的数学模型，。在这项工作中，数学模型的计算机模拟了连 滑块 铰链 销的动态行为。结果表明，在操作过程中进行 的 销两个条件。在某些时刻 销 进行流体动力润滑条件和别人是在与轴承的表面接触。数值结果表明， 不论 初始条件轴承参数 多少， 销断裂的油膜，进入与轴承的表面接触， 如前所述的规定，达到一个高的偏心销，使油膜压力甚至分裂它。这种行为表明，弹流润滑模型的混合模型接触条件的使用（ 3 DOF。）可能更适合本文的分析，在这种情况下，直接接触可以防止甚至当销不动压润滑。然而，评估的动态响应和压力分布润滑时，所提出的混合模型可 以应用。的弹流模型的实现需要从流体的混合模型弹流状态过渡的进一步研究。 对于轴承的径向间隙，可以得出结论，在径向间隙的增加会导致油膜厚度随之增加，降低水动力和促进与轴承表面引脚的接触。相反，在初始角速度的增加导致的流体动力作用于销增加，因为这些力量的初始角速度成正比。这些较强的 液压 动力增加销 的轴承表面的距离，防止触碰这面。 连杆滑块 铰链 是相对复杂的流体动力轴承由于使用的两种流体动力润滑和滑块曲柄机构 模型的非线性方程组 滑块曲柄机构问题的解决方案。这组方程提出了高数值的刚度，使数值积分方法在微分方程的成功解 决问题的一个重要因素的选择。 致谢 作者希望感谢 FAPESP， CAPES 和 CNPQ 提供 的资金支持这项工作的。 参考文献 1 M. Takiguchi, M. Oguri, T. Someya,，旋转在汽油发动机活塞销的运动研究， SAE 论文 938142，底特律，美国， 1993. 2 M. Takiguchi, K. Nagasawa, T. Suhara, M. Hiruma,摩擦小端连接一个汽车发动机连杆轴承润滑特性，秋天技术会议 ASME 2 卷（ 1996） 1 6。 T. Suhara, S. Ato, M. Takiguchi, S. Furuhama,，本文对古浜庄，活塞销座轴承的汽车发动机的摩擦与润滑特性， SAE 论文 970840，底特律，美国， 1997。 4 C. Zhang, H.S. Cehng, L. Qiu, K.W. Knipstein, J. Bolyard,，活塞销 /行为划伤孔轴承的混合润滑，部分。 1，实验研究，赛宝反。 46 卷（ 2004）193 199。 5 C. Zhang, H.S. Cehng, J.O. Wang，活塞销 /行为划伤孔轴承 的混合润滑，部分。 2，实验研究，赛宝。反。 47 卷（ 2004） 149 156。丹尼尔的学生， ,机械原理 46（ 2011） 1434 14521451 6 J.L. Ligier, P. Ragot，活塞 销 ：磨损和旋转运动， SAE纸 2005-01-1651，美国，底特律， 2005。 7 J.L. Ligier, P. Ragot，小端连杆润滑， SAE 纸 2006-01-1101，美国，底特律， 2006。 8 A.L. Schwab, J.P. Meijaard, P. Meijers，在刚性 和弹性力学系统的动态分析的旋转关节间隙模型的比较，机械原理（ 9） 37（ 2002） 895 913。 9 P. Flores, J. Ambrosio, J.P. Claro，润滑关节的平面多体机械系统的动态分析，多体系统动力学 12卷（ 2004） 47 74。 10 P. Flores, J. Ambrosio, J.C.P. Claro, H.M. Lankarani, C.S. Koshy，在包括间隙和润滑关节的机械系统动力学研究，机械原理 41卷（ 2006） 247 261。 11 S. Erkaya, S. Su, I. Uzmay 的滑块曲柄机构， 偏心连接器和行星齿轮动态分析，机械原理 42 卷（ 2007） 393 408。 12 I. Khemili, L. Romdhane，一个灵活的曲柄滑块 含间隙机构的动力学分析，欧洲 /固体力学学报 27卷（ 2008） 882 898。 13 E.A. Estupian, I.F. Santos，建模的封闭式压缩机使用不同的约束方程，以适应多体动力学和流体动力学润滑，对机械科学与工程第三十一号 1巴西社会杂志（ 2009） 35 46。 14 S. Doughty，力学的机器，约翰威利父子，美国， 1988。 Analysis of the dynamics of a slidercrank mechanism with hydrodynamic lubrication in the connecting rodslider joint clearance Gregory B. Daniel, Katia L. Cavalca Department of Mechanical Design, University of Campinas, Faculty of Mechanical Engineering, Postal Box 6122, 13083-970 Campinas, SP, Brazil a b s t r a c t The conventional slider-connecting rodcrank mechanism is widely applied in mechanical systems. The use of hydrodynamic bearings in the mechanism joints is of particular interest in reducing friction, mainly in special conditions of lubrication such as the connecting rodslider joint. This bearing belongs to a class of bearings with alternating rotational motion. This paper proposes a mathematical model for this particular problem, considering the dynamics of the slider-connecting rodcrank system interacting with the lubrication phenomenon in bearings with alternating motion. Two models were used to analyze the dynamics of the system. The first model (by Eksergian Equation of Motion) represents the system when the connecting rod end is in contact with the bearing surface, assuming, in this condition, the same behavior as that of rigid bearings (without clearance). The second model (by Lagrange Method) represents the system when the connecting rod end is in the hydrodynamic lubrication mode in the slider bore clearance. In this condition, the slider moves in relation to the connecting rod, presenting a problem of multi-degrees-of-freedom. The mathematic model of hydrodynamic lubrication was introduced to obtain more realistic results of the systems dynamic behavior. 2011 Elsevier Ltd. All rights reserved. 1. Introduction The lubrication system is a crucial element in the optimal performance of machines in general in view of its numerous functions, e.g., lubricating and protecting components, reducing friction, and cleaning and cooling internal mechanisms. The operation of this system requires proper calibration. Therefore, excessive or insufficient lubrication influences the dynamic behavior of the machine and can seriously damage the mechanisms involved. The mechanisms of many machines contain numerous hydrodynamic bearings, but the hydrodynamic bearing of the connecting rodslider joint is particularly important because it is part of a new class of hydrodynamic bearing called a Hydrodynamic Bearing with Alternating Motion. Unlike traditional hydrodynamic