WO2019084948A1 - 一种卧式数控车床的主轴径向热漂移误差建模及补偿方法 - Google Patents

一种卧式数控车床的主轴径向热漂移误差建模及补偿方法 Download PDF

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WO2019084948A1
WO2019084948A1 PCT/CN2017/109492 CN2017109492W WO2019084948A1 WO 2019084948 A1 WO2019084948 A1 WO 2019084948A1 CN 2017109492 W CN2017109492 W CN 2017109492W WO 2019084948 A1 WO2019084948 A1 WO 2019084948A1
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thermal
spindle
error
attitude
headstock
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PCT/CN2017/109492
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English (en)
French (fr)
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刘阔
王永青
刘海波
李特
刘海宁
厉大维
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大连理工大学
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Priority to PCT/CN2017/109492 priority Critical patent/WO2019084948A1/zh
Priority to US16/325,984 priority patent/US10838392B2/en
Publication of WO2019084948A1 publication Critical patent/WO2019084948A1/zh

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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23Q—DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING; MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT
    • B23Q11/00—Accessories fitted to machine tools for keeping tools or parts of the machine in good working condition or for cooling work; Safety devices specially combined with or arranged in, or specially adapted for use in connection with, machine tools
    • B23Q11/0003—Arrangements for preventing undesired thermal effects on tools or parts of the machine
    • B23Q11/0007—Arrangements for preventing undesired thermal effects on tools or parts of the machine by compensating occurring thermal dilations
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23Q—DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING; MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT
    • B23Q11/00—Accessories fitted to machine tools for keeping tools or parts of the machine in good working condition or for cooling work; Safety devices specially combined with or arranged in, or specially adapted for use in connection with, machine tools
    • B23Q11/14—Methods or arrangements for maintaining a constant temperature in parts of machine tools
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23Q—DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING; MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT
    • B23Q15/00—Automatic control or regulation of feed movement, cutting velocity or position of tool or work
    • B23Q15/007—Automatic control or regulation of feed movement, cutting velocity or position of tool or work while the tool acts upon the workpiece
    • B23Q15/18—Compensation of tool-deflection due to temperature or force
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23Q—DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING; MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT
    • B23Q17/00—Arrangements for observing, indicating or measuring on machine tools
    • B23Q17/22—Arrangements for observing, indicating or measuring on machine tools for indicating or measuring existing or desired position of tool or work
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/404—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by control arrangements for compensation, e.g. for backlash, overshoot, tool offset, tool wear, temperature, machine construction errors, load, inertia
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/35—Nc in input of data, input till input file format
    • G05B2219/35408—Calculate new position data from actual data to compensate for contour error
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/49—Nc machine tool, till multiple
    • G05B2219/49219—Compensation temperature, thermal displacement

Definitions

  • the invention belongs to the technical field of error compensation of numerical control machine tools, and specifically relates to a method for modeling and compensating the radial thermal drift error of a spindle of a horizontal numerical control lathe.
  • the thermal error of the machine tool is a problem that has plagued the machine tool industry for decades. Due to the existence of thermal error of the machine tool, the problems are: the machining accuracy of the single piece is unsatisfactory; the consistency of the batch processing parts is poor, and the scrap rate is high; in order to reduce the thermal error, the machine needs a heat machine after the machine is turned on, and the energy loss is large; The processing accuracy is high, and a constant temperature workshop is also required. These problems indicate that thermal errors have a number of adverse effects on the machine.
  • the error prevention method eliminates or reduces the heat source of the machine tool by design and manufacturing, but the biggest disadvantage is the high cost.
  • the cost of improving the accuracy of the machine tool increases exponentially.
  • Thermal error compensation technology as a method to improve the accuracy and stability of CNC machine tools, has many advantages, such as relatively low cost and wide application range.
  • the thermal error of CNC machine tools mainly includes two parts: feed axis thermal error and spindle thermal error.
  • the thermal error of the feed axis can be greatly reduced by the closed-loop feedback of the grating, but the thermal error of the spindle lacks effective suppression means.
  • Spindle thermal errors include axial thermal elongation errors and radial thermal drift errors.
  • scholars have carried out research on the compensation of axial axial thermal elongation error of the spindle, and tried various modeling methods such as multiple regression method, neural network method, thermal mode method, time series method and support vector machine.
  • TJKo analyzed the thermal bending deformation caused by the thermal gradient of the vertical machining center spindle system and established the thermal error prediction model of the spindle radial direction in "Particular behavior of spindle thermal deformation by thermal bending".
  • the spindle is not compensated, and the judgment of the hot deformation posture of the spindle is not given. Then, and the impact of the machine structure size on the model prediction results.
  • the radial thermal error of the spindle of a CNC lathe is very important because the Z-direction accuracy of the lathe is more concerned with respect to the accuracy of the Z-axis.
  • the invention aims at the problem of radial thermal error compensation of the numerical control lathe spindle, and proposes a radial thermal drift error modeling and compensation method for the horizontal numerical control lathe main shaft.
  • the object of the present invention is to provide an effective radial thermal drift error modeling and compensation method for a horizontal CNC lathe spindle, and solve the problem of radial thermal error compensation of the numerical control lathe spindle.
