, 倪陈兵11. 东北大学 机械工程与自动化学院, 辽宁 沈阳 110819;
2. 东北大学 航空动力装备振动及控制教育部重点实验室, 辽宁 沈阳 110819
收稿日期：2017-11-23
基金项目：国家自然科学基金资助项目(51775092)；中央高校基本科研业务费专项资金资助项目(N170306002)。
作者简介：高峰(1990-)，男，山东临沂人，东北大学博士研究生;
孙伟(1975-), 男，辽宁营口人，东北大学教授，博士生导师。
摘要：对叶片涂敷NiCoCrAlY+YSZ硬涂层, 研究具有硬涂层的大小叶盘的动力学模型和振动特性, 主要研究硬涂层对大小叶盘的阻尼减振性能的影响.首先, 利用复模量理论和Oberst梁理论, 建立了涂层大小叶盘结构的解析模型; 然后, 基于Gram-Schmidt法和Rayleigh-Ritz法, 求解涂层大小叶盘结构的模态特性和受迫响应.结果表明:硬涂层阻尼对大小叶盘固有频率的影响很小, 但能使模态损耗因子提高4倍, 说明硬涂层可以有效提高大小叶盘的阻尼性能; 此外, 涂敷硬涂层可以有效抑制其受迫响应.
关键词：大小叶盘硬涂层阻尼被动减振解析建模受迫响应
Analytical Modeling and Vibration Analysis of Hard-Coated Splitter Blisk
GAO Feng1,2, SUN Wei1,2
, NI Chen-bing11. School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819, China;
2. Key Laboratory of Vibration and Control of Aero-Propulsion System(Ministry of Education), Northeastern University, Shenyang 110819, China
Corresponding author: SUN Wei, E-mail: weisun@mail.neu.edu.cn
Abstract: The blades were coated with NiCoCrAlY+YSZ, then the dynamic model and vibration characteristics of the hard-coated splitter blisk were investigated, as well as the effect of hard coating on damping vibration reduction of the splitter blisk. Firstly, analytical model of the hard-coated splitter blisk was obtained using the complex modulus and the Oberst beam theories. The mode characteristics and forced responses of the hard-coated splitter blisk were then solved using the Gram-Schmidt process and Rayleigh-Ritz method. The results reveal that hard coating has a weak effect on natural frequencies of splitter blisk but makes the modal loss factors increase about four times, which explains that hard coating can effectively improve the damping performance of the splitter blisk. Moreover, the forced response can be suppressed remarkably.
Key words: splitter bliskhard coatingpassive vibration reductionanalytical modelingforced response
在常规叶片通道的后部加入分流叶片, 既可抑制甚至消除大弯度叶片叶背处由于惯性作用造成的气流分离, 又可避免过多的全弦长叶片造成的叶片槽道前部气流堵塞[1].近年来, 学者对大小叶盘开展了较深入的研究.Yang等[2]研究了分流叶片对双吸式离心泵汽蚀性能的影响; Kassanos等[3]研究了带分流叶片的Francis水轮机对引流管汽蚀性能的作用; Guo等[4]研究了带分流叶片的离心泵在高速状态下的汽蚀性能; Korkmaz等[5]研究了分流叶片对深井泵工作性能的影响, 并求解了最优化组合参数.Torshizi等[6]研究了分流叶片轮廓对压气机的影响, 并利用基因优化完成了叶轮结构的多层优化.
大小叶盘的结构阻尼较小, 容易发生疲劳失效, 因此, 需要引入附加阻尼器实现有效减振.由于结构一体化, 传统的干摩擦阻尼器[7-9]很难应用在大小叶盘结构上.研究中常用的黏弹性阻尼[10]、约束层阻尼[11]和压电阻尼[12]的阻尼性能往往因无法承受高温、高压和强腐蚀等而失效.硬涂层具有较高硬度[13]，耐高温、耐摩擦、耐腐蚀[14-16], 还具有较好的阻尼能力.Yang等[17]基于有限元法求解了涂敷NiCrAlY硬涂层的Mg-Al合金板的固有特性.Sun等[18]利用能量法求解了涂敷MgO+Al2O3涂层的悬臂梁的振动特性; Zhang等[19]分析了涂敷硬涂层的薄壁圆柱壳在不同约束下的振动特性.
本文对叶片涂敷NiCoCrAlY+YSZ硬涂层, 并分析大小叶盘-硬涂层阻尼结构的振动特性.首先, 引入气动弹簧来考虑叶片与轮盘的耦合关系, 并基于复模量和Oberst梁理论建立了涂层大小叶盘的解析模型；然后, 基于Gram-Schmidt法和Rayleigh-Ritz法求解了涂层大小叶盘的模态特性和受迫响应, 重点研究了硬涂层对大小叶盘的阻尼减振性能.