bearings, this type of hydrodynamic bearing does not make a complete rotation. Therefore, recent years have seen a growing need to study the behavior of this specific type of hydrodynamic bearing and its influence on the dynamic behavior of the machine. Most of the research into the hydrodynamic bearing of the connecting rodslider joint has so far been conducted by the Musashi Institute of Technology. In fact, this institute has developed and constructed devices to investigate lubrication and friction in the hydrodynamic bearing of slider pins for more than 15 years. Takiguchi et al. 1, for example, studied the rotating motion of a floating piston pin applied in gasoline-powered automotive engines. Three years later, Takiguchi et al. 2 developed a measuring device that determines the status of lubrication in the hydrodynamic bearing of the connecting rodslider joint based on measured friction force. In another study conducted at the Musashi Institute of Technology, Suhara et al. 3 examined lubrication conditions in the hydrodynamic bearing of the slider pin of gasoline-powered automotive engines, considering as parameters of analysis the length, internal diameter and material of the piston pin. More recently, Zhang et al. 4,5 developed tools to investigate wear in hydrodynamic bearings of the piston pin. In 2005, Ligier and Ragot 6 analyzed the behavior of the hydrodynamic bearing of the piston pin. The following year, these authors 7 presented a general view of the operation of the hydrodynamic bearing of the connecting rodpiston joint in four-stroke engines, emphasizing the oil feed in the bearing. As mentioned earlier, hydrodynamic bearings with alternating motion are part of slidercrank mechanisms. Hence, a reliable mathematical model of this mechanism must take into account the behavior of the bearing in the connecting rodslider joint. Owing to the wide application of the slidercrank mechanism, many studies have focused on devising a mathematical model and analyzing the dynamic behavior of this mechanical system. Schwab, Meijaard and Meijers 8 compared the dynamic behavior of the slidercrank mechanism, considering different connecting rodslider joint models such as a Hertz contact model with dissipation, an impact model and a hydrodynamic bearing model. In the aforementioned work, the authors also compared the dynamic behavior of the mechanism considering the connecting rod as an elastic and rigid component. The results obtained by those authors show that the assumption of a connecting rod as an elastic component, as well as the lubrication condition in the joint, tends to significantly decrease the vibrations in the dynamic response of the mechanism. Flores et al. 9 analyzed the dynamic behavior of the slidercrank mechanism, modeling the connecting rodslider joint as a dry contact without friction, a dry contact with friction and as a hybrid model which considers hydrodynamic lubrication for small eccentricity and dry contact with friction for high eccentricity. Their results show that the dry contact model with friction is more realistic than the dry contact model without friction, due to the magnitude of the oscillations in the dynamic response. As for the hybrid model, the results they obtained showed the smallest oscillation in the dynamic response, but there are no results in the literature that can support this observation. Soon thereafter, Flores et al. 10 analyzed the dynamics of the slidercrank mechanism considering the effects of dry contact, friction and hydrodynamic lubrication on the connecting rodslider joint. The results obtained with the dry contact model without friction showed high oscillations in the dynamic response of the mechanism, making this model less realistic than the dry contact model with friction. When the model with hydrodynamic bearing was considered, the results were very similar to those obtained in the slidercrank mechanism with ideal joints. Erkaya, Su and Uzmay 11 analyzed the kinematics and dynamics of a modified slidercrank mechanism with an additional eccentric link between the connecting rod and crank pin. The results they obtained with the slidercrank mechanism they developed were evaluated and compared to that of a conventional