Prediction method for reverse transmission failure of ball screw pair

Abstract

The present invention discloses a method for predicting reverse transmission failure of a ball screw pair. The present invention performs dynamic modeling on the ball motion under the reverse transmission working condition of the ball screw pair, combines the roller control theory, describes the sliding-rolling ratio formula between the ball and the two rollers under the reverse transmission working condition, and obtains the conditions satisfied by various parameters when the rolling-slip composite motion degenerates into pure sliding motion. It can more accurately describe the actual situation when it degenerates into pure sliding motion, and more accurately predict the reverse transmission failure of the ball screw pair. In actual engineering applications, in addition to taking corresponding measures to prevent reverse transmission and protect the ball screw pair, the present invention theoretically provides a method for predicting reverse transmission failure to avoid the occurrence of self-locking.

Description

Technical Field

[0001] The invention belongs to the technical field of research on the failure mechanism of a ball screw pair, and in particular to a method for predicting reverse transmission failure of a ball screw pair. Background Art

[0002] A ball screw reverse drive converts the linear motion of the nut into rotational motion of the screw through the rolling action of the balls. This method offers high efficiency, precision, load capacity, and long life. However, under reverse drive conditions, when subjected to heavy axial loads, the ball motion degenerates from a combined rolling-sliding motion to a pure sliding motion, affecting the accuracy and life of the ball screw and posing the risk of self-locking. This self-locking can damage the entire system, impacting overall equipment performance. Therefore, developing a failure prediction method for ball screw reverse drives is crucial for improving performance.

[0003] Currently, there is very little research on the reverse transmission of ball screw pairs. However, in actual engineering applications, reverse transmission is inevitable and reverse transmission failure should be avoided as much as possible. Summary of the Invention

[0004] The purpose of the present invention is to address the problems existing in the above-mentioned prior art and provide a method for predicting the failure of the reverse transmission of a ball screw pair. Theoretically, it provides a theoretical range of certain process parameters to avoid self-locking, which has a certain effect on the design and selection of ball screw transmission devices.

[0005] The technical solution to achieve the objectives of the present invention is: a method for predicting reverse transmission failure of a ball screw pair. In practical engineering applications, in addition to taking appropriate measures to prevent reverse transmission and protect the ball screw pair, a theoretical method for predicting reverse transmission failure is provided to avoid the occurrence of self-locking. The method includes the following steps:

[0006] Step 1: When the ball screw pair is in reverse transmission condition, for the case where the ball center is fixed in space and a pure rolling point appears on the contact ellipse, for the ball-nut contact, the linear velocity of the ball is equal to the linear velocity of the nut raceway, and the ball rotation angular velocity ω is obtained. R The relationship between the nut movement speed v0 and the nut relative to the ball rotation angular velocity ω0; for ball screw contact, the linear velocity of the ball is equal to the linear velocity of the screw raceway, and the ball rotation angular velocity ω is obtained R Angular velocity of the ball relative to the screw ω i relationship;

[0007] Step 2: For the case where the ball center is not fixed in space but the nut raceway is fixed, obtain the angular velocity ω of the ball revolution m The relationship between the angular velocity ω0 of the nut relative to the ball; based on the angular velocity ω of the ballm The component of the screw rotation angular velocity ω on the b-axis in the Frenet coordinate system is used to obtain the screw angular velocity ω relative to the ball i Ignoring the pivot motion caused by the gyroscopic torque, obtain ω at this time R The projection component ω on the three coordinate axes t, n, and b in the Frenet coordinate system t 、ω n 、ω b ;

[0008] Step 3: Let the radius of the pure rolling point on the contact ellipse where the ball contacts the nut be r0′ = the radius of the pure rolling point on the contact ellipse where the ball contacts the screw be r i ′ = ball radius r b Based on the above relationships, the ball rotation angular velocity ω is obtained R The angular velocity of the nut relative to the ball ω0, the angular velocity of the screw relative to the ball ω i The ratio of the ball's revolution angular velocity ω m The calculation formula of tanβ is obtained by using the outer ring control theory, where β is the angle between the U axis and its projection axis on the tb plane, which is called the rotation angle.

