Stiffness characteristics and life prediction of rail pads of subway damping fasteners
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摘要: 为了研究地铁减振扣件弹性垫在服役过程中刚度敏感性和对线路环境振动的影响，以南京地铁多条运营线路中抽取的压缩型减振扣件为研究对象，开展了压缩型扣件轨下弹性垫服役刚度特性、常温下疲劳特性和热加速疲劳老化特性等多环境室内综合测试；基于测试结果对比分析了新旧压缩型扣件轨下弹性垫使用时间与刚度变化的相关性，得到了轨下弹性垫的时间-寿命特性曲线，提出了轨下弹性垫刚度变化百分比与使用时间的寿命预测模型。研究结果表明：在周期性轮轨载荷和线路温湿碱环境等综合作用下，地铁减振扣件轨下弹性垫的服役刚度随使用时间呈线性增加趋势，其弹性发生了性能退化；新的轨下弹性垫热加速疲劳老化刚度曲线与服役抽样轨下弹性垫的刚度曲线趋势基本一致，即轨下弹性垫的热加速循环老化试验能够模拟或演化轨道交通线路的热机械循环载荷等现场条件；基于Arrhenius寿命-应力热加速老化模型，轨下弹性垫服役应力和加速老化应力下的加速因子分别为1.99和1.36，进而可通过加速因子预测减振扣件轨下弹性垫的更换周期。Abstract: In order to study the stiffness sensitivity of rail pads of subway damping fasteners in service and their influence on line environmental vibration, the compression-type damping fasteners extracted from several operating lines of Nanjing Metro were taken as the research object, and the multi-environmental indoor comprehensive tests of the stiffness characteristics, fatigue characteristics at room temperature, and thermal accelerated fatigue aging characteristics of rail pads of the compression-type fasteners in service were carried out. Based on the test results, the correlations between service time and stiffness change of rail pads of new and old compression-type fasteners were compared and analyzed. The time-life characteristic curves of rail pads were obtained, and the life prediction model for the percentage change in stiffness and service time of the rail pads was proposed. Research results indicate that under the combined effect of periodic wheel-rail loads and the temperature, humidity, and alkaline environment of the line, the service stiffness of rail pads of the subway damping fasteners increases linearly with service time, and the elasticity of rail pads undergoes performance degradation. The stiffness curve trend of the new rail pads under thermal accelerated fatigue aging is basically consistent with the stiffness curve trend of the sampled rail pads in service. In other words, the thermal accelerated cyclic aging test of the rail pads can simulate or evolve on-site conditions such as the thermal mechanical cyclic load of the line. Based on the Arrhenius life-stress thermal accelerated aging model, the acceleration factors of rail pads under service stress and accelerated aging stress are 1.99 and 1.36, respectively, which can be used to predict the replacement cycle of rail pads of the damping fasteners.
