Ball screw pair starting torque calculation method

Abstract

The application provides a ball screw pair starting torque calculation method, comprising the following steps: calculating the stress of each ball in the ball screw pair under the action of load; calculating the tangential and normal displacement of the ball and the contact point of the raceway; using the boundary element method to disperse the contact area of the ball and the raceway, and constructing the discrete convolution relationship between the tangential and normal displacement and the shear stress; using the fast Fourier transform and the conjugate gradient method to solve the discrete convolution relationship between the tangential and normal displacement and the shear stress, and obtaining the shear stress distribution; judging the ball sticking and sliding area according to the friction coefficient and the shear stress distribution, and tracking the evolution process of the stick-slip interface; when all the balls enter the sliding state, the corresponding critical tangential load is recorded, and the screw starting torque is calculated through the distributed prediction calculation model. The application can accurately predict the ball screw pair starting torque without relying on a large number of physical tests, and is suitable for the fast evaluation of the starting torque under multiple working conditions and multiple parameters.

Description

Technical Field

[0001] This invention relates to the field of ball screw pair technology, and specifically to a method for calculating the starting torque of a ball screw pair. Background Technology

[0002] Starting torque, a key indicator of the initial driving performance of a ball screw pair, directly reflects its internal contact friction state and significantly impacts critical performance aspects such as high-precision positioning, low-speed starting, and energy consumption control. However, due to the complex internal structure and highly nonlinear raceway contact behavior of ball screw pairs, starting torque is influenced by multiple coupled factors, including preload, lubrication status, number of balls, surface roughness, and material stiffness, making accurate prediction difficult through simple experimental methods. Current experimental measurement methods are not only costly and have poor repeatability, but also cannot capture the actual shear stress state of the raceway during structural closure or fretting stages, failing to meet the requirements of high-precision modeling and control. Therefore, establishing a starting torque prediction method based on contact mechanics is of great significance.

[0003] Existing studies mostly employ empirical formulas like NSK or single-factor experiments to analyze the influencing factors of starting torque, but they neglect the evolution of the stick-slip region and the non-uniformity of the tangential stress response of each ball, and also struggle to describe the spatial distribution of the raceway shear stress field. Therefore, this invention proposes a method for calculating the starting torque of ball screw pairs based on a combination of boundary element method and fast Fourier convolution. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a method for calculating the starting torque of a ball screw pair. This method can accurately identify the contact state and critical slip load of each ball, and systematically considers the coupling mechanism between microscopic contact behavior and macroscopic transmission performance. This provides a theoretical basis and technical support for modeling and optimizing the friction performance of ball screw pairs.

[0005] The technical solution adopted in this invention is as follows:

[0006] A method for calculating the starting torque of a ball screw pair includes the following steps: calculating the force on each ball in the ball screw pair under load; calculating the tangential and normal displacements at any contact point between the ball and the raceway; discretizing the contact area between the ball and the raceway using the boundary element method, and constructing the discrete convolution relationship between the tangential and normal displacements and shear stress using the Boussinesq-Cerruti equation; efficiently solving the discrete convolution relationship between the tangential and normal displacements and shear stress using the fast Fourier transform and conjugate gradient method to obtain the shear stress distribution; determining the ball adhesion and slippage regions based on the friction coefficient and shear stress distribution, and tracking the evolution of the stick-slip interface; when all balls enter the slippage state, recording the corresponding critical tangential load, and calculating the starting torque of the screw using a distributed prediction calculation model for the starting torque.

[0007] Furthermore, the force on each ball is calculated based on the deformation coordination relationship between the inner screw, nut, and ball of the ball screw pair after deformation. The calculation formula is as follows:

[0008] ,

[0009] In the formula, This represents the preload force on the leadscrew, which is equal to the sum of the forces acting on each ball in the axial direction. Indicates the first Each ball is subjected to normal force on the nut side; Indicates the first The contact angle of each ball; This indicates the normal force on the i-th ball on the nut side; Indicates each The contact angle of each ball; Indicates the helix angle; and Indicates material parameters; and These represent the stiffness coefficients of the nut and the lead screw, respectively.

[0010] Furthermore, at any point of contact between the ball and the raceway The calculation methods for the tangential and normal displacements at the location are as follows:

[0011] Constructing the Green function The normal and tangential displacements can be expressed as the convolution of shear stress and normal pressure with the Green's function:

[0012] ,

[0013] in,

[0014] ,

[0015] ,

[0016] ,

[0017] ,

[0018] In the formula, Represent the Green's function; and These represent the elastic moduli of the balls and raceways, respectively. and These represent the Poisson's ratios of the balls and raceways, respectively.

