赵军 , 方文勇 , 韩式国 . 自由曲面数控加工表面形貌仿真 [J ] . 工具技术 , 2007 , 41 ( 8 ): 17 - 19 .

ZHAO J , FANG W Y , HAN S G . Topography simulation of NC milling of free-form surface [J ] . Tool Engineering , 2007 , 41 ( 8 ): 17 - 19 . (in Chinese)

LIU X B , SOSHI M , SAHASRABUDHE A , et al . A geometrical simulation system of ball end finish milling process and its application for the prediction of surface micro features [J ] . Journal of Manufacturing Science & Engineering , 2006 , 128 ( 1 ): 74 - 85 .

LAVERNHE S , QUINSAT Y , LARTIGUE C . Model for the prediction of 3D surface topography in 5-axis milling [J ] . International Journal of Advanced Manufacturing Technology , 2010 , 51 ( 9/10/11/12 ): 915 - 924 . DOI: 10.1007/s00170-010-2686-3 http://doi.org/10.1007/s00170-010-2686-3 http://link.springer.com/10.1007/s00170-010-2686-3 http://link.springer.com/10.1007/s00170-010-2686-3

FERNANDO R P , SOLDANI X , LOYA J , et al . A new approach for time-space wear modeling applied to machining tool wear [J ] . Wear , 2017 , 390 - 391 : 125 - 134 .

SCHMITZ T L , COUEY J , MARSH E , et al . Runout effects in milling: Surface finish, surface location error, and stability [J ] . International Journal of Machine Tools & Manufacture , 2007 , 47 ( 5 ): 841 - 851 . DOI: 10.1016/j.ijmachtools.2006.06.014 http://doi.org/10.1016/j.ijmachtools.2006.06.014 https://linkinghub.elsevier.com/retrieve/pii/S0890695506001672 https://linkinghub.elsevier.com/retrieve/pii/S0890695506001672

CHEN W Q , SUN Y Z , HUO D H , et al . Modelling of the influence of tool runout on surface generation in micro milling [J ] . Chinese Journal of Mechanical Engineering , 2019 , 32 ( 1 ): 1 - 9 . DOI: 10.1186/s10033-018-0313-7 http://doi.org/10.1186/s10033-018-0313-7

ZHANG X , PAN X D , WANG G L . Influence factors of surface topography in micro-side milling [J ] . International Journal of Advanced Manufacturing Technology , 2019 , 105 ( 12 ): 5239 - 5245 . DOI: 10.1007/s00170-019-04527-y http://doi.org/10.1007/s00170-019-04527-y

ARIZMENDI M , JIM NEZ A . Modelling and analysis of surface topography generated in face milling operations [J ] . International Journal of Mechanical Sciences , 2019 , 163 : 105061 - 1 -9. DOI: 10.1016/j.ijmecsci.2019.105061 http://doi.org/10.1016/j.ijmecsci.2019.105061 https://linkinghub.elsevier.com/retrieve/pii/S0020740319313104 https://linkinghub.elsevier.com/retrieve/pii/S0020740319313104

WANG L P , GE S Y , SI H , et al . Elliptical model for surface topography prediction in five-axis flank milling [J ] . Chinese Journal of Aeronautics , 2020 , 33 ( 4 ): 1361 - 1374 . DOI: 10.1016/j.cja.2019.06.007 http://doi.org/10.1016/j.cja.2019.06.007 https://linkinghub.elsevier.com/retrieve/pii/S100093611930250X https://linkinghub.elsevier.com/retrieve/pii/S100093611930250X

ZEROUDI N , FONTAINE M . Prediction of machined surface geometry based on analytical modelling of ball-end milling [J ] . Procedia CIRP , 2012 , 1 : 108 - 113 . DOI: 10.1016/j.procir.2012.04.017 http://doi.org/10.1016/j.procir.2012.04.017 https://linkinghub.elsevier.com/retrieve/pii/S2212827112000182 https://linkinghub.elsevier.com/retrieve/pii/S2212827112000182

