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The selection program and life analysis of linear rolling guide rail pair

the rolling linear guide rail pair has been used for about 20 years. As a new rolling functional component, it has been widely used in precision instruments, CNC machine tools and so on. The most important thing for the rolling linear guide pair is to understand its load performance. If the load distribution state is known, the static and dynamic load capacity, operation life and reliability of the rolling linear guide pair can be estimated; At the same time, when the guide rail pair system is installed and assembled on site, the allowable installation error of the guide rail pair, the accuracy and service life of the system can be predicted with the characteristics of wide measurement limitation, high accuracy and fast response, and the service life and reliability under these conditions, and the driving force required by the guide rail pair system can be determined as the design basis for the ball screw and other driving systems

1 load calculation of guide rail pair

1.1 relationship between force and moment

as shown in Figure 1, take two guide rail four slide worktables as an example, and set X, y, Z coordinate systems; The force component is in the plane perpendicular to the X axis, and there are five kinds of force and moment loads acting on the system

(1) FY: vertical load; (2) FZ: horizontal load; (3) MZ: overturning moment; (4) MX: swinging moment; (5) My: shaking moment; In order to simplify the analysis, the structures of the workbench, the guide rail and the sliding block, except the groove part, are regarded as rigid bodies. Set the coordinate origin at o

Figure 1 workbench system bearing force and moment

Figure 2 simplified schematic diagram of force

equivalent treatment method can be adopted for force load, and its principle is shown in Figure 2; There is no need to simplify the moment action

1.2 the calculation of slider reaction

is shown in Figure 1. Let K be the number of the slide block, and its reaction forces in the Y-axis and z-axis directions are: fryk, frzk, as shown in equations (1) - (8)

(1) slider k=1

(1)

(2)

(2) slider k=2

(3)

2. Making electronic and electrical parts

(4)

(3) slider k=3

(5)

(6)

(4) slider k=4

(7)

(8)

1.3 displacement calculation of workbench

the displacement form of workbench is shown in Figure 3. The action corresponding to force and moment can be divided into the following five components, namely:

(1) 1=displacement in the y-axis direction; (2) 2= overturning angle; (3) 3= swing angle; (4) 4 = displacement in z-axis direction; (5) 2= rocking angle; The displacement of any point m (x, y, z) on the workbench in the Y-axis and z-axis directions is set as y and Z, which can be expressed by the following formula:

y= 1+ 2x+ 3Z (9)

z= 4+ 5x- 3Y (10)

1.4 statically indeterminate sliding block reaction

in statically indeterminate sliding block, the displacement component corresponding to external load and moment has 1 ~ 5 as unknowns. Given appropriate initial values, the elastic deformation and load of each steel ball in each sliding block can be obtained by numerical method

in order to improve the motion accuracy of about 30 provincial-level new material R & D and utilization demonstration projects that are leading in the domestic industry, and to use as many effective steel balls to bear the load as possible, transition curves with radius R are designed at both ends of the slider groove, as shown in Figure 4. Therefore, the influence of transition curve on load and elastic deformation must be considered. The width XR, C ［ 3 ］ shall be selected according to the following calculation:

xr=3da; c=0.002Da；

Figure 3 table displacement form

Figure 4 groove surface and steel ball matching state

the clearance x given to the steel ball at different points on the transition curve is different. With reference to literature [3], it can be calculated according to the following formula:

x=r (1-cos)

｜ XZ ｜ UX XR ｜ (11)

Figure 5 shows the state of elastic deformation ijk and distributed load pijk of K sliding block, j-row groove and steel ball I on one side of the guide rail. When there is no external load acting on the workbench, the slider and ball are represented by dotted lines, and the curvature center points of the guide rail and slider groove and the center points of the steel ball are represented by AR, Ag and O respectively; The guide rail is regarded as unable to move, so the workbench moves y and Z according to formula (9) (10), the point Ag moves to Ag, the initial contact angle becomes ijk, and the elastic deformation ijk of the steel ball can be expressed as follows:

then the elastic deformation on the I steel ball of the j-th row groove of the k-th slider can be expressed as follows:

(12)

where: is the preload, i.e. the interference, f is the groove curvature ratio, f=r/da, X is determined by formula (11), vy VX is the projection value of the distance between the curvature centers of the slider groove arc and the guide groove arc in the Y and Z directions after bearing

cb is the Hertz coefficient, which can be calculated by reference [1]. The contact angle of the steel ball is ijk, which can be obtained from the Hertz elastic contact theory and figure 5:

pijk=cb ijk3/2 (13)