  • the technical solution of the present invention is to first test the two-point thermal drift error of the numerical control lathe main shaft along the radial direction and the corresponding key point temperature. Then, the thermal tilt angle of the main shaft is obtained based on the hot tilt deformation mechanism of the main shaft, and the correlation analysis method is used to analyze the correlation between the thermal dip angle and the temperature difference between the left and right sides of the headstock. According to the positive and negative thermal drift errors of the two points tested and the extension or shortening of the left and right sides of the headstock, the thermal deformation of the main shaft is classified and the thermal drift error model under various thermal deformation attitudes is established. Then, the asymptotic integration method is used to analyze the influence of the machine structure size on the model prediction results. In the real-time compensation, the hot deformation attitude of the spindle is automatically judged according to the temperature of the key point, and the corresponding thermal drift error model is automatically selected to compensate the spindle.
  • a method for modeling and compensating the radial thermal drift error of a horizontal CNC lathe the steps are as follows:
  • the first step is the radial thermal drift error and key point temperature test of the CNC lathe spindle.
  • thermal error e i of the spindle 1 in the vertical direction produces a thermal error component e i,x in the X direction
  • thermal errors e 1,x and e 2 ,x of the spindle 1 along the X direction are calculated as follows:
  • the second step is the correlation analysis between the thermal inclination angle of the main shaft and the temperature difference.
  • the thermal tilt angle of the spindle 1 after heating is calculated by the following formula:
  • L snr is the distance between the two error measuring points
  • R is a correlation matrix with ⁇ T, for a covariance matrix between and ⁇ T;
  • the third step is the radial thermal drift error model of the main shaft under different thermal deformation attitudes.
  • the thermal deformation of the main shaft 1 is divided into three categories and ten races; l is the amount of thermal change on the left side of the headstock 2, ⁇ r is the amount of thermal change on the right side of the headstock 2, ⁇ l and ⁇ r are both positive for thermal expansion, negative for shrinkage; d crs is for deformation
  • l is the amount of thermal change on the left side of the headstock 2
  • ⁇ r is the amount of thermal change on the right side of the headstock 2
  • ⁇ l and ⁇ r are both positive for thermal expansion, negative for shrinkage
  • d crs is for deformation
  • d spl is the distance between the left and right end faces of the headstock 2
  • d ss is the horizontal end of the right end face of the headstock 2 and the left displacement sensor 7
  • the distance d snr is the distance between the left displacement
  • ⁇ l (t) ⁇ l1 ⁇ (T 1 (t)-T 1 (0))+ ⁇ l2 (5)
  • ⁇ r (t) ⁇ r1 ⁇ (T 2 (t)-T 2 (0))+ ⁇ r2 (6)
  • ⁇ l1 , ⁇ l2 , ⁇ r1 and ⁇ r2 are the coefficients to be identified;
  • the fourth step is the analysis of the influence of the machine tool size on the model prediction results.
  • X is a random vector consisting of d spl and d ss
  • ⁇ is the allowed deviation index
  • ⁇ a is the fluctuation value of the prediction residual and is defined as:
  • R is the prediction residual for d spl and d ss as random variables, which is a function of d spl and d ss
  • R n is the prediction residual of the model when d spl and d ss are true values
  • N is the thermal error. The number of sampling points at the time of testing
  • the first-order moment method is used to calculate the failure probability index of the predicted residual fluctuation value according to equation (17):
  • the reliability of calculating the fluctuation value of the prediction residual by the asymptotic integration method belongs to a certain allowable deviation range according to the formula (21):
  • the fifth step the determination of the hot deformation posture of the main shaft and the model selection
  • the thermal deformation amounts ⁇ l , ⁇ r and d ⁇ of the two sides of the headstock 2 are used to determine the hot deformation posture of the spindle 1 which changes irregularly during the machining process.
  • d ⁇ is the distance from the intersection of the deformed main shaft 1 and the original initial state spindle 1 to the right side surface of the headstock 2; in various thermal deformation attitudes, the calculation formula of d ⁇ is calculated by the formula (22):
  • Attitude (6) ⁇ l ⁇ r ⁇ 0, d ss ⁇ d ⁇ ⁇ d ss +d snr
  • Attitude (8) ⁇ r ⁇ ⁇ l ⁇ 0
  • d wp is the distance from the workpiece to the end face of the chuck 9 and d s is the displacement sensor 7 from the left end to the end face of the chuck 9 Distance; in various thermal deformation attitudes, whether d wp ⁇ d s , d s ⁇ d wp ⁇ d s +d snr or d wp >d s +d snr , the thermal error compensation amount e wp of the workpiece to be machined Calculated according to formula (23):
  • the predicted value e wp of the thermal error is input into the numerical control system of the machine tool in real time, and the thermal error compensation of the spindle of the numerical control lathe at any position and time is realized.
  • the thermal error prediction model is established based on the hot tilt deformation mechanism of the main shaft, and the robustness of the model is strong;
  • the thermal error prediction model considers 10 kinds of thermal deformation attitudes obtained by theoretical analysis, so the model is suitable for any spindle speed change and ambient temperature change;
  • the model can automatically determine the thermal deformation attitude of the main shaft according to the temperature of the left and right sides of the headstock, and adopt the corresponding thermal error model.
  • Figure 1 shows the structure of the spindle system and the temperature sensor layout.
  • Figure 2 shows the error test instrument and its installation diagram.
  • Figure 3 is an exploded view of the radial thermal drift error of the spindle.
  • Figure 4 is a schematic illustration of the spindle in an initial thermal equilibrium state.
