Method for determining the preload value of the screw based on thermal error and temperature rise weighting

Abstract

A method for determining the preload value of the screw based on thermal error and temperature rise weighting. Firstly, thermal behavior test of the feed shaft under typical working conditions is carried out to obtain the maximum thermal error and the temperature rise at the key measuring points in each preloaded state. Then, a mathematical model of the preload value of the screw and the maximum thermal error is established; meanwhile, another mathematical model of the preload value of the screw and the temperature rise at the key measuring points is also established. Finally, the optimal preload value of the screw is obtained. The thermal error of the feed shaft and the temperature rise of the moving components are comprehensively considered, improving the processing accuracy and accuracy stability of the machine tool, and ensuring the service life of the moving components such as bearings.

Claims

1. A method for determining a preload value of a screw based on thermal error and temperature rise weighting, wherein, firstly, under a different preload states of the screw, a thermal behavior test of a feed shaft under typical working conditions is carried out to obtain a maximum thermal error and a temperature rise at a key measuring points in each preload state; then, a mathematical model of the preload values of the screw and the maximum thermal error is established and, meanwhile, a mathematical model of the preload value of the screw and the temperature rise at the key measuring points is also established; finally, a optimal preload value of the screw is obtained by optimizing a weighting function of the maximum thermal error and the temperature rise at each measuring point as an objective function; specific steps are as follows: first step is a thermal behavior test of the feed shaft under typical working conditions; a first temperature sensor is located on a front bearing of a feed system, a second temperature sensor is located on a nut, a third temperature sensor is located on a rear bearing of the feed system, and a fourth temperature sensor is located on a bed near the screw; a motion trajectory of a machine tool is analyzed when machining workpieces, and a motion information of the feed shaft is extracted, including a travel range, a feed speed and a running frequency; in a different preload states of the screw, the preload value of the screw is measured by a preload angle of the preload nut, and a thermal behavior test of the feed shaft is performed: in an initial thermal steady state, a full-range positioning error of the feed shaft is measured by a laser interferometer, and a temperature value of the first temperature sensor, the second temperature sensor, the third temperature sensor and the fourth temperature sensor are recorded; the feed shaft is heated under a motion process, and the full-range positioning error of the feed shaft and a temperature of each measuring point under the motion process is tested regularly, that is a testing process and it stop until the screw reaches thermal balance; second step is to calculate the maximum thermal error of the feed shaft and the temperature rise at the key measuring points; based on the temperature value collected in the first step, the maximum thermal error of the feed shaft is calculated according to equation (1) for each preload condition: thermal error of the feed shaft is calculated according to equation (1) for each preload condition:
E.sub.max_i=E.sub.i(M.sub.i,N)−E.sub.i(1,N)  (1) where: E.sub.max_i is the maximum thermal error when an ith preload value is used; M.sub.i is a number of positioning error tests when the ith preload value is used; N is a number of points for the positioning error test; E.sub.i(M.sub.i, N) is a Nth point data of a M.sub.ith positioning error test when the ith preload value is used; E.sub.i(1,N) is the Nth point data of a first positioning error test when the ith preload value is used; the temperature rise of each measuring point under each preload value is calculated according to equation (2):
ΔT.sub.i,j=[T.sub.i,j(M.sub.i)−T.sub.i,j(1)]−[T.sub.i,4(M.sub.i)−T.sub.i,4(1)]  (2) wherein: ΔT.sub.i,j is the temperature rise of a jth temperature sensor when an ith preload value is used; T.sub.i,j(M.sub.i) is a M.sub.ith measurement value of the jth temperature sensor when the ith preload value is used; T.sub.i,j(1) is a first measurement value of the jth temperature sensor when the ith preload value is used; T.sub.i, 4(M.sub.i) is the M.sub.ith measurement value of the fourth temperature sensor when the ith preload value is used, and T.sub.i,4 (1) is a first measurement value of the fourth temperature sensor when the ith preload value is used; third step is to establish a mathematical model of the preload values of the screw and the maximum thermal error and another mathematical model of the preload values of the screw and the temperature rise at the key measuring points; a relationship between the preload value of the screw and the maximum thermal error of the feed shaft is as shown in equation (3):
E.sub.max=a.sub.0−a.sub.1×A  (3) where: E.sub.max is the maximum thermal error of the feed shaft, A is the preload value of the screw, that is, a locking angle of the preload nut, and a.sub.0 and a.sub.1 are coefficients; a mathematical model of the preload value of the screw and the temperature rise of the jth temperature sensor is shown in equation (4): $\begin{array}{cc}\Delta T_j=b_{j,0}+b_{j,1}\times e^{\left(b_{j,2}\times A\right)}& \left(4\right)\end{array}$ where: ΔT.sub.j is the temperature rise of the jth temperature sensor, and b.sub.j,0, b.sub.j,1, and b.sub.j,2 are coefficients; according to the maximum thermal error and temperature rise data of the screw under different preload values obtained in the second step, the coefficients a.sub.0, a.sub.1, b.sub.j,0, b.sub.j,1, and b.sub.j,2 in formula (3) and formula (4) are identified by a least squares method; fourth step is to calculate the optimal preload value of the screw; the weighting functions of the maximum thermal error and temperature rise at key measuring points are shown in equation (5): $\begin{array}{cc}F\left(E_{\max },\Delta T_j\right)={\lambda }_0\times E_{\max }+\underset{j=1}{\overset{3}{.Math.}}{\lambda }_j\Delta T_j& \left(5\right)\end{array}$ where: λ.sub.0 is a weight coefficient of the maximum thermal error of the feed shaft, and λ.sub.j is a weight coefficient of the temperature rise of the jth temperature sensor; rewriting equation (5) according to equations (3) and (4) gives: $\begin{array}{cc}F\left(A\right)={\lambda }_0\times \left(a_0-a_1\times A\right)+\underset{j=1}{\overset{3}{.Math.}}{\lambda }_j\left(b_{j,0}+b_{j,1}\times e^{\left(b_{j,2}\times A\right)}\right)& \left(6\right)\end{array}$ through automatic optimization based on equation (7), the optimal preload value of the screw can be obtained; $\begin{array}{cc}\min \left[F\left(A\right)\right]=\min \left[{\lambda }_0\times \left(a_0-a_1\times A\right)+\underset{j=1}{\overset{3}{.Math.}}{\lambda }_j\left(b_{j,0}+b_{j,1}\times e^{\left(b_{j,2}\times A\right)}\right)\right]& \left(7\right)\\ A_{\min }\le A\le A_{\max }& \end{array}$ where: A.sub.min and A.sub.max are a lower and upper limits of the preload value A of the screw respectively in the automatic optimization process.

