A method for calculating the coupling characteristics of wear and vibration of a screw nut pair

Abstract

The application discloses a calculation method for the wear and vibration coupling characteristics of a screw nut pair, and provides technical support for the coupling characteristics research of the wear and vibration of the screw nut pair by means of the geometric structure parameters, material parameters and working condition parameters of the screw nut pair, a dynamic wear depth prediction model of the screw nut pair, a stiffness model of the screw nut pair considering the wear influence and a nonlinear dynamic model of the screw nut pair based on the Hertz contact theory and the Archard wear theory. The dynamic wear depth prediction model can replace the wear experiment which is time-consuming and high in cost. The nonlinear dynamic model can be used for solving the dynamic response of the screw nut pair under the wear action, and has important theoretical significance and engineering application value for enriching the dynamic characteristic research of the screw nut pair.

Description

Technical Field

[0001] This invention relates to the field of mechanical design technology, and in particular to a method for calculating the coupling characteristics of wear and vibration of a lead screw and nut pair. Background Technology

[0002] Lead screw and nut assemblies are widely used in various precision equipment such as CNC machine tools, medical devices, and robots. Wear between rolling elements has a significant impact on the vibration characteristics of lead screw and nut assemblies and is one of the main reasons for the decrease in the accuracy of kinematic pairs or even their failure. Typical wear models are mainly represented by the Archard model, Bayer model, and Krelisky model. The introduction of these models laid the theoretical foundation for studying the wear characteristics of lead screw and nut assemblies. Vibration is the external reflection of the internal state of the lead screw and nut assemblies. In the initial operating state, the vibration of the lead screw and nut assemblies is stable and the various frequency components are relatively weak. As the wear condition worsens, the preload gradually decreases while the clearance gradually increases, resulting in an increase in the vibration amplitude and frequency components of the lead screw and nut assemblies. The contact stress between the rolling elements increases with the increase in vibration amplitude, which in turn further aggravates the wear condition of the lead screw and nut assemblies. Therefore, the wear and vibration of lead screw and nut assemblies are combined, interfering with and influencing each other.

[0003] Due to the complexity of loading conditions and wear mechanisms, the influence of wear is rarely considered when studying the vibration characteristics of lead screw and nut pairs, resulting in poor accuracy of the obtained vibration characteristics. Summary of the Invention

[0004] Therefore, the purpose of this invention is to provide a method for calculating the coupling characteristics of wear and vibration of a lead screw and nut pair, so as to improve the accuracy of vibration characteristics.

[0005] A method for calculating the coupling characteristics of wear and vibration of a lead screw and nut pair includes the following steps:

[0006] Step S11: Based on Archard's wear theory, calculate the wear volume W of the nut raceway. i And by the wear volume W i and wear area A i Calculate the normal wear depth h of the nut raceway. i The expression is given by i, where i represents the installation orientation of the ball bearing, i = L or R, where L is left and R is right;

[0007] Step S12: Considering wear factors, based on Hertz contact theory and according to the given geometric parameters, material parameters, and operating parameters of the lead screw and nut pair, the ball bearings under the preload F are calculated sequentially. n Normal contact force P of the ball under action Fn and normal deformation δ Fn, the normal contact force P of the ball under the residual pre-tightening force F' n and the load F i , the normal deformation δ i and the force component f i exerted on the ball, the geometric deformation compatibility equation of the ball and the load-displacement (F-x) relationship equation of the screw nut pair, and the load-displacement (F-x) relationship equation of the screw nut pair is combined with the wear depth equation h i to construct a system of simultaneous equations, the system of simultaneous equations is decoupled by using the implicit function derivation method, and then a segmented nonlinear stiffness model k(x,t) of the screw nut pair in the wear state is obtained;

[0008] In step S13, the screw nut pair is simplified as a single degree of freedom system with variable stiffness, a nonlinear dynamic equation of the screw nut pair under harmonic excitation is constructed and solved to obtain the dynamic response of the screw nut pair in the wear state, and the nonlinear dynamic equation is as follows:

[0009]

[0010] wherein x, and respectively represent the displacement, velocity and acceleration of the nut in the axial direction, F and w respectively represent the excitation force and excitation frequency; and c represents the damping coefficient.