  • Figure 5 is a thermal deformation attitude diagram of a spindle of a numerically controlled lathe
  • Figure 5 (a) shows the thermal deformation attitude (1) under the condition of e 1 >0 and e 2 >0;
  • Figure 5 (b) shows the thermal deformation attitude (2) - (4) under the condition of e 1 > 0 and e 2 >0;
  • Figure 5 (c) is a thermal deformation attitude (6) under the condition of e 1 >0, and e 2 ⁇ 0, thermal deformation attitude (5); e 1 ⁇ 0, and e 2 >0;
  • Fig. 5(d) shows the thermal deformation postures (7)-(10) under the condition of e 1 < 0 and e 2 < 0.
  • Figure 6 is a flow chart showing the modeling and compensation of the radial thermal drift error of the spindle.
  • Figure 7 is a plot of the error and temperature of the spindle at different speeds
  • Figure 7 (a) is the spindle error value in the X direction
  • Fig. 7(b) shows the temperature values on the left and right sides of the headstock.
  • Figure 8 is a graph showing the relationship between the thermal tilt angle of the spindle and the temperature difference.
  • Fig. 9 is a diagram showing a state of thermal deformation state at 4000 rpm.
  • Figure 10 is a simulation effect diagram of the spindle at each speed
  • Figure 10 (a) is an effect diagram at 2000 rpm
  • Figure 10 (b) is an effect diagram at 3000 rpm
  • Fig. 10 (c) is an effect diagram at 4000 rpm.
  • Figure 11 is a graph of data before and after compensation of the spindle at 4000 rpm.
  • Figure 12 is a graph of data before and after compensation of the spindle at 3500 rpm.
  • the X-axis saddle of the horizontal CNC lathe has an inclination angle of 60°.
  • the mechanical spindle 1 is horizontally mounted on the bed and driven by a belt with a maximum speed of 5000 rpm.
  • the distance between the two side faces of the headstock 2 is 356 mm.
  • the distance from the right side of the headstock 2 to the left displacement sensor 7 during the test is 251 mm, and the distance between the left displacement sensor 7 and the right displacement sensor 8 is 76.2 mm.
  • the first step is the radial thermal drift error and key point temperature test of the CNC lathe spindle.
  • the temperature T 1 and T 2 of the left and right sides of the headstock 2 are respectively tested by two temperature sensors (Fig. 1), and the spindle 1 is tested by two displacement sensors respectively.
  • the spindle 1 was first rotated at 4000 rpm for 4 hours, then the spindle 1 was stopped for 3 hours, and error and temperature data were collected. In the same way, the error and temperature data of the spindle at 3000 rpm and 2000 rpm were collected.
  • the second step is the correlation analysis between the thermal inclination angle of the main shaft and the temperature difference.
  • the correlation between the two is calculated according to the equation (4).
  • Thermal dip at 4000, 3000 and 2000 rpm The correlation coefficients with the temperature difference ⁇ T are: 0.898, 0.940, and 0.992, respectively. It can be further seen from these results that the thermal dip at different speeds.
  • the correlation with the temperature difference ⁇ T is relatively strong, which fully demonstrates that the thermal tilt of the main shaft is mainly due to the temperature difference between the two sides of the headstock.
  • the third step is the radial thermal drift error model of the main shaft under different thermal deformation attitudes.
  • d crs (t) at any time t is calculated by the equation (7).
  • the thermal drift errors e 1,x (t) and e 2,x (t) of the main axis 1 in the X direction at any time t are calculated by the equations (8) and (9).
  • the thermal change amounts ⁇ l and ⁇ r of the left and right end faces of the headstock 2 are obtained by applying the error test values e 1, x, t and e 2, x, t of 4000 rpm.
  • the independent variables T 1 and T 2 , the dependent variables ⁇ l and ⁇ r are known, and their parameters are identified by least squares method. The parameters identified are shown in Table 1.
  • the fourth step is the analysis of the influence of the machine tool size on the model prediction results.
  • the reliability of the predicted residual error value less than 1 ⁇ m is calculated by the asymptotic integration method.
  • p r is approximately equal to 1, which means that the fluctuations of d spl and d ss have little effect on the prediction results. Therefore, although the d spl and d ss measured at the test site have errors, they do not affect the prediction accuracy of the model.
  • the fifth step the determination of the hot deformation posture of the main shaft and the model selection
  • the thermal deformation amounts ⁇ l , ⁇ r and d ⁇ of the two sides of the headstock 2 are used to determine the hot deformation posture of the spindle 1 which changes irregularly during the machining process.
  • d ⁇ is the distance from the intersection of the deformed main shaft 1 and the original initial state spindle 1 to the right side surface of the headstock 2 .
  • equation (22) the calculation formula of d ⁇ is calculated by equation (22).
  • FIG. 9 shows a thermal attitude switching diagram obtained by the spindle 1 at 4000 rpm according to the above-described determination criterion. It can be seen that the spindle is in the attitude (1) in the time range of 0 to 0.57 h; the spindle is in the attitude (5) in the time range of 0.58 to 0.71 h; the spindle is in the time range of 0.72 to 4.93 h. Attitude (7); The spindle is in the attitude (8) in the time range of 4.94 to 7h.
  • d wp be the distance from the workpiece to the end face of the chuck 9 and d s be the distance from the left displacement sensor 7 to the end face of the chuck 9.
  • Wp is calculated according to formula (23).
  • Figure 10 shows the simulation results for the spindle (1) at different speeds.
  • e 1, x, t represents e 1
  • c represents e 1
  • r represents e 1
  • E represents T 2
  • test the value of x, E 2 represents a C
  • calculates the value of x, e 2, x, r represents 2, E simulation residual values of x.