Description

DRAWINGS

(1) FIG. 1 shows a schematic diagram of the location of the temperature measuring points of the feed shaft.

(2) FIG. 2 shows the effect of modeling the maximum thermal error.

(3) FIG. 3(a) shows the effect of modeling the temperature rise of the first temperature sensor.

(4) FIG. 3(b) shows the effect of modeling the temperature rise of the second temperature sensor.

(5) FIG. 3(c) shows the effect of modeling the temperature rise of the third temperature sensor.

(6) In the figures: 1 feed shaft motor; 2 front bearings of the screw; 3 the first temperature sensor; 4 screw; 5 work table; 6 nut; 7 the second temperature sensor; 8 bed; 9 the fourth temperature sensor; 10 the third temperature sensor; 11 rear bearings; 12 preload nuts; 13 a laser interferometer.

DETAILED DESCRIPTION

(7) In order to make the objects, technical solutions, and the advantages of the present invention clearer, the present invention is described in detail with reference to the accompanying drawings.

(8) The embodiment of the present invention is described in detail, taking the X-axis of a vertical machining center as an example. The machining center has an X-axis travel range of 0˜−500 mm and the maximum feed speed is 32000 mm/min.

(9) The first step is the thermal behavior test of the feed shaft under typical working conditions.

(10) The first temperature sensor 3 is located on the front bearing 2 of the feed system, the second temperature sensor 7 is located on the nut 6, the third temperature sensor 10 is located on the rear bearing 11 of the feed system, and the fourth temperature sensor 9 is located on the bed 8 near the screw.

(11) The measured machining center is oriented towards the consumer electronics industry. The typical workpieces processed are aluminum casings for mobile phones and tablet computers. The typical working conditions are determined according to the processing process: the common travel range is −100˜−400 mm; the common feed speed is 2000 mm/min; the average machining time of a single workpiece is 90 s, and the machining interval of the workpieces is 15 s.

(12) The thermal behavior test of the feed shaft is carried out respectively under the conditions that the preload nuts have locking angles of 0°, 60°, 120°, 180° and 270°:

(13) In the initial thermal steady state of the feed shaft, the full-range positioning error of the feed shaft is tested by a laser interferometer, and the temperature values from the first temperature sensor 3, the second temperature sensor 7, the third temperature sensor 10, and the fourth temperature sensor 9 are recorded. Then, the feed shaft is heated under typical motion information. The heating engine program is shown in Table 1.

(14) TABLE-US-00001 TABLE 1 CNC program for heating engine AAA: G4F1 BBB: REPEAT BBBP = 14 G90 G1 X-400 F2000 G4F15 G4F1 goto AAA X-100 M30 Down to the second column Finished

(15) The full-range positioning error is tested every 15 minutes during the movement, and the temperature values of the first temperature sensor 3, the second temperature sensor 7, the third temperature sensor 10, and the fourth temperature sensor 9 are recorded. The heating engine process is run for 2 hours, when the feed shaft reaches thermal balance, the test is stopped.

(16) The second step is to calculate the maximum thermal error of the feed shaft and the temperature rise at the key measuring points.

(17) Based on the thermal error and the temperature data collected in the first step, the maximum thermal error of the feed shaft in each preload condition is calculated according to equation (1):
E.sub.max_i=E.sub.i(M.sub.i,N)−E.sub.i(1,N)  (1)
where: E.sub.max_i is the maximum thermal error when the ith preload value is used. M.sub.i is the number of positioning error tests when the ith preload value is used, and N is the number of points for the positioning error test. E.sub.i(M.sub.i,N) is the Nth point data of the M.sub.ith positioning error test when the ith preload value is used. E.sub.i(1,N) is the Nth point data of the first positioning error test when the ith preload value is used.

(18) The temperature rise for each measuring point under each preload value is calculated according to equation (2):
ΔT.sub.i,j=[T.sub.i,j(M.sub.i)−T.sub.i,j(1)]−[T.sub.i,4(M.sub.i)−T.sub.i,4(1)]  (2)
where: ΔT.sub.i,j is the temperature rise of the jth temperature sensor when the ith preload value is used. T.sub.i,j(M.sub.i) is the M.sub.ith measurement value of the jth temperature sensor when the ith preload value is used. T.sub.i,j(1) is the first measurement value of the jth temperature sensor when the ith preload value is used. T.sub.i,4(M.sub.i) is the M.sub.ith measurement value of the fourth temperature sensor 9 when the ith preload value is used. T.sub.i,4(1) is the first measurement value of the fourth temperature sensor 9 when the ith preload value is used.

(19) According to equation (1) and equation (2), the maximum thermal error and the temperature rise of each measuring point under each preload value of the screw are then calculated. The specific results are shown in Table 2.