1 理论分析1.1 大小叶盘-硬涂层阻尼结构分析模型图 1所示为涂层大小叶盘结构的动力学模型.轮盘内、外径分别为Ri和Ro, 厚度为hd; 大、小叶片的长度分别为lb和lv, 宽度分别为wb和wv, 厚度分别为hd和hv, 叶片数量为P, 安装角均为φ; 叶片两侧的硬涂层阻尼涂敷厚度相等, 总厚度为hc; 大小叶盘转速为Ω.引入了具有平动刚度KT, p和转动刚度KR, p的气动弹簧来充分考虑轮盘与叶片的边界条件和连续条件.特别地, 本文仅研究了轮盘结构在z方向的位移wd和涂层叶片在zp方向的位移wb, p.此外, 以轮盘的几何中心Od为圆心建立全局圆柱坐标系(r, θ, z), 以第p个叶片的几何中心Op为圆心建立局部笛卡尔坐标系(xp, yp, zp).
图 1(Fig. 1)
 | 图 1 大小叶盘-硬涂层阻尼结构的解析模型Fig.1 Analytical model of the hard-coated splitter blisk |
1.2 叶片-硬涂层阻尼结构的等效参数求解将复合结构等效成属性均匀的单一结构，可以在不降低计算精度的前提下极大地简化计算过程[20].图 2所示为涂层叶片原理图和复合Oberst梁截面图.
图 2(Fig. 2)
 | 图 2 叶片-硬涂层阻尼结构Fig.2 Structure of hard-coated blade (a)—原理图；(b)—截面图. |
叶片和硬涂层的复弹性模量Eb*和Ec*分别表示为
 | (1) |
 | (2) |
当复合梁在纯弯曲状态时, 其平衡方程表示为
 | (3) |
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 | (8) |
 | (9) |
 | (10) |
 | (11) |
1.3 大小叶盘-硬涂层阻尼结构的振动分析在旋转态下, 气动弹簧的势能Vs表示为
 | (12) |
 | (13) |
 | (14) |
 | (15) |
 | (16) |
 | (17) |
 | (18) |
 | (19) |
 | (20) |
令无量纲量a=r/Ro, bp=xp/l, 当大小叶盘以频率ω自由振动时, 根据小变形理论的运动叠加原理, 轮盘和叶片阻尼结构的位移方程分别为
 | (21) |
 | (22) |
 | (23) |
 | (24) |
将能量公式(11)~(20)代入拉格朗日方程, 可得
 | (25) |
 | (26) |
 | (27) |
 | (28) |
 | (29) |
 | (30) |
大小叶盘-硬涂层阻尼结构的频域响应函数为Z：
 | (31) |
2 实例研究2.1 大小叶盘-硬涂层阻尼结构的基本参数图 3a为大小叶盘的连续参数试件, 轮盘内径25 mm, 外径100 mm, 厚度3 mm; 大叶片长度80 mm, 宽度24 mm; 小叶片长度40 mm, 宽度24 mm; 叶片厚度均为3 mm; 气动弹簧平动和转动刚度均为106 N/m.硬涂层单层涂敷厚度0.1 mm, 如图 3b所示.表 1所列为大小叶盘和硬涂层的材料参数.
图 3(Fig. 3)
 | 图 3 大小叶盘和大小叶盘-硬涂层阻尼结构Fig.3 Splitter blisk with or without hard-coating damper (a)—无涂层叶盘；(b)—带涂层叶盘. |
表 1(Table 1)
 表 1 大小叶盘-硬涂层阻尼结构的力学参数Table 1 Mechanical parameters of hard-coated splitter blisk
| 表 1 大小叶盘-硬涂层阻尼结构的力学参数 Table 1 Mechanical parameters of hard-coated splitter blisk |
2.2 大小叶盘-硬涂层阻尼结构的振动特性为了校核解析模型的有效性, 本文还进行了有限元分析和试验测试.其中, 有限元模型采用SOLID185单元划分, 共有53 316个单元和68 436个节点.
图 4所示为不同途径得到的大小叶盘-硬涂层阻尼结构的固有频率.可以看到, 三组数据之间存在误差, 主要因为:①有限元法精度受到网格类型和单元属性等因素影响; ②试验测试精度受到测试方法和环境的影响.但是, 这些误差保持在一定范围内.此外, 三组数据在整体上的走势非常相似, 这说明解析分析的数据精度是有保证的.