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Abstract

一种卧式数控车床的主轴(1)径向热漂移误差建模及补偿方法。先测试数控车床主轴(1)沿径向的两点热漂移误差及对应的关键点温度；再基于主轴(1)的热倾斜变形机理获取主轴的热倾角，并分析热倾角与主轴箱(2)左右两侧温度差的相关性。根据被测两点的热漂移误差的正负及主轴箱(2)左右侧伸长或缩短的情况，将主轴(1)热变形情况进行分类并建立各种热变形姿态下的热漂移误差模型。然后分析机床结构尺寸对模型预测结果的影响。在实时补偿时，根据关键点的温度自动判断主轴(1)的热变形姿态，并自动选择相应的热漂移误差模型对主轴(1)进行补偿。该方法实现加工过程中数控车床主轴(1)热变形姿态的判别，并根据热变形机理实现对主轴(1)径向热漂移误差的预测。

Description

一种卧式数控车床的主轴径向热漂移误差建模及补偿方法 技术领域
本发明属于数控机床误差补偿技术领域，具体为一种卧式数控车床的主轴径向热漂移误差建模及补偿方法。
背景技术
机床的热误差是困扰机床行业几十年的难题。由于机床热误差的存在，导致的问题在于：单件的加工精度不合格；批量加工零件的一致性差、废品率高；为了减少热误差，机床开机后需要热机，能耗损失大；若对工件的加工精度要求高，还需要建立恒温车间。这些问题说明热误差对机床造成了众多不良影响。
目前减小机床热误差的方法主要有两种：误差防止法和误差补偿法。误差防止法是通过设计和制造方式消除或减少机床的热源，但是最大的缺点是成本高。当机床精度达到一定程度后，提高机床精度所花费的成本呈指数型增长。而热误差补偿技术作为一种提高数控机床精度稳定性的方法有很多优点，如相对低的成本、应用范围广等。
数控机床的热误差主要包括进给轴热误差和主轴热误差两部分。进给轴的热误差可以通过光栅尺闭环反馈的方式得到极大地降低，但是主轴热误差却缺乏有效的抑制手段。主轴热误差包括轴向热伸长误差和径向热漂移误差。目前学者们对主轴轴向热伸长误差补偿的研究开展得比较多，并尝试了多元回归法、神经网络法、热模态法、时间序列法和支持向量机等多种建模方法。但是，对于主轴径向热漂移误差建模和补偿的研究却非常少。唯一比较相似的是T.J.Ko在《Particular behavior of spindle thermal deformation by thermal bending》中分析了由立式加工中心主轴系统的热梯度引起的热弯曲变形并建立了主轴径向的热误差预测模型，但是未对主轴进行补偿，更没有给出主轴热变形姿态的判定准 则以及机床结构尺寸对模型预测结果的影响分析。
然而，数控车床的主轴径向热误差是非常重要的，因为相对于Z轴精度来说，人们更关心车床的X向精度。本发明针对数控车床主轴径向热误差补偿的难题，提出一种卧式数控车床主轴的径向热漂移误差建模及补偿方法。
发明内容
本发明的目的为提供一种有效的卧式数控车床主轴的径向热漂移误差建模及补偿方法，解决数控车床主轴径向热误差补偿的难题。
为解决上述技术问题，本发明的技术方案为：首先测试数控车床主轴沿径向的两点热漂移误差及对应的关键点温度。然后，基于主轴的热倾斜变形机理获取主轴的热倾角，并采用相关性分析方法分析热倾角与主轴箱左右两侧温度差的相关性。根据被测两点的热漂移误差的正负及主轴箱左右侧伸长或缩短的情况，将主轴热变形情况进行分类并建立各种热变形姿态下的热漂移误差模型。然后，采用渐近积分法分析机床结构尺寸对模型预测结果的影响。在实时补偿时，根据关键点的温度自动判断主轴的热变形姿态，并自动选择相应的热漂移误差模型对主轴进行补偿。
本发明的技术方案：
一种卧式数控车床的主轴径向热漂移误差建模及补偿方法，步骤如下：
第一步，数控车床主轴径向热漂移误差和关键点温度测试
在数控车床主轴1的径向热漂移误差和温度测试时，采用2个温度传感器分别测试主轴箱2左右两侧的温度T1和T2，采用2个位移传感器分别测试主轴1夹持的检棒5的两个位置点沿X向的误差；测试时先让主轴1以某一转速转动几个小时(如4小时)而升温，然后让主轴1停止转动几个小时(如3小时)而降温；
主轴1沿竖直方向的热误差ei产生X向的热误差分量ei,x，主轴1沿X向的热误差e1,x和e2,x按如下公式计算：
e2,x＝sin(αxdir)×e2 (1)
e1,x＝sin(αxdir)×e1 (2)
式中，αxdir为车床X轴的倾斜角度；i＝1或2，1表示右侧，2表示左侧；
第二步，主轴热倾角与温度差的相关性分析
主轴1受热后的热倾角通过如下公式计算：
Figure PCTCN2017109492-appb-000001
式中，
Figure PCTCN2017109492-appb-000002
为主轴1的热倾角，Lsnr为两个误差测点之间的距离；
确定主轴1的热倾角
Figure PCTCN2017109492-appb-000003
与两个温度之差ΔT之间的关系图，ΔT＝T1-T2，分析两个曲线的相似程度；进一步地，按照如下公式计算两者的相关性：
Figure PCTCN2017109492-appb-000004
式中，R为
Figure PCTCN2017109492-appb-000005
与ΔT的相关性矩阵，
Figure PCTCN2017109492-appb-000006
为
Figure PCTCN2017109492-appb-000007
和ΔT之间的协方差矩阵；
第三步，不同热变形姿态下的主轴径向热漂移误差模型
