RESEARCH ON THE CALCULATION METHOD OF THE SPACE PULLEY ROPE CONTACT SECTION OF NUCLEAR RING CRANE BASED ON K-T CONDITION
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摘要: 滑轮绳索系统是一类可以利用内嵌于其中的绳索控制的多体系统, 一般存在大量绳索接触段, 随着机械系统的复杂化和智能化, 对这类系统的精确性和可靠性提出了高要求. 针对核环吊起升机构中空间滑轮绳索接触段, 推导了接触段绳索微元体平衡方程, 得到了接触力密度的解析表达式. 将绳索应变求解转化为优化问题, 利用库恩塔克(K-T)条件, 建立了绳索轴向应变以及应变对弧长导数满足的非线性方程, 并求出了内部应变对滑轮两端参数的导数, 计算了滑轮与绳索接触段的应变分布, 推导了绳索方位角与弧长应满足的协调方程. 同时结合接触段滑轮槽截面的几何特点, 推导了切向和法向接触力密度与绳索轴向应变之间的关系, 提出了滑轮两侧绳索应满足的边界条件, 利用边界点处绳索与滑轮物质速度相等的条件, 建立了约束方程. 数值算例表明, 计算结果符合绳索受力变形规律和接触力变化趋势. 提供的方法为包含空间滑轮绳索机构的核环吊机构以及其他大型机械系统分析提供了新的思路.Abstract: The pulley rope system is a type of multi body system that can be controlled by ropes embedded within it , generally, there are a large number of contact segments,with the complexity and intelligence of mechanical systems, high demands have been placed on the accuracy and reliability of such systems . This paper mainly focuses on the space pull rope contact section of the nuclear ring lifting mechanism. Firstly,the equilibrium equation of the element body of the rope in the contact section is derived, and the analytical expression of the contact force density is obtained. Secondly,the solution of rope strain is transformed into an optimization problem.The nonlinear equation of strain and arc length derivative of strain is established by using Kuhntak condition. The derivative of internal strain to parameters at both ends of pulley is obtained. The strain distribution in contact section and the coordination equation between azimuth Angle and arc length are calculated. At the same time, the relationship between tangential and normal contact force density is derived based on the geometric characteristics of the pulley groove section, and the boundary conditions that the rope on both sides of the pulley should meet are proposed. Based on the condition that the material velocity of rope and pulley at the boundary point is equal, the constraint equation is established. In this paper, the contact forces of pulleys with different radius and different types of pulleys are analyzed, and the strain distribution rules of the contact section are summarized.The numerical examples show that the calculated results are consistent with the law of stress deformation and the trend of contact force change of the rope. The method presented in this paper provides a new idea for the analysis of large-scale mechanical systems including pulley and rope mechanisms. Moreover, it also provides theoretical preparation for the analysis of pulley-rope systems.
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引 言
滑轮绳索系统广泛应用在航天器、卫星等机构中, 在系统运动中起重要的动力牵引作用. 在很多机械系统中, 绳索有着不可替代的功能. 比如核反应堆搭建和操作的核环吊系统, 必须在提升几百吨重物的过程中使重物的水平位移偏差控制在每10 m小于5 mm的范围. 这些机械系统若要达到精确可靠的要求, 必须对起升机构系统进行精确可靠的计算.
核环吊安装和运行于核电站反应堆的正上方, 在核电站建造和运行阶段, 为重要机械设备的安装维修以及运输提供吊运服务. 核环吊一般分为主起升机构和副起升机构[1], 但是不管是哪一类机构, 其本质都是由钢丝绳、滑轮、卷筒等部件组合而成的一组滑轮钢丝绳系统. 核环吊中的吊装部件一般都需要高精度的定位, 所以对于这类系统的计算, 提出了更高的要求. 钢丝绳系统中存在大量的滑轮, 如果忽略滑轮尺寸, 不考虑滑轮的作用, 会造成计算的误差偏大, 无法精确地计算系统运动, 更无法准确得出系统中绳索总长度. 但想要高精度地分析此类系统, 滑轮与绳索接触段的计算就尤为重要. 谭春林等[2]对绳索动力学做了很多基础研究, 王郡等[3]对弹性绳进行了轨迹模拟. 滑轮绳索系统不仅在核环吊机构[4]中大量存在, 在大型空间展开机构[5]中, 以及航母中的拦阻索[6]也被广泛应用. 接触力计算的准确性将直接影响整个系统的动力学行为. 传统的商业软件在做滑轮绳索系统时有局限性, 不能得到准确的轮索接触力. 滑轮与绳索的接触段的求解尚没有成熟而统一的方法, 目前主要有3种方法. 第1种采用经典的绞盘模型[7], 这种方法可以得到滑轮两端拉力关系, 但是一般仅适用于绳索与滑轮处于滑动状态或者临界滑动状态, 处理的构件为吊环情况较多, 采用此方法无法得到接触段内部应变. 第2种方法采用接触模型, 以赫兹接触为基础, 很多学者建立了不同的接触力模型, 比如Kelvin-Voigt模型[8]和Hunt-Crossley模型[9]. Flores等[10]对连续接触力模型进行了改进, 提出了新的计算方法, Lankarani等[11]针对两球中心碰撞提出了一种接触力模型, Zhang等[12]针对具有复杂几何形状的物体接触, 提出了一种计算模型. Bulin等[13]采用Hunt-Crossley接触模型以及LuGre微变模型建立钢丝绳与滑轮局部的接触力模型, 从中计算出切向接触力和法向接触力, 但是基于此类方法, 摩擦力计算的结果依赖于参数的选择, 而这些模型中的参数设定需要大量的工程试验验证, 如果参数选择不合适, 接触模型与柔性体的本构关系会相互干扰, 造成数值不稳定. 第3种方法采用工程中的经验公式计算接触应力. 一般思路是将绳索拉力、滑轮直径和绳索直径带入经验公式得到接触应力[14], 这种计算方式得到的接触应力在整个接触区域均相同, 这与实际情况有差别. 这种方式在处理动力学问题时也无法判断绳索与滑轮是否存在相对滑动, 描述变量不能应用于整个滑轮绳索系统的分析中.
Ju等[15]针对绳索滑轮系统, 研究了静态结构分析方法, 结果说明不考虑摩擦会产生错误的分析结论. 李春明[16]将弹性绳作为特殊的体, 研究了弹性绳力学行为与振动特性. 胡建峰等[17]采用理论力学以及摩擦学原理提出了滑轮损耗模型, 同时也说明了滑轮的摩擦对整个系统动力学行为的影响. 魏建东[18]针对静力学问题提出了索-轮单元法. 此方法在不考虑滑轮摩擦力的情况下具有较广泛的应用. 阚子云[19]对含绳索的张拉结构进行了深入研究; Peng等[20]对绳索驱动的机械臂做了大量分析; Zhao等[21]对含滑轮绳索系统的空间展开结构和可回收火箭中的绳索系统进行了详细分析[22], 也说明了过线滑轮的摩擦对展开系统有着重要的影响. Qi等[23]针对起重机中的滑轮绳索系统, 基于空间描述, 细致分析了动滑轮、定滑轮的描述参数, 并应用在了多个场景中[24]. Du等[25]对考虑滑轮尺寸的绳索驱动可展开天线进行了动力学分析.
在核环吊机构工作状态中, 如果滑轮方位不合理, 可能会出现打滑现象. 由于滑轮与绳索之间的状态不容易确定, 如果仅将绳子对滑轮的作用力整体分析, 而不对内部应变的分布规律做详细计算和讨论, 则无法判断起升机构中的滑轮运动规律. 传统的计算方法一般将系统中所有的绳索离散为多个绳索单元, 随着系统运动, 跟踪每个单元节点的运动以判断绳索是否与滑轮接触, 对处于接触状态的绳索单元, 用弹簧阻尼模型计算对应的接触力. 但形函数不能高精度描述滑轮轮廓, 只能通过缩小单元尺寸逼近滑轮上的一段圆弧, 往往需要很多单元才能模拟滑轮绳索的运动. 其次, 接触力计算需要实时判断绳索滑轮间接触条件, 单元形函数稍稍复杂一点就会大幅度地增加这一过程的难度, 也经常发生不收敛现象. 实际核环吊中的滑轮之间是相互影响的, 需要判断整个系统中哪个滑轮最有可能出现打滑现象, 以及什么时候可能滑动. 这就需要对接触段进行定量分析, 得到接触力的分布规律.
一些研究假设各点处摩擦系数为常数, 利用微分方程推出经过滑轮两端的拉力关系[26]. 但是不能计算出接触段的应变变化, 为了保证各点处的摩擦系数小于最大静摩擦系数, 需要对变量进行约束, 将此问题转化为一个优化问题. 严格意义上讲, 滑轮绳索之间的接触问题处于面面接触, 接触局部可能有蠕滑, 根据经验公式或者摩擦模型均无法有效地判断蠕滑出现的位置. 目前商业软件的处理方法是将绳索看成梁单元或者实体单元[27], 如果采用梁单元分析, 无法得到接触段的切向接触力, 如果采用实体单元分析, 当大部分工况为静摩擦时, 计算出的内部接触力误差很大[28], 而且对于含有滑轮绳索系统的大型机械结构, 实体单元计算量非常庞大[29]. ADMAS中采用Bushing来模拟绳节与绳节之间的受力[30], 采用冲击函数和泊松模型计算接触力, 需要实时判断两个实体单元区域的接触情况, 分析异常耗时, 而且准确度较低.
本文结合绳索受力特点, 首先推导了接触段绳索微元体平衡方程, 得到了接触力表达式, 利用动态节点单元[31]建立了系统的整体平衡方程. 根据库恩塔克条件, 建立了接触段应变需要满足的条件, 并求解了非线性方程的雅可比矩阵, 求出了系统的平衡状态以及滑轮上绳索的接触力密度分布. 通过数值算例, 与商业软件进行了对比, 验证了计算方法的可行性. 本文提出的计算方法可以适用于含有滑轮绳索的机械系统. 当滑轮半径较小时商业软件计算不准确, 文中方法仍旧可以准确地计算接触段. 最后, 研究了起升机构滑轮接触力, 总结了滑轮接触段应变分布规律, 分析了可能出现打滑的位置.
1. 滑轮绳索之间的接触力
对于滑轮与绳索之间的接触段绳索, 根据微元体平衡分析, 如图1所示, 其中${{\boldsymbol{e}}_x}$,${{\boldsymbol{e}}_y}$,${{\boldsymbol{e}}_z}$为滑轮参照坐标系, ${{\boldsymbol{e}}_z}$为滑轮轴方向, ${\boldsymbol{r}}$为接触段绳索截面形心矢径.