CHEN Z T , YUE C X , LIU X L , et al . Surface topography prediction model in milling of thin-walled parts considering machining deformation [J ] . Materials , 2021 , 14 ( 24 ): 7679 . DOI: 10.3390/ma14247679 http://doi.org/10.3390/ma14247679 https://www.mdpi.com/1996-1944/14/24/7679 https://www.mdpi.com/1996-1944/14/24/7679 With the continuous improvement of the performance of modern aerospace aircraft, the overall strength and lightweight control of aircraft has become a significant feature of modern aerospace parts. With the wide application of thin-walled parts, the requirements for dimensional accuracy and surface quality of workpieces are increasing. In this paper, a numerical model for predicting surface topography of thin-walled parts after elastic deformation is proposed. In view of the geometric characteristics in the cutting process, the cutting force model of thin-walled parts is established, and the meshing relationship between the tool and the workpiece is studied. In addition, the influence of workpiece deformation is considered based on the beam deformation model. Cutting force is calculated based on deformed cutting thickness, and the next cutting–meshing relationship is predicted. The model combines the radial deflection of the workpiece in the feed direction and the changing meshing relationship of the tool–workpiece to determine the three-dimensional topography of the workpiece. The error range between the experimental and the simulation results of surface roughness is 7.45–13.09%, so the simulation three-dimensional morphology has good similarity. The surface topography prediction model provides a fast solution for surface quality control in the thin-walled parts’ milling process.

PENG Z X , JIAO L , YAN P , et al . Simulation and experimental study on 3D surface topography in micro-ball-end milling [J ] . International Journal of Advanced Manufacturing Technology , 2018 , 96 ( 5 ): 1943 - 1958 . DOI: 10.1007/s00170-018-1597-6 http://doi.org/10.1007/s00170-018-1597-6

宋文刚 , 刘战强 , 吕文军 . 工件初始表面几何状态对铣削力及表面形貌的影响 [J ] . 工具技术 , 2021 , 55 ( 7 ): 28 - 32 .

SONG W G , LIU Z Q , LÜ W J . Influence of initial surface geometric states of workpiece on milling force and surface topography [J ] . Tool Engineering , 2021 , 55 ( 7 ): 28 - 32 . (in Chinese)

白利娟 , 张建华 , 陶国灿 , 等 . 振动辅助铣削加工仿生表面研究 [J ] . 中国机械工程 , 2016 , 27 ( 9 ): 1229 - 1233 ,1242.

BAI L J , ZHANG J H , TAO G C , et al . Vibration assisted milling for bionic surface manufacturing [J ] . China Mechanical Engineering , 2016 , 27 ( 9 ): 1229 - 1233 ,1242. (in Chinese)

LU X H , HU X C , JIA Z Y , et al . Model for the prediction of 3D surface topography and surface roughness in micro-milling Inconel 718 [J ] . The International Journal of Advanced Manufacturing Technology , 2018 , 94 ( 5 ): 2043 - 2056 . DOI: 10.1007/s00170-017-1001-y http://doi.org/10.1007/s00170-017-1001-y

梁鑫光 , 姚振强 . 基于动力学响应的球头刀五轴铣削表面形貌仿真 [J ] . 机械工程学报 , 2013 , 49 ( 6 ): 171 - 178 . 针对自由曲面球头刀五轴铣削中刀具与工件复杂的位姿关系，利用球头铣削刃的三维次摆线轨迹，提出一种在工件表面等距离间隔缓存残留高度、在刀具端等时间间隔缓存振动响应的双缓存离散机制，分别实现五轴铣削中的名义加工表面形貌建模与系统动态响应轨迹仿真。在此基础上，以切削刃经过工件表面残留区域的时间信息为纽带，利用系统瞬时动态位移响应的插值信息对已加工表面法向残留高度进行修正，建立考虑铣削系统动力学响应的已加工表面形貌预测模型。以变切深、变转速、恒定的刀具倾角与每齿进给量所进行的验证试验表明，提出的球头刀五轴铣削表面形貌建模方法可有效预测颤振工况与稳定铣削工况的加工表面形貌与纹理特征。