图 4(Fig. 4)
 | 图 4 大小叶盘-硬涂层阻尼结构固有频率Fig.4 Natural frequency of hard-coated splitter blisk |
图 5所示为解析分析与有限元分析(A&F)以及解析分析与试验测试(A&T)间的模态置信因子(MAC).可以看到, A&F的MAC值普遍大于0.975, 说明两者具有非常好的相关性; 由于不理想的试验环境, A&T的MAC值均小于A&F, 但仍大于0.920, 两者间的相关性较好.
图 5(Fig. 5)
 | 图 5 大小叶盘-硬涂层阻尼结构的模态置信因子Fig.5 MAC of hard-coated splitter blisk |
3 硬涂层阻尼对大小叶盘的影响表 2和表 3分别是由解析分析和试验测试得到的大小叶盘和大小叶盘-硬涂层阻尼结构的固有频率及其变化率.从中可以发现, 大小叶盘的固有频率在涂敷硬涂层阻尼后全都普遍变小, 但是其变化量并不是很大, 变化率绝对值分别在1.27 % ~2.61 %和2.24 % ~6.52 %，说明涂敷硬涂层阻尼并不会对大小叶盘的固有频率造成显著的影响.
表 2(Table 2)
表 2 解析分析得到的固有频率Table 2 Natural frequencies of splitter blisk by analytical approach
| 表 2 解析分析得到的固有频率 Table 2 Natural frequencies of splitter blisk by analytical approach | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
表 3(Table 3)
表 3 试验测试得到的固有频率Table 3 Natural frequency of splitter blisk by the test
| 表 3 试验测试得到的固有频率 Table 3 Natural frequency of splitter blisk by the test | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
图 6所示是大小叶盘和大小叶盘-硬涂层阻尼结构的模态损耗因子.可以发现, 由于外部干扰, 试验模态损耗因子略大于解析模态损耗因子, 但是, 试验结果与解析结果具有相似的整体变化趋势.重要的是, 涂敷硬涂层阻尼后大小叶盘的模态损耗因子增大了约4倍, 说明硬涂层可以有效提高大小叶盘的阻尼性能.
图 6(Fig. 6)
 | 图 6 大小叶盘和大小叶盘-硬涂层阻尼结构模态损耗因子Fig.6 Modal loss factor of the splitter blisk with or without hard coating |
图 7是由解析分析得到的大小叶盘和大小叶盘-硬涂层阻尼结构在阶次激励下的受迫响应.可以看到, 涂敷硬涂层后, 大小叶盘的受迫响应峰值全都明显减小, 说明大小叶盘在共振区域的受迫振动响应被硬涂层显著抑制.
图 7(Fig. 7)
 | 图 7 结构在阶次激励下的受迫响应Fig.7 Forced response of splitter blisk under engine order excitation |
4 结论1) 利用复模量理论和能量法建立了大小叶盘-硬涂层阻尼结构的解析模型, 并利用Ritz法和Rayleigh阻尼模型求解其在阶次激励下的受迫响应.
2) 发现NiCoCrAlY+YSZ硬涂层不会对大小叶盘的固有频率造成很大影响, 但会使其模态损耗因子提高约4倍, 能有效提高其阻尼性能, 并使其受迫响应峰值明显减小, 实现大小叶盘的有效减振.
参考文献
| [1] | 王东艺, 彭正华, 詹洪飞, 等. 航空发动机压气机大小叶片技术[J].海军航空工程学院学报, 2005, 20(2): 205–207. ( Wang Dong-yi, Peng Zheng-hua, Zhan Hong-fei, et al. The magnitude blade technology of the aviation engine compressor[J].Journal of Naval Aeronautical and Astronautical University, 2005, 20(2): 205–207.) |
| [2] | Yang W, Xiao R F, Wang F J, et al. Influence of splitter blades on the cavitation performance of a double suction centrifugal pump[J].Advances in Mechanical Engineering, 2014, 6: 963197. |
| [3] | Kassanos I, Anagnostopoulos J, Papantonis D.Numerical analysis of the effect of splitter blades on draft tube cavitation of a low specific speed Francis turbine[C/OL]//6th IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems.Ljubljana, Slovenia, 2015[2017-10-21].http://iahrwg2015.si/files/papers/1_Numerical_and_experimental_investigation_on_cavitating_flows/IAHR__WG_15_1_7_KASSANOS.pdf. |
| [4] | Guo X M, Zhu L H, Zhu Z C, et al. Numerical and experimental investigations on cavitation characteristics of a high-speed centrifugal pump with a splitter-blade inducer[J].Journal of Mechanical Science and Technology, 2015, 29(1): 259–267.DOI:10.1007/s12206-014-1232-x |
| [5] | Korkmaz E, G?lcü M, Kurbano?lu C. Effects of blade discharge angle, blade number and splitter blade length on deep well pump performance[J].Journal of Applied Fluid Mechanics, 2017, 10(2): 529–540. |
| [6] | Torshizi S A M, Benisi A H, Durali M. Multilevel optimization of the splitter blade profile in the impeller of a centrifugal compressor[J].Scientia Iranica, 2017, 24(2): 707–714.DOI:10.24200/sci.2017.4055 |
| [7] | Zhao P F, Zhang Q, Wu J, et al. Experimental study of dynamic characteristics of dry friction damping of turbine blade steel[J].Advanced Materials Research, 2013, 690/691/692/693: 1979–1982. |