[0009] Step 4: Assume that the ball contacts the nut at point A and the lead screw at point B. Obtain the coordinates of the ball center O and the two contact points in the Frenet coordinate system.

[0010] Step 5: Obtain the position vector based on the coordinate transformation relationship between the fixed coordinate system and the Frenet coordinate system The expression of and its derivative to obtain the instantaneous velocity of the ball center The expression of the linear velocity of the contact points A and B on the ball and Based on the nut movement speed v0 and the screw rotation angular velocity ω, the linear velocity of the contact point A on the nut is obtained and the linear velocity of contact point B on the ball screw

[0011] Step 6: Change the linear velocity of contact point A on the ball and the linear velocity of contact point A on the nut Substituting into the relationship of the sliding-rolling ratio, the sliding-rolling ratio S of the contact point between the single ball and the nut raceway under the reverse transmission condition is obtained. n Similarly, the sliding-rolling ratio S of the contact point between the single ball and the screw raceway under reverse transmission conditions can be obtained s ;

[0012] Step 7: When the ball screw pair degenerates from rolling transmission to sliding transmission, the ball and the nut are relatively stationary, and the ball and the screw are in pure sliding motion. Let S n =0, S s =1, obtain the nut movement speed v0 and ball rotation angular velocity ω when the ball screw pair is in reverse transmission condition and degenerates from rolling transmission to sliding transmission R With the rotation angle β, contact angle α0, α i , and the helix angle λ, and the helix angle λ is used to determine whether the ball screw pair has reverse transmission failure.

[0013] Compared with the prior art, the present invention has the following significant advantages:

[0014] (1) By modeling the ball motion under the reverse transmission condition of the ball screw pair, the contact situation between the ball and the two raceways can be described more accurately. The relationship between the various motion parameters (ball rotation and revolution angular velocity, nut rotation angular velocity relative to the ball, screw rotation angular velocity relative to the ball) when the ball screw pair is in the reverse transmission condition can be obtained more intuitively and accurately, as well as their changes with process parameters (contact angle, rotation angle), which fills the gap in the industry's analysis of the reverse transmission motion of the ball screw pair.

[0015] (2) A calculation formula for the sliding-rolling ratio between the ball and the two rollers under reverse transmission conditions is proposed, which provides a reference for judging at which stage of the sliding-rolling motion the current movement of the ball and the two rollers is.

[0016] (3) Combined with the outer ring raceway control theory, the conditions that various parameters must satisfy when the rolling-sliding composite motion degenerates into pure sliding motion are obtained, which can more accurately describe the actual situation when it degenerates into pure sliding motion.

[0017] (4) Based on the above results, a dynamic model of the ball screw pair under reverse transmission working conditions was established, and the change of the slip-roll ratio when the reverse transmission fails was analyzed, which made the prediction of the reverse transmission failure of the ball screw pair more accurate.

[0018] The present invention is further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 This is a flow chart of the method for predicting reverse transmission failure of a ball screw pair according to the present invention.

[0020] Figure 2 1 is a diagram showing the position of the ball center in the rectangular coordinate system and the Frenet coordinate system in one embodiment.

[0021] Figure 3 Schematic diagram of the ball rotation axis U and the rotation angle in one embodiment.

[0022] Figure 4 Schematic diagram of the contact between the ball and the nut in one embodiment.

[0023] Figure 5 Schematic diagram of the contact between the ball and the screw in one embodiment.