[0011] Further, the step S11 specifically comprises:

[0012] In step S111, based on the Archard wear theory, the wear volume W i of the nut race is calculated as follows:

[0013]

[0014] wherein N represents the number of balls in the screw nut pair, K represents the wear coefficient, P i (L) represents the contact pressure between the ball and the nut race, V represents the linear velocity of the nut, w represents the angular velocity of the ball, R represents the distance from the contact point of the ball and the nut race to the center of the ball, L represents the running distance of the nut, and H represents the material hardness;

[0015] In step S112, the wear volume W i is divided by the wear area A i of the nut race to obtain the normal wear depth h i of the nut race as follows:

[0016]

[0017] wherein k n represents the load-displacement coefficient, δi Let Σρ represent the normal deformation of the ball, Σρ represent the curvature of the contact area between the ball and the nut raceway, and m represent the normal deformation of the ball. a Let d be the semi-major axis of the ellipse in which the ball and nut raceway contact, with dimensionless length 1, and d be the diameter of the nut raceway.

[0018] Further, step S12 specifically includes:

[0019] Step S121: Calculate the preload F based on the given geometric parameters, material parameters, and operating parameters of the lead screw and nut pair. n Normal contact force P of the ball under action Fn and normal deformation δ Fn for:

[0020]

[0021] Where, β Fn For the lead screw and nut pair under preload F n The contact angle under action, where λ is the lead angle of the leadscrew;

[0022] Step S122: Wear will lead to a decrease in preload. Taking wear into account, the residual preload F′ will be reduced. n As boundary conditions for load partitioning, the left and right balls are calculated under the residual preload F′. n Normal contact force P under load F i Normal deformation δ i and the component force f it bears i The normal contact force P of the left ball bearing. L and normal deformation δ L It can be represented as:

[0023]

[0024] Where, β Lw This is the contact angle of the left ball bearing.

[0025] Component force f L Represented as:

[0026]

[0027] Normal contact force P of the right ball R and normal deformation δ R It can be represented as:

[0028]

[0029] Where, β Rw This is the contact angle of the right ball bearing.

[0030] Component force f R Represented as:

[0031]

[0032] Step S123, after the screw nut pair runs a distance, the left and right raceways of the screw nut will be worn to different degrees. According to the positions of the ball center and the curvature center of the screw nut pair raceway, the normal deformation geometry equation of the left and right balls can be obtained as follows:

[0033]

[0034] wherein s Lw and s Rw respectively represent the curvature center distances of the left and right raceways of the screw nut and the screw under the residual pre-tightening force F n ′ and the load F, and s o is the curvature center distance between the screw nut raceway and the screw raceway under no load.

[0035] Step S124, based on the normal contact force equation and the normal deformation geometry equation of the balls, the load-displacement (F-x) relationship equation of the screw nut pair can be obtained as follows:

[0036]

[0037] wherein x′ FnL =(s FnLw sinβ FnLw -s o sinβ o )cosλ represents the critical displacement of the left ball under the residual pre-tightening force F′ n , s FnLw represents the curvature center distance of the left raceways of the screw nut and the screw under the residual pre-tightening force F′ n , β FnLw represents the contact angle of the left ball under the residual pre-tightening force F′ n .