[0019] , , These are intermediate parameters used to simplify expressions; It represents the reciprocal of the equivalent elastic modulus.

[0020] Furthermore, the expression for the Boussinesq-Cerruti equation is:

[0021] ,

[0022] In the formula, Indicates continuous convolution; Indicates in Tangential displacement in the direction; Indicates in Tangential displacement in the direction; Indicates in Normal displacement in the direction; Represent the Green's function; , They represent in and Shear stress in the direction, Indicates normal force.

[0023] Furthermore, the expression for the shear stress distribution is as follows:

[0024] ,

[0025] In the formula, , and Representing the rectangular unit on , and Contact displacement in the direction; Indicates the influence coefficient; , They represent in and Shear stress in the direction; Indicates normal force.

[0026] Furthermore, the normal and tangential displacements satisfy the following relationship:

[0027] ,

[0028] In the formula, , and Represents the rigid body displacement components. and Indicates tangential and normal displacement. This indicates the contact gap under normal loading conditions. Indicates the initial gap; , and Representing the rectangular unit on , , Contact displacement in the direction.

[0029] Furthermore, the contact area between the ball and the raceway is divided into contact areas. Non-contact area And satisfy the following conditions:

[0030] The contact area meets the requirements of positive pressure and zero clearance; the non-contact area meets the requirements of zero pressure and has a clearance.

[0031] The condition equations it satisfies are as follows:

[0032] ,

[0033] In the formula, Indicates the subscript of a rectangular cell; Indicates normal force; Indicates the contact gap; This indicates the normal force acting on the ball.

[0034] Furthermore, the contact area Divided into adhesion zones and slip zone And satisfy the following conditions:

[0035] In the adhesion zone Within the slip zone, the shear stress is less than the static friction limit, and no relative sliding occurs; Internally, the direction of the tangential friction force is opposite to the direction of local slip, and the shear stress is equal to the friction limit; the condition equations it satisfies are as follows:

[0036] ,

[0037] In the formula, , They represent in and Shear stress in the direction; Indicates the subscript of a rectangular cell; Indicates normal force; Indicates the static friction coefficient; and This indicates the tangential displacement.

[0038] The criteria for determining the starting torque are as follows:

[0039] With increasing tangential load, the adhesion zone... Gradually shrinking, slip zone Gradually expand until the entire contact area is reached. Full slip occurs; define critical tangential load. For the first The tangential load that enables the ball to slip as a whole, and the starting torque is the sum of the torques when all the balls simultaneously enter the fully slipped state;

[0040] Among them, the critical tangential load The expression is:

[0041] ,

[0042] In the formula, Indicates the critical tangential load; express Shear stress in the direction; Indicates the size of the rectangular unit;

[0043] The formula for the distributed prediction calculation model of the starting torque is:

[0044] ,

[0045] In the formula, Indicates the starting torque; Indicates the lead screw radius; Indicates the radius of the ball bearing; This indicates the contact angle of each ball; Indicates the helix angle; This indicates the total number of balls.

[0046] The beneficial effects of this invention are:

[0047] (1) It realizes accurate prediction of the starting torque of ball screw pair without relying on a large number of physical tests, and is suitable for rapid evaluation of starting torque under multiple working conditions and multiple parameters;

[0048] (2) The system takes into account the contact response of each ball under three-dimensional tangential load, and can simulate the evolution of the stick-slip region during the generation of starting torque;

[0049] (3) The efficient algorithm combining fast Fourier convolution and conjugate gradient method is adopted, which greatly improves the solution efficiency and accuracy of tangential contact problems;

[0050] (4) By combining the critical slip states of each ball, a distributed prediction calculation model for the starting torque was established, which can truly reflect the influence mechanism of the cooperative sliding of multiple balls in the ball screw pair on the starting torque, making the calculation results more accurate and more physically meaningful. Attached Figure Description

[0051] Figure 1 This is a flowchart of the method for calculating the starting torque of a ball screw pair according to an embodiment of the present invention;

[0052] Figure 2 This is a block diagram of the friction torque test bench according to an embodiment of the present invention;

[0053] Figure 3 This is a schematic diagram illustrating the deformation coordination relationship between the ball and the raceway according to an embodiment of the present invention;

[0054] Figure 4 A comparison graph showing the relationship between theoretical and experimental results;

[0055] Figure 5 This is a schematic diagram illustrating the evolution of the normalized surface tangential traction force distribution of ball number 20 in Sample 1.