绳索动力学方程可以写作[32]
$$ \frac{{\partial {\boldsymbol{f}}}}{{\partial s}} + {\boldsymbol{q}} = \rho \left(\frac{{{\partial ^2}{\boldsymbol{r}}}}{{\partial {t^2}}} - {\boldsymbol{g}}\right) $$ (1) 其中, $\rho $为绳索质量线密度; ${\boldsymbol{f}}$为绳索内力; ${\boldsymbol{q}}$为绳索所受分布外力密度, $s$为标志物质点的弧长坐标, 如图2所示, 绳索截面连体基矢量为${{\boldsymbol{e}}_s},{{\boldsymbol{e}}_t},{{\boldsymbol{e}}_b}$
$$ \left. \begin{gathered} {{\boldsymbol{e}}_b} = \cos \theta {{\boldsymbol{e}}_x} + \sin \theta {{\boldsymbol{e}}_y} \\ {{\boldsymbol{e}}_s} = \cos \theta {{\boldsymbol{e}}_y} - \sin \theta {{\boldsymbol{e}}_x} \\ {{\boldsymbol{e}}_t} = {{\boldsymbol{e}}_b} \times {{\boldsymbol{e}}_s} \\ \end{gathered} \right\} $$ (2) 其中, $\theta $为描述绳索横截面位置的方位角, 如图3所示.
绳索内力可以表示为
$$ {\boldsymbol{f}} = {s_g}{E_s}A{\varepsilon _s}{{\boldsymbol{e}}_s} $$ (3) 式中, ${E_s}$为绳索弹性模量, $A$为绳索横截面面积, ${s_g}$为绕绳方式的类型, ${s_g} = 1$为逆时针绕绳, ${s_g} = - 1$为顺时针绕绳, ${\varepsilon _s}$为绳索轴向应变, 绳索内力的弧长导数
$$ \frac{{\partial {\boldsymbol{f}}}}{{\partial s}} = {s_g}{E_s}{\varepsilon '_s}{{\boldsymbol{e}}_s} + {s_g}{E_s}{\varepsilon _s}{\boldsymbol{\kappa }} \times {{\boldsymbol{e}}_s} $$ (4) 其中曲率[33]
$$ {\boldsymbol{\kappa }} = {\kappa _s}{{\boldsymbol{e}}_s} + {\kappa _t}{{\boldsymbol{e}}_t} + {\kappa _b}{{\boldsymbol{e}}_b} $$ (5) 其连体基中的分量
$$ {\kappa _s} = {{\boldsymbol{e'}_t}} \cdot {{\boldsymbol{e}}_b} = 0 $$ (6) $$ {\kappa _t} = {{\boldsymbol{e'}_b}} \cdot {{\boldsymbol{e}}_s} = \theta ' $$ (7) $$ {\kappa _b} = {{\boldsymbol{e'}_s}} \cdot {{\boldsymbol{e}}_t} = 0 $$ (8) 由式(1)可得接触段接触力密度
$$ {\boldsymbol{q}} = {q_s}{{\boldsymbol{e}}_s} + {q_t}{{\boldsymbol{e}}_t} + {q_b}{{\boldsymbol{e}}_b} $$ (9) 其中
$$ {q_s} = \rho {\boldsymbol{e}}_s^{\mathrm{T}}({{{\partial ^2}{\boldsymbol{r}}} \mathord{\left/ {\vphantom {{{\partial ^2}{\boldsymbol{r}}} {\partial {t^2}}}} \right. } {\partial {t^2}}} - {\boldsymbol{g}}) + {\bar q_s} $$ (10) $$ {q_t} = \rho {\boldsymbol{e}}_t^{\mathrm{T}}({{{\partial ^2}{\boldsymbol{r}}} \mathord{\left/ {\vphantom {{{\partial ^2}{\boldsymbol{r}}} {\partial {t^2}}}} \right. } {\partial {t^2}}} - {\boldsymbol{g}}) + {\bar q_t} $$ (11) $$ {q_b} = \rho {\boldsymbol{e}}_b^{\mathrm{T}}({{{\partial ^2}{\boldsymbol{r}}} \mathord{\left/ {\vphantom {{{\partial ^2}{\boldsymbol{r}}} {\partial {t^2}}}} \right. } {\partial {t^2}}} - {\boldsymbol{g}}) + {\bar q_b} $$ (12) 滑轮上绳索为一段圆弧, 初始弧长的微分${\mathrm{d}}{s_0}$, 现弧长的微分${\mathrm{d}}s = r{s_g}{\mathrm{d}}\theta $, 则轴向应变
$$ {\varepsilon _s} = \frac{{{\mathrm{d}}s - {\mathrm{d}}{s_0}}}{{{\mathrm{d}}{s_0}}} = r{s_g}\theta ' - 1 $$ (13) 再结合式(1)和式(4), 可得接触力主要部分为
$$\qquad\qquad\quad {\bar q_s} = - {s_g}{E_s}{\varepsilon '_s} $$ (14) $$\qquad\qquad\quad {\bar q_t} = 0 $$ (15) $$\qquad\qquad\quad {\bar q_b} = {r^{ - 1}}{E_s}{\varepsilon _s}(1 + {\varepsilon _s}) $$ (16) 以上3式对于含有绳索驱动的机构分析有重要意义, 它们揭示了接触段法向力与绳索轴向应变之间, 以及切向接触力和应变对弧长导数之间的关系. 表明如果想要绳索缠绕力变大, 则需要的应变越大, 应变对弧长的导数与切向接触力成正相关. 在含有机械臂抓手的机构中, 能控制住物体的前提是绳索产生应变, 如果想要接触段切向力平稳, 则应变对弧长导数应该在一个小范围波动. 根据上文推导得到的式(14) ~ 式(16), 可计算不同接触力下的轴向应变.
2. 第三类边界条件与接触分析
绳索与滑轮间的摩擦系数为常量$\mu $, 如图4所示, 假设滑轮槽和绳索的接触横截面为一段圆弧, 绳索受到滑轮槽的作用力, 法向接触力密度为${q_n}$. 滑轮槽直径为$d$, 绳索直径为${d_0}$. 如图5所示, 假设滑轮槽与绳索接触段为一段圆弧, 同时绳索运动过程中, 绳索横截面始终为圆形截面, 滑轮槽与绳索接触横截面的接触角$\alpha $两端分别为
$$ {\vartheta _1} = \text{π} + {\alpha _0} $$ (17) $$ {\vartheta _n} = 2\text{π} - {\alpha _0} $$ (18) 如图6所示, 绳索嵌入在滑轮槽中, 这里认为滑轮槽半径与绳索半径相同, 即两者之间为面面接触. 绳索截面圆心为0, $AC$和$BD$为圆截面的切线, 切点为C和D. 滑轮槽直径为A点与B点之间的长度$d$, 绳索直径为${d_0}$, 红色区域C点到D点为接触部分, 利用三角形$\Delta OAC$, 可得
$$ {{\alpha }_{0}}={a}\cos \left( {{d}_{0}}{{d}^{-1}} \right) $$ (19) 假设绳索形心在截面内的微位移
$$ {\boldsymbol{u}} = {u_1}{{\boldsymbol{e}}_t} + {u_2}{{\boldsymbol{e}}_b} $$ (20) 通过分析绳索径向及其位移, 在绳索的横截面上, 沿表面分布力密度与径向位移成正比, 绳索分布力密度
$$ {q_t} = {r_0}\int\limits_{{\vartheta _1}}^{{\vartheta _n}} {{{\boldsymbol{q}}_n} \cdot {{\boldsymbol{e}}_t}{\text{d}}\alpha } $$ (21) $$ {q_b} = {r_0}\int\limits_{{\vartheta _1}}^{{\vartheta _n}} {{{\boldsymbol{q}}_n} \cdot {{\boldsymbol{e}}_b}{\text{d}}\alpha } $$ (22) 式中, $ {{\boldsymbol{q}}_{{n}}} = - k\delta {{\boldsymbol{e}}_n} $, $ k $为接触刚度.
滑轮绳槽要求[34]
$$ {\alpha _0} \leqslant \frac{1}{6}\text{π} $$ (23) 因此$ \text{π} - 2{\alpha _0} + \sin (2{\alpha _0}) > 0 ,$ 从而${u_2} < 0$ ; 由于$ \left| {{q_b}} \right| \gg \left| {{q_t}} \right| $, $ \left| {{u_2}} \right| \gg \left| {{u_1}} \right| $, 再考虑到$\sin \alpha < 0$, 这里由于假设绳索形心在截面内的微位移${\boldsymbol{u}}$, 绳索截面径向为$ {{\boldsymbol{e}}_n} = \cos \alpha {{\boldsymbol{e}}_t} + \sin \alpha {{\boldsymbol{e}}_b} $, 则径向位移(嵌入量)为
$$ \delta {\text{ = }}{\boldsymbol{u}} \cdot {{\boldsymbol{e}}_n} = {u_1}\cos \alpha + {u_2}\sin \alpha$$ 综上可以推断$\delta > 0$. 将入绳点到出绳点间的绳索按弧长坐标等分为$n - 1$段, 根据节点处截面的状态确定${\bar q_s}$.