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0. 引言
高速列车受长波长轨道不平顺、车辆轻量化和空气动力影响, 列车横向振动明显加大, 乘坐舒适性恶化, 半主动控制技术正是针对这一问题发展起来的新技术, 是提高列车稳定性和舒适性的有效方法。在以往的研究中, 以天棚阻尼原理计算车体需要的期望阻尼力, 即用与车体横向振动速度成比例的阻尼力来抑制车体的横向振动, 控制策略上采用了门限开关控制、预测控制与优化控制等[1-4]。这些控制结构只能抑制车体横移和侧滚引起的横向振动, 对摇头振动引起的横向振动没有抑制作用。从控制方法看, 开关控制速度快, 但是门限的选择困难, 预测和优化控制计算量大, 实时性差, 对模型依靠性强, 而悬挂系统本身是一个模型时变、强耦合、非线性的动力系统[5]。基于此, 本文提出一种新的控制结构, 从根本上抑制由车体摇头振动引起的横向振动, 考虑悬挂模型时变的特点, 将模糊控制[5]引入半主动控制中, 对高速列车的半主动悬挂系统进行研究。
1. 模型建立
1.1 17自由度的半主动悬挂动力学模型
本文假设列车直线行驶, 不考虑轮轨非线性蠕滑力、钢轨弹性、轮对扰动等因素, 建立反应由轨道的水平不平顺和方向不平顺引起车体横向振动的17自由度动力学模型[4, 6]
\begin{aligned} \boldsymbol{M} & \ddot{X}+\boldsymbol{C} \dot{X}+\boldsymbol{K} X=\boldsymbol{G} \mathit{\pmb{ω}} \\ X=&\left(y_{\mathrm{w} 1}, y_{\mathrm{w} 2}, y_{\mathrm{w} 3}, y_{\mathrm{w} 4}, \Psi_{\mathrm{w} 1}, \Psi_{\mathrm{w} 2}, \Psi_{\mathrm{w} 3}, \Psi_{\mathrm{w} 4}\right.\\ &\left.y_{\mathrm{t} 1}, y_{\mathrm{t} 2}, \phi_{\mathrm{t} 1}, \phi_{\mathrm{t} 2}, \Psi_{\mathrm{t} 1}, \Psi_{\mathrm{t} 2}, y_{\mathrm{c}}, \phi_{\mathrm{c}}, \Psi_{\mathrm{c}}\right) \end{aligned} 式中: yw1、yw2、yw3、yw4分别为4个轮对的横移运动; Ψw1、Ψw2、Ψw3、Ψw4分别为4个轮对的摇头运动; yt1、yt2分别为2个转向架的横移运动; ϕt1、ϕt2分别为2个转向架的侧滚运动; Ψt1、Ψt2分别为2个转向架的摇头运动; yc、ϕc、Ψc分别为车体的横移、侧滚和摇头运动; M、C、K分别为整车质量矩阵、阻尼矩阵和刚度矩阵; G为轨道输入分布矩阵; ω为轨道方向和水平不平顺输入。
1.2 可调减振器模型
C=54.368\ 7 u^3-65.227\ 8 u^2+29.931\ 7 u+2.474\ 7\\ \ \ \ \ \ \ \ \ 0 \leqslant u \leqslant 1.4 式中: u为控制电流(A); C为可调减振器在不同控制电流下的输出阻尼(kN·s·m-1)。
2. 控制的基本原理、控制结构与模糊控制器
2.1 基本原理
车体的横向振动加速度来自3个方面, 车体横移振动、车体侧滚引起的横向振动和车体摇头引起的横向振动。车体横移和侧滚引起的横向振动, 车体前后端方向一致, 车体摇头引起的横向振动, 车体前后端方向相反。要抑制横向振动, 就必须同时抑制这3个振动。对此, 在计算期望阻尼力时, 对前后减振器的控制电流进行独立调节, 使得半主动悬挂前后阻尼力之和等于横移与侧滚引起的横向振动加速度与车体质量的乘积, 前后阻尼力之差与阻尼器之间距离的乘积等于车体摇头角加速度与车体摇头转动惯量的乘积, 用这2个理想的阻尼力去调节减振器工作, 达到从根本上抑制车体摇头运动引起的横向振动的目的。在前后构架中心对应的车体位置处, 安装加速度传感器1和2, 见图 1。
加速度传感器1、2关于y轴对称安装, 检测车体的横向振动加速度, 感应由车体的横向平移振动、摇头振动和侧滚振动所引起的横向振动的总和, 方程式分别为