LIANG X G , YAO Z Q . Dynamic-based simulation for machined surface topography in 5-axis ball-end milling [J ] . Journal of Mechanical Engineering , 2013 , 49 ( 6 ): 171 - 178 . (in Chinese) Focusing on the complexity of the relative motion between cutter and workpiece in five-axis milling, a 3D trochoid mechanism is adopted to describe the actual tooth trajectory of the ball-end cutting edge. A kind of two-buffer mechanism is proposed to calculate residual height. One is established to determine each spatial buffer point which is discretized by a constant interval on workpiece surface, the other is built in cutter centre to simulate system dynamic response for each time interval. Thus, the nominal machined surface topography and system vibration trajectory can be calculated respectively. According to the exact time when cutting edge passing by the discretized point of the residual surface, the corresponding instantaneous vibration response can be interpolated and used to update the nominal residual height so that the prediction model of surface topography can be established with the consideration of milling system dynamic response. Validation tests are carried out with the varying cutting depth, increasing spindle speed, constant lead angle and fixed feed rate. It indicates that the proposed method can be used to predict surface topography accurately and distinguish stable milling status from chatter effectively.

张国华 , 李咚咚 , 李茂伟 , 等 . 超声椭圆振动车削三维形貌形成研究 [J ] . 兵工学报 , 2017 , 38 ( 10 ): 2002 - 2009 . DOI: 10.3969/j.issn.1000-1093.2017.10.017 http://doi.org/10.3969/j.issn.1000-1093.2017.10.017 为研究椭圆振动车削表面三维几何形貌形成规律及影响因素，采用金属切削理论对椭圆振动切削过程和三维切削模型进行了分析。研究结果发现：切削过程中相邻两转之间不同的相位差特征值对最终表面形貌有着重要的影响；通过已有的试验结果表明，相位差特征值分别为0和0.5时所形成的表面形貌截然不同，与理论分析结果相吻合。

ZHANG G H , LI D D , LI M W , et al . Research on formation of 3-D micro-surface in ultrasonic elliptical vibration cutting [J ] . Acta Armamentarii , 2017 , 38 ( 10 ): 2002 - 2009 . (in Chinese)

COSTES J P , MOREAU V . Surface roughness prediction in milling based on tool displacements [J ] . Journal of Manufacturing Processes , 2011 , 13 ( 2 ): 133 - 140 . DOI: 10.1016/j.jmapro.2011.02.003 http://doi.org/10.1016/j.jmapro.2011.02.003 https://linkinghub.elsevier.com/retrieve/pii/S1526612511000089 https://linkinghub.elsevier.com/retrieve/pii/S1526612511000089

秦国华 , 何志芬 , 王华敏 , 等 . 基于动力学与遗传算法的工件位置偏离预测与控制方法 [J ] . 机械工程学报 , 2017 , 53 ( 1 ): 110 - 120 . DOI: 10.3901/JME.2017.01.110 http://doi.org/10.3901/JME.2017.01.110 装夹布局的合理规划是影响工件加工质量与生产安全的关键因素，为此系统地提出了工件位置偏离的动力学预测模型及其控制方法。首先通过将工件与夹具之间的接触模式等价地转化为线性弹簧-阻尼系统，利用运动学和弹性力学理论推导出工件的振动微分方程，以确定工件在装夹布局中的位置偏离。根据工件与夹具之间的接触为单向这一实际情况，运用胡克定律建立了工件位置偏离的约束条件。通过坐标转换方法，提出在约束条件下振动微分方程的模态分析求解方法，实现工件位置偏离的预测，试验结果表明所提出预测结果与试验结果完全吻合。其次，为使工件位置偏离达到最小，构建工件位置偏离的最小2-范数为目标的装夹布局优化模型，通过以合理装夹布局中工件位置偏离的2-范数为自变量构造个体适应度，提出装夹布局优化模型的遗传算法求解技术。提出的工件位置偏离预测与控制方法，能够避免工件处于非稳定状态下优化模型的求解过程，为复杂工件装夹布局方案的合理设计提供了基础理论。