如果截面处于滑动状态, 利用$ {q_s} = \mu {{{q}}_n} $, $ {\bar q_n} = - {{\boldsymbol{q}}_{{n}}} \cdot {{\boldsymbol{e}}_n} = k\delta $, 横截面的法向力合力为法向力密度$ {\bar q_n} $沿着接触截面区域C到D积分
$$ {{{q}}_n} = \displaystyle\int\limits_C^D {{{\bar q}_n}{\mathrm{d}}s} {\text{ = }} k\displaystyle\int\limits_C^D {({u_1}\cos \alpha + {u_2}\sin \alpha ){\mathrm{d}}s}$$ 可得到
$$ {q_s} = {c_{qs}}{q_b} $$ (24) 式中
$$ {c_{qs}} = 4\mu {\text{sign}}({v_r}){[\text{π} - 2{\alpha _0} + \sin (2{\alpha _0})]^{ - 1}}\cos {\alpha _0} $$ (25) 其中$ {\text{sign}}({v_r}) $是绳索横截面与滑轮相对速度的符号函数. 当局部出现相对蠕滑时, 绳索截面速度大于滑轮速度为正, 绳索截面速度小于滑轮速度为负. 由式(13)、式(16)和式(24)可得节点处
$$ {\bar q_{s,i}} = {c_{qs}}{q_{b,i}}{\text{ = }}{s_g}{c_{qs}}{E_s}{\varepsilon _i}{\theta '_i} $$ (26) 如果截面处于相对静止状态, ${q_s}$与${q_b}$无明确关系, 此时应按式(14)确定${q_s}$, 即
$$ {\bar q_{s,i}} = - {E_s}{\varepsilon '_{s,i}} $$ (27) 得到节点处的轴向接触力后, 采用第三类边界条件的样条插值近似切向接触力密度
$$ {q_s} = {N_1}{q_{s,i}} + {N_2}{q_{s,i + 1}} + {N_3}\frac{{\partial {q_{s,}}_i}}{{\partial \xi }} + {N_4}\frac{{\partial {q_{s,i + 1}}}}{{\partial \xi }} $$ (28) 其中$\xi $为归一化参数, 形函数
$$\begin{split} &{N_1} = 2{\xi ^3} - 3{\xi ^2} + 1, \qquad {N_2} = - 2{\xi ^3} + 3{\xi ^2}\\ &{N_3} = {\xi ^3} - 2{\xi ^2} + \xi, \qquad {N_4} = {\xi ^3} - {\xi ^2} \end{split}$$ 第$i$个节点处轴向接触力对归一坐标的$\xi $导数为
$$ \frac{{\partial {q_{s,}}_i}}{{\partial \xi }} = \left[ {{q_{s,1}},{q_{s,2}}, \cdots ,{q_{s,n}}} \right]{{\boldsymbol{\bar h}}_i} $$ (29) 因为$ {{\boldsymbol{\bar h}}_i} $为矩阵${\boldsymbol{\bar H}}$的第$i$列, 切向接触力密度可表示为
$$ {q_s} = ({N_1}{\boldsymbol{e}}_i^{\mathrm{T}} + {N_2}{\boldsymbol{e}}_{i + 1}^{\mathrm{T}} + {N_3}{\boldsymbol{h}}_i^{\mathrm{T}} + {N_4}{\boldsymbol{h}}_{i + 1}^{\mathrm{T}}){\boldsymbol{\eta }} $$ (30) 其中, $ {\boldsymbol{\eta }} = \left[ {{q_{s,1}},{q_{s,2}}, \cdots ,{q_{s,n}}} \right] $, 当样条端部导数未知时, 可采用第2节点与第n−1节点处3阶导数连续(not-a-knot)作为补充方程, 与节点处二阶导数连续性条件共同组集为矩阵方程
$$ \left[ {\frac{{\partial {q_s}_{,1}}}{{\partial \xi }},\frac{{\partial {q_s}_{,2}}}{{\partial \xi }}, \cdots ,\frac{{\partial {q_s}_{,n}}}{{\partial \xi }}} \right]{\boldsymbol{U}} = \left[ {{q_s}_{,1},{q_s}_{,2}, \cdots ,{q_s}_{,n}} \right]{\boldsymbol{V}} $$ (31) 其中
$$ {\boldsymbol{U}} = \left[ {\begin{array}{*{20}{r}} 1&2&{}&{}&{}& \cdots &0 \\ 0&8&2&{}&{}&{}&{} \\ { - 1}&2&8& \ddots &{}&{}&{} \\ \vdots &{}&2& \ddots &2&{}& \vdots \\ {}&{}&{}& \ddots &8&2&1 \\ {}&{}&{}&{}&2&8&0 \\ 0& \cdots &{}&{}&{}&2&{ - 1} \end{array}} \right] $$ (32) 为$n \times n$维常数矩阵
$$ {\boldsymbol{V}} = \left[ {\begin{array}{*{20}{r}} { - 2}&{ - 6}&{}& \cdots &{}&0 \\ 4&0&{ - 6}&{}&{}&{} \\ { - 2}&6&0& \ddots &{}&{} \\ \vdots &{}&6& \ddots &{}& \vdots \\ {}&{}&{}& \ddots &{ - 6}&{ - 2} \\ {}&{}&{}&{}&0&4 \\ {0}&{\cdots}&{}&{}&6&{ - 2} \end{array}} \right] $$ (33) 也为$n \times n$维常数矩阵, 利用这两个矩阵可以得到
$$ {\boldsymbol{\bar H}} = {\boldsymbol{V}}{{\boldsymbol{U}}^{ - 1}} $$ (34) 3. 应变与方位角
接触段绳索左右两端弧长坐标分别为${s_1}$和${s_n}$, 轴向应变分别为${\varepsilon _1}$和${\varepsilon _n}$, 两节点间的应变可用赫米特插值近似为
$$ \varepsilon = {N_1}{\varepsilon _i} + {N_2}{\varepsilon _{i + 1}} + L({N_3}{\varepsilon '_i} + {N_4}{\varepsilon '_{i + 1}}) $$ (35) 其中, 形函数
$$ {N}_{1} = 2{\xi }^{3}-3{\xi }^{2} + 1,\quad N'_{1} = 6{\xi }^{2}-6\xi $$ $$ {N}_{2} = -2{\xi }^{3} + 3{\xi }^{2},\quad N'_{2} = -6{\xi }^{2} + 6\xi $$ $$ {N}_{3} = {\xi }^{3}-2{\xi }^{2} + \xi ,\quad N'_{3} = 3{\xi }^{2}-4\xi + 1 $$ $$ {N}_{4} = {\xi }^{3}-{\xi }^{2},\quad N'_{4} = 3{\xi }^{2}-2\xi $$ 其中, $L = ({s_n} - {s_1}){(n - 1)^{ - 1}}$, $n$为节点数, 归一化坐标
$$ \xi = {L^{ - 1}}(s - {s_i}) $$ (36) 应变的弧长导数
$$ \varepsilon ' = {L^{ - 1}}({N'_1}{\varepsilon _i} + {N'_2}{\varepsilon _{i + 1}}) + {N'_3}{\varepsilon '_i} + {N'_4}{\varepsilon '_{i + 1}} $$ (37) 单元内的方位角
$$ \theta = {\theta _i} + {s_g}{r^{ - 1}}L\int\limits_0^\xi {(1 + \varepsilon ){\text{d}}\xi } $$ (38) 将式(35)代入式(38)可得方位角
$$ \theta = {\theta _i} + {s_g}{r^{ - 1}}L(\xi + {\bar N_1}{\varepsilon _i} + {\bar N_2}{\varepsilon _{i + 1}} + L{\bar N_3}{\varepsilon '_i} + L{\bar N_4}{\varepsilon '_{i + 1}}) $$ (39) $$ \begin{split} &{\theta _{i + 1}} = {\theta _i} + {s_g}{r^{ - 1}}L\left(1 + \frac{1}{2}{\varepsilon _i} + \frac{1}{2}{\varepsilon _{i + 1}}\right) + \\ &\qquad {s_g}{r^{ - 1}}{L^2}\left(\frac{1}{{12}}{\varepsilon '_i} - \frac{1}{{12}}{\varepsilon '_{i + 1}}\right) \end{split}$$ (40) 应变与方位角满足协调条件, 即绳索右端方位角${\theta _n}$满足约束方程
$$ \Delta s = r{s_g}({\theta _n} - {\theta _1}) - L\left[\sum\limits_{i = 1}^n {{\varepsilon _i}} - \frac{1}{2}({\varepsilon _1} + {\varepsilon _n}) + \frac{1}{{12}}L({\varepsilon '_1} - {\varepsilon '_n})\right] $$ (41) 4. 应变求解的数学模型
在系统运动中, 两者之间的接触大部分情况为静摩擦, 法向接触力与切向接触力需要满足库伦定律, 同时两端应变等于边界应变, 现弧长等于接触段应变的积分. 如果将应变当作系统变量, 代入整体平衡方程, 方程是自然满足的, 这是因为应变与接触力的关系本身就是根据微元体平衡得到的, 所以无法通过系统平衡得到应变分布. 为了解决这些困难, 将此问题转化为优化求解, 将应变需要满足的关系转化为不等式约束, 求得内部应变的分布. 滑轮接触段应变应满足约束条件如下
$$ {\left. {\bar \varepsilon } \right|_{\xi = 0}} = {\varepsilon _1} $$ (42) $$ {\left. {\bar \varepsilon } \right|_{\xi = 1}} = {\varepsilon _n} $$ (43) $$ \int\limits_0^1 {(1 + \bar \varepsilon ){\mathrm{d}}\xi } = {s_g}r\bar \theta {\bar s^{ - 1}} $$ (44) 节点处应变
$$ {\bar \varepsilon _i} = {c_{i,1}} + {c_{i,2}}{\bar \varepsilon _1} + {c_{i,3}}{\bar \varepsilon _n} $$ (45) $$ {\bar \varepsilon '_i} = {c_{i,4}} + {c_{i,5}}{\bar \varepsilon _1} + {c_{i,6}}{\bar \varepsilon _n} $$ (46) 其中系数
$$\qquad\qquad {c_{i,1}} = ({s_g}r\bar \theta {\bar s^{ - 1}} - 1{\text{ )}}{\eta _i}(6 - 6{\eta _i}) $$ $$\qquad\qquad {c_{i,2}} = 3\eta _i^2 - 4{\eta _i} + {\text{ }}1 $$ $$\qquad\qquad {c_{i,3}} = 3\eta _i^2 - 2{\eta _i} $$ $$\qquad\qquad {c_{i,4}} = ({s_g}r\bar \theta {\bar s^{ - 2}} - {\bar s^{ - 1}}{\text{ )(6}} - 12{\eta _i}) $$ $$\qquad\qquad {c_{i,5}} = {\bar s^{ - 1}}(6{\eta _i} - 4) $$ $$\qquad\qquad {c_{i,6}} = {\bar s^{ - 1}}(6{\eta _i} - 2) $$ 其中, $ {\eta _i} = (i - 1){(n - 1)^{ - 1}} $, $\bar \theta $为接触段两端方位角之差, $\bar s$为接触段两端弧长坐标之差. 应变求解可转化为如下优化问题, 求解变量
$$ {\varepsilon _i}{\text{ }}(n > i > 1)\text{, }{\varepsilon '_i} $$ (47) 组成求解变量列阵$ {\boldsymbol{\varepsilon }} = \left[ {{\varepsilon _2},{\varepsilon _3}, \cdots ,{\varepsilon _{n - 1}}} \right] $和$ {\boldsymbol{\varepsilon '}} = \left[ {{{\varepsilon '_1}},{{\varepsilon '_2}}, \cdots ,{{\varepsilon '_n}}} \right] $.