(14)

taking the whole sliding block workbench as the research object, for the force and torque balance conditions at the origin o, the following equation can be obtained, namely

(15)

(16)

(17)

(18)

.(19)

where FJK is: for guide rail 1; F1k=F2k=-1; F3k=F4k=1; For guide rail 2:f1k=f2k=1; F3k=F4k=-1; Aijk is the arm length from the origin to the action point of pijk, aijk=zrsin ijk yrcos continues to apply ijk with axial deformation, where Zr and yr are the curvature centers of the guide rail groove; According to the above theory, five displacements 1 ~ 5 are taken as unknowns, and the nonlinear equations can be solved by Newton rupson method from equations (15) ~ (19)

2 life and reliability

2.1 rated life and reliability analysis

Figure 5 groove, steel ball elastic deformation and load distribution

Figure 6 K deformation of a steel ball on the sliding block

(1) life distribution

the life distribution of the rolling linear guide is the relationship between the cumulative damage rate f (L) and the life value L when a group of devices operate under the same condition. According to the relevant literature [2], the life distribution is Weibull distribution, that is,

(20)

where l> 0, M> 0,> and the specific meanings of the three parameters are as follows:

m: shape parameter or Weibull slope Dimension parameters;: Location parameters or minimum life

m value: for steel ball, m=10/9; For rollers m=9/8 or 3/2

dimension parameter: it has the feature that when the life value L is, the distribution function f (L) =0.63

between the rated dynamic load C and the load f acting on the device, P is taken as the acceleration index of the load, which has the following relationship:

(21)

the position parameter is the minimum life of the device without peeling damage, which is generally about 5% of the 90% rated life L10 of the rolling bearing. The specific value has no literature to refer to, so =0 can be generally treated; However, the life test of the specimen is related to the size of the load, so the value must be set; Therefore, the three parameter Weibull =0 is included for life analysis. Figure 7 shows the relationship between life value L and probability density function f (L) and the relationship between cumulative distribution function f (L) and reliability R (L). These graphs move equally where l=0, i.e. =0

Figure 7 probability density function, cumulative distribution function, reliability

(2) when the rated life

the radial load acting on the device is f, the 90% existence rate, that is, the reliability is 90% of the residual life value L10 of the device, expressed by the following formula:

where: p=3 unit: 50km (the rolling body is a steel ball)

p=10/3 unit: 100km (the rolling body is a roller)

damage rate (f (L) is n%, which is determined by formula (20), (21) (22) the life value of any reliability R (L) =1-f (L) =1-n%:

(23)

(3) LG system life

the LG system device shown in Figure 1 can calculate the life ln of the system by probability, ln (k) is the life of each slider

ln=ln (1) -m+ln (2) -m+ (24)

2 calculation example

example 1: the rated dynamic load is c=3800kgf when there is a radial load of 750kgf acting on the guide rail pair, calculate the rated life and 50% life. At this time, the minimum life is unknown, Assume 0

solution: from equations (22) and (23) and taking m=10/9 and p=3, we can get

from this example, under the same working conditions, the life value of 50% reliability is 5.45 times that of 90% reliability

example 2: calculate 90% life of LG system as shown in Figure 1, where the external load acts on the center of slider k=1. And the work is not impacted and the installation error is very small. 1x=1z=125mm, radial force fy=1000kgf

solution: calculate the reaction force of each sliding block of statically indeterminate system by using the program. The results are as follows:

fry1=739kgf, fry2=255kgf, fry3=-253kgf, fry4=249kgf

Frz1=-8.8kgf,Frz2=-6.5kgf,Frz3=6.3kgf,Frz4=9.1kgf;

therefore, it can be seen that the load distribution on each sliding block is uneven and there are several reaction forces in the z-axis direction. Therefore, it is necessary to consider the safety factor [3] st=1 ~ 2, which is taken as 1.2 at this time

calculation of rated dynamic load c1=c2=c3=c4=3800/1.2=3167kgf

90% rated life of each slider is

l10 (2) = (3167/255) 3=1916 50=9784km

l10 (3) = (3167/253) 3=1961 50=78074km

from equation (24) and taking m=10/9, the overall service life of the system at this time is

l10= (1.0976) -0.9=3660km

therefore, when there are 100 sets of this system, Ten of them have reached the stripping life, and the running distance is 3660km. The life value smaller than the 3935km life value of the smallest slider k=1 in the system is predictable

3 conclusion

the load and life calculation of the rolling linear guide pair are derived comprehensively, which can provide a theoretical basis for the more in-depth and comprehensive study of the rolling linear guide

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