[0056] Figure 6 This is a schematic diagram showing the distribution evolution of the sticky and slippery area of ​​ball bearing number 20 in sample 1. Detailed Implementation

[0057] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0058] like Figure 1 As shown in the figure, an embodiment of the present invention provides a method for calculating the starting torque of a ball screw pair, which includes the following steps:

[0059] S1: Calculate the force on each ball in the ball screw pair under load, that is, the sum of the forces on the balls in the axial direction.

[0060] In this invention, the force on each ball inside the nut is calculated based on the deformation coordination relationship between the inner screw, nut, and ball of the ball screw pair after deformation. The calculation formula is as follows:

[0061] ,

[0062] In the formula, This represents the preload force on the leadscrew, which is equal to the sum of the forces acting on each ball in the axial direction. Indicates the first Each ball is subjected to normal force on the nut side; Indicates the first The contact angle of each ball; Indicates the first Each ball is subjected to normal force on the nut side; Indicates each The contact angle of each ball; Indicates the helix angle; Indicates the total number of balls; and Indicates material parameters; and These represent the stiffness coefficients of the nut and the lead screw, respectively.

[0063] S2: Calculate the tangential and normal displacements at any contact point between the ball and the raceway.

[0064] Under axial load, the elliptical contact area formed between the ball and the raceway is typically much smaller than the dimensions or radius of the ball and raceway. The maximum pressure occurs at the center of this contact area and gradually decreases radially outwards. Therefore, this contact interaction can be idealized as the interaction between a sphere and an elastic half-space. Figure 2 As shown, the balls bear a normal load Qi and a tangential load Ti in orthogonal directions, thus generating a normal pressure distribution. Shear stress and .

[0065] Specifically, the calculation methods for normal and tangential displacements are as follows:

[0066] First, construct the Green function. The tangential and normal displacements can be expressed as the convolution of shear stress and normal pressure with the Green's function:

[0067] ,

[0068] In the formula:

[0069] ,

[0070] ,

[0071] ,

[0072] ,

[0073] in, Represent the Green's function; and These represent the elastic moduli of the balls and raceways, respectively. and These represent the Poisson's ratios of the balls and raceways, respectively.

[0074] , , These are intermediate parameters used to simplify expressions; It represents the reciprocal of the equivalent elastic modulus.

[0075] S3: The contact area between the ball and the raceway is discretized using the boundary element method to construct the discrete convolution relationship between normal and tangential displacements and shear stress.

[0076] Specifically, the computational region is discretized into The system uses rectangular elements, and treats the pressure (the force on a single ball, discretized to the force on each rectangular element) and tangential and normal displacements as constants at the center of the rectangular element, defining influence coefficients. It is used to characterize the response of unit pressure to tangential and normal displacement, and the boundary element method (BEM) is used for numerical solution.

[0077] Specifically, let a be the minor axis and b be the major axis of the ellipse in the contact area between the ball and the raceway, and let the calculation region be defined as... And discretize the computational domain into A rectangular unit.

[0078] Among them, the influence coefficient The expression is:

[0079] ,

[0080] in, ;

[0081] In the formula, express , , direction; and Represents the subscript of a rectangular cell; Represents rectangular unit and exist Distance in direction; Represents rectangular unit and exist Distance in a direction.

[0082] The expression for the Boussinesq-Cerruti equation is:

[0083] ,

[0084] In the formula, Indicates continuous convolution; Indicates in Tangential displacement in the direction; Indicates in Tangential displacement in the direction; Indicates in Normal displacement in the direction; Represent the Green's function; , They represent in and in Shear stress in the direction, Indicates normal force.

[0085] S4: Solve the tangential contact problem using Fast Fourier Transform (FFT) and Conjugate Gradient Method (CGM), that is, solve the discrete convolution relationship between tangential and normal displacements and shear stress, thereby obtaining the shear stress distribution.

[0086] The expression for shear stress distribution is as follows:

[0087] ,

[0088] In the formula, , and Representing the rectangular unit on , , Contact displacement in the direction; Indicates the influence coefficient; , They represent in and Shear stress in the direction; Indicates normal force.

[0089] At the same time, the tangential and normal displacements also satisfy the following relationship:

[0090] ,

[0091] In the formula, , and Represents the rigid body displacement components. and Indicates tangential and normal displacement. This indicates the contact gap under normal loading conditions. Indicates the initial gap; , and Representing the rectangular unit on , , Contact displacement in the direction.