目标函数
$$\qquad\qquad\qquad f = \sum {(f_i^2} + g_i^2) $$ (48) $$\qquad\qquad\qquad {f_i} = {\varepsilon _i} - {\bar \varepsilon _i}, \qquad {g_i} = {\varepsilon '_i} - {\bar \varepsilon '_i} $$ (49) 由式(42) ~ 式(44)可得到等式约束
$$ \begin{split} &w = \bar s\Biggr[1 + \displaystyle\sum\limits_{i = 1}^n {{\varepsilon _i}} - \dfrac{1}{2}({\varepsilon _1} + {\varepsilon _n}) +\\ &\qquad \dfrac{1}{{12}}{{(n - 1)}^{ - 2}}\bar s({{\varepsilon '_1}} - {{\varepsilon '_n}})/ {{n - 1}}\Biggr]{\text{ = }} r{s_g}\bar \theta \end{split}$$ (50) $$ {\varepsilon _1} - {\bar \varepsilon _1} = 0 $$ (51) $$ {\varepsilon _n} - {\bar \varepsilon _n} = 0 $$ (52) 不等式约束
$$ r{\varepsilon '_i} + \mu {\varepsilon _i} \geqslant 0 $$ (53) $$ \qquad\qquad \mu {\varepsilon _i} - r{\varepsilon '_i} \geqslant 0 $$ (54) $$\qquad\qquad {\varepsilon _i} \geqslant 0 $$ (55) 根据库恩塔克条件(K-T条件)可得
$$\begin{split} &\mu ({\lambda _i} + {\bar \lambda _i}) + {\lambda _w}{p_i} + {\lambda _{{\varepsilon _1}}}({{\partial {\varepsilon _1}} \mathord{\left/ {\vphantom {{\partial {\varepsilon _1}} {\partial {\varepsilon _i}}}} \right. } {\partial {\varepsilon _i}}}) +\\ &\qquad {\lambda _{{\varepsilon _n}}}({{\partial {\varepsilon _n}} \mathord{\left/ {\vphantom {{\partial {\varepsilon _n}} {\partial {\varepsilon _i}}}} \right. } {\partial {\varepsilon _i}}}) + {\hat \lambda _i} = 2{f_i} \end{split}$$ (56) $$ r({\lambda _i} - {\bar \lambda _i}) + {\lambda _w}{q_i} = 2{g_i} $$ (57) 其中${\lambda _w}$,${\lambda _{{\varepsilon _1}}}$,${\lambda _{{\varepsilon _n}}}$,${\lambda _i}$,${\bar \lambda _i}$和${\hat \lambda _i}$分别为库恩塔克乘子.
$$ {\boldsymbol{p}}\; {\overset{\Delta}{\mathop{=}}}\; \frac{{\partial w}}{{\partial {\boldsymbol{\varepsilon }}}} = {(n - 1)^{ - 1}}\bar s\left[\frac{1}{2};1; \cdots ;1;\frac{1}{2}\right] $$ (58) $$ {\boldsymbol{q}}\; {\overset{\Delta}{\mathop{=}}}\;\frac{{\partial w}}{{\partial {\boldsymbol{\varepsilon '}}}} = \frac{1}{{12}}{(n - 1)^{ - 2}}{\bar s^2}\left[ {1;0; \cdots ;0; - 1} \right] $$ (59) 式中$ {\boldsymbol{p}} $和$ {\boldsymbol{q}} $为式(50)对求解变量$ {\boldsymbol{\varepsilon }} $和$ {\boldsymbol{\varepsilon '}} $求偏导数. $ {p_i} $和$ {q_i} $分为$ {\boldsymbol{p}} $和$ {\boldsymbol{q}} $中的第$i$个元素.
接触段的切向力与法向力需要满足库伦摩擦定律, 将其变为约束方程, 即为式(53) ~ 式(54), 根据互补理论, 将此不等式约束转化为如下方程
$$ \sqrt {\lambda _i^2 + {{(r{{\varepsilon '}_i} + \mu {\varepsilon _i})}^2}} = {\lambda _i} + r{\varepsilon '_i} + \mu {\varepsilon _i} $$ (60) $$ \sqrt {\bar \lambda _{\text{i}}^2 + {{(\mu {\varepsilon _i} - r{{\varepsilon '}_i})}^2}} = {\bar \lambda _i} + \mu {\varepsilon _i} - r{\varepsilon '_i} $$ (61) $$ \sqrt {\hat \lambda _i^2 + {{({\varepsilon _i} - {{10}^{ - 8}})}^2}} = {\hat \lambda _i} + {\varepsilon _i} - {10^{ - 8}} $$ (62) 同时, 接触段内还应满足方位角协调方程以及边界条件, 即如下方程
$$\qquad\qquad\quad w = 0 $$ (63) $$\qquad\qquad\quad {\varepsilon _1} - {\bar \varepsilon _1} = 0 $$ (64) $$\qquad\qquad\quad {\varepsilon _n} - {\bar \varepsilon _n} = 0 $$ (65) 上式共同组成了一组关于应变和应变弧长导数的非线性方程, 其中接触段绳索的边界参数
$$ {\boldsymbol{x}} = \left[ {{\varepsilon _1};{\varepsilon _n};{s_1};{s_n};{\theta _1};{\theta _n}} \right] $$ (66) 将库恩塔克条件得到的方程对边界参数求导数后, 可得到如下方程
$$ {\boldsymbol{AX}} = {\boldsymbol{B}} $$ (67) 其中系数矩阵
$$ {\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} { - 2{{\boldsymbol{\varGamma }}_{f,\varepsilon }}}&{{{\boldsymbol{0}}_{n \times n}}}&{\mu {\boldsymbol{E}}}&{\mu {\boldsymbol{E}}}&{\boldsymbol{p}}&{{{\partial {\varepsilon _1}} \mathord{\left/ {\vphantom {{\partial {\varepsilon _1}} {\partial {{\boldsymbol{\varepsilon }}^{\mathrm{T}}}}}} \right. } {\partial {{\boldsymbol{\varepsilon }}^{\mathrm{T}}}}}}&{{{\partial {\varepsilon _n}} \mathord{\left/ {\vphantom {{\partial {\varepsilon _n}} {\partial {{\boldsymbol{\varepsilon }}^{\mathrm{T}}}}}} \right. } {\partial {{\boldsymbol{\varepsilon }}^{\mathrm{T}}}}}}&{\boldsymbol{E}} \\ { - 2{{\boldsymbol{\varGamma }}_{g,\varepsilon }}}&{ - 2{{\boldsymbol{\varGamma }}_{g,\varepsilon '}}}&{r{\boldsymbol{E}}}&{ - r{\boldsymbol{E}}}&{\boldsymbol{q}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times n}}} \\ {\mu {{\boldsymbol{D}}_\lambda }}&{r{{\boldsymbol{D}}_\lambda }}&{\mu {{\boldsymbol{D}}_\varepsilon }{\text{ + }}r{{\boldsymbol{D}}_{\varepsilon '}}}&{{{\boldsymbol{0}}_{n \times n}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times n}}} \\ {\mu {{\boldsymbol{D}}_{\bar \lambda }}}&{ - r{{\boldsymbol{D}}_{\bar \lambda }}}&{{{\boldsymbol{0}}_{n \times n}}}&{\mu {{\boldsymbol{D}}_\varepsilon } - r{{\boldsymbol{D}}_{\varepsilon '}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times n}}} \\ {{{\partial w} \mathord{\left/ {\vphantom {{\partial w} {\partial {\boldsymbol{\varepsilon }}}}} \right. } {\partial {\boldsymbol{\varepsilon }}}}}&{{{\partial w} \mathord{\left/ {\vphantom {{\partial w} {\partial {\boldsymbol{\varepsilon '}}}}} \right. } {\partial {\boldsymbol{\varepsilon '}}}}}&{{{\boldsymbol{0}}_{1 \times n}}}&{{{\boldsymbol{0}}_{1 \times n}}}&0&0&0&{{{\boldsymbol{0}}_{1 \times n}}} \\ {{{\partial {\varepsilon _1}} \mathord{\left/ {\vphantom {{\partial {\varepsilon _1}} {\partial {\boldsymbol{\varepsilon }}}}} \right. } {\partial {\boldsymbol{\varepsilon }}}}}&{{{\boldsymbol{0}}_{1 \times n}}}&{{{\boldsymbol{0}}_{1 \times n}}}&{{{\boldsymbol{0}}_{1 \times n}}}&0&0&0&{{{\boldsymbol{0}}_{1 \times n}}} \\ {{{\partial {\varepsilon _n}} \mathord{\left/ {\vphantom {{\partial {\varepsilon _n}} {\partial {\boldsymbol{\varepsilon }}}}} \right. } {\partial {\boldsymbol{\varepsilon }}}}}&{{{\boldsymbol{0}}_{1 \times n}}}&{{{\boldsymbol{0}}_{1 \times n}}}&{{{\boldsymbol{0}}_{1 \times n}}}&0&0&0&{{{\boldsymbol{0}}_{1 \times n}}} \\ {{{\boldsymbol{D}}_{\hat \lambda }}}&{{{\boldsymbol{0}}_{n \times n}}}&{{{\boldsymbol{0}}_{n \times n}}}&{{{\boldsymbol{0}}_{n \times n}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{D}}_\varepsilon } - {{10}^{ - 8}}} \end{array}} \right] $$ $$ {\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{B}}_{1,3}}}&{{{\boldsymbol{B}}_{1,4}}} \\ {{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{B}}_{2,3}}}&{{{\boldsymbol{B}}_{2,4}}} \\ {{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}} \\ {{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}} \\ 0&0&{{{\boldsymbol{B}}_{5,3}}}&{{{\boldsymbol{B}}_{5,4}}} \\ 1&0&0&0 \\ 0&1&0&0 \\ {{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}}&{{{\boldsymbol{0}}_{n \times 1}}} \end{array}} \right]$$ ${\boldsymbol{A}}$和${\boldsymbol{B}}$矩阵中的元素为
$$ {{\boldsymbol{\varGamma }}_{f,\varepsilon }} = {\boldsymbol{E}} + \left[ {\begin{array}{*{20}{l}} { - {{\boldsymbol{c}}_2}}&{{{\boldsymbol{0}}_{n \times 1}}}& \cdots &{{{\boldsymbol{0}}_{n \times 1}}}&{ - {{\boldsymbol{c}}_3}} \end{array}} \right] $$ $$ {{\boldsymbol{\varGamma }}_{g,\varepsilon '}} = {\boldsymbol{E}} $$ $$ {{\boldsymbol{\varGamma }}_{g,\varepsilon }} = \left[ {\begin{array}{*{20}{l}} { - {{\boldsymbol{c}}_5}}&{{{\boldsymbol{0}}_{n \times 1}}}& \cdots &{{{\boldsymbol{0}}_{n \times 1}}}&{ - {{\boldsymbol{c}}_6}} \end{array}} \right] $$ $$ {{\boldsymbol{D}}_\lambda } = {\text{diag}}({\boldsymbol{\lambda }}) \text{, } {{\boldsymbol{D}}_{\bar \lambda }} = {\text{diag}}({\boldsymbol{\bar \lambda }}) \text{, } {{\boldsymbol{D}}_{\hat \lambda }} = {\text{diag}}({\boldsymbol{\hat \lambda }}) $$ $$ {{\boldsymbol{D}}_\varepsilon } = {\text{diag}}({\boldsymbol{\varepsilon }}) \text{, } {{\boldsymbol{D}}_{\varepsilon '}} = {\text{diag}}({\boldsymbol{\varepsilon '}}) $$ $$ {{\boldsymbol{B}}_{1,3}} = - 2\frac{{\partial {{\boldsymbol{c}}_1}}}{{\partial \bar s}} - {\lambda _w}\frac{{\partial {\boldsymbol{p}}}}{{\partial \bar s}}, \quad {{\boldsymbol{B}}_{1,4}} = - 2\frac{{\partial {{\boldsymbol{c}}_1}}}{{\partial \bar \theta }} $$ $$ {{\boldsymbol{B}}_{2,3}} = - 2\left(\frac{{\partial {{\boldsymbol{c}}_4}}}{{\partial \bar s}} + {\varepsilon _1}\frac{{\partial {{\boldsymbol{c}}_5}}}{{\partial \bar s}} + {\varepsilon _n}\frac{{\partial {{\boldsymbol{c}}_6}}}{{\partial \bar s}}\right) - {\lambda _w}\frac{{\partial {\boldsymbol{q}}}}{{\partial \bar s}} $$ $$ {{\boldsymbol{B}}_{2,4}} = - 2\frac{{\partial {{\boldsymbol{c}}_4}}}{{\partial \bar \theta }}$$ $$ {{\boldsymbol{B}}_{5,3}} = - \frac{{\partial w}}{{\partial \bar s}} \text{, }\quad {{\boldsymbol{B}}_{5,4}} = - \frac{{\partial w}}{{\partial \bar \theta }} $$ 其中${{\boldsymbol{c}}_1},{{\boldsymbol{c}}_2}, \cdots, {{\boldsymbol{c}}_6}$为式(45) ~ 式(46)中的系数组成的列阵, ${\boldsymbol{\lambda }},{\boldsymbol{\bar \lambda }}和{\boldsymbol{\hat \lambda }}$为式(53) ~ 式(55)对应库恩塔克乘子组成的列阵.