根据两个误差数据e1,x和e2,x的正负号及主轴箱2左右侧伸长或缩短的情况，将主轴1的热变形情况分为3大类、10小种；设δl为主轴箱2左侧面的热变化量，δr为主轴箱2右侧面的热变化量，δl和δr都是热膨胀时为正，收缩时为负；dcrs为变形后的主轴1与初始状态的主轴1的交点到主轴箱2右侧面的距离，dspl为主轴箱2的左右端面的距离，dss为主轴箱2右端面与左侧位移传感器7沿水平方向的距离，dsnr为左侧位移传感器7和右侧位移传感器8沿水平方向的距离；假设δl>δr≥0且检棒5靠近左侧位移传感器7和右侧位移传感器8的热变形姿态，建立主轴1的径向热漂移误差与温度之间的关系；通过式(5)和(6)表征主 轴箱2左右两侧的热膨胀量与温度的线性关系：
δl(t)＝ζl1×(T1(t)-T1(0))+ζl2 (5)
δr(t)＝ζr1×(T2(t)-T2(0))+ζr2 (6)
其中，ζl1，ζl2，ζr1和ζr2为待辨识系数；
对于δl>δr≥0且检棒5靠近左侧位移传感器7和右侧位移传感器8的热变形姿态，任意时刻t的dcrs(t)通过式(7)计算：
Figure PCTCN2017109492-appb-000008
任意时刻t主轴1沿X方向的热漂移误差e1,x(t)和e2,x(t)通过式(8)和(9)计算：
Figure PCTCN2017109492-appb-000009
Figure PCTCN2017109492-appb-000010
第四步，机床结构尺寸对模型预测结果的影响分析
分析热漂移误差模型中dspl和dss的测量偏差对模型预测结果的影响，采用渐近积分法分析预测残差的波动值属于某一允许的偏差范围的可靠度；针对该问题的功能函数Z的表达式描述为：
Z＝gx(X)＝δ-δa(dspl,dss) (10)
式中，X为dspl和dss组成的随机向量，δ为允许的偏差指标，δa为预测残差的波动值且定义为：
Figure PCTCN2017109492-appb-000011
式中，R为dspl和dss作为随机变量时的预测残差，是dspl和dss的函数，Rn为dspl和dss为真实值时模型的预测残差，N为热误差测试时的采样点数；
设X的联合概率密度函数为fx(x)，预测残差的波动值不属于某一允许偏差范围的概率按式(12)计算：
pf＝∫gx(x)≤0exp[h(x)]dx (12)
式中，h(x)＝lnfx(x)；
设
Figure PCTCN2017109492-appb-000012
为极限状态面上的一点，在该点将h(x)展开成Taylor级数并取至二次项：
Figure PCTCN2017109492-appb-000013
式中，
υ＝▽h(x*) (14)
B＝-[▽2h(x*)]-1 (15)
将极限状态面Z＝gx(X)＝0以点x*处的超平面代替，以实现对预测残差波动值超出允许范围概率的渐近积分；采用一次二阶矩法按式(16)计算预测残差的波动值属于某一允许偏差范围的可靠性指标：
Figure PCTCN2017109492-appb-000014
采用一次二阶矩法按式(17)计算预测残差的波动值属于某一允许偏差范围的失效概率指标：
Figure PCTCN2017109492-appb-000015
根据解最优化问题的Lagrange乘子法，引入乘子λ，由泛函L(x,λ)＝h(x)+λgx(x)的驻值条件之一
Figure PCTCN2017109492-appb-000016
得到
Figure PCTCN2017109492-appb-000017
将式(18)代入式(16)，得
Figure PCTCN2017109492-appb-000018
将式(19)代入式(17)，得
Figure PCTCN2017109492-appb-000019
采用渐近积分法计算预测残差的波动值属于某一允许的偏差范围的可靠度根据式(21)得到：
pr＝1-pfL (21)
第五步，主轴热变形姿态的判定及模型选择
采用主轴箱2两侧面的热变形量δl、δr和dσ判定加工过程中主轴1无规律变化的热变形姿态。其中，dσ为变形后的主轴1与原初始状态主轴1的交点到主轴箱2右侧面的距离；在各种热变形姿态下，dσ的计算公式均通过式(22)计算：
Figure PCTCN2017109492-appb-000020
对10种主轴热变形姿态的判定准则设定为：
姿态(1)：δl>δr≥0,dσ≤dss
姿态(2)：δr<0<δl
姿态(3)：δr≤δl<0
姿态(4)：δl<δr<0,dss+dsnr<dσ
姿态(5)：δl>δr≥0,dss<dσ≤dss+dsnr
姿态(6)：δl<δr<0,dss<dσ≤dss+dsnr
姿态(7)：δl>δr≥0,dss+dsnr<dσ
姿态(8)：δr≥δl≥0
姿态(9)：δl<0<δr
姿态(10)：δl<δr<0,dσ≤dss
最后，考虑到主轴1的热倾斜，对不同长度的工件补偿不同的误差；设dwp为工件被加工点到卡盘9端面的距离，ds为左侧位移传感器7到卡盘9端面的距离；在各种热变形姿态下，无论dwp<ds、ds<dwp<ds+dsnr还是dwp>ds+dsnr，工件被加工点的热误差补偿量ewp均按照式(23)计算：
Figure PCTCN2017109492-appb-000021
将热误差的预测值ewp实时输入到机床的数控系统中，实现对数控车床主轴在任意位置和时间的热误差补偿。
本发明的有益效果：
(1)为卧式数控车床主轴的径向热误差补偿提供一种新方法，解决卧式数控车床径向热误差补偿的难题。
(2)提高数控车床主轴的精度稳定性。