slidercrank mechanism. This comparison indicated that although the conventional and modified slidercrank mechanisms have the same stroke and the same gas pressure in the cylinder, the modified mechanism has a higher torque output than the conventional mechanism. Khemili and Romdhane 12 analyzed the dynamic behavior of a planar flexible slidercrank mechanism with clearance, performing simulated and experimental tests and comparing the experimental results with the numerical results of the simulations using ADAMS software. They found that the presence of clearance affects the dynamic response of the system, and that the couplers flexibility acts as a suspension system in the mechanism. Estupian and Santos 13 developed a mathematical model for a linear reciprocating compressor. They checked the dynamic behavior of the mechanical system, considering the dynamics of the mechanical components based on multibody dynamics (rigid components) and the finite element method (flexible components). They also evaluated the influence of the hydrodynamic bearing, describing the hydrodynamic forces in the joints using the Reynolds equation. Their results showed that maximum forces and minimum film thickness are obtained when the piston is close to the top dead center and that the nonlinear behavior of the orbits increases due to the tilting oscillations of the crank, which are influenced by the length of the crank pin. This work involves an analysis of the dynamic behavior of the slidercrank mechanism and the hydrodynamic lubrication condition of the bearing in the connecting rodslider joint. For this reason, the analysis is based on two different models, whose application depends on the lubrication conditions (eccentricity value) of the bearing. The first model (1dof) considers the connecting rodslider joint as an ideal joint and is applied when the connecting rod end is in contact with the bearing surface 14. The second model (3dof) is applied when the connecting rod end is in hydrodynamic lubrication in the slider bore clearance. In this condition, the slider moves in relation to the connecting rod, characterizing a problem of multiple degrees of freedom in which the interaction between the connecting rod and the slider occurs through the hydrodynamic forces of the lubricant fluid. Therefore, the solution is obtained from a hybrid model composed of a model of the slidercrank mechanism with ideal joints and a model of the slidercrank mechanism with hydrodynamic bearing in the connecting rodslider joint. Thus, an important parameter in this analysis is the minimum oil film thickness considered in the hydrodynamic lubrication. According to Flores 9, for high eccentricity (low minimum oil film thickness), the pressure causes elastic deformation of the surfaces, which can be of the same order as the lubricant film thickness. These circumstances differ from those obtained in hydrodynamic lubrication and a more realistic analysis can be made based on the theory of elastohydrodynamic lubrication. For this reason, this work considered the hydrodynamic lubrication condition for a minimum oil film thickness exceeding 10% of the radial clearance, which represents an eccentricity ratio of 0.9. The dynamic responses of the slidercrank mechanism obtained from the hybrid model are compared with those obtained from the conventional model (slidercrank mechanism with ideal joints). Moreover, the pressure distribution in the bearing is obtained during one period of lubrication condition. It is important to emphasize that the hydrodynamic model used in the present work was developed previously by the authors in cooperation with Bannwart 15. 2. Methodology In this work, a planar slidercrank mechanism was modeled in order to determine its dynamic behavior. However, instead of the conventional slidercrank mechanism, a hydrodynamic bearing was considered in the connecting rodslider joint. This assumption considers that the pin is not restricted to only one direction of motion, which is the case when the bearing is considered without clearance. This section describes the hybrid mathematical model (3DOF) of the planar slidercrank mechanism considering a hydrodynamic bearing in the connecting rodslider joint, which is used to analyze the dynamic behavior in the lubrication condition. The mathematical model used to analyze the dynamic behavior in the contact condition was the conventional slidercrank mechanism (1 DOF) proposed by Doughty 14. 