[0092] S5: Determine the adhesion and slip regions based on the friction coefficient and shear stress distribution, and track the evolution of the stick-slip interface.

[0093] In solving the contact problem, the contact area between the ball and the raceway can be divided into contact areas. Non-contact area And satisfy the following conditions:

[0094] The contact area between the ball and the raceway satisfies positive pressure and zero clearance; the non-contact area satisfies zero pressure and has clearance.

[0095] The condition equations it satisfies are as follows:

[0096] ,

[0097] In the formula, Indicates the subscript of a rectangular cell; Indicates normal force; Indicates the contact gap; This indicates the normal force acting on the ball.

[0098] In addition, the contact area It can be further divided into adhesion zones. and slip zone It satisfies the following conditions:

[0099] Within the adhesion zone, the shear stress is less than the static friction limit, and no relative sliding occurs. Within the slip zone, the direction of the tangential friction force is opposite to the direction of local slip, and the shear stress is equal to the friction limit. The condition equations they satisfy are as follows:

[0100] ,

[0101] In the formula, , They represent in and The shear stress in the direction is obtained by solving a system of linear equations that satisfy the corresponding relationship between the normal and tangential displacements, in the adhesive region. and slip zone The satisfied condition equations serve as constraints; Indicates the subscript of a rectangular cell; Indicates normal force; Indicates the static friction coefficient; and This indicates the tangential displacement.

[0102] S6: When all balls enter the slip state, record the corresponding critical tangential load, and calculate the screw starting torque through the distributed prediction calculation model of starting torque.

[0103] The criteria for determining the starting torque are as follows:

[0104] With increasing tangential load, the adhesion zone... Gradually shrinking, slip zone Gradually expand until the entire contact area is reached. A full slip occurs. Define the critical tangential load. For the first The tangential load required for the entire ball to slip is the sum of the starting torques when all balls simultaneously enter the fully slipped state.

[0105] Among them, the critical tangential load The expression is:

[0106] ,

[0107] In the formula, Indicates the critical tangential load; express Shear stress in the direction; Indicates the size of the rectangular unit.

[0108] The formula for the distributed prediction calculation model of the starting torque is:

[0109] ,

[0110] In the formula, Indicates the starting torque; Indicates the lead screw radius; Indicates the radius of the ball bearing; This indicates the contact angle of each ball; Indicates the helix angle; This indicates the total number of balls.

[0111] experiment:

[0112] To verify the method proposed in this invention, three sets of ball screws (model 2516, manufactured by China Laine Corporation) were used. Sample parameters are detailed in Table 1. The starting torque test bench is as follows... Figure 3 As shown. In the test configuration, the ball screw shaft is mounted on the support unit and connected to the worktable. Relative movement is effectively prevented by constraining its rotation through fixing bolts inserted into the nut flange. Each ball screw reciprocates for three cycles at a constant speed of 20 rpm to ensure measurement consistency.

[0113] Table 1: Ball Screw Parameters

[0114] parameter numerical values unit ball diameter 3.969 millimeters Thread lead 16 millimeters Pitch circle diameter 25 millimeters Lead angle 4.55 Spend Number of cycles 4 Ambient temperature 20±1 ℃

[0115] Starting torque verification:

[0116] like Figure 4 The image shows the test results for three representative samples: a is sample 1, b is sample 2, and c is sample 3. Figure 4 a(1), Figure 4 b(1) and Figure 4 c(1) presents the curve of frictional torque variation, and also records the starting torque. and operating friction torque . Figure 4 a(2), Figure 4 b(2) and Figure 4 c(2) provides a magnified view of the starting torque region and marks the 95% confidence interval of the prediction.

[0117] exist Figure 4 a(1), Figure 4 b(1) and Figure 4 In c(1), the blue curve represents the experimentally measured friction torque, and the green square represents the predicted value. The red asterisk indicates the experimentally measured starting torque value. The starting torque value estimated using the NSK empirical formula is marked with a green circle. The trend of the blue curve indicates that the friction torque rapidly rises to its peak value within a short period of time. The friction torque then dropped slightly to a stable operating level.