QIN G H , HE Z F , WANG H M , et al . Prediction and control of workpiece position deviation based on dynamics and genetic algorithm [J ] . Journal of Mechanical Engineering , 2017 , 53 ( 1 ): 110 - 120 . (in Chinese) DOI: 10.3901/JME.2017.01.110 http://doi.org/10.3901/JME.2017.01.110 The reasonable planning of the fixturing layout is crucial to guarantee the machining quality and safety manufacturing. Therefore, the prediction and control method is systematically proposed for the workpiece position error. Above all, by transforming the contacts between the workpiece and fixture into the linear spring-damp system, the vibration differential equation of the workpiece, which can determine the position movement of the workpiece in the fixturing layout, is concluded to obtain the workpiece position deviation according to the theory of kinematics and elastic mechanics. Based on the fact that the contacts between the workpiece and fixture are unidirectional, the hook law is used to formulate the constraint conditions of the workpiece position deviation. According to coordinate transformation method, the modal analysis solution technology of the vibration differential equation is proposed to predict the workpiece position error. A numerical test is demonstrated to validate the prediction results are in good agreement with the experimental data in the reference. And then, in order to minimize the workpiece position deviation, the optimal model with the objective of minimizing the 2-norm of the workpiece position deviation is proposed for the fixturing layout. According to the workpiece position deviation in the reasonable fixturing layout, the individual fitness is defined by taking its 2-norm as independent variable. Thus, the genetic algorithm can be skillfully developed to solve the optimal model of fixturing layout. The presented prediction and control method of workpiece position error can avoid the solution of the optimal model under the condition of the workpiece in unstable fixturing. It can provide a basic theory of fixturing layout design for the complex workpiece.<p>&nbsp;</p>

TAO Z J , KUMAR A S , NEE A Y C , et al . Modelling and experimental investigation of a sensor-integrated workpiece-fixture system [J ] . International Journal of Computer Applications in Technology , 1997 , 10 ( 3/4 ): 236 - 250 .

QIN G H , ZHANG W H , WU Z X , et al . Systematic modeling of workpiece-fixture geometric default and compliance for the prediction of workpiece machining error [J ] . ASME Journal of Manufacturing Science and Engineering , 2007 , 129 : 789 - 801 . DOI: 10.1115/1.2336260 http://doi.org/10.1115/1.2336260 https://asmedigitalcollection.asme.org/manufacturingscience/article/129/4/789/468388/Systematic-Modeling-of-Workpiece-Fixture-Geometric https://asmedigitalcollection.asme.org/manufacturingscience/article/129/4/789/468388/Systematic-Modeling-of-Workpiece-Fixture-Geometric Control of workpiece machining error (WME) is a key concern in the design of a fixture system. In this paper, source errors, which are categorized into workpiece-fixture geometric default and workpiece-fixture compliance, are systematically investigated to reveal their effects upon the WME. The underlying mechanism is that source errors lead to the workpiece position error (WPE), the workpiece elastic deformations (WED), and the inconsistent datum error (IDE), and all of them will contribute together to the WME. Here, the IDE refers to the dimension deviation of the processing datum from the locating datum once two references do not coincide. An overall quantitative formulation is proposed for the computing of WME in terms of WPE, WED, and IDE for the first time. In detail, the WPE raised in the workpiece-locating and clamping process is evaluated based on the geometric defaults and local deformations of workpiece-fixture in the contact region. The WED relative to the workpiece-clamping process is determined by solving a nonlinear mathematical programming problem of minimizing the total complementary energy of the frictional workpiece-fixture system. Some numerical tests are finally demonstrated to validate the proposed approach on the basis of both theoretical and experimental data given in the references.