[0024] Figure 6 Graph showing the predicted results of reverse transmission failure of a ball screw pair in one embodiment. DETAILED DESCRIPTION

[0025] In order to make the purpose and technical solution of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0026] In one embodiment, combined Figure 1 , provides a method for predicting reverse transmission failure of a ball screw pair, the method comprising the following steps:

[0027] Step 1: When the ball screw pair is in reverse transmission condition, for the case where the ball center is fixed in space and a pure rolling point appears on the contact ellipse, for the ball-nut contact, the linear velocity of the ball is equal to the linear velocity of the nut raceway, and the ball rotation angular velocity ω is obtained. R The relationship between the nut movement speed v0 and the nut relative to the ball rotation angular velocity ω0; for ball screw contact, the linear velocity of the ball is equal to the linear velocity of the screw raceway, and the ball rotation angular velocity ω is obtained R Angular velocity of the ball relative to the screw ω i relationship;

[0028] Step 2: For the case where the ball center is not fixed in space but the nut raceway is fixed, obtain the angular velocity ω of the ball revolution m The relationship between the angular velocity ω0 of the nut relative to the ball; based on the angular velocity ω of the ball m The component of the screw rotation angular velocity ω on the b-axis in the Frenet coordinate system is used to obtain the screw angular velocity ω relative to the ball i Ignoring the pivot motion caused by the gyroscopic torque, obtain ω at this time R The projection component ω on the three coordinate axes t, n, and b in the Frenet coordinate system t 、ω n 、ω b ;

[0029] Step 3, because r0′ and r i ′ is very close to the ball radius, so that the radius of the pure rolling point on the contact ellipse where the ball contacts the nut is r0′ = the radius of the pure rolling point on the contact ellipse where the ball contacts the screw is r i′ = ball radius r b , substitute into the above relationships to obtain the ball rotation angular velocity ω R The angular velocity of the nut relative to the ball ω0, the angular velocity of the screw relative to the ball ω i The ratio of the ball's revolution angular velocity ω m The calculation formula of tanβ is obtained by using the outer ring control theory, where β is the angle between the U axis and its projection axis on the tb plane, which is called the rotation angle.

[0030] Step 4: Generally, assume that the ball contacts the nut at point A and the screw at point B. Obtain the coordinates of the ball center O and the two contact points in the Frenet coordinate system.

[0031] Step 5: Obtain the position vector based on the coordinate transformation relationship between the fixed coordinate system and the Frenet coordinate system The expression of and its derivative to obtain the instantaneous velocity of the ball center The expression of the linear velocity of the contact points A and B on the ball and Based on the nut movement speed v0 and the screw rotation angular velocity ω, the linear velocity of the contact point A on the nut is obtained and the linear velocity of contact point B on the ball screw

[0032] Step 6: Change the linear velocity of contact point A on the ball and the linear velocity of contact point A on the nut Substituting into the relationship of the sliding-rolling ratio, the sliding-rolling ratio S of the contact point between the single ball and the nut raceway under the reverse transmission condition is obtained. n Similarly, the sliding-rolling ratio S of the contact point between the single ball and the screw raceway under reverse transmission conditions can be obtained s ;

[0033] Step 7: When the ball screw pair degenerates from rolling transmission to sliding transmission, the ball and the nut are relatively stationary, and the ball and the screw are in pure sliding motion. Let S n =0, S s =1, obtain the nut movement speed v0 and ball rotation angular velocity ω when the ball screw pair is in reverse transmission condition and degenerates from rolling transmission to sliding transmission R With the rotation angle β, contact angle α0, α i , and the helix angle λ, and the helix angle λ is used to determine whether the ball screw pair has reverse transmission failure.

[0034] Furthermore, when the ball screw pair is in reverse transmission, the force and velocity acting on the nut are active, and the torque and angular velocity on the screw are passive. In order to analyze the motion and force of the ball in the reverse transmission condition of the ball screw pair, the following is established: Figure 2 The coordinate system shown and Figure 3 The diagram of ball rotation axis U and rotation angle is shown. Figure 3 Get ω R The projection components on the three coordinate axes t, n, and b in the Frenet coordinate system are:

[0035] ω t =ω R cosβsinβ′

[0036] ω b =-ω R cosβcosβ′

[0037] ω n =-ω R sinβ

[0038] Where, ω R is the ball rotation angular velocity, ω t 、ω b 、ω n are ω R The projection components on the t, n, and b axes; β is the angle between the U axis and its projection axis on the tb plane, called the rotation angle; β′ is the angle between the projection axis of the U axis on the tb plane and the b axis.