[0038] Step S125, the load-displacement (F-x) relationship equation of the screw nut pair is combined with the wear depth equation h i to construct a coupled simultaneous equation group as follows:

[0039]

[0040] wherein x′ FnR =(s FnRw sinβ FnRw -s o sinβ o )cosλ is the critical displacement of the right ball under the residual pre-tightening force F′ n , s FnRwRight raceway of nut, screw under residual pre-tightening force F' n center distance under the action of curvature, β FnRw Right contact angle of ball under residual pre-tightening force F' n

[0041] Step S126, the implicit function derivation method is used to decouple and derive the coupled simultaneous equations, and then the segmented nonlinear stiffness model k(x, t) of the screw nut pair under the wear state is obtained, the stiffness k(x, t) of the screw nut pair is a segmented nonlinear function about displacement x, and can be represented as:

[0042]

[0043] Wherein, symbols A, B, C, D are:

[0044]

[0045]

[0046]

[0047]

[0048] The beneficial effects of the application are:

[0049] (1) The dynamic wear model of the screw nut pair established by the application can obtain the influence of running distance, load and structure parameters on the wear performance of the screw nut pair. The model can replace the time-consuming and high-cost wear experiment, and provides a theoretical basis and a feasible method for failure estimation of the screw nut pair.

[0050] (2) The coupling model of wear and vibration of the screw nut pair established by the application can solve the dynamic response of the screw nut pair under the wear state, which has important theoretical significance and engineering application value for enriching the dynamic characteristic research of the screw nut pair. BRIEF DESCRIPTION OF DRAWINGS

[0051] Figure 1 It is a structure schematic diagram of the screw nut pair of the application;

[0052] Figure 2 It is a structure schematic diagram of the screw nut pair under load of the application;

[0053] Figure 3 It is a position schematic diagram of the ball center and the curvature center of the raceway of the screw nut pair of the application;

[0054] Figure 4 It is a position schematic diagram of the ball center and the curvature center of the raceway of the screw nut pair of the application (considering the wear factor).​

[0055] Figure 5 A single degree of freedom dynamic model of the lead screw-nut pair with variable stiffness of the present application;

[0056] Figure 6 A load-displacement relationship experiment diagram of the lead screw-nut pair of the present application;

[0057] Figure 7 The calculated contact force and contact angle of the lead screw-nut pair change trend with running distance;

[0058] Figure 8 The calculated nut race wear depth change trend with running distance;

[0059] Figure 9 The calculated stiffness of the left and right sides of the lead screw-nut pair change trend with running distance;

[0060] Figure 10 The calculated influence of different wear conditions on the stiffness of the lead screw-nut pair;

[0061] Figure 11 The calculated influence of different wear conditions on the amplitude-frequency response of the lead screw-nut pair;

[0062] Figure 12 The calculated influence of different wear conditions on the time domain response of the lead screw-nut pair.

[0063] The following specific embodiments will further illustrate the present application in conjunction with the above figures. DETAILED DESCRIPTION

[0064] In order to facilitate the understanding of the present application, the present application will be described more fully below with reference to the relevant drawings. The drawings show several embodiments of the present application. However, the present application can be realized in many different forms and is not limited to the embodiments described herein. On the contrary, the purpose of providing these embodiments is to make the disclosure of the present application more thorough and comprehensive.

[0065] Please refer to Figures 1 to 11 , a calculation method of the wear and vibration coupling characteristics of the lead screw-nut pair provided in an embodiment of the present application, the steps are as follows:

[0066] Step 1. Based on the Archard wear theory, the wear depth of the lead screw-nut pair is calculated

[0067] Step 1.1, the wear volume can be expressed as:

[0068]

[0069] where K is the wear coefficient; P is the normal contact force; LS L represents the sliding distance, and H represents the material hardness. Since the sliding distance between the ball and the nut raceway is much greater than the sliding distance between the ball and the screw raceway, wear mainly occurs on the nut raceway. S It can be represented as:

[0070]

[0071] Where V is the linear velocity of the nut; w is the angular velocity of the ball; R is the distance from the contact point between the ball and the nut raceway to the center of the ball; and L is the running distance of the nut. Since the wear of the nut raceway increases with the running distance L, the volume W of the nut raceway is... i With ball deformation δ i The relationship can be expressed as an integral expression, where W is the wear volume of the nut raceway. i for:

[0072]

[0073] Where N is the number of balls in the lead screw and nut assembly, P i (L) is the contact pressure between the ball and the nut raceway, which is a function related to the running distance L.

