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## Mass matching design of machine tool parts based on spatial dynamics optimization

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Abstract

A multi-objective matching design method based on the whole machine space dynamics prediction model was proposed by taking the moving structure mass as design variables. The whole machine workspace was constructed based on the working stroke of the moving parts of the machine tool. Then the orthogonal test method was used to design the spatial pose, and the prediction model of the spatial natural frequency was established. The sensitivity analysis of the machining path was conducted in the workspace. The best and worst position was identified. Then the optimal distribution of the mass of moving parts was designed by multi-objective mass matching method by taking the natural frequency of the machine tool with the worst position as the optimization objective. The natural frequency of the optimized machine tool was calculated. The dynamic characteristics of the best and worst position before and after optimization were analyzed and compared. Results show that the natural frequency of the machine tool was improved with the optimization of multi-objective mass matching, and the maximum frequency response amplitude of the tool tip was significantly reduced. The dynamic performance of the machine tool was greatly improved.

Keywords： spatial pose ; prediction model ; dynamic characteristics ; multi-objective optimization ; mass matching

HUANG Hua, DENG Wen-qiang, LI Yuan, GUO Run-lan. Mass matching design of machine tool parts based on spatial dynamics optimization. Journal of Zhejiang University(Engineering Science)[J], 2020, 54(10): 2009-2017 doi:10.3785/j.issn.1008-973X.2020.10.019

## 1. 机床空间动态特性定义

${{M\ddot q}} + {{C\dot q}} + {{K}}{{q}} = {{0}}.$

MCK发生改变，因此机床的动态特性会随着移动件位置的变化而改变. 在加工空间内，MCK均可以视为各移动件空间位姿(xyz)的函数，故机床空间动态特性的本构方程可以描述为

${{M}}(x,y,z){\ddot{ q}} + {{C}}(x,y,z){{\dot q}} + {{K}}(x,y,z){{q}} = {{0}}.$

$\left( {{{M}}(x,y,z) + \Delta {{m}}(x,y,z)} \right){\ddot{ q}} + {{C}}(x,y,z){{\dot q}} + {{K}}(x,y,z){{q}} = {{0}}.$

${\omega _{ir}} ={({{1 + {\phi}_r^{\rm T}\Delta m(x,y,z){\phi} _{r}}}}) ^{-1/2} {\omega _{0r}}.$

### 图 1

Fig.1   HMC630 Horizontal Machining Center

Tab.1  Orthogonal experimental factors and levels

 水平 因素 x/m y/m z/m 1 0.02 0.02 0.02 2 0.2 0.175 0.1375 3 0.4 0.35 0.275 4 0.6 0.525 0.4125 5 0.78 0.68 0.53

Tab.2  Orthogonal experiment of position

 试验号 位姿坐标 x/m y/m z/m 1 0.02 0.02 0.02 2 0.02 0.175 0.1375 3 0.02 0.35 0.275 4 0.02 0.525 0.4125 5 0.02 0.68 0.53 6 0.2 0.02 0.1375 7 0.2 0.175 0.275 8 0.2 0.35 0.4125 9 0.2 0.525 0.53 10 0.2 0.68 0.02 11 0.4 0.02 0.275 12 0.4 0.175 0.4125 13 0.4 0.35 0.53 14 0.4 0.525 0.02 15 0.4 0.68 0.1375 16 0.6 0.02 0.4125 17 0.6 0.175 0.53 18 0.6 0.35 0.02 19 0.6 0.525 0.1375 20 0.6 0.68 0.275 21 0.78 0.02 0.53 22 0.78 0.175 0.02 23 0.78 0.35 0.1375 24 0.78 0.525 0.275 25 0.78 0.68 0.4125

### 2.2. 整机有限元建模及结合面的处理

Tab.3  Material parameters for steel and cast iron

 材料 ρ/(kg·m−3) E/GPa μ ξ 钢 7850 200 0.30 0.28 铸铁 7200 110 0.28 0.006

Tab.4  Equivalent stiffness and damping parameters of joints

 结合部类型 k / (N·m−1) ζ / (N·s·m−1) 导轨滑块法向 4.9×106 3150 导轨滑块切向 4.2×106 1400 丝杆螺母轴向 1.7×106 3200 螺栓法向 9.0×106 6860 螺栓切向 7.8×106 5500

Tab.5  Results of orthogonal experiment

 试验号 f1/Hz f2/Hz f3/Hz f4/Hz f5/Hz 1 64.215 71.125 148.50 199.19 212.03 2 65.631 72.928 147.55 205.56 217.52 3 65.630 73.673 145.08 211.32 221.69 4 63.995 71.216 139.55 210.05 224.00 5 69.530 80.051 139.90 239.68 243.26 6 64.211 70.475 146.21 201.19 215.28 7 65.098 71.842 143.10 206.99 220.27 8 64.254 72.413 140.08 211.84 224.32 9 62.146 70.023 135.10 209.57 224.63 10 64.607 71.649 150.37 201.22 213.29 11 63.682 69.430 141.86 202.19 218.09 12 64.573 71.680 139.53 210.28 225.33 13 70.307 85.978 140.70 238.65 252.55 14 65.857 74.416 151.69 206.07 215.94 15 64.571 70.841 147.78 203.02 216.33 16 62.404 68.259 137.13 202.42 220.20 17 61.911 69.460 133.94 207.20 223.36 18 66.171 75.681 152.24 206.73 216.89 19 65.849 73.548 149.12 208.59 218.43 20 64.069 69.794 143.28 203.77 219.17 21 60.671 67.130 132.92 202.07 220.87 22 65.646 73.803 150.05 203.48 214.90 23 66.163 74.787 149.68 209.48 219.10 24 65.322 72.449 144.50 209.89 221.29 25 62.833 68.567 138.40 203.60 221.10