(3)解决批量盘轴类零件加工时一致性差的问题，降低废品率。
(4)只需要在某一主轴转速下一次性采集误差和温度数据，试验过程简单、快速；
(5)热误差预测模型基于主轴的热倾斜变形机理而建立，模型的鲁棒性强；
(6)热误差预测模型考虑了理论分析得到的10种热变形姿态，因此模型适合于任意主轴转速变化和环境温度变化的情况；
(7)模型可自动根据主轴箱左右侧的温度判断主轴的热变形姿态，并采用相应的热误差模型。
附图说明
图1为主轴系统的结构和温度传感器布置图。
图2为误差测试仪器及其安装图。
图3为主轴径向热漂移误差的分解图。
图4为主轴在初始热平衡状态的示意图。
图5为数控车床的主轴热变形姿态图；
图5(a)为e1＞0，且e2＞0条件下，热变形姿态(1)；
图5(b)为e1＞0，且e2＞0条件下，热变形姿态(2)-(4)；
图5(c)为e1＞0，且e2＜0条件下，热变形姿态(5)；e1＜0，且e2＞0条件下，热变形姿态(6)；
图5(d)为e1＜0，且e2＜0条件下，热变形姿态(7)-(10)。
图6为主轴径向热漂移误差建模及补偿的流程图。
图7为主轴在不同转速下的误差和温度图；
图7(a)为X向的主轴误差值；
图7(b)为主轴箱左右侧的温度值。
图8为主轴热倾角与温度差的关系图。
图9为4000转时的热变形状态切换图。
图10为主轴在各转速的仿真效果图；
图10(a)为在2000rpm条件下的效果图；
图10(b)为在3000rpm条件下的效果图；
图10(c)为在4000rpm条件下的效果图。
图11为对主轴在4000rpm的补偿前后的数据图。
图12为对主轴在3500rpm的补偿前后的数据图。
图中：1主轴；2主轴箱；3左侧温度传感器；4右侧温度传感器；5检棒；
6位移传感器支架；7左侧位移传感器；8右侧位移传感器；9卡盘。
具体实施方式
为了使本发明的目的、技术方案和优点更加清晰明了，下面结合主轴径向热漂移误差的测试、建模及补偿的具体实施方式并参照附图，对本发明作详细说明。本实施例以本发明的技术方案为前提给出了详细的实施方式和具体的操 作过程，但本发明的保护范围不限于下述的实施例。
卧式数控车床的X轴床鞍的倾斜角度为60°，机械式主轴1水平安装在床身上，通过皮带进行传动，最高转速为5000rpm。主轴箱2两侧面的距离为356mm，试验时主轴箱2右侧面到左侧位移传感器7的距离为251mm，左侧位移传感器7和右侧位移传感器8间的距离为76.2mm。
实施的具体步骤如下：
第一步，数控车床主轴径向热漂移误差和关键点温度测试
在数控车床主轴1的径向热漂移误差和温度测试时，采用2个温度传感器分别测试主轴箱2左右两侧的温度T1和T2(图1)，采用2个位移传感器分别测试主轴1所夹持检棒5的两个位置点沿X向的误差(图2)。测试时先让主轴1以4000rpm转动4小时，然后让主轴1停止转动3小时，并采集误差和温度数据。采用相同的方式，采集主轴在3000rpm和2000rpm的误差和温度数据。
这样，得到主轴1在不同转速下升降温过程中被测两点的X向热漂移误差e1,x和e2,x，以及主轴箱2左右两侧的温度(T1和T2)，见图7。
第二步，主轴热倾角与温度差的相关性分析
根据式(3)计算主轴1的热倾角，并绘制主轴1在不同转速下的热倾角
Figure PCTCN2017109492-appb-000022
与温度差(ΔT＝T1-T2)之间的关系图(图8)。可以看出，在不同转速下两者的相关性都比较强。
进一步地，按照式(4)计算两者的相关性。在4000、3000和2000rpm转速下，热倾角
Figure PCTCN2017109492-appb-000023
与温度差ΔT的相关系数分别为：0.898、0.940和0.992。通过这些结果进一步可以看出，在不同转速下热倾角
Figure PCTCN2017109492-appb-000024
与温度差ΔT的相关性都比较强，这充分说明主要是由于主轴箱两侧的温度差导致了主轴的热倾斜。
第三步，不同热变形姿态下的主轴径向热漂移误差模型
对主轴1所有可能出现的热变形姿态进行分析。根据两个误差数据e1,x和e2,x的正负号及主轴箱2左右侧伸长或缩短的情况，将主轴1的热变形情况分为3大类、10小种，如图5所示。以图5中的热变形姿态(1)为例，建立主轴1的径向热漂移误差与温度之间的关系。尽管主轴箱2左右两侧的温度都是不均匀的，但是它的温度场却是连续且近似线性变化的。因此，建立主轴箱2左右两侧的热膨胀量与温度的线性关系，通过温度表征主轴箱2两侧的热膨胀量的动态变化，其关系模型表示为式(5)和(6)。
对于图5(1)的热变形姿态，任意时刻t的dcrs(t)通过式(7)计算。
任意时刻t主轴1沿X方向的热漂移误差e1,x(t)和e2,x(t)通过式(8)和(9)计算。
图5中热变形姿态(2)～(10)的热误差与温度之间的模型参考图5(1)得到。
应用4000rpm的误差测试值e1,x,t和e2,x,t，反推得到主轴箱2左右两端面的热变化量δl和δr。这样，对于式(5)和(6)来说，其自变量T1和T2、因变量δl和δr均为已知，采用最小二乘法对其参数进行辨识。辨识得到的参数如表1所示。
表1.辨识得到的参数
Figure PCTCN2017109492-appb-000025
第四步，机床结构尺寸对模型预测结果的影响分析
对于该卧式数控车床，dspl＝356mm，dss＝251mm，dsnr＝76.2mm。设dspl和dss的测量值在一定范围内波动，且分别满足均值
Figure PCTCN2017109492-appb-000026