a_{y 1}=\left(a_{y 0}-a_{y \phi}\right)+a_{y \Psi} (1) a_{y 2}=\left(a_{y 0}-a_{y \phi}\right)-a_{y \Psi} (2) 式中: ay1、ay2为横向加速度传感器1、2的检测值; ay0为车体横向平移振动加速度; ayϕ为车体侧滚引起的横向振动加速度; ayΨ为车体摇头引起的横向振动加速度。
由式(1)、(2)变形可得车体横移和侧滚引起的横向振动为
\left(a_{y 0}-a_{y \phi}\right)=\left(a_{y 1}+a_{y 2}\right) / 2 车体摇头引起的横向振动为
a_y \Psi=\left(a_{y 1}-a_{y 2}\right) / 2 根据力学的动静平衡原理, 前后悬挂所需要的理想阻尼力分别为
\begin{aligned} &F_{\mathrm{q}}=\frac{m\left(a_{y 0}-a_{y \phi}\right) L^2+J_{\mathrm{c}x} a_y \Psi}{2 L^2} \\ &F_{\mathrm{h}}=\frac{m\left(a_{y 0}-a_{y \phi}\right) L^2-J_{\mathrm{c} x} a_y \Psi}{2 L^2} \end{aligned} 式中: m为车体质量; Jcx为车体摇头的转动惯量; L为前后转向架中心的距离。
2.2 控制结构
利用车体前后加速度, 计算前后悬挂所期望的阻尼力与实际阻尼力之差和差的变化率, 并将此作为误差和误差的变化率, 通过模糊控制器得到前后减振器控制电流, 其具体过程见图 2。从图 2中可以看出, 由加速度反馈和减振器实际阻尼力反馈构成双环反馈控制。
2.3 模糊控制器
模糊控制器的设计见图 3, 查询表通过离线计算获得。离线计算由模糊化、模糊推理和反模糊化组成。模糊化把误差e和误差变化率v由非模糊量转化模糊量E和V, 通过模糊推理得到模糊控制变量U, 反模糊化得到减振动器精确控制量u。模糊控制器采用二维结构, 以e和v为输入变量, u为输出变量。E的模糊集为
\{\mathrm{NB}, \mathrm{NM}, \mathrm{NS}, \mathrm{NO}, \mathrm{PO}, \mathrm{PS}, \mathrm{PM}, \mathrm{PB}\} V的模糊集为
\{\mathrm{NB}, \mathrm{NM}, \mathrm{NS}, \mathrm{ZO}, \mathrm{PS}, \mathrm{PM}, \mathrm{PB}\} U的模糊集为
\{\mathrm{NB}, \mathrm{NM}, \mathrm{NS}, \mathrm{ZO}, \mathrm{PS}, \mathrm{PM}, \mathrm{PB}\} 为了提高控制器的稳态精度, 误差的模糊量为8。E和V的论域为
\{-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6\} U的论域为
\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\} 输入变量以钟形函数为模糊隶属函数, 输出变量以三角函数为模糊变量, 模糊变量误差、误差变化率和减振器的控制电流的赋值见表 1~3, 其中x、y、z分别对应模糊集误差、误差变化率和控制电流在各自论域上的取值。
表 1 误差赋值Table 1. Assignments of errorμE（x） x －6 －5 －4 －3 －2 －1 0 1 2 3 4 5 6 E NB 1.0 0.8 0.7 0.4 0.1 0 0 0 0 0 0 0 0 NM 0.2 0.7 1.0 0.7 0.2 0 0 0 0 0 0 0 0 NS 0 0 0.1 0.5 1.0 0.8 0.3 0 0 0 0 0 0 NO 0 0 0 0 0.1 0.6 1.0 0 0 0 0 0 0 PO 0 0 0 0 0 0 1.0 0.6 0.1 0 0 0 0 PS 0 0 0 0 0 0 0.3 0.8 1.0 0.5 0.1 0 0 PM 0 0 0 0 0 0 0 0 0.2 0.7 1.0 0.7 0.2 PB 0 0 0 0 0 0 0 0 0.1 0.4 0.7 0.8 1.0 表 2 误差变化率赋值Table 2. Assignments of error rate\mu_{\dot{E}}(y) y －6 －5 －4 －3 －2 －1 0 1 2 3 4 5 6 \dot{E} NB 1.0 0.8 0.7 0.4 0.1 0 0 0 0 0 0 0 0 NM 0.2 0.7 1.0 0.7 0.2 0 0 0 0 0 0 0 0 NS 0 0 0.2 0.7 1.0 0.9 0 0 0 0 0 0 0 ZO 0 0 0 0 0 0 0.5 1.0 0.5 0 0 0 0 PS 0 0 0 