### 2.3. 求解整机空间固有频率预测模型

${{\omega }}{\rm{ = }}\left[ {{\omega _{\rm{1}},}\;\;{\omega _{\rm{2}},}\;\;{\omega _{\rm{3}},}\;\;{\omega _{\rm{4}},}\;\;{\omega _{\rm{5}}}} \right] = {{X\beta }}.$

${R^2} = 1 - \frac{{{S_{\rm{SE}}}}}{{{S_{\rm{ST}}}}},$

${S_{{\rm{SE}}}} = \sum\limits_{i = 1}^N {({Y_i} - {y_i}} {)^2},$

${S_{{\rm{ST}}}} = \sum\limits_{i = 1}^N {{Y_i}^2 - } {{{{\left(\sum\limits_{i = 1}^N {{y_i}} \right)}^2}} \bigg/ N}.$

Tab.6  Accuracy test of prediction model

 ω R2 ω R2 ω1 0.996 5 ω4 0.998 4 ω2 0.998 6 ω5 0.998 7 ω3 0.999 2 − −

### 图 2

Fig.2   Spatial inherent frequency prediction model of machine tools

## 3. 坐标轴灵敏度分析及机床位姿识别

### 图 3

Fig.3   Global effect of inherent frequency on coordinate axis

### 图 4

Fig.4   Iterative history diagram of worst position coordinates

### 图 5

Fig.5   Iterative history diagram of best position coordinates

## 4. 移动件多目标质量匹配优化

Tab.7  Levels for mass test of moving parts

 kg 水平 m1 m2 m3 −1 1209.115 2245.745 293.764 0 1511.400 2807.200 367.200 1 1813.672 3368.618 440.646

Tab.8  Test table for mass distribution of moving parts

 kg 试验号 移动件质量 m1 m2 m3 1 1209.115 2245.745 293.764 2 1209.115 3368.618 367.200 3 1813.672 2245.745 367.200 4 1813.672 3368.618 367.200 5 1209.115 2807.200 293.764 6 1209.115 2807.200 440.646 7 1813.672 2807.200 293.764 8 1813.672 2807.200 440.646 9 1511.400 2245.745 293.764 10 1511.400 2245.745 440.646 11 1511.400 3368.618 293.764 12 1511.400 3368.618 440.646 13 1511.400 2807.200 367.200

### 图 6

Fig.6   Main effect of moving parts mass on natural frequencies of machine tools

Tab.9  Comparison of mass changes of moving parts before and after optimization

 kg 状态 m1 m2 m3 优化前 1511.400 2807.200 367.200 优化后 1639.916 2245.745 293.801

## 5. 优化结果分析

Tab.10  First five natural frequencies of worst position of machine tools before and after optimization

 状态 f1/Hz f2/Hz f3/Hz f4/Hz f5/Hz 优化前 59.534 65.709 133.986 197.342 220.331 优化后 68.056 77.534 150.375 229.642 249.632

Tab.11  First five natural frequencies of best position of machine tools before and after optimization

 状态 f1/Hz f2/Hz f3/Hz f4/Hz f5/Hz 优化前 64.397 70.327 145.61 203.57 217.91 优化后 71.946 78.541 162.53 226.94 243.00

### 图 7

Fig.7   Frequency response curve of worst position

### 图 8

Fig.8   Frequency response curve of best position

## 6. 结　论

(1) 采用正交试验法将机床的工作空间进行离散化，得到各移动件位姿正交试验表；运用有限元分析软件计算各试验的响应值，通过最小二乘法建立机床的空间固有频率预测模型. 对机床沿xyz方向的工作路径进行灵敏度分析，获得机床加工过程中的最佳工艺路线；采用遗传算法，识别出机床工作空间内的最差和最优位姿.

(2) 以机床最差位姿的固有频率为优化目标，采用Box-Behnken方法对工作台、立柱和主轴箱的质量进行试验设计，建立机床固有频率与移动件质量间的响应面模型，基于NSGA-II算法进行匹配优化. 结果表明：优化后移动件的总质量减少了约10.8%，在最差位姿状态下，机床前5阶固有频率分别提高了14.3%、18.0%、12.2%、16.4%和13.3%；在最优位姿状态下，前5阶固有频率分别提高了11.7%、11.7%、11.5%、11.5%和10.3%.

(3) 以机床刀尖为研究对象，对优化后的机床进行频率响应分析. 结果表明：在最差位姿状态下，刀尖节点的最大响应振幅降低了81.9%；在最优位姿状态下，刀尖节点的最大响应振幅降低了56.9%. 机床工作空间内的动力学性能得到了有效提高，为后续面向平稳加工的工艺路径规划提供了保证.

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