$$ {\boldsymbol{X}} = \left[ {\begin{array}{*{20}{l}} {{{\partial {\boldsymbol{\varepsilon }}} / {\partial {\varepsilon _1}}}}&{{{\partial {\boldsymbol{\varepsilon }}} / {\partial {\varepsilon _n}}}}&{{{\partial {\boldsymbol{\varepsilon }}} / {\partial \bar s}}}&{{{\partial {\boldsymbol{\varepsilon }}} / {\partial \bar \theta }}} \\ {{{\partial {\boldsymbol{\varepsilon '}}} / {\partial {\varepsilon _1}}}}&{{{\partial {\boldsymbol{\varepsilon '}}} / {\partial {\varepsilon _n}}}}&{{{\partial {\boldsymbol{\varepsilon '}}} / {\partial \bar s}}}&{{{\partial {\boldsymbol{\varepsilon '}}} / {\partial \bar \theta }}} \\ {{{\partial {\boldsymbol{\lambda }}} / {\partial {\varepsilon _1}}}}&{{{\partial {\boldsymbol{\lambda }}} \mathord{\left/ {\vphantom {{\partial {\boldsymbol{\lambda }}} {\partial {\varepsilon _n}}}} \right. } {\partial {\varepsilon _n}}}}&{{{\partial {\boldsymbol{\lambda }}} / {\partial \bar s}}}&{{{\partial {\boldsymbol{\lambda }}} / {\partial \bar \theta }}} \\ {{{\partial {\boldsymbol{\bar \lambda }}} / {\partial {\varepsilon _1}}}}&{{{\partial {\boldsymbol{\bar \lambda }}} / {\partial {\varepsilon _n}}}}&{{{\partial {\boldsymbol{\bar \lambda }}} / {\partial \bar s}}}&{{{\partial {\boldsymbol{\bar \lambda }}} / {\partial \bar \theta }}} \\ {{{\partial {\lambda _w}} / {\partial {\varepsilon _1}}}}&{{{\partial {\lambda _w}} / {\partial {\varepsilon _n}}}}&{{{\partial {\lambda _w}} / {\partial \bar s}}}&{{{\partial {\lambda _w}} / {\partial \bar \theta }}} \\ {{{\partial {\lambda _{{\varepsilon _1}}}} / {\partial {\varepsilon _1}}}}&{{{\partial {\lambda _{{\varepsilon _1}}}} / {\partial {\varepsilon _n}}}}&{{{\partial {\lambda _{{\varepsilon _1}}}} / {\partial \bar s}}}&{{{\partial {\lambda _{{\varepsilon _1}}}} / {\partial \bar \theta }}} \\ {{{\partial {\lambda _{{\varepsilon _n}}}} / {\partial {\varepsilon _1}}}}&{{{\partial {\lambda _{{\varepsilon _n}}}} / {\partial {\varepsilon _n}}}}&{{{\partial {\lambda _{{\varepsilon _n}}}} / {\partial \bar s}}}&{{{\partial {\lambda _{{\varepsilon _n}}}} / {\partial \bar \theta }}} \\ {{{\partial {\boldsymbol{\hat \lambda }}} / {\partial {\varepsilon _1}}}}&{{{\partial {\boldsymbol{\hat \lambda }}} / {\partial {\varepsilon _n}}}}&{{{\partial {\boldsymbol{\hat \lambda }}} / {\partial \bar s}}}&{{{\partial {\boldsymbol{\hat \lambda }}} / {\partial \bar \theta }}} \end{array}} \right] $$ (68) 将式(67)再对边界参数求导
$$ {\boldsymbol{A}}{{\partial {\boldsymbol{X}}} \mathord{\left/ {\vphantom {{\partial {\boldsymbol{X}}} {\partial x}}} \right. } {\partial x}} = {{\partial {\boldsymbol{B}}} \mathord{\left/ {\vphantom {{\partial {\boldsymbol{B}}} {\partial x - }}} \right. } {\partial x - }}{{\partial {\boldsymbol{A}}} \mathord{\left/ {\vphantom {{\partial {\boldsymbol{A}}} {\partial x}}} \right. } {\partial x}}{\boldsymbol{X}} $$ (69) 可得到求解变量对边界参数的二阶导数.
5. 绳索两端边界条件与平衡方程
静力学分析时, 弹性节点力
$$ {{\boldsymbol{f}}_e} = L{E_s}\int\limits_0^1 {{\boldsymbol{\bar T}}_{\varepsilon ,q}^{\text{T}}{\varepsilon _s}{\mathrm{d}}\xi } $$ (70) 节点体力为
$$ {{\boldsymbol{f}}_g} = \rho L\int\limits_0^1 {{\boldsymbol{\bar T}}_{r,q}^{\text{T}}{\boldsymbol{g}}{\mathrm{d}}\xi } $$ (71) 其中$ {\boldsymbol{\bar T}}_{\varepsilon ,q}^{\text{T}} $,$ {\boldsymbol{\bar T}}_{r,q}^{\text{T}} $为系统描述变量与应变变化率和矢径变化率的转换阵, $L$为单元长度, ${\boldsymbol{g}}$为重力加速度. 对式(70) ~ 式(71)求系统变量的导数可得雅可比矩阵. 滑轮边界点矢径
$$ {\boldsymbol{r}} = {{\boldsymbol{r}}_c} + r{\boldsymbol{n}} $$ (72) 其中${{\boldsymbol{r}}_c}$为滑轮形心矢径, $r$为滑轮半径. 接触边界点法向和切向矢量分别为
$$ {\boldsymbol{n}} = {\mathrm{cos}}\theta {{\boldsymbol{e}}_x} + \sin \theta {{\boldsymbol{e}}_y} $$ (73) $$ {\boldsymbol{t}} = {\mathrm{cos}}\theta {{\boldsymbol{e}}_y} - \sin \theta {{\boldsymbol{e}}_x} $$ (74) 如图7所示, ${\boldsymbol{t}}$为滑轮槽切线方向, ${{\boldsymbol{e}}_z}$为滑轮面法线方向, 当绳索进入滑轮时, 绳索切向不一定沿着滑轮槽切向${\boldsymbol{t}}$, 可能会有角度偏差, 实际绳索的方向为$ {\boldsymbol{\bar t}} = \cos \varphi {\boldsymbol{t}} + \sin \varphi {{\boldsymbol{e}}_z} $.
于是边界点处矢径对弧长的导数
$$ {\boldsymbol{r'}} = (1 + {\boldsymbol{\varepsilon}} ){\boldsymbol{\bar t}} $$ (75) 绳索在边界点处的物质速度
$$ {{\partial {\boldsymbol{r}}} \mathord{\left/ {\vphantom {{\partial {\boldsymbol{r}}} {\partial t}}} \right. } {\partial t}} = {\boldsymbol{\dot r}} - {\boldsymbol{r'}}\dot s $$ (76) 滑轮上与绳索边界点重合点的物质速度
$$ {{\partial {\boldsymbol{\hat r}}} \mathord{\left/ {\vphantom {{\partial {\boldsymbol{\hat r}}} {\partial t}}} \right. } {\partial t}} = {{\boldsymbol{\dot r}}_c} + r{\boldsymbol{\bar \omega }} \times {\boldsymbol{n}} + r\dot \gamma {\boldsymbol{t}}$$ (77) 当绳索在边界点处相对于滑轮无滑动时
$$ ({{\partial {\boldsymbol{r}}} \mathord{\left/ {\vphantom {{\partial {\boldsymbol{r}}} {\partial t}}} \right. } {\partial t}} - {{\partial {\boldsymbol{\hat r}}} \mathord{\left/ {\vphantom {{\partial {\boldsymbol{\hat r}}} {\partial t}}} \right. } {\partial t}}) \cdot {\boldsymbol{t}} = 0 $$ (78) 其中, 虚速度关系
$$ \delta {\dot \theta _1} - \delta \dot \gamma - (\delta {\dot s_1}){r^{ - 1}}(1 + {\varepsilon _1})\cos {\varphi _1} = 0 $$ (79) $$ \delta {\dot \theta _2} - \delta \dot \gamma - (\delta {\dot s_2}){r^{ - 1}}(1 + {\varepsilon _2})\cos {\varphi _2} = 0 $$ (80) 式(79) ~ 式(80)表示了绳索运动与滑轮自转间的关系, 求解平衡方程时, 补充方程为滑轮自转角为
$$ \gamma = 0 $$ (81) 6. 算例
算例1: 首先讨论一个平面起升机构, 系统只包含一个动滑轮, 一端为核环吊驱动卷筒, 一端为吊装机构固定点, 滑轮绳索基本参数: 滑轮半径分别为0.1, 0.3和0.5 m. 绳索半径0.01 m, 绳索抗拉模量2.1 × 105 MPa, 密度7800 kg/m3. 系统的原点位于卷筒中心, 动滑轮下方有4000 kg的吊载. 系统模型的长度(x方向)大概为5 m, 高度(z方向)为5 m. 动滑轮在不同半径下接触段绳索长度约为0.23, 0.8和1.2 m. 以下算例中数据曲线的x轴均为归一化弧长坐标. 系统模型见图8. 系统平衡后, 不同半径的滑轮两端边界计算结果见表1.