[0074] Step 1.2, based on the characteristic that the wear of the lead screw and nut pair mainly occurs on the nut raceway, the normal wear depth h of the nut raceway... i It can be represented as:

[0075]

[0076] Among them, the wear area A of the nut raceway i It can be represented as A i =2e i πd, where d is the raceway diameter of the nut, e i e is the major semi-axis of the ellipse in which the ball and nut raceway contact. i It can be represented as:

[0077]

[0078] Where, m a Let Σρ be the semi-major axis of the ellipse in which the ball and nut raceway contact, with dimension 1; Σρ represents the curvature of the ball-nut raceway contact.

[0079] nut raceway normal wear depth h i The expression can be rewritten as:

[0080]

[0081] Where, k nLoad-displacement coefficient.

[0082] Step 2. Establish the contact stiffness model of the screw-nut pair

[0083] Step 2.1, the structural diagram of the screw-nut pair is shown in Figure 1 , wherein O BL and O BR represent the centers of the left and right balls, respectively; O NL , O NR and O SL , O SR represent the left and right raceway curvature centers of the nut and the screw, respectively; r N and r S are the raceway radii of the nut and the screw, respectively; β Fn is the contact angle of the screw-nut pair under the pre-tightening force F n ; the nut pitch (P+ε) is greater than the screw pitch (P), resulting in the axial pre-tightening force F n experienced by the balls due to extrusion. When the screw-nut pair only bears the pre-tightening force F n , the normal contact force P Fn and the normal deformation δ Fn of each ball can be represented as:

[0084]

[0085] wherein λ is the lead angle of the screw.

[0086] Step 2.2, the structural diagram of the screw-nut pair under load is shown in Figure 2 , wherein β L , β R and P L , P R represent the normal contact angle and the normal contact force of the left and right balls, respectively; x represents the relative displacement of the nut and the screw under the load F; f L and f R are the components of the left and right balls under the load F, respectively; D s is the diameter of the ball. The pre-tightening force can be used as the boundary condition of the load partition: (a) f L <F n and (b) f L ≥ F n . The positions of the ball center and the raceway curvature center of the screw-nut pair are shown in Figure 3 , wherein β0 is the initial contact angle of the screw-nut pair; λ Fn is the deformation amount of the ball along the spiral line under the pre-tightening force F n ; λ nis the relative displacement of the nut and screw along the helix direction under the load F. The screw-nut pair is under the pre-tightening force F n , point O NL and O NR will move to new positions O' NL and O' NR respectively. Again, the load F is applied to the screw-nut pair, points O' NL and O' NR will move to new positions O' N ' L and O' N ' R respectively.s o is the center distance of curvature between the nut raceway and the screw raceway when the load is zero, s Fn is the center distance of curvature of the nut and screw raceways under the pre-tightening force F n , s L and s R represent the center distance of curvature of the left and right raceways of the nut and screw under the pre-tightening force F n and the load F respectively.

[0087] s o is expressed as:

[0088] s o = r N + r s - D s

[0089] s Fn is expressed as:

[0090]

[0091] Step 2.3, wear will cause the pre-tightening force to decrease, considering the influence of the wear factor, the residual pre-tightening force F n ' is taken as the boundary condition of the load partition, which is divided into two categories of residual pre-tightening force F n ' and load F in the same direction and in the opposite direction, therefore, the normal contact force P i , the normal deformation δ i and the force component f i of the two types of balls are solved respectively. Specifically, the ball load can be determined according to the geometric deformation coordination of the ball.

[0092] The normal contact force P L and the normal deformation δ L of the left ball can be expressed as:

[0093]

[0094] where β LwFor the left ball contact angle.

[0095] Force f L is expressed as:

[0096]

[0097] Considering the effect of wear, the normal contact force P R of the right ball is: R and the normal deformation δ Rw is expressed as:

[0098]

[0099] where β R is the right ball contact angle.