[0039] When the ball screw pair is in reverse transmission condition, the nut moves linearly under the action of axial force F, and the nut has an axial movement speed v0 relative to the ball.

[0040] Assume that the center of the ball is fixed in space, that is, the ball only rotates but does not revolve. The ball and the nut are deformed under the action of axial load. Assume that the radius of the pure rolling point on the contact ellipse is r0′. In addition, at the position of the rolling radius r0′ on the ball surface, the linear velocity of the ball is equal to the linear velocity of the nut raceway. Combined with Figure 4 Determine the relationship in step 1 as:

[0041]

[0042] Where v0 is the nut speed; λ is the helix angle of the ball screw; ω0 is the angular velocity of the nut relative to the ball; d m is the nominal diameter of the ball screw pair; r0′ is the radius of the pure rolling point on the contact ellipse at the contact point between the ball and the nut; α0 is the contact angle value at the contact point between the ball and the nut raceway.

[0043] Assume that the center of the ball is fixed in space, and there is a pure rolling point between the ball and the screw, and its distance from the center of the ball is r i ′, rolling radius r on the ball surface i', the linear velocity of the ball is equal to the linear velocity of the screw raceway, combined with Figure 5 Determine the other relationship in step 1 as:

[0044]

[0045] Where, ω i is the angular velocity of the screw relative to the ball; r i ′ is the radius of the pure rolling point on the contact ellipse where the ball contacts the screw; α i is the contact angle value at the contact point between the ball and the screw raceway.

[0046] If the ball center is not fixed in space, but the nut raceway is fixed, then the angular velocity ω0 of the nut relative to the ball is the angular velocity ω of the ball's revolution m , the screw raceway has an absolute angular velocity ω=ω i +ω m However, the anti-rotation device in the ball screw pair allows the nut to move only along the axis of the screw and not rotate. Therefore, the relationship in step 2 is:

[0047] ω0=-ω m cosλ

[0048] Where, ω m is the angular velocity of the ball.

[0049] The component of the ball's angular velocity parallel to the b-axis is ω m cosλ, the component of the screw's angular velocity parallel to the b-axis is ωcosλ, so the screw's angular velocity relative to the ball is ω i The calculation formula is:

[0050] ω i =(ω-ω m )cosλ

[0051] Where ω is the angular velocity of the screw.

[0052] In addition, ignoring the pivot motion caused by the gyroscopic torque, β′=0 at this time, so determine the ω in step 2 at this time R The projection component ω on the three coordinate axes t, n, and b in the Frenet coordinate system t 、ω n 、ω R The calculation formula is:

[0053] ω t =0

[0054] ω b =-ω R cosβ

[0055] ω n =-ω R sinβ

[0056] Because r0′ and r i ' is very small compared to the ball radius, so r0'=r i ′=r b Substituting the above equations, we can get the ball rotation angular velocity ω mentioned in step 3 R The angular velocity of the nut relative to the ball ω0, the angular velocity of the screw relative to the ball ω i The ratio of the ball's angular velocity ω m The calculation formula is:

[0057]

[0058]

[0059]

[0060] Where r b is the ball radius.

[0061] In the dynamic analysis of rolling bearings, according to Jones's simplified assumption, the steel ball in contact with both the inner and outer rings only rolls and spins on one raceway, while performing pure rolling on the opposite raceway. The motion of a ball screw pair is very similar to the motion of a rolling bearing with a rotating inner ring and a fixed outer ring. Therefore, the outer ring control theory is adopted, which assumes that only pure rolling occurs on the outer ring raceway. so =0:

[0062] ω so =-ω0sinα0+ω b sinα0-ω n cosα0

[0063] =ω R (sinβcosα0-cosβsinα0)-ω0sinα0=0

[0064] Where, ω so is the sliding angular velocity of the ball on the outer ring raceway.