和
Figure PCTCN2017109492-appb-000027
方差
Figure PCTCN2017109492-appb-000028
和
Figure PCTCN2017109492-appb-000029
由于dspl和dss的测量值的分布类型未知，采用渐近积分法分析预测 残差的波动值小于1μm的可靠度。对于该问题，其功能函数Z的表达式定义为式(10)。
采用渐近积分法按照式(12)～(20)，计算得到预测残差的波动值小于1μm的可靠度为
Figure PCTCN2017109492-appb-000030
可以看出，pr近似等于1，这表示dspl和dss的波动对预测结果的影响很小。因此，尽管在试验现场测量的dspl和dss有误差，但是不影响模型的预测精度。
第五步，主轴热变形姿态的判定及模型选择
采用主轴箱2两侧面的热变形量δl、δr和dσ判定加工过程中主轴1无规律变化的热变形姿态。其中，dσ为变形后的主轴1与原初始状态主轴1的交点到主轴箱2右侧面的距离。在各种热变形姿态下，dσ的计算公式均通过式(22)计算。
根据10种主轴1热变形姿态的判定准则，图9给出了主轴1在4000rpm时根据上述判定准则得到的热姿态切换图。可以看出，在0～0.57h的时间范围内，主轴处于姿态(1)；在0.58～0.71h的时间范围内，主轴处于姿态(5)；在0.72～4.93h的时间范围内，主轴处于姿态(7)；在4.94～7h的时间范围内，主轴处于姿态(8)。
由于主轴(1)产生了热倾斜误差，因此对于不同长度工件的补偿量是有差别的。设dwp为工件被加工位置到卡盘9端面的距离，ds为左侧位移传感器7到卡盘9端面的距离。在图5所示的10种热变形姿态下，dwp<ds,ds<dwp<ds+dsnranddwp>ds+dsnr时，工件被加工位置的热误差补偿量ewp均按式(23)计算。
图10给出了对主轴(1)在不同转速的仿真结果。其中，e1,x,t表示e1,x的测试值，e1,x,c表示e1,x的计算值，e1,x,r表示e1,x的仿真残差值，e2,x,t表示e2,x的测试 值，e2,x,c表示e2,x的计算值，e2,x,r表示e2,x的仿真残差值。
在补偿和未补偿状态下，在卧式数控车床上分别以4000rpm和3500rpm再次执行试验，同时用两个温度传感器和两个位移传感器采集主轴1的温度和热误差。补偿前后的对比结果如图11和12所示。
应该说明的是，本发明的上述具体实施方式仅用于示例性阐述本发明的原理和流程，不构成对本发明的限制。因此，在不偏离本发明精神和范围的情况下所做的任何修改和等同替换，均应包含在本发明的保护范围内。

Claims (1)

  1. 一种卧式数控车床的主轴径向热漂移误差建模及补偿方法，其特征在于，步骤如下：
    第一步，数控车床主轴径向热漂移误差和关键点温度测试
    在数控车床主轴(1)的径向热漂移误差和温度测试时，采用2个温度传感器分别测试主轴箱(2)左右两侧的温度T1和T2，采用2个位移传感器分别测试主轴(1)夹持的检棒(5)的两个位置点沿X向的误差；测试时先让主轴(1)以某一转速转动升温，然后让主轴(1停止转动而降温；
    主轴(1)沿竖直方向的热误差ei产生X向的热误差分量ei,x，主轴(1)沿X向的热误差e1,x和e2,x按如下公式计算：
    e2,x＝sin(αxdir)×e2  (1)
    e1,x＝sin(αxdir)×e1  (2)
    式中，αxdir为车床X轴的倾斜角度；i＝1或2，1表示右侧，2表示左侧；
    第二步，主轴热倾角与温度差的相关性分析
    主轴(1)受热后的热倾角通过如下公式计算：
    Figure PCTCN2017109492-appb-100001
    式中，
    Figure PCTCN2017109492-appb-100002
    为主轴(1)的热倾角，Lsnr为两个误差测点之间的距离；
    确定主轴(1)的热倾角
    Figure PCTCN2017109492-appb-100003
    与两个温度之差ΔT之间的关系图，ΔT＝T1-T2，分析两个曲线的相似程度；进一步地，按照如下公式计算两者的相关性：
    Figure PCTCN2017109492-appb-100004
    式中，R为
    Figure PCTCN2017109492-appb-100005
    与ΔT的相关性矩阵，
    Figure PCTCN2017109492-appb-100006
    为
    Figure PCTCN2017109492-appb-100007
    和ΔT之间的协方差矩阵；
    第三步，不同热变形姿态下的主轴径向热漂移误差模型
    根据两个误差数据e1,x和e2,x的正负号及主轴箱(2)左右侧伸长或缩短的情况， 将主轴(1)的热变形情况分为3大类、10小种；设δl为主轴箱(2)左侧面的热变化量，δr为主轴箱(2)右侧面的热变化量，δl和δr都是热膨胀时为正，收缩时为负；dcrs为变形后的主轴(1)与初始状态的主轴(1)的交点到主轴箱(2)右侧面的距离，dspl为主轴箱(2)的左右端面的距离，dss为主轴箱(2)右端面与左侧位移传感器(7)沿水平方向的距离，dsnr为左侧位移传感器(7)和右侧位移传感器(8)沿水平方向的距离；假设δl>δr≥0且检棒(5)靠近左侧位移传感器(7)和右侧位移传感器(8)的热变形姿态，建立主轴(1)的径向热漂移误差与温度之间的关系；通过式(5)和(6)表征主轴箱(2)左右两侧的热膨胀量与温度的线性关系：