[0100] Force f L is expressed as:

[0101]

[0102] Step 2.4, after the screw nut pair runs a distance, the positions of the ball center and the curvature centers of the left and right screw nut raceways are shown in FIG. 4, where h R and h FnLw represent the wear depths of the left and right screw nut raceways, respectively; β FnRw and β n represent the contact angles of the left and right balls under the residual preload F′ n . Under the residual preload F′ FnLw , the distances s FnRw and s n between the curvature centers of the left and right screw nut raceways are expressed as:

[0103]

[0104] Under the residual preload F′ Lw and the load F, the distances s Rw and s Lw between the curvature centers of the left and right screw nut raceways are expressed as:

[0105]

[0106]

[0107] The contact angles β Rw and β L of the left and right balls are expressed as:

[0108]

[0109]

[0110] Normal deformation of left and right balls δ L and δ R Can be rewritten as:

[0111]

[0112] Step 2.5, based on the normal contact force equation of the ball and the normal deformation geometry equation, the relationship between load F and displacement x can be obtained as:

[0113]

[0114] Where x' FnL = (s FnLw sinβ FnLw -s o sinβ o )cosλ represents the axial deformation of the left ball under the action of residual preload F' n

[0115] Step 2.6, wear depth h L and h R Is expressed as:

[0116]

[0117]

[0118] Where the symbol J is:

[0119] Wear depth h i and load F are functions of running distance L, a system of simultaneous equations needs to be established, which is expressed as:

[0120]

[0121] The derivative of the system of simultaneous equations with respect to the running distance L is solved by the implicit function differentiation method to decouple and calculate the normal wear depth value h i , which is expressed as:

[0122]

[0123] Where the symbol Can be expressed as:

[0124]

[0125] The symbol Can be expressed as:

[0126] ​

[0127] The symbols A, B, C, and D are represented as follows:

[0128]

[0129]

[0130]

[0131]

[0132] Step 2.7: Based on the different force conditions of the balls on the left and right sides of the lead screw nut pair, when the load F is along the positive x-axis, the left ball will be in the position where x ≥ x′. FnL (x′ FnL It loses its function when ≥0); when the load F is along the negative x-axis, the right ball bearing loses its function when x ≤ x′. FnR (x′ FnR It loses its function when the residual preload F is ≤0. Specifically, the ball bearings are held in place by the residual preload F. n Axial deformation x′ under action Fni This serves as the boundary point for the nonlinear stiffness k(x,t).

[0133] Therefore, the stiffness k(x,t) of the lead screw and nut assembly can be expressed as:

[0134]

[0135] Wherein, the critical displacement x′ FnR =(s PRw sinβ PRw -s o sinβ o )cosλ.

[0136] Step 3. Establish a nonlinear dynamic model of the lead screw and nut pair.

[0137] To study the coupling characteristics of wear and vibration in a leadscrew and nut assembly, the leadscrew and nut assembly can be simplified as a single-degree-of-freedom system with variable stiffness, such as... Figure 5 As shown. The dynamic equation of the lead screw and nut pair under harmonic excitation can be expressed as:

[0138]

[0139] Where, x, and represents the axial displacement, velocity, and acceleration of the nut, respectively; F and w represent the excitation force and excitation frequency, respectively; and c is the damping coefficient.

[0140] Furthermore, the technical solution of the present invention will be described in further detail below with reference to the accompanying drawings.

[0141] 1. In order to obtain the wear depth of the screw nut pair, the geometric parameters, material parameters and working condition parameters of the screw nut pair should be given in advance.

[0142] 2. According to the step 1, the expression of the nut race wear depth h i is obtained.

[0143] 3. The expression of the wear depth h i obtained in the step 1 is combined with the expression of the relationship between the load F and the displacement x obtained in the step 2.5, and is substituted into the step 2.7 to calculate the segmented nonlinear stiffness k(x, t) of the screw nut pair considering the influence of wear.