[0065] The ball's rotational angular velocity ω R Substituting the ratio of the angular velocity of the nut relative to the ball ω0 into the above formula, the calculation formula for tanβ described in step 3 is:

[0066]

[0067] In general, assuming that the ball contacts the nut at point A and the screw at point B, the expression of the ball center and the two contact points in the Frenet coordinate system described in step 4 can be obtained as follows:

[0068] O′=(0,0,0) T

[0069] A=(0,-r b cosα0,r b sinα0) T

[0070] B=(0,r b cosα i , -r b sinα i ) T

[0071] According to the coordinate transformation relationship between the fixed coordinate system and the Frenet coordinate system, the position vector can be obtained for:

[0072]

[0073] Taking the first derivative of the above formula, we can get the instantaneous velocity of the ball center. for:

[0074]

[0075] Where r m =d m / 2,d m is the nominal diameter of the ball screw pair; It is the angular velocity of the ball.

[0076] The linear velocity of the contact points A and B on the ball is expressed as:

[0077]

[0078]

[0079] The linear velocity of contact point A on the nut is:

[0080]

[0081] The linear velocity of contact point B on the ball screw is:

[0082]

[0083] Where, It is the angular velocity of the screw.

[0084] In the external contact surface of the ball and raceway in relative sliding, the external velocity along the sliding contact line is inconsistent. The relative amount of rolling and sliding can be expressed by the sliding-rolling ratio at the contact point:

[0085]

[0086] Where S is the sliding-rolling ratio of the contact point, V1 is the velocity of the active point at the contact point, and V2 is the velocity of the driven point.

[0087] Therefore, the sliding-rolling ratio of the contact point between the single ball and the nut raceway under the reverse transmission condition described in step 6 can be obtained:

[0088]

[0089] Where, is the linear velocity of the contact point A on the ball; is the linear velocity of contact point A on the nut.

[0090] The sliding-rolling ratio of the contact point between a single ball and the screw raceway is expressed as:

[0091]

[0092] Where, is the linear velocity of the contact point B on the ball; is the linear velocity of contact point B on the ball screw.

[0093] When the ball screw pair is in reverse transmission condition, the axial load will affect the sliding of the two solid surfaces. When the applied axial load is large, the ball motion in the ball screw degenerates from a composite rolling and sliding motion to a pure sliding motion, and there is a risk of self-locking.

[0094] When the ball and the nut are relatively stationary and the ball and the screw are in pure sliding motion, that is, S n =0, S s =1, or the movement between the ball and the nut is pure sliding, and the ball and the screw are relatively stationary, that is, S n =1, S s = 0, the ball screw pair degenerates into a trapezoidal screw. Because the outer raceway control theory is used, assuming that the balls only roll in contact with the outer raceway, the only possible situation here is that the balls and nut are relatively stationary, while the balls and screw experience pure sliding motion. The following explores the relationship between the nut travel speed, the ball rotational angular velocity, and the screw rotational angular velocity in this case.

[0095] When the ball and the nut are relatively stationary and the ball and the screw are in pure sliding motion, that is, S n =0, S s =1.

[0096]

[0097] Right now have to

[0098]

[0099]

[0100] Right now or

[0101] when When

[0102]

[0103] when When

[0104]

[0105] When the ball screw pair is in reverse transmission condition under the action of axial load F,

[0106]

[0107] Where F is the axial load, a0 is the axial acceleration of the nut, and m is the total weight of the nut and the workbench.

[0108] and:

[0109] v0=a0t

[0110] Where t is the action time, unit is s.