    δl(t)＝ζl1×(T1(t)-T1(0))+ζl2  (5)
    δr(t)＝ζr1×(T2(t)-T2(0))+ζr2  (6)
    其中，ζl1，ζl2，ζr1和ζr2为待辨识系数；
    对于δl>δr≥0且检棒(5)靠近左侧位移传感器(7)和右侧位移传感器(8)的热变形姿态，任意时刻t的dcrs(t)通过式(7)计算：
    Figure PCTCN2017109492-appb-100008
    任意时刻t主轴(1)沿X方向的热漂移误差e1,x(t)和e2,x(t)通过式(8)和(9)计算：
    Figure PCTCN2017109492-appb-100009
    Figure PCTCN2017109492-appb-100010
    第四步，机床结构尺寸对模型预测结果的影响分析
    分析主轴径向热漂移误差模型中dspl和dss的测量偏差对模型预测结果的影响，采用渐近积分法分析预测残差的波动值属于某允许的偏差范围的可靠度；针对该问题的功能函数Z的表达式描述为：
    Z＝gx(X)＝δ-δa(dspl,dss)  (10)
    式中，X为dspl和dss组成的随机向量，δ为允许的偏差指标，δa为预测残差的波动值且定义为：
    Figure PCTCN2017109492-appb-100011
    式中，R为dspl和dss作为随机变量时的预测残差，是dspl和dss的函数；Rn为dspl和dss为真实值时模型的预测残差；N为热误差测试时的采样点数；
    设X的联合概率密度函数为fx(x)，预测残差的波动值不属于某允许偏差范围的概率按式(12)计算：
    pf＝∫gx(x)≤0exp[h(x)]dx  (12)
    式中，h(x)＝lnfx(x)；
    设
    Figure PCTCN2017109492-appb-100012
    为极限状态面上的一点，在该点将h(x)展开成Taylor级数并取至二次项：
    Figure PCTCN2017109492-appb-100013
    式中，
    Figure PCTCN2017109492-appb-100014
    Figure PCTCN2017109492-appb-100015
    将极限状态面Z＝gx(X)＝0以点x*处的超平面代替，以实现对预测残差波动值超出允许范围概率的渐近积分；采用一次二阶矩法按式(16)计算预测残差的波动值属于某允许偏差范围的可靠性指标：
    Figure PCTCN2017109492-appb-100016
    采用一次二阶矩法按式(17)计算预测残差的波动值属于某允许偏差范围的失效概率指标：
    Figure PCTCN2017109492-appb-100017
    根据解最优化问题的Lagrange乘子法，引入乘子λ，由泛函L(x,λ)＝h(x)+λgx(x)的驻值条件之一
    Figure PCTCN2017109492-appb-100018
    得到
    Figure PCTCN2017109492-appb-100019
    将式(18)代入式(16)，得
    Figure PCTCN2017109492-appb-100020
    将式(19)代入式(17)，得
    Figure PCTCN2017109492-appb-100021
    采用渐近积分法计算预测残差的波动值属于某允许的偏差范围的可靠度根据式(21)得到：
    pr＝1-pfL  (21)
    第五步，主轴热变形姿态的判定及模型选择
    采用主轴箱(2)两侧面的热变形量δl、δr和dσ判定加工过程中主轴(1)无规律变化的热变形姿态；其中，dσ为变形后的主轴(1)与原初始状态主轴(1)的交点到主轴箱(2)右侧面的距离；在各种热变形姿态下，dσ的计算公式均通过式(22)计算：
    Figure PCTCN2017109492-appb-100022
    对10种主轴热变形姿态的判定准则设定为：
    姿态(1)：δl>δr≥0,dσ≤dss
    姿态(2)：δr<0<δl
    姿态(3)：δr≤δl<0
    姿态(4)：δl<δr<0,dss+dsnr<dσ
    姿态(5)：δl>δr≥0,dss<dσ≤dss+dsnr
    姿态(6)：δl<δr<0,dss<dσ≤dss+dsnr
    姿态(7)：δl>δr≥0,dss+dsnr<dσ
    姿态(8)：δr≥δl≥0
    姿态(9)：δl<0<δr
    姿态(10)：δl<δr<0,dσ≤dss；
    最后，考虑到主轴(1)的热倾斜，对不同长度的工件补偿不同的误差；设dwp为工件被加工点到卡盘(9)端面的距离，ds为左侧位移传感器(7)到卡盘(9)端面的距离；在各种热变形姿态下，无论dwp<ds、ds<dwp<ds+dsnr还是dwp>ds+dsnr，工件被加工点的热误差补偿量ewp均按照式(23)计算：
    Figure PCTCN2017109492-appb-100023
    将热误差的预测值ewp实时输入到机床的数控系统中，实现对数控车床主轴在任意位置和时间的热误差补偿。
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