表 1 各滑轮端部的弧长坐标、方位角、轴向应变Table 1. The arc length coordinates, azimuth angle and axial strain of boundary of the pulleypulley (r = 0.1 m) pulley (r = 0.3 m) pulley (r = 0.5 m) ${s_1}$/m 5.6733 5.6474 5.6144 ${s_n}$/m 5.9063 6.3679 6.8512 ${\theta _1}$/(°) 3.5464 3.5104 3.4743 ${\theta _n}$/(°) 5.8784 5.9143 5.9503 ${\varepsilon _1}$ 0.0009 0.0009 8.8936 × 10−4 ${\varepsilon _n}$ 0.0005 0.0006 6.5274 × 10−4 ![]()
图 9 不同滑轮半径接触段轴向应变随弧长的变化Figure 9. The axial strain of the contact section with different pulley radii varies with arc length针对以上算例, 分别采用了本文方法和Ansys进行了对比计算, 如果使用Ansys里面的梁单元, 不能计算出接触段绳索的切向接触力, 只能采用实体单元solid45, 然后提取单元应变, 实体单元大小为3 mm, 从图9中可以看到, 当滑轮半径为0.5 m和0.3 m时, 本文计算出的接触段应变分布与Ansys大致相同, 但是Ansys耗时时间43 s, 本文用时仅5 s, 且Ansys计算出的数据波动很大, 当滑轮半径为0.1 m时, 软件计算出的应变明显不合理, 呈现类似于正弦曲线的趋势, 说明在处理滑轮绳索接触段时, 软件有局限性, 对于小半径的滑轮, 应变变化率快, 计算可能不准确.
从应变计算结果可知, 当滑轮半径为0.5 m时, 滑轮接触段应变左侧大于右侧, 随着滑轮半径的减小, 滑轮右侧应变逐步增加. 从图10和图11可知, 滑轮半径越小, 切向接触力密度变化越大, 应变与法向力的峰值越大. 随着滑轮半径的增大, 切向接触与法向接触力反而减小, 接触力越均匀, 这也说明了滑轮内部接触力并不是均匀分布的, 当切向接触力与法向接触力的比值大于一定数值后, 局部区域可能会有打滑现象, 即蠕滑现象.
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图 10 切向接触力密度${q_s}$随弧长的变化Figure 10. Tangential contact force density varies with arc length![]()
图 11 法向接触力密度${q_b}$随弧长的变化Figure 11. Normal contact force density varies with arc length算例2: 核环吊中空间滑轮绳索基本参数: 滑轮半径均为0.1 m, 绳索半径0.01 m, 绳索抗拉模量1.8 × 105 MPa, 密度7800 kg/m3, 系统包含3个滑轮, 两个动滑轮, 一个定滑轮. 动滑轮吊载质量为18000 kg. 系统坐标系原点位于卷筒中心处, 系统模型的长度(x方向)大概为15 m, 高度(z方向)为5 m. 两个定滑轮接触段长度约为0.2 m, 动滑轮接触段长度约为0.4 m. 计算结果中定义切向接触力与法向接触力的比值为摩擦系数. 系统模型如图12所示. 3个滑轮的两端边界计算结果如表2所示.
表 2 各滑轮端部的弧长坐标、方位角、轴向应变Table 2. The arc length coordinates, azimuth angle and axial strain of boundary of the pulleysFixed pulley1 Fixed pulley2 Movable1 ${s_1}$/m 4.7385 15.8054 10.1547 ${s_n}$/m 4.9497 16.0384 10.5995 ${\theta _1}$/(°) 1.5113 2.7400 3.5412 ${\theta _n}$/(°) 0.3990 1.5166 5.8836 ${\varepsilon _1}$ 0.0533 0.0507 0.0532 ${\varepsilon _n}$ 0.0533 0.0016 0.0532 表2给出了系统中两个定滑轮和一个动滑轮的边界参数. 卷筒出绳点弧长坐标为0, 弧长坐标从左到右依次增加, 右侧固定点弧长坐标为最大值.
表3给出了动滑轮在总体坐标系下初始和平衡后的位置, x坐标变化了0.01 m, y坐标保持不变, z坐标由−5 m变为了−4.85 m, 即动滑轮向上提升了0.15 m. 由图13可知, 滑轮两端的方位角基本与弧长呈线性函数关系, 两个定滑轮的绳索方位角均随着弧长递减, 且变化率基本相同, 动滑轮的绳索方位角随着弧长递增.
表 3 初始时刻和平衡后动滑轮的位置Table 3. The position of the moving pulley at the initial moment and balancingx/m y/m z/m initial position 7.5 −0.5 −5 balance position 7.49 −0.5 −4.85 由图14可知, 动滑轮绳索接触段的应变并不是线性变化的, 峰值大概在整个区域的1/3处, 左侧的应变略大于右侧的应变, 滑轮与绳索之间的材料摩擦系数[34]为0.28, 图15中已用虚线标出, 结合图15不同位置处的摩擦系数, 判断此系统中的动滑轮接触段可能在1/3处发生蠕滑. 两个定滑轮的摩擦系数均远小于动滑轮的摩擦系数, 所以两个定滑轮的接触段绳索与滑轮相对不容易发生滑动, 见图16. 表4给出了在总体坐标系下各个滑轮左端绳索拉力F1和右端绳索拉力F2的各分量.
表 4 滑轮的左端拉力与右端拉力Table 4. Left tension and right tension on the pulleyUnit/kN Fixed pulley1 Fixed pulley2 Movable1 Fx1 −95.71 −35.67 −37.25 Fy1 0 0 0 Fz1 5.69 −84.00 88.219 Fx2 37.25 28.78 37.25 Fy2 0 0 0 Fz2 −88.34 −1.5618 88.2141 由图17可知, 切向接触力密度随弧长坐标的变化较为剧烈, 尤其是靠近固定点端滑轮的切向接触力密度有明显的非线性变化. 越靠近固定点的滑轮, 切向接触力密度变化越为剧烈. 由图14和图18可知, 定滑轮和动滑轮的轴向应变在各接触段分布区别很大. 由图18可知, 轴向应变与法向接触力密度变化趋势相同.
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图 17 定滑轮2接触段切向接触力密度的随弧长的变化Figure 17. The variation of tangential contact force density with arc length in the contact section of fixed pulley 2![]()
图 18 定滑轮接触段绳索的轴向应变和法向接触力密度${q_b}$Figure 18. Axial strain and normal contact force density of the ropein the contact section of the fixed pulley算例3: 核环吊中的卷筒一般为同一侧, 在起吊上方会有两个导向定滑轮, 防止钢丝绳与滑轮面外角过大, 进而出现磨损, 影响绳索寿命和起升过程中的稳定性. 本算例讨论了面外角对接触段接触力的影响. 系统基本参数: 吊载重量4000 kg, 滑轮半径0.2 m, 绳索半径0.01 m, 绳索抗拉模量2.1 × 105 MPa, 密度7800 kg/m3, 系统包含5个定滑轮, 以及4个动滑轮组成的滑轮组, 总体坐标系建在左端卷筒的中心, 见图19.
表5和表6给出了系统各定滑轮和动滑轮的边界参数的计算结果. 由端部弧长坐标可以大概判断接触段的长度.
表 5 各定滑轮端部的弧长坐标、方位角、轴向应变Table 5. The arc length coordinates, azimuth angle and axial strain of boundary of the pulleyPulley1 Pulley2 Pulley3 Pulley 4 Pulley5 ${s_1}$/m 5.1045 25.9961 47.307 68.634 89.9972 ${s_n}$/m 5.3237 26.6390 47.959 69.276 90.3167 ${\theta _1}$/(°) 1.5119 3.1814 3.1833 3.1853 0.005 ${\theta _n}$/(°) 0.1866 0.4254 0.4273 0.4295 1.6031 ${\varepsilon _1}$ 2.71 × 10−4 2.60 × 10−4 2.64 × 10−4 2.636 × 10−4 2.507 × 10−4 ${\varepsilon _n}$ 2.65 × 10−4 2.63 × 10−4 2.64 × 10−4 2.666 × 10−4 2.184 × 10−4 表 6 各动滑轮端部的弧长坐标、方位角、轴向应变Table 6. The arc length coordinates, azimuth angle and axial strain of boundary of the pulleyPulley1 Pulley2 Pulley3 Pulley 4 ${s_1}$/m 15.3129 36.6389 57.9812 79.3419 ${s_n}$/m 15.9566 37.2829 58.6251 79.9772 ${\theta _1}$/(°) −3.17 −3.1769 −3.1745 −3.172 ${\theta _n}$/(°) −0.419 0.048 0.0457 0.0052 ${\varepsilon _1}$ 2.52 × 10−4 2.524 × 10−4 2.534 × 10−4 2.563 × 10−4 ${\varepsilon _n}$ 2.54 × 10−4 2.529 × 10−4 2.524 × 10−4 2.527 × 10−4 由图20可知, 起升机构中的滑轮接触段应变分布是有规律的, 随着滑轮号的增大, 变化趋势为左侧应变逐步减小, 右侧应变逐步增大, 位于中间位置的滑轮, 应变分布趋向于均匀, 靠近左侧驱动卷筒的滑轮, 则左侧应变大于右侧应变, 靠近驱动卷筒右侧的滑轮, 则右侧应变大于左侧应变, 这也符合绳索受力规律.