0 0 0 0 0.9 1.0 0.7 0.1 0 0 PM 0 0 0 0 0 0 0 0 0.2 0.7 1.0 0.7 0.2 PB 0 0 0 0 0 0 0 0 0.1 0.4 0.7 0.8 1.0 表 3 控制电流赋值Table 3. Assignments of control currentμU（z） z 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 U NB 1.00 0.75 0.50 0.25 0 0 0 0 0 0 0 0 0 0 0 NM 0 0.25 0.75 1.00 0.75 0.50 0.25 0 0 0 0 0 0 0 0 NS 0 0 0 0 0 0.33 0.66 1.00 0.66 0.33 0 0 0 0 0 ZO 0 0 0 0 0 0 0 0 0 0.50 1.00 0 0 0 0 PS 0 0 0 0 0 0 0 0 0 0 0 1.00 0.50 0 0 PM 0 0 0 0 0 0 0 0 0 0 0 0 0.50 1.00 0.20 PB 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 根据半主动悬挂的振动特性, 提炼出半主动悬挂的模糊控制规则, 见表 4。表 4中的每条规则都可以用语言描述, 如表中最后一条控制规则可表述为如果E= PB和V= PB, 则U= PB。自然语言描述为: 如果理想阻尼力远大于减振器实际阻尼力, 且继续增大的趋势很强, 那么应向减振器提供最大的控制电流, 获得最大的阻尼系数。
表 4 半主动悬挂模糊控制规则Table 4. Fuzzy control rules of semi-active suspensionU V NB NM NS ZO PS PB PB E NB NB NB NB NB NM NS NS NM NB NB NB NB NM NS NS NS NM NM NM NM ZO PS PS NO NM NM NS ZO PS PM PM PO NM NM NS ZO PS PM PM PS NS NS ZO PM PM PM PM PM PS PS PM PB PB PB PB PB PS PS PM PB PB PB PB 本文采用Mandani推理算法实现模糊推理, 得
U=(E \times V)^{\circ} {\rm{R}} 式中: “\circ”为模糊关系合成运算; R为运算关系。
采用加权平均法进行反模糊化
u=\frac{\sum\limits_{j=1}^n k_j u_j}{\sum\limits_{j=1}^n k_j} 式中: kj为控制变量在不同论域的加权系数; uj为控制变量在不同论域中的取值。
采用此方法计算的查询表见表 5。
表 5 半主动悬挂模糊控制查询表Table 5. Quering table of fuzzy control for semi-active suspensionz y －6 －5 －4 －3 －2 －1 0 1 2 3 4 5 6 x －6 0 1 0 1 1 1 3 4 4 5 5 6 7 －5 2 2 2 2 2 2 3 4 5 5 6 7 7 －4 3 3 3 3 3 3 4 5 6 6 7 8 8 －3 3 3 3 3 3 4 5 5 6 7 7 8 9 －2 3 4 4 4 5 5 6 6 7 7 8 9 9 －1 4 5 5 5 6 6 7 7 7 9 9 9 10 0 5 5 5 6 7 7 7 7 8 11 11 12 12 1 6 6 6 6 7 8 8 8 8 12 12 12 13 2 6 7 7 7 8 9 10 13 13 12 13 13 13 3 7 7 7 8 8 9 10 13 13 13 13 13 13 4 8 8 9 9 9 10 11 13 13 13 13 13 13 5 10 10 10 10 12 12 13 13 14 14 14 14 14 6 11 11 12 12 13 13 13 14 14 14 14 14 14 3. 计算结果分析
以某型号高速客车为例, 列车技术参数见表 6, 计算速度为270 km·h-1, 以美国6级轨道谱为输入, 在被动悬挂和半主动悬挂2种情况下, 分别计算车体前后横向振动、车体的摇头振动和对应的傅立叶振动谱。车体前端横向振动和FFT振动谱见图 4, 车体后端横向振动和FFT振动谱见图 5, 车体摇头横向振动和FFT振动谱见图 6。表 6中, Mc、Jcx、Jcz分别为车体的质量、摇头和侧滚的转动惯量；Mt、Jtx、Jtz分别为转向架的质量、摇头和侧滚的转动惯量; Mw、lc、Ksy、Kpy分别为轮对的质量、转向架中心距、二系和一系刚度。
表 6 悬挂系统部分参数Table 6. Some parameters of suspension systemMc=40 000 kg, Jcx=89 396 kg·m2, Jcz=2485 756 kg·m2 Mt=2 280 kg, Jtx=2 650 kg·m2, Jtz=3 000 kg·m2 Mw=1 920 kg, 2lc=18 m, Ksy=208 kN·m－1, Kpy=7.5 MN·m－1 