| [8] | Li G, Zhang Q, Zhao W, et al.Dynamic response analysis of blades with damping structures of shroud and snubber[C]//First International Conference on Information Sciences, Machinery, Materials and Energy.[S.l.]: Atlantis Press, 2015: 281-284. |
| [9] | He B, Ouyang H, Ren X, et al.Dynamic response of a simplified turbine blade model with under-platform dry friction dampers considering normal load variation[J/OL].Applied Sciences, 2017, 7(3): 228[2017-9-25].https://www.mdpi.com/2076-3417/7/3/228/htm. |
| [10] | Ghaderi A A, Mohammadzadeh A, Bahrami M N. Optimum design of damped vibration absorber for viscoelastic bladed disk assemblies[J].Mechanics, 2015, 21(6): 465–471. |
| [11] | Meng J, Sun D. Research on vibration suppression of wind turbine blade based on bamboo wall three-layer damping structure[J].Journal of Vibroengineering, 2017, 19(1): 87–99.DOI:10.21595/jve.2016.17378 |
| [12] | Zhou B, Thouverez F, Lenoir D. An adaptive control strategy based on passive piezoelectric shunt techniques applied to mistuned bladed disks[J].Journal of Computational and Applied Mathematics, 2013, 246: 289–300.DOI:10.1016/j.cam.2012.06.023 |
| [13] | Alsayed E Z, Hariri I, Nakashima S, et al. Effects of coating materials on nano-indentation hardness of enamel and adjacent areas[J].Dental Materials, 2016, 32(6): 807–816.DOI:10.1016/j.dental.2016.03.023 |
| [14] | Shao F, Yang K, Zhao H, et al. Effects of inorganic sealant and brief heat treatments on corrosion behavior of plasma sprayed Cr2O3-Al2 O3 composite ceramic coatings[J].Surface and Coatings Technology, 2015, 276: 8–15.DOI:10.1016/j.surfcoat.2015.06.045 |
| [15] | Stathopoulos V, Sadykov V, Pavlova S, et al. Design of functionally graded multilayer thermal barrier coatings for gas turbine application[J].Surface and Coatings Technology, 2016, 295: 20–28.DOI:10.1016/j.surfcoat.2015.11.054 |
| [16] | Ganvir A, Markocsan N, Joshi S. Influence of isothermal heat treatment on porosity and crystallite size in axial suspension plasma sprayed thermal barrier coatings for gas turbine applications[J].Coatings, 2016, 7(1): 4–27.DOI:10.3390/coatings7010004 |
| [17] | Yang Z X, Han Q K, Jin Z H, et al. Solution of natural characteristics of a hard-coating plate based on Lindstedt-Poincaré perturbation method and its valedictions by FEM and measurement[J].Nonlinear Dynamics, 2015, 81(3): 1207–1218.DOI:10.1007/s11071-015-2063-8 |
| [18] | Sun W, Liu Y. Vibration analysis of hard-coated composite beam considering strain dependent characteristic of coating material[J].Acta Mechanica Sinica, 2016, 32(4): 731–742.DOI:10.1007/s10409-016-0564-4 |
| [19] | Zhang Y, Sun W, Yang J, et al. Nonlinear vibration analysis of a hard-coating cylindrical shell with elastic constraints by finite element method[J].Nonlinear Dynamics, 2017, 90(4): 2879–2891.DOI:10.1007/s11071-017-3849-7 |
| [20] | Sanliturk K Y, Koruk H. Development and validation of a composite finite element with damping capability[J].Composite Structures, 2013, 97: 136–146. |
| [21] | 王娇.叶片阻尼结构的振动分析方法及其阻尼抑振效果研究[D].沈阳: 东北大学, 2013. ( Wang Jiao.Research on vibration analysis methods and vibration suppression effect of damping structure of blade[D].Shenyang: Northeastern University, 2013.http://cdmd.cnki.com.cn/Article/CDMD-10145-1016011711.htm) |
| [22] | Jin G, Yang C, Liu Z. Vibration and damping analysis of sandwich viscoelastic-core beam using Reddy's higher-order theory[J].Composite Structures, 2016, 140: 390–409.DOI:10.1016/j.compstruct.2016.01.017 |