[0111] Substitute the v0 calculation formula into S n =0, S s = 1, and the tanβ calculation formula can be used in parallel to obtain the relationship between the various parameters when the ball screw pair is in the reverse transmission condition and degenerates into pure sliding motion as described in step 7:

[0112] when hour

[0113]

[0114]

[0115] when hour

[0116]

[0117] When the helix angle at this point is less than the equivalent friction angle, the ball screw pair is at risk of reverse transmission failure. However, the helix angle of the ball screw pair is fixed during design and generally does not change. Furthermore, the friction coefficient during ball screw movement is uncertain, making the equivalent friction angle difficult to determine. Therefore, the present invention considers that reverse transmission failure has occurred when the helix angle at this point is significantly less than the initial helix angle.

[0118] For example, in one of the embodiments, the method of the present invention is further verified and explained. Taking the GZ3206T-5-P2 ball screw pair as an example, the reverse transmission failure prediction model of the ball screw pair is calculated under the same rotation angle and different axial load conditions. The results are as follows: Figure 6 As shown in the figure, the screw speed increases with the increase of axial load when the ball screw pair degenerates into pure sliding motion. When the axial load is less than 36.5KN, the difference between the helix angle of the ball screw pair when it degenerates into pure sliding motion and the initial helix angle is very small, and the variation range is within 0.1 degrees. It is believed that no reverse transmission failure has occurred. When the axial load is 56KN, the difference between the helix angle of the ball screw pair when it degenerates into pure sliding motion and the initial helix angle increases to 0.25 degrees. It is believed that at this time, the plastic deformation of the ball and the raceway increases, and a small indentation appears between the ball and the raceway that does not affect the reverse transmission, resulting in a decrease in the helix angle. When the axial load is 125KN, the plastic deformation between the ball and the raceway increases sharply, the indentation depth gradually increases, and the helix angle of the ball screw pair decreases sharply. It is believed that reverse transmission failure has occurred.

[0119] In summary, the present invention has carried out dynamic modeling of the ball motion under the reverse transmission working condition of the ball screw pair, and combined with the roller control theory, described the sliding-rolling ratio formula between the ball and the two rollers under the reverse transmission working condition, and obtained the conditions that various parameters must satisfy when the rolling-slip composite motion degenerates into pure sliding motion. It can more accurately describe the actual situation when it degenerates into pure sliding motion, and more accurately predict the reverse transmission failure of the ball screw pair.

[0120] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are only illustrative of the principles of the present invention. Without departing from the spirit and scope of the present invention, any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included in the scope of protection of the present invention.