2.1. Kinematic analysis of the planar slidercrank mechanism The kinematics of the planar slidercrank mechanism was developed considering the scheme in Fig. 1. Fig. 1a shows the planar slidercrank mechanism and Fig. 1b depicts a free body diagram of the connecting rod pin and the slider bore. R is the length of the crank, L is the length of the connecting rod, q is the angular displacement of the crank, A is the angular displacement of the connecting rod, XP and YP are the linear displacements of the connecting rod pin, XPT and YPT are the linear displacements of the slider, and FXP and FYP are the hydrodynamic forces in the X and Y directions, respectively. Fig. 1c illustrates the hydrodynamic bearing in the connecting rodslider joint of the planar slidercrank mechanism. OH is the center of the bearing, OP is the center of the connecting rod pin, RH is the radius of the bearing, RP is the radius of the connecting rod pin, e is the eccentricity, hmin is the minimum oil film thickness, and hmax is the maximum oil film thickness. Moreover, the eccentricity ratio () is the ratio between the eccentricity (e) and the radial clearance (Cr), where the radial clearance is the difference between the radius of the bearing and the radius of the connecting rod pin. According to this figure, the position of the slider pin can be represented as shown in Eq. 1: Xp = R:cos（ q ） + L:cos（ A） (1) Yp = R:sin（ q） L:sin（ A） : The derivatives of Eq. 1 give the velocity and acceleration of the connecting rod pin, respectively, obtained as: where KP is the matrix of the velocity coefficients of the slider pin, and LqP and LAP are the partial derivative matrices of the velocity coefficient of the slider pin, defined as: 2.2. Kinematic analysis of the center of mass of the crankconnecting rod subsystem The purpose of this analysis is to determine the dynamic behavior of the center of mass of the components, which enables one to find the equation of motion of the slidercrank mechanism 14. Fig. 2a and b shows the crank-connecting rod subsystem with mass center in the crank and in the connecting rod, respectively. Upm and Vpm are the mass center coordinates of the crank (PM) in the referential system located in the crank (UM,VM), Upb and Vpb are the mass center coordinates of the connecting rod (PB) in the referential system located in the connecting rod (UB,VB), XPm and YPm are the linear displacements of the mass center of the crank (PM) in the inertial referential system (X,Y) and XPb and YPb are the linear displacements of the mass center of the connecting rod (PB) in the inertial referential system (X,Y). According to Fig. 2a, the center of mass position of the crank can be described as: The velocity can be written from the derivative form of Eq. (4) as: The displacement and velocity of the center of mass of the connecting rod were obtained similarly, considering the scheme in Fig. 2b. In order to determine the dynamic behavior of the crank-connecting rod subsystem, the linear and angular velocities of all the components of this subsystem must be grouped 14. These velocities are necessary to determine the kinetic energy (rotational and translational) of the subsystems and, consequently, to obtain the equations of motion by the Lagrange method. Thus, based on Eq. 7 and Eq. 9, it is possible to write: where KC represents the generalized velocity coefficient matrix of the crank-connecting rod subsystem. Deriving KC withrespect to the independent variables q and A, one has: where Lq and LA are the partial derivative matrices of the generalized velocity coefficients. Finally, considering MM, MB, IM e IB as crank mass, connecting rod mass, crank rotation inertia and connecting rod rotation inertia, respectively, the mass matrix can be obtained as: 2.3. Generalized forces in the crank-connecting rod subsystem The generalized excitation forces are obtained by the virtual work concept, in the case of the hybrid model presented here, applied to the dissipative viscous forces due to the hydrodynamic lubrication at the connecting rod pinslider joint clearance. According to Doughty 14, the generalized forces can be determined using the generalized velocity coefficients matrix for the connecting point where the external forces are applied. Thus, the generalized forces are given by Eq. 14: These generalized forces occur because of the hydrodynamic forces applied on the slider joint (FXP and FYP). In this work, the hydrodynamic forces are determined from a lubrication model of hydrodynamic bearings with oscillatory motion, as proposed by Bannwart 15. 