(20) TABLE-US-00002 TABLE 2 Summary of the Maximum Thermal Error and Temperature Rise Data Temperature Temperature Temperature rise of rise of rise of Maximum the first the second the third Preload thermal temperature temperature temperature value/° error/μm sensor/° C. sensor/° C. sensor/° C. 0 29.6 3.24 2.89 2.76 60 27.2 3.35 2.97 2.98 120 20.6 3.47 3.11 3.36 180 13.5 4.01 2.94 3.97 270 8.8 5.23 3.05 4.98

(21) The third step is to establish one mathematical model of the preload values of the screw and the maximum thermal error and another mathematical model of the preload values of the screw and the temperature rise at the key measuring points.

(22) The relationship between the preload value of the screw and the maximum thermal error of the feed shaft is as shown in equation (3):
E.sub.max=a.sub.0−a.sub.1×A  (3)
where: E.sub.max is the maximum thermal error of the feed shaft, A is the preload value of the screw (i.e., the locking angle of the preload nut 12), and a.sub.0 and a.sub.1 are coefficients.

(23) The mathematical model of the preload values of the screw and the temperature rise of the jth temperature sensor is shown in equation (4):
ΔT.sub.1=b.sub.j,0+b.sub.j,1×e.sup.(b.sup.j,2.sup.×A)  (4)
where: ΔT.sub.j is the temperature rise of the jth temperature sensor, b.sub.j,0, b.sub.j,1 and b.sub.j,2 are coefficients.

(24) According to the maximum thermal error and the temperature rise data, under the different preload values of the screw obtained in the second step, and based on the least squares method, the coefficients in the model can be obtained according to equation (3) and equation (4). The coefficients are as follows: a.sub.0=30.418, a.sub.1=0.083, b.sub.1,0=3.073, b.sub.1,1=0.15, b.sub.1,2=0.010, b.sub.2,0=0.718, b.sub.2,1=2.220, b.sub.2=0.0002, b.sub.3,0=1.814, b.sub.3,1=0.912 and b.sub.3=0.005. The effect of modeling the maximum thermal error is shown in FIG. 2, and the effect of modeling the temperature rise of the first to the third temperature sensors is shown in FIG. 3(a)˜FIG. 3(c).

(25) The fourth step is to calculate the optimal preload value of the screw.

(26) The weighting function of the maximum thermal error and the temperature rise at the key measuring points is shown in equation (5):

(27) $\begin{array}{cc}F\left(E_{\max },\Delta T_j\right)={\lambda }_0\times E_{\max }+\underset{j=1}{\overset{3}{.Math.}}{\lambda }_j\Delta T_j& \left(5\right)\end{array}$
where: λ.sub.0 is the weight coefficient of the maximum thermal error of the feed shaft, and λ.sub.j is the weight coefficient of the temperature rise of the jth temperature sensor.

(28) Rewriting equation (5) according to equations (3) and (4) gives:

(29) $\begin{array}{cc}F\left(A\right)={\lambda }_0\times \left(a_0-a_1\times A\right)+\underset{j=1}{\overset{3}{.Math.}}{\lambda }_j\left(b_{j,0}+b_{j,1}\times e^{\left(b_{j,2}\times A\right)}\right)& \left(6\right)\end{array}$

(30) Considering the suppression effect on the thermal error and the control over the temperature rise, the weight coefficients in the equation (6) are set as: λ.sub.0=0.15, λ.sub.1=0.8, λ.sub.2=0.1, and λ.sub.3=0.8.

(31) Then automatic optimization is carried out based on equation (7):

(32) $\begin{array}{cc}\min \left[F\left(A\right)\right]=\min \left[{\lambda }_0\times \left(a_0-a_1\times A\right)+\underset{j=1}{\overset{3}{.Math.}}{\lambda }_j\left(b_{j,0}+b_{j,1}\times e^{\left(b_{j,2}\times A\right)}\right)\right]& \left(7\right)\\ A_{\min }\le A\le A_{\max }& \end{array}$

(33) The optimal preload value of the X-axis screw of the vertical machining center can be obtained as 156°.