图 4 车体前端横向振动和振动谱比较Figure 4. Comparison of lateral vibrations and vibration spectrums in front of car body
图 5 车体后端横向振动和振动谱比较Figure 5. Comparison of lateral vibrations and vibration spectrums at back end of car body本文参照标准TB/T2360—93[9], 计算半主动和被动悬挂列车的横向平稳性指标。半主动和被动悬挂系统的横向平稳性指标、振动幅值比较结果见表 7。
表 7 平稳性指标和振动幅值比较Table 7. Comparison of ride indexes and vibration amplitudes悬挂类型 平稳性指标 横向振动幅值/（m·s－2） 摇头振动幅值/（rad·s－2） 车体前 车体后 车体前 车体后 被动悬挂 2.093 3 2.085 5 0.40 0.38 0.002 0 半主动悬挂 1.830 8 1.803 9 0.22 0.19 0.001 3 改善情况/% 12.54 13.50 48.45 50.00 35.00 从表 7中可看出: 车体前后的平稳性改善明显, 改善情况分别为12. 54%、13. 50%, 半主动悬挂的平稳性达到良好; 车体的摇头振动抑制效果好, 振幅减少了35. 00%;车体前后横向振动减振明显, 振幅减少分别为48. 45%和50. 00%。从图 4(b)、5(b)和6(b)中半主动悬挂与被动悬挂的振动谱相比较可以看出, 在车体的低频共振频率段(1~ 6 Hz), 其横向振动和摇头振动都抑制了50. 00%, 达到了半主动悬挂所期望的控制目的。
图 6 车体摇头振动和摇头振动谱比较Figure 6. Comparison of shaking vibrations and vibration spectrums for car body4. 结语
(1) 通过抑制车体摇头振动来抑制车体横向振动, 采用车体前后阻尼力独立差异调节, 结合模糊控制技术, 能够很好解决列车横向悬挂振动问题, 所提出的控制结构和控制方法适合悬架的半主动控制。
(2) 在实际应用中, 由于加速度反馈对高频敏感, 需要高性能的可调阻尼, 提高作动频率, 可以考虑选择一定的频段进行控制。
(3) 本文是在直线路上计算控制效果, 通过曲线时, 传感器还要感应离心力, 因此, 需要通过信号处理的办法将离心加速度滤出, 将移出离心加速度后的振动信号作为控制输入。
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图 7 轨下弹性垫动静刚度比随使用时间变化曲线
Figure 7. Curve of stiffness ratio with service time of rail pad
图 8 扣件与轨下弹性垫刚度随使用时间关系
Figure 8. Relationships between stiffness and service time of fastener and rail pad
图 9 轨下弹性垫高温下时间与刚度关系
Figure 9. Relationship between time and stiffness of rail pad at high temperature
图 12 热加速疲劳老化循环次数与刚度关系
Figure 12. Relationship between thermal accelerated fatigue aging cycles and stiffness
图 13 刚度变化率与循环次数、使用时间关系
Figure 13. Rate of stiffness change in relation to cycles and service time
表 1 试验方法与参数
Table 1. Test methods and parameters
试验方法 样件类型 服役时间/月 试验类型 数量/件 载荷参数 1 线路垫板 24~66 动静刚度 3 每天388列，15万人次 2 新垫板 0 静态刚度 3 割线静刚度载荷为15、45 kN 3 新垫板 0 加热静刚度 3 割线静刚度载荷为15、45 kN，周期为30 d，温度为80 ℃ 4 新垫板 0 常温循环疲劳 3 疲劳幅值为8~45 kN，加载频率为4 Hz，循环次数为6.0×106 5 新垫板 0 热加速疲劳老化 3 温度为80 ℃，加载频率为4 Hz，循环次数为6.0×106，刚度取值为15、45 kN 
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