2.4. Kinetic and potential energy of the crank-connecting rod subsystem The kinetic energy of the crank-connecting rod subsystem is determined for the mass matrix and the matrix of the generalized velocity coefficient, according to Eq. 15: Fig. 3a shows the subsystem composed of the crank and the connecting rod, considering the center of mass of these components. According to Fig. 3a, the potential energy can be written as: 3. Results The purpose of this analysis was to ascertain the models applicability. Computer simulations were performed in Fortran, allowing for an analysis of the lubrication state in the hydrodynamic bearing. In this work, the contact condition was considered when the eccentricity ratio exceeded the value of 0.9. Otherwise, the lubrication condition was considered. The flowchart in Fig. 4 shows the solution process applied in the dynamic analysis. First, the hydrodynamic forces in the bearing are evaluated for the initial condition. The connecting rod pin and slider accelerations are then determined and the hydrodynamic forces are evaluated. Thus, the equation of motion is solved for the initial condition, and the hydrodynamic forces and accelerations are also evaluated. Therefore, the displacements and velocities of the connecting rod pin and slider are calculated up to the next time step (t + t). After determining the new velocities and displacements, the eccentricity of the pin in the bearing must be verified. If the eccentricity ratio is smaller than 0.9, the pin is in lubrication condition. In this case, the new hydrodynamic forces are evaluated, the accelerations of the pin and slider are determined again, and so on successively. During the hydrodynamic lubrication condition, the systems dynamic behavior is evaluated considering the lubrication model presented in Section 2.7. However, when the slider pin exceeds the eccentricity threshold (=0.9), it is considered to be in contact with the surface of the bearing. In this case, it is assumed that the systems dynamic behavior is governed by the conventional slidercrank mechanism, with no clearance in the connecting rodslider joint. Therefore, the conventional model of the slidercrank mechanism is used until a reverse motion occurs in the slider. After the reverse motion in the slider, the slider pin leaves the bearing surface, shifting the lubrication condition. As for the resolution of the differential equations of motion, the conventional model is not complex, and can therefore be solved relatively quickly (10 minutes in a computer system of 2.0 GHz processor and 3.0 GB RAM). However, the equations of motion for the lubrication condition are highly complex due to the high numerical stiffness. Hence, a particular numerical integration method must be chosen according to the characteristics of these equations. In this work, the solution for the 3 dof. hybrid model was found by using a multi-step method for the integration of stiff initial value problems, the Rowmap method 16. However, despite the efficiency of this method, the solution of the differential equations still involves a high computational cost (approximately 48 h in a computer system of 2.0 GHz processor and 3.0 GB RAM). The parameters listed in Tables 1 and 2 represent the geometry of the hydrodynamic bearing and the slidercrank mechanism,respectively. These parameters were obtained from a conventional internal combustion engine. The initial conditions presented inTable 3 were considered in the computer simulations and are in agreement with the literature and with some experimental values found during operation. Simulation We began by considering the physical parameters presented in Tables 1 