[0144] 4. The segmented nonlinear stiffness k(x, t) of the screw nut pair obtained in the step 2.7 is substituted into the step 3, and the differential equation of the system is solved by the fourth-order Runge-Kutta method.

[0145] Taking the SBN4016 screw nut pair as an example, the detailed geometric and material parameters of the screw nut pair are given in Table 1.

[0146] Table 1 Geometric structure and material parameters of SBN4016 screw nut pair

[0147]

[0148] When the influence of wear factor is considered (h L ≠ 0 and h R ≠ 0), in order to ensure the accuracy of the calculation of the stiffness k(x, t), the relationship between the load F and the displacement x needs to be verified by experiments. After the screw nut pair runs a distance under the action of the load F, the nut race will be worn to different degrees. Since the structure of the screw nut pair is compact and inconvenient to disassemble, it is difficult to directly measure the wear depth h i of the nut race. In order to ensure the accuracy of the dynamic wear prediction model, the load-displacement (F-x) relationship of the screw nut pair needs to be verified by experiments, and the experimental device is shown in Figure 6 The load F is applied to the screw nut pair in a static state after wear by a weight, and the nut displacement value x is measured by a laser displacement sensor. The theoretical value of the nut displacement x can be obtained by step 3.4. The comparison between the theoretical value and the experimental value of the displacement x is shown in Table 2, and the results show that the dynamic wear prediction model established by the present application is accurate.

[0149] Table 2 Comparison of theoretical and experimental displacement values

[0150]

[0151]

[0152] The contact state, wear depth, stiffness of the screw-nut pair with the running distance are shown in Figure 7 、 Figure 8 and Figure 9 . It can be seen from Figure 7 that the contact force P i and the contact angle β iw decrease with the increase of the running distance L. It can be seen from Figure 8 that the right raceway wear depth h R of the nut increases accordingly with the increase of the load F, while the left raceway wear depth h L of the nut decreases gradually. It can be seen from Figure 9 that the left stiffness k L of the screw-nut pair decreases with the increase of the load F or the running distance L. When the displacement x≥x′ FnL , k L decreases to 0, and the right stiffness k R of the screw-nut pair increases with the increase of the load F, but decreases with the increase of the running distance L.

[0153] The influence of wear on the stiffness and nonlinear response of the screw-nut pair is shown in Figure 10 、 Figure 11 and Figure 12 . It can be seen from Figure 10 that the stiffness k of the screw-nut pair and the absolute value of the critical displacement |x′ Fni | decrease with the increase of the running distance L. It can be seen from Figure 11 that the amplitude-frequency response curve shows soft and hard nonlinear phenomena, and the jump phenomenon occurs at the critical displacement x′ Fni . The amplitude increases with the increase of the running distance L, and the resonance frequency decreases with the increase of the running distance L. It can be seen from Figure 12 that the time-domain response curves show significant differences with the increase of the wear degree.

[0154] In the specification, each embodiment is described in a progressive manner, and each embodiment focuses on the difference from other embodiments. The same or similar parts between each embodiment can be referred to each other. The above-described embodiments only express several embodiments of the present application, and the description is more specific and detailed, but it cannot be understood as the limitation of the scope of the present application. It should be pointed out that, for those skilled in the art, without departing from the concept of the present application, some modifications and improvements can be made, which are within the scope of the present application. Therefore, the protection scope of the present application should be subject to the appended claims.​