由图21可知, 靠近驱动卷筒两侧的动滑轮1和4, 摩擦系数大于中间位置的两个动滑轮2和3. 图22为定滑轮5的接触段摩擦系数随归一化弧长坐标的变化, 滑轮与绳索之间的材料摩擦系数[34]为0.28, 图中已用虚线标出. 当面外角为0.18°时, 定滑轮5的接触段中间区域的绳索会出现蠕滑. 图22中改变滑轮的方位, 使滑轮出绳点处的面外角由0.18°减小为0.08°, 接触段摩擦系数整体会下降, 即切向接触力与法向接触力的比值降低, 所以通过改变边界处与卷筒连接的导向定滑轮的方位, 可以改善摩擦系数, 减少发生打滑的概率.
由算例1可知随着滑轮半径的增大, 接触段的接触力变化越平稳, 接触力越均匀, 靠近固定点端的滑轮, 接触力比远离边界处的滑轮大, 同时接触段接触力变化越剧烈. 在起升机构吊载过程中, 由于左右端固定点滑轮与卷筒相连接, 卷筒在工作过程中, 出绳点的位置一直在变化, 导致绳索方向在变化, 会造成左右端固定点处滑轮的绳索面外角的改变. 如果安装方位不合适, 面外角一直增大, 则绳索会磨损滑轮沿, 更容易出现打滑, 所以需要更加注意左右端固定点滑轮的安装方位, 且适当增加滑轮半径, 同时减小驱动卷筒与导向滑轮的面外角, 此处的接触段受力不均匀, 更易出现打滑现象.
7. 结论
本文从接触段绳索微元体平衡出发, 推导了绳索动力学方程, 继而得到了接触力密度的表达式. 利用几何关系计算出了离散的接触力密度, 并通过结合绳索接触段受力特点, 将接触力密度离散, 利用库恩塔克条件以及接触力需要满足的互补方程, 建立了数学模型, 提出了一种处理接触段的计算方法, 得到了接触段的应变分布. 通过计算得到结果如下.
(1) 绳索接触力受到多个因素的影响, 分布不均匀, 当滑轮与绳索处于静摩擦时, 应变微幅变化, 当系统中滑轮两侧连接卷筒与重物时, 应变变化较为明显, 靠近驱动卷筒, 靠近固定点, 且半径越小的滑轮, 接触段接触力变化剧烈, 应变峰值较大.
(2) 动滑轮接触段应变为非线性变化, 两端应变基本相同, 但是在靠近固定点的滑轮, 两端应变有差异, 容易出现打滑现象, 当接触段为定滑轮, 且一段连接固定点时, 应变分布不一致, 可根据计算出的摩擦系数判断可能出现蠕滑的区域.
(3) 滑轮内部方位角基本成线性变化, 包角越大, 接触力越均匀, 绳索容易平稳过渡. 通过控制导向滑轮的面外角, 可以降低接触段摩擦系数, 有利于起吊过程稳定.
文中分析了不同半径的滑轮和不同类型滑轮的接触力, 总结了接触段应变分布规律, 为核环吊起升机构以及含有滑轮绳索机构的大型机械系统分析提供了理论准备.
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图 9 不同滑轮半径接触段轴向应变随弧长的变化
Figure 9. The axial strain of the contact section with different pulley radii varies with arc length
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图 10 切向接触力密度${q_s}$随弧长的变化
Figure 10. Tangential contact force density varies with arc length
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图 11 法向接触力密度${q_b}$随弧长的变化
Figure 11. Normal contact force density varies with arc length
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图 17 定滑轮2接触段切向接触力密度的随弧长的变化
Figure 17. The variation of tangential contact force density with arc length in the contact section of fixed pulley 2
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图 18 定滑轮接触段绳索的轴向应变和法向接触力密度${q_b}$
Figure 18. Axial strain and normal contact force density of the ropein the contact section of the fixed pulley
表 1 各滑轮端部的弧长坐标、方位角、轴向应变
Table 1 The arc length coordinates, azimuth angle and axial strain of boundary of the pulley
pulley (r = 0.1 m) pulley (r = 0.3 m) pulley (r = 0.5 m) ${s_1}$/m 5.6733 5.6474 5.6144 ${s_n}$/m 5.9063 6.3679 6.8512 ${\theta _1}$/(°) 3.5464 3.5104 3.4743 ${\theta _n}$/(°) 5.8784 5.9143 5.9503 ${\varepsilon _1}$ 0.0009 0.0009 8.8936 × 10−4 ${\varepsilon _n}$ 0.0005 0.0006 6.5274 × 10−4 
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表 2 各滑轮端部的弧长坐标、方位角、轴向应变
Table 2 The arc length coordinates, azimuth angle and axial strain of boundary of the pulleys
Fixed pulley1 Fixed pulley2 Movable1 ${s_1}$/m 4.7385 15.8054 10.1547 ${s_n}$/m 4.9497 16.0384 10.5995 ${\theta _1}$/(°) 1.5113 2.7400 3.5412 ${\theta _n}$/(°) 0.3990 1.5166 5.8836 ${\varepsilon _1}$ 0.0533 0.0507 0.0532 ${\varepsilon _n}$ 0.0533 0.0016 0.0532
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表 3 初始时刻和平衡后动滑轮的位置
Table 3 The position of the moving pulley at the initial moment and balancing
x/m y/m z/m initial position 7.5 −0.5 −5 balance position 7.49 −0.5 −4.85
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表 4 滑轮的左端拉力与右端拉力
Table 4 Left tension and right tension on the pulley
Unit/kN Fixed pulley1 Fixed pulley2 Movable1 Fx1 −95.71 −35.67 −37.25 Fy1 0 0 0 Fz1 5.69 −84.00 88.219 Fx2 37.25 28.78 37.25 Fy2 0 0 0 Fz2 −88.34 −1.5618 88.2141
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表 5 各定滑轮端部的弧长坐标、方位角、轴向应变
Table 5 The arc length coordinates, azimuth angle and axial strain of boundary of the pulley
Pulley1 Pulley2 Pulley3 Pulley 4 Pulley5 ${s_1}$/m 5.1045 25.9961 47.307 68.634 89.9972 ${s_n}$/m 5.3237 26.6390 47.959 69.276 90.3167 ${\theta _1}$/(°) 1.5119 3.1814 3.1833 3.1853 0.005 ${\theta _n}$/(°) 0.1866 0.4254 0.4273 0.4295 1.6031 ${\varepsilon _1}$ 2.71 × 10−4 2.60 × 10−4 2.64 × 10−4 2.636 × 10−4 2.507 × 10−4 ${\varepsilon _n}$ 2.65 × 10−4 2.63 × 10−4 2.64 × 10−4 2.666 × 10−4 2.184 × 10−4
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表 6 各动滑轮端部的弧长坐标、方位角、轴向应变
Table 6 The arc length coordinates, azimuth angle and axial strain of boundary of the pulley
Pulley1 Pulley2 Pulley3 Pulley 4 ${s_1}$/m 15.3129 36.6389 57.9812 79.3419 ${s_n}$/m 15.9566 37.2829 58.6251 79.9772 ${\theta _1}$/(°) −3.17 −3.1769 −3.1745 −3.172 ${\theta _n}$/(°) −0.419 0.048 0.0457 0.0052 ${\varepsilon _1}$ 2.52 × 10−4 2.524 × 10−4 2.534 × 10−4 2.563 × 10−4 ${\varepsilon _n}$ 2.54 × 10−4 2.529 × 10−4 2.524 × 10−4 2.527 × 10−4
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[1] 陈松涛, 高宁, 王翰涛. 核岛内部环行桥式起重机设计. 起重运输机械, 2011, 6: 33-37 (Chen Songtao, Gao Ning, Wang Hantao. Design of a circular bridge crane inside the nuclear island. Lifting and Transportation Machinery, 2011, 6: 33-37 (in Chinese) Chen Songtao, Gao Ning, Wang Hantao. Design of a circular bridge crane inside the nuclear island. Lifting and Transportation Machinery, 2011, 6: 33-37 (in Chinese)
[2] 谭春林, 魏承. 绳索动力学基础研究. 北京: 科学出版社, 2018 (Tan Chunlin, Wei Cheng. Basic Research of Rope Dynamics. Beijing: Science Press, 2018 (in Chinese) Tan Chunlin, Wei Cheng. Basic Research of Rope Dynamics. Beijing: Science Press, 2018 (in Chinese)
[3] 王郡, 朱永宁, 徐鉴. 自主泳动弹性绳的轨迹模拟. 力学学报, 2019, 51(1): 198-208 (Wang Jun, Zhu Yongning, Xu Jian. Trajectory simulation of self-propelled elastic rods in fluid. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(1): 198-208 (in Chinese) doi: 10.6052/0459-1879-18-230 Wang Jun, Zhu Yongning, Xu Jian . Trajectory simulation of self-propelled elastic rods in fluid. Chinese Journal of Theoretical and Applied Mechanics,2019 ,51 (1 ):198 -208 (in Chinese) doi: 10.6052/0459-1879-18-230[4] 齐朝晖, 宋慧涛, 张志刚. 核环吊平衡路径与吊重轨迹偏差. 工程力学, 2014, 31(1): 209-217 (Qi Zhaohui, Song Huitao, Zhang Zhigang. Equilibrium paths and load trajectory deflections of nuclear ring cranes. Engineering Mechanics, 2014, 31(1): 209-217 (in Chinese) doi: 10.6052/j.issn.1000-4750.2012.09.0673 Qi Zhaohui, Song Huitao, Zhang Zhigang . Equilibrium paths and load trajectory deflections of nuclear ring cranes. Engineering Mechanics,2014 ,31 (1 ):209 -217 (in Chinese) doi: 10.6052/j.issn.1000-4750.2012.09.0673[5] 胡海岩, 田强, 张伟等. 大型网架式可展开空间结构的非线性动力学与控制. 力学进展, 2013, 43(4): 390-414 (Hu Haiyan, Tian Qiang, Zhang Wei, et al. Nonlinear dynamics and control of large deployable space structures composed of trusses and meshes. Advances in Mechanics, 2013, 43(4): 390-414 (in Chinese) doi: 10.6052/1000-0992-13-045 Hu Haiyan, Tian Qiang, Zhang Wei, et al . Nonlinear dynamics and control of large deployable space structures composed of trusses and meshes. Advances in Mechanics,2013 ,43 (4 ):390 -414 (in Chinese) doi: 10.6052/1000-0992-13-045[6] 沈文厚, 赵治华, 任革学等. 拦阻索冲击的多体动力学仿真研究. 振动与冲击, 2015, 34(5): 73-77 (Shen Wenhou, Zhao Zhihua, Ren Gexue, et al. Multi-body dynamic simulation of impact on cross deck pendant. Journal of Vibration and Shock, 2015, 34(5): 73-77 (in Chinese) Shen Wenhou, Zhao Zhihua, Ren Gexue, et al. Multi-body dynamic simulation of impact on cross deck pendant. Journal of Vibration and Shock, 2015, 34(5): 73-77 (in Chinese)
[7] Beer FP. Vector Mechanics for Engineers: Statics and Dynamics. Wiley, 1962
[8] Lankarani HM. Canonical equations of motion and estimation of parameters in the analysis of impact problems. [PhD Thesis]. Tucson, Arizona: University of Arizona, 1988
[9] Hunt KH, Crossley FR. Coeficient of restitution interpreted as damping in vibroimpact. Journal of Applied Mechanics, 1975, 42(2): 440-445 doi: 10.1115/1.3423596