and 2, the initial conditions listed in Table 3 and a radial clearance of 20 m for the bearing. The dynamics of the slidercrank mechanism was analyzed with the hydrodynamic bearing in the connecting rodslider joint. The computer simulations in this work considered a threshold condition for hydrodynamic lubrication. The hydrodynamic lubrication condition thus involved an eccentricity ratio of up to 0.9 because of the extremely low film thickness, which is an inadequate condition for hydrodynamic lubrication. Therefore, the maximum simulated value for the eccentricity ratio is 0.9, which indicates that the pin is not in the hydrodynamic condition and assumes it is in contact with the bearing surface. Fig. 5a and b shows the orbits obtained in the first and second lap, respectively. As Fig. 5a indicates, the pins initial eccentricity ratio is 0.6. When the motion begins, the pin moves to the left until it exceeds the eccentricity limit, after which it presumably remains in contact with the bearing surface until the sliders motion is inverted. In this case, i.e., when the pin is in contact with the bearing surface, the dynamics are analyzed based on the mathematical model of the conventional slidercrank mechanism 14. When the motion is inverted, the pin moves from the bearing surface to the upper region of the bearing. Table 1 Parameters of the hydrodynamic bearing. Table 2 Parameters of the slidercrank mechanism. Table 3 Initial conditions employed in the computer simulations. approaches the surface of the bearing, pin and slider are pushed into opposite directions by the hydrodynamic forces, causing them move in relation to each other. This causes the pin to return to the lower region of the bearing, once again exceeding the eccentricity limit. The pin then remains in contact with the surface bearing until the end of the lap. In the second lap, the dynamic behavior of the piston pin is similar to the first lap. Fig. 6 illustrates the dynamic behavior of the crank. As can be seen, the crank velocity and acceleration are similar in the two systems, i.e., the slidercrank mechanism with journal bearing (3 dof) and the conventional slidercrank mechanism (1 dof). However, note that both velocity and acceleration tend to increase as a function of time in the 3 dof model due to the transition between the two applied models. Fig. 5. Orbit of the slider joint in Simulation 1, (a) First Lap, (b) Second Lap. Fig. 6. Crank analysis in Simulation 1, (a) Angular Velocity, (b) Angular Acceleration. Fig. 7. Connecting rod analysis in Simulation 1, (a) Angular Displacement, (b) Angular Velocity. When the pin exceeds the eccentricity limit, the last point analyzed in the hydrodynamic condition is the initial condition of the 1 dof. Conventional mechanism(contact condition). Thus, at this point of themotion, the dynamic behavior of the systemis described by the initial condition given in the aforementioned 3 dof.model. Consequently, in the transition of the hybrid model to the conventional one, the velocity input differs from the initial condition given for both models at t=0.0. Therefore, the main effect is increase in the cranks angular velocity in each transition of the model, which is shifted with respect to the 1 dof. conventional model. As can be seen in Fig. 6b, the angular acceleration of the crank has small discontinuities, unlike the conventional model. Although the crank is not directly connected to the hydrodynamic bearing, these discontinues are caused by the dynamic response of the pin during the lubrication condition. The same behavior was reported by Schwab, Meijaard and Meijers 8. According to Schwab 8, the dynamic behavior of the slidercrank mechanism with hydrodynamic bearing in the connection rodslider joint is similar to the conventional mechanism, in which the dynamic responses are represented by the smoother curves. However, the responses differ slightly only when pin crossing into the bearing occurs. This crossing involves high journal center velocities and, consequently, it increases the peak forces. Fig. 7 depicts the displacement and velocity of the connecting rod. Note that the connecting rods displacement and velocity in the slidercrank mechanism with journal bearing (3 