Claims

1. A method for calculating the wear and vibration coupling characteristics of a ball screw pair, characterized by, Comprise the following steps: Step S11, based on Archard wear theory, calculate the wear volume of the nut race , and the normal wear depth of the nut race is calculated by the wear volume and the wear area , wherein i represents the installation orientation of the ball, i = L or R, L is left, and R is right.​ Step S12, based on the Hertz contact theory, and according to the geometric parameters, material parameters and working condition parameters of the given screw nut pair, the normal contact force and the normal deformation of the ball under the action of the pre-tightening force F0 are sequentially calculated and the expression of the normal contact force and the normal deformation of the ball under the action of the residual pre-tightening force F0 and the load F , the normal deformation and the force component borne by the ball , the geometric deformation coordination equation of the ball and the load-displacement (F-x) relationship equation of the screw nut pair are combined to construct a system of simultaneous equations, the system of simultaneous equations is decoupled by using the implicit function derivation method, and then a segmented nonlinear stiffness model k(x, t) of the screw nut pair in the wear state is obtained. Step S121, based on the geometric parameters, material parameters and working condition parameters of the given screw nut pair, the pre-tightening force is calculated The normal contact force of the ball under the action And the normal deformation Is: wherein is the contact angle of the ball screw nut pair under the pretension force and λ is the lead angle of the ball screw. Step S122, the wear will cause the preload to decrease, and the residual preload As the boundary condition of the load partition, the normal contact force of the left and right balls under the residual preload and the normal contact force under the load F , the normal deformation and the force component borne , the normal contact force of the left ball and the normal deformation are: wherein is the left ball contact angle; component force is represented as: Normal contact force of right ball and normal deformation may be expressed as: wherein, is the right ball contact angle; component forces is represented as: Step S123, after the screw nut pair runs a distance, the left and right raceways of the nut will be worn to different degrees, according to the positions of the ball center and the curvature center of the screw nut pair raceway, the normal deformation geometric equation of the left and right balls can be obtained as: wherein, , respectively represent the center of curvature distance of the left and right raceways of the nut and the screw under the residual pretension force and the load F, is the center of curvature distance between the raceways of the nut and the screw when empty. Step S124, based on the normal contact force equation and the normal deformation geometric equation of the balls, the load-displacement (F-x) relationship equation of the screw nut pair can be obtained as: wherein, represents the critical displacement of the left ball under the residual pre-tightening force F n ′, represents the center of curvature distance of the left raceway of the nut, screw under the residual pre-tightening force F n ′, represents the contact angle of the left ball under the residual pre-tightening force F n ′. Step S125, the load-displacement (F-x) relationship equation of the screw-nut pair is combined with the wear depth equation are combined to construct a coupled system of equations: wherein is the critical displacement of the right ball under the residual pre-tightening force F n is the center of curvature distance of the right raceway of the nut, screw under the residual pre-tightening force F n is the contact angle of the right ball under the residual pre-tightening force F n ​​​ Step S126, the implicit function derivation method is used to decouple and derive the coupled simultaneous equations, and then a segmented nonlinear stiffness model under the wear state of the screw nut pair is obtained , the stiffness of the screw nut pair is a segmented nonlinear function about displacement x, which can be expressed as: Wherein, symbols A, B, C, D are: ； Step S13, the screw nut pair is simplified as a single degree of freedom system with variable stiffness, the nonlinear dynamics equation of the screw nut pair under harmonic excitation is constructed and solved to obtain the dynamics response of the screw nut pair under the wear state, and the nonlinear dynamics equation is: ； wherein, , and respectively represent the displacement, velocity and acceleration of the nut along the axial direction, F and w are respectively the exciting force and the exciting frequency; c is the damping coefficient.

2. The method of claim 1, wherein, The step S11 specifically comprises: Step S111, based on Archard wear theory, calculate the wear volume of the nut race is: ； wherein N is the number of balls in the ball screw nut pair, K is the wear coefficient, is the contact pressure between the ball and the nut race, V is the linear speed of the nut, w is the angular speed of the ball, R is the distance from the contact point of the ball and the nut race to the center of the ball, L is the running distance of the nut, and H is the material hardness. Step S112, the wear volume The normal wear depth of the nut race can be obtained by dividing the wear volume The normal wear depth of the nut race can be obtained by dividing the wear volume is: ； wherein, is the load-displacement coefficient, is the ball normal deformation, is the curvature of the ball and nut race contact, is the ball and nut race contact ellipse dimensionless major axis, d is the nut race diameter.

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