Claims

1. A method for predicting reverse transmission failure of a ball screw pair, characterized in that: The method comprises the following steps: Step 1: When the ball screw pair is in reverse transmission condition, for the case where the ball center is fixed in space and a pure rolling point appears on the contact ellipse, for the ball-nut contact, the linear velocity of the ball is equal to the linear velocity of the nut raceway, and the ball rotation angular velocity ω is obtained. R The relationship between the nut movement speed v0 and the nut relative to the ball rotation angular velocity ω0; for ball screw contact, the linear velocity of the ball is equal to the linear velocity of the screw raceway, and the ball rotation angular velocity ω is obtained R Angular velocity of the ball relative to the screw ω i relationship; Step 2: For the case where the ball center is not fixed in space but the nut raceway is fixed, obtain the angular velocity ω of the ball revolution m The relationship between the angular velocity ω0 of the nut relative to the ball; based on the angular velocity ω of the ball m The component of the screw rotation angular velocity ω on the b-axis in the Frenet coordinate system is used to obtain the screw angular velocity ω relative to the ball i Ignore the pivot motion caused by the gyroscopic torque and obtain ω at this time. R The projection component ω on the three coordinate axes t, n, and b in the Frenet coordinate system t 、ω n 、ω b ; Step 3: Let the radius of the pure rolling point on the contact ellipse where the ball contacts the nut be r0′ = the radius of the pure rolling point on the contact ellipse where the ball contacts the screw be r i ′ = ball radius r b Based on the above relationships, the ball rotation angular velocity ω is obtained R The angular velocity of the nut relative to the ball ω0, the angular velocity of the screw relative to the ball ω i The ratio of the ball's revolution angular velocity ω m The calculation formula of tanβ is obtained by using the outer ring control theory, where β is the angle between the U axis and its projection axis on the tb plane, which is called the rotation angle. Step 4: Assume that the ball contacts the nut at point A and the lead screw at point B. Obtain the coordinates of the ball center O and the two contact points in the Frenet coordinate system. Step 5: Obtain the position vector based on the coordinate transformation relationship between the fixed coordinate system and the Frenet coordinate system The expression of and its derivative to obtain the instantaneous velocity of the ball center The expression of the linear velocity of the contact points A and B on the ball and Based on the nut movement speed v0 and the screw rotation angular velocity ω, the linear velocity of the contact point A on the nut is obtained and the linear velocity of contact point B on the ball screw Step 6: Change the linear velocity of contact point A on the ball and the linear velocity of contact point A on the nut Substituting into the relationship of the sliding-rolling ratio, the sliding-rolling ratio S of the contact point between the single ball and the nut raceway under the reverse transmission condition is obtained. n Similarly, the sliding-rolling ratio S of the contact point between the single ball and the screw raceway under reverse transmission conditions can be obtained s ; Step 7: When the ball screw pair degenerates from rolling transmission to sliding transmission, the ball and the nut are relatively stationary, and the ball and the screw are in pure sliding motion. Let S n =0, S s =1, obtain the nut movement speed v0 and ball rotation angular velocity ω when the ball screw pair is in reverse transmission condition and degenerates from rolling transmission to sliding transmission R With the rotation angle β, contact angle α0, α i , and the helix angle λ, and the helix angle λ is used to determine whether the ball screw pair has reverse transmission failure.

2. The method for predicting reverse transmission failure of a ball screw pair according to claim 1, characterized in that: The ball rotation angular velocity ω in step 1 R The relationship between the nut movement speed v0 and the nut relative to the ball rotation angular velocity ω0 is: Where v0 is the nut speed; λ is the helix angle of the ball screw; ω0 is the angular velocity of the nut relative to the ball; d m is the nominal diameter of the ball screw pair; r0′ is the radius of the pure rolling point on the contact ellipse at the contact point between the ball and the nut; α0 is the contact angle value at the contact point between the ball and the nut raceway; β is the angle between the U axis and its projection axis on the tb plane, called the rotation angle; β′ is the angle between the projection axis of the U axis on the tb plane and the b axis; The ball rotation angular velocity ω R Angular velocity of the ball relative to the screw ω i The relationship is: Where, ω i is the angular velocity of the screw relative to the ball; r i ′ is the radius of the pure rolling point on the contact ellipse where the ball contacts the screw; α i is the contact angle value at the contact point between the ball and the screw raceway.

3. The method for predicting reverse transmission failure of a ball screw pair according to claim 1, characterized in that: The angular velocity ω of the ball revolution in step 2 m The relationship between the angular velocity ω0 of the nut relative to the ball is: ω0=-ω m cosλ Where λ is the helix angle of the ball screw pair; The angular velocity of the screw relative to the ball in step 2 is ω i for: oh i =(ω-ω m )cosλ Where ω is the angular velocity of the screw, and the component of the ball's angular velocity parallel to the b-axis is ω m cosλ, the component of the screw rotation angular velocity parallel to the b-axis is ωcosλ.