[10] Flores P, Machado M, Silva MT. On the continuous contact force models for soft materials in muhibody dynamics. Muhibody SystemDynamics, 2011, 25(3): 357-375
[11] Lankarani HM, Nikravesh PE. A contact force model with hysteresis damping for impact analysis of muhibody systems. Journal of Mechanical Design, 1990, 112(3): 369-376 doi: 10.1115/1.2912617
[12] Zhang J, Li W, Zhao L, et al. A continuous contact force model for impact analysis in multibody dynamics. Mechanism and Machine Theory, 2020, 153: 103946
[13] Bulín R, HajMan M, Polach P. Nonlinear dynamics of a cable-pulley system using the absolute nodal coordinate formulation. Mechanics Research Communications, 2017, 82: 21-28
[14] 孙冠, 沈志军, 张东昱. 钢丝绳与滑轮间接触应力计算. 广东第二师范学院学报, 2019, 39(5): 74-78 (Sun Guan, Shen Zhijun, Zhang Dongyu. Calculation of contact stress between rope and pulley. Journal of Guangdong Second Normal College, 2019, 39(5): 74-78 (in Chinese) Sun Guan, Shen Zhijun, Zhang Dongyu. Calculation of contact stress between rope and pulley. Journal of Guangdong Second Normal College, 2019, 39(5): 74-78 (in Chinese)
[15] Ju F, Choo YS. Super element approach to cable passing through multiple pulleys. International Journal of Solids and Structures, 2005, 42(11-12): 3533-3547
[16] 李春明. 弹性绳系统的动力学建模与计算机仿真. 系统仿真学报, 2008, 20(1): 62-64, 168 (Li Chunming. Dynamic modeling and computer simulation of elastic rope system. Journal of System Simulation, 2008, 20(1): 62-64, 168 (in Chinese) Li Chunming. Dynamic modeling and computer simulation of elastic rope system. Journal of System Simulation, 2008, 20(1): 62-64, 168 (in Chinese)
[17] 胡建峰, 肖勇, 赵志华. 大型周边桁架式天线展开过程动力学建模仿真. 科学技术与工程, 2016, 16(12): 171-176 (Hu Jianfeng, Xiao Yong, Zhao Zhihua. Dynamic modeling and simulation of large peripheral truss antenna deployment process. Science Technology and Engineering, 2016, 16(12): 171-176 (in Chinese) Hu Jianfeng, Xiao Yong, Zhao Zhihua. Dynamic modeling and simulation of large peripheral truss antenna deployment process. Science Technology and Engineering, 2016, 16(12): 171-176 (in Chinese)
[18] 魏建东. 索结构分析的滑移索单元法. 工程力学, 2004, 21(6): 179-183, 217 (Wei Jiandong. Sliding cable element method for cable structure analysis. Engineering Mechanics, 2004, 21(6): 179-183, 217 (in Chinese) Wei Jiandong. Sliding cable element method for cable structure analysis. Engineering Mechanics, 2004, 21(6): 179-183, 217 (in Chinese)
[19] 阚子云. 张拉整体结构动力学响应分析的和算法研究. [博士论文]. 大连: 大连理工大学, 2019 (Kan Ziyun. Research on modeling and numerical method for dynamic response analysis of tensegrity structure. [PhD Thesis]. Dalian: Dalian University of Technology, 2019 (in Chinese) Kan Ziyun. Research on modeling and numerical method for dynamic response analysis of tensegrity structure. [PhD Thesis]. Dalian: Dalian University of Technology, 2019 (in Chinese)
[20] Peng H, Li F, Kan Z, et al. Symplectic instantaneous optimal control of deployable structures driven by sliding cable actuators. Journal of Guidance, Control, and Dynamics, 2020, 43(6): 1114-1128
[21] Peng Y, Zhao Z, Zhou M, et al. Flexible multibody model and the dynamics of the deployment of mesh antennas. Journal of guidance, Control and Dynamics, 2017, 40(6): 1499-1510
[22] Zhang H, Zhao Z, Ren G, et al. Arresting-cable system for robust terminal landing of reusable rockets. Journal of Spacecraft and Rockets, 2021, 58(2): 425-443
[23] Qi Z, Wang J, Wang G. An efficient model for dynamic analysis and simulation of cable-pulley systems with time-varying cable lengths. Mechanism and Machine Theory, 2017, 116: 383-403
[24] Wang J, Qi Z, Wang G. Hybrid modeling for dynamic analysis of cable-pulley systems with time-varying length cable and its application. Journal of Sound & Vibration, 2017, 406: 227-294
[25] Du X, Du J, Bao H, et al. Dynamic analysis of the deployment for mesh reflector antennas driven with variable length cables. Journal of Computational and Nonlinear Dynamics, 2019, 14(11): 111006
[26] 赵孟良. 空间可展结构展开过程动力学理论分析仿真及试验. [博士论文]. 杭州:浙江大学, 2007 (Zhao Mengliang. Dynamic analysis, simulation and experiment of spatial developable structure development process. [PhD Thesis]. Hangzhou: Zhejiang University, 2007 (in Chinese) Zhao Mengliang. Dynamic analysis, simulation and experiment of spatial developable structure development process. [PhD Thesis]. Hangzhou: Zhejiang University, 2007 (in Chinese)
[27] 马惠珠, 卞智慧, 邓振伟. 一种钢丝绳有限元模型的参数化建模方法. 山西建筑, 2023, 49(04): 55-58 (Ma Huizhu, Bian Zhizhi, Deng Zhenwei. A parameterized modeling method for finite element models of steel wire ropes. Shanxi Architecture, 2023, 49(04): 55-58 (in Chinese) Ma Huizhu, Bian Zhizhi, Deng Zhenwei . A parameterized modeling method for finite element models of steel wire ropes. Shanxi Architecture,2023 ,49 (04 ):55 -58 (in Chinese)[28] 潘荣安, 杜文正, 马保珠. 摩擦系数对钢丝绳有限元分析结果的影响. 计算机仿真, 2019, 36(8): 218-221 (Pan Rong'an, Du Wenzheng, Ma Baozhu. The influence of friction coefficient on the finite element analysis results of steel wire ropes. Computer Simulation, 2019, 36(8): 218-222 (in Chinese) Pan Rong'an, Du Wenzheng, Ma Baozhu. The influence of friction coefficient on the finite element analysis results of steel wire ropes. Computer Simulation, 2019, 36(8): 218-222 (in Chinese)
[29] 曹旭阳, 付林生, 邢烨. 起升机构钢丝绳缠绕系统建模及摇摆仿真分析. 起重运输机械, 2020, 18: 43-48 (Cao Xuyang, Fu Linsheng, Xing Ye. Modeling and swaying simulation analysis of the steel wire rope winding system of the lifting mechanism. Lifting and Transportation Machinery, 2020, 18: 43-48 (in Chinese) doi: 10.3969/j.issn.1001-0785.2020.18.012 Cao Xuyang, Fu Linsheng, Xing Ye . Modeling and swaying simulation analysis of the steel wire rope winding system of the lifting mechanism. Lifting and Transportation Machinery,2020 ,18 :43 -48 (in Chinese) doi: 10.3969/j.issn.1001-0785.2020.18.012[30] 李永波, 魏禹. 基于虚拟样机滑轮-绳索机构的建模及仿真分析. 应用科技, 2013, 40(3): 1-5 (Li Yongbo, Wei Yu. Modeling and simulation analysis of pulley-rope mechanism based on virtual prototype. Applied Science and Technology, 2013, 40(3): 1-5 (in Chinese) Li Yongbo, Wei Yu. Modeling and simulation analysis of pulley-rope mechanism based on virtual prototype. Applied Science and Technology, 2013, 40(3): 1-5 (in Chinese)
[31] 齐朝晖, 国树东, 卓英鹏. 滑轮绳索系统中动态节点绳索单元. 力学学报, 2019, 51(6): 1857-1871 (Qi Zhaohui, Guo Shudong, Zhuo Yingpeng. Rope elements with moving nodes in rope-pulley systems. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(6): 1857-1871 Qi Zhaohui, Guo Shudong, Zhuo Yingpeng . Rope elements with moving nodes in rope-pulley systems. Chinese Journal of Theoretical and Applied Mechanics,2019 ,51 (6 ):1857 -1871 [32] 齐朝晖. 多体系统动力学. 北京: 科学出版社, 2008 (Qi Zhaohui. Dynamics of Multibody Systems. Beijing: Science Press, 2008 (in Chinese) Qi Zhaohui. Dynamics of Multibody Systems. Beijing: Science Press, 2008 (in Chinese)
[33] 齐朝晖, 方慧青, 张志刚等. 基于曲率离散的几何非线性空间梁单元. 应用数学和力学, 2014, 35(5): 498-509 (Qi Chaohui, Fang Huiqing, Zhang Zhigang, et al. Geometrically nonlinear spatial beam element based on curvature dispersion. Applied mathematics and Mechanics, 2014, 35(5): 498-509 (in Chinese) Qi Chaohui, Fang Huiqing, Zhang Zhigang, et al. Geometrically nonlinear spatial beam element based on curvature dispersion. Applied mathematics and Mechanics, 2014, 35(5): 498-509 (in Chinese)
[34] 成大先. 机械设计手册(第六版). 北京: 化学工业出版社, 2017 (Cheng Daxian. Handbook of Mechanical Design. 6th edn. Beijing: Chemical Industry Press, 2017 (in Chinese) Cheng Daxian. Handbook of Mechanical Design. 6th edn. Beijing: Chemical Industry Press, 2017 (in Chinese)
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