dof) and in the conventional slidercrank mechanism (1 dof) are similar. As Fig. 8 indicates, the dynamic behavior of the slider is similar in the two systems (1dof and 3 dof). Fig. 9 shows the angular acceleration of the connecting rod and the linear acceleration of the slider. Note that the connecting rod and slider acceleration attained in the slidercrank mechanism with the journal bearing (3 dof) are similar to the acceleration achieved with the conventional slidercrank mechanism (1 dof). However, the accelerations reached in the slidercrank mechanism with journal bearing show oscillations due to the behavior of the joint in the hydrodynamic lubrication condition. Fig. 8. Slider analysis in Simulation 1, (a) Linear Displacement, (b) Linear Velocity. Fig. 9. Acceleration analysis in Simulation 1, (a) Connecting rod Acceleration, (b) Slider Acceleration As Fig. 9 indicates, the discontinuities caused during the lubrication condition are more significant in the acceleration of the slider and the connecting rod, because the hydrodynamic bearing causes these components to interact. When the pin is not in the lubrication condition, it is considered to be in contact with the bearing surface. The mathematical model used to analyze the dynamic behavior in the contact condition is the conventional slidercrank mechanism model. Thus, the dynamic behavior of the mechanism is the conventional one. 4. Conclusions This work involved an analysis of the dynamic behavior of the slidercrank mechanism and the lubrication condition in the connecting rodslider joint. To this end, a mathematical model of the slidercrank mechanism was developed considering a hydrodynamic bearing in the connecting rodslider joint. In this work, the computer simulations of the mathematical model described the dynamic behavior of the pin in the connecting rodslider joint. The results indicated that the pin is subjected to two conditions during the operation. At some moments the pin is subjected to the hydrodynamic lubrication condition and at others it is in contact with the surface of the bearing. The numerical results indicated that the pin breaks the oil film, thus coming into contact with the surface of the bearing, regardless of the initial conditions or the parameters of the bearing. As described earlier herein, the pin reaches a high eccentricity, causing compression of the oil film or even splitting it. This behavior suggests that the use of the elastohydrodynamic lubrication model in the contact condition of the hybrid model (3 dof.) may be more suitable for this analysis and, in this case, direct contact could be prevented even when the pin is not in hydrodynamic lubrication. However, to evaluate the dynamic response and the pressure distribution during the lubrication condition, the proposed hybrid model can be applied. The implementation of the elastohydrodynamic model requires further investigation of the transition from the hydrodynamic to the elastohydrodynamic condition of the hybrid model. With regard to the bearings radial clearance, it can be concluded that an increase in the radial clearance causes a consequent increase in the oil film thickness, reducing the hydrodynamic forces and facilitating the pins contact with the bearing surface. Conversely, an increase in the initial angular velocity causes the hydrodynamic forces acting on the pin to increase, since these forces are directly proportional to the initial angular velocity. These stronger hydrodynamic forces increase the distance of the pins approach to the bearing surface, preventing it from touching this surface. The solution of the problem of the slidercrank mechanism with hydrodynamic bearing in the connecting rodslider joint is relatively complex due to the nonlinear equations of the models used in both hydrodynamic lubrication and the slidercrank mechanism. This set of equations presents high numerical stiffness, making the choice of the numerical integration method an important factor in the successful solution of differential equations. Acknowledgments The authors would like to thank FAPESP, CAPES and CNPQ for their financial support of this work. References 1 M. Takiguchi, M. Oguri, T. 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