4. The method for predicting reverse transmission failure of a ball screw pair according to claim 1, characterized in that: ω R The projection component ω on the three coordinate axes t, n, and b in the Frenet coordinate system t 、ω n 、ω b Respectively expressed as: oh t =ω R cosβsinβ′ oh n =-ω R sinβ oh b =-ω R cosβcosβ′ Where, ω R is the angular velocity of the ball rotation; β is the angle between the U axis and its projection axis on the tb plane, which is called the rotation angle; β′ is the angle between the projection axis of the U axis on the tb plane and the b axis; Ignoring the pivot motion caused by the gyroscopic torque, β′=0, so ω in step 2 R The projection component ω on the three coordinate axes t, n, and b in the Frenet coordinate system t 、ω n 、ω R They are: oh t =0 oh n =-ω R sinβ oh b =-ω R cosβ.

5. The method for predicting reverse transmission failure of a ball screw pair according to claim 2, characterized in that: The ball rotation angular velocity ω in step 3 R The angular velocity of the nut relative to the ball ω0, the angular velocity of the screw relative to the ball ω i The ratio of the ball's revolution angular velocity ω m The calculation formulas are: Where ω is the angular velocity of the screw.

6. The method for predicting reverse transmission failure of a ball screw pair according to claim 5, characterized in that: In step 3, the outer ring control theory is used to obtain the calculation formula of tanβ, which specifically includes: Assume that only pure rolling occurs on the outer ring raceway, that is, ω so =0: oh so =-ω0sinα0+ω b sinα0-ω n cosα0 =ω R (sinβcosα0-cosβsinα0)-ω0sinα0=0 Where, ω so is the sliding angular velocity of the ball on the outer ring raceway; The ball's rotational angular velocity ω R Substituting the ratio of the angular velocity ω0 of the nut relative to the ball into the above formula, the calculation formula for tanβ is:

7. The method for predicting reverse transmission failure of a ball screw pair according to claim 1, characterized in that: The coordinates of the ball center and the two contact points in the Frenet coordinate system in step 4 are: O′＝(0，0，0) T A=(0,-r b cosα0,r b sinα0) T B=(0,r b cosα i ,-r b sinα i ) T 。 8. The method for predicting reverse transmission failure of a ball screw pair according to claim 1, characterized in that: The specific process of step 5 includes: According to the coordinate transformation relationship between the fixed coordinate system and the Frenet coordinate system, the position vector is obtained for: Taking the first derivative of the above formula, we can get the instantaneous velocity of the ball center. for: Where r m =d m / 2,d m is the nominal diameter of the ball screw pair; It is the angular velocity of the ball revolution; Linear velocity of contact points A and B on the ball and Respectively expressed as: Linear velocity of contact point A on the nut for: Linear velocity of contact point B on the ball screw for: Where, It is the angular velocity of the screw.

9. The method for predicting reverse transmission failure of a ball screw pair according to claim 8, characterized in that: The relative amount of rolling and sliding is expressed as the slide-to-roll ratio of the contact point: Where S is the sliding-rolling ratio of the contact point, V1 is the velocity of the active point at the contact point, and V2 is the velocity of the driven point; Then, the sliding-rolling ratio S of the contact point between the single ball and the nut raceway under the reverse transmission condition described in step 6 is n for: Sliding-rolling ratio S of the contact point between a single ball and the screw raceway under reverse transmission conditions s Expressed as:

10. The method for predicting reverse transmission failure of a ball screw pair according to claim 9, characterized in that: Step 7 specifically includes: S n =0, that is: Right now have to Right now or when Then, we get: when Then, we get: When the ball screw pair is in reverse transmission condition under the action of axial load F, there are: Where F is the axial load, a0 is the axial acceleration of the nut, and m is the total weight of the nut and the workbench; and v0=a0t Where, t is the action time, unit is s; Substitute v0 into S n =0, S s = 1, and tanβ is paralleled in the relationship obtained to obtain the relationship between the various parameters when the ball screw pair is in the reverse transmission condition and degenerates into pure sliding motion as described in step 7: when hour when hour When the helix angle is less than the equivalent friction angle, it indicates that the ball screw pair is in danger of reverse transmission failure. When the helix angle at this time is smaller than the initial helix angle, it indicates that the ball screw pair has failed in reverse transmission.

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