[0118] Enlarged curve ( Figure 4 a(2), Figure 4 b(2) and Figure 4 c(2) indicates that the predicted starting torque value is in high agreement with the measured result. Taking sample 1 as an example, the predicted value of 0.1347 Nm deviates from the measured torque value of 0.1380 Nm by only 2.4%. The prediction errors of samples 2 and 3 are 1.78% and 1.20% respectively, which fully verifies the accuracy and robustness of the method proposed in this invention. In contrast, the NSK formula systematically underestimates the starting torque, especially in sample 1, where the difference is most significant. Its predicted value of 0.1252 Nm is about 9.2% lower than the measured value. This highlights the limitations of the empirical model, which ignores the local stick-slip transition of the ball along the lead and the uneven force distribution.

[0119] Furthermore, the 95% confidence interval confirms that the model predictions all fall within a narrow range of uncertainty, covering the measured torque values ​​for all samples. This highlights the model's ability to capture key factors influencing the starting torque of the ball screw, including micro-slippage behavior and deformation effects in the contact area.

[0120] Evolution of tangential load distribution:

[0121] Figure 5 This demonstrates how sphere number 20 in Sample 1 is subjected to a gradually increasing external tangential load. The evolution of the normalized surface tangential traction force distribution q / p0 under the action of load. The load range is from to ,in The coefficient of friction is the local friction coefficient. This is a local normal load. Each subgraph contains... The three-dimensional surface diagram is shown, along with its contact plane projection and a dashed line that indicates the cross-sectional profile along the major axis of the elliptical contact area.

[0122] During the initial loading phase ( Figure 5 (1), The tangential traction force distribution exhibits a distinct antisymmetric pattern and relatively low amplitude. The peak value is consistently below 0.0015. Its shape extends along a contact ellipse, with the central region primarily maintaining an adhesive state. When the tangential load increases to... ( Figure 5 (3)) when The amplitude increased significantly, exceeding ±0.003, while the slip zone gradually expanded outwards. As loading progressed, a gradual but noticeable increase in the high traction region was observed. ( Figure 5 (5) and above, the tangential traction force distribution forms a sharp peak. The maximum value is close to 0.005. The gradually steepening change in the surface contour lines reflects the increasing local frictional resistance.

[0123] These dashed cross-sectional profiles further clarify the tangential behavior characteristics. Under low load conditions, the profile exhibits a wavy shape, with the central region showing a value close to zero and the sides showing an upward slope, which is typical of elastic micro-slip. As the tangential load increases, the slope gradually becomes steeper, and the plateau region continuously widens, indicating that the contact surface slippage phenomenon is gradually expanding. It is worth noting that when... hour( Figure 6 (8) The phenomenon of a continuously rising peak value suggests that the local contact is approaching the Coulomb limit. This indicates that overall slippage is about to occur. At this point, the distribution of the traction field tends to be uniform, which corresponds to the initial stage of global slippage and marks a critical transition node closely related to the starting torque of the ball screw.

[0124] Evolution of the distribution of sticky-slippery regions:

[0125] To supplement the analysis of the surface tangential traction force distribution, Figure 6 It shows that with external tangential force The evolution of the adhesion and slip regions on the elliptical contact interface as the size increases. Each sub-figure presents the evolution from... arrive Contact states under different load levels: The dark blue adhesive area (Astick) contrasts sharply with the light blue slip area (Aslip).

[0126] Under initial load conditions ( and The contact area is primarily in an adhesive state, with the adhesive zone almost covering the entire elliptical contact surface. As the tangential load increases, a noticeable reduction in the central adhesive zone can be observed. This contraction develops symmetrically radially outwards until… At this point, the adhesion area is clearly concentrated in the central part, and slippage begins to appear around the periphery. At that time, the adhesion zone shrinks into a narrow elliptical band along the major axis; eventually, in The fact that the contact state almost completely disappeared indicates that the contact state has essentially transformed into overall slip. This transformation process is related to... Figure 5 This is consistent with the tangential traction saturation phenomenon shown, where the peak value is reached when the friction limit is fully activated. The ratio began to stabilize.

[0127] The evolution of the adhesion and slip zones is related to the tangential traction force distribution characteristics discussed earlier. Figure 5 Directly related. In the initial stage of loading, The bimodal characteristic corresponds to the double-lobed slip zone formed at the contact edge. As the load increases, the peak value of the tangential traction force shifts inwards, which is related to... Figure 6 Contraction occurs synchronously in the adhesion zone. Negative contraction under high load. The disappearance of the component ( Figure 5 (7)-(9)) further confirms the dominance of forward slip. At this point, the adhesion zone almost completely disappears. Figure 5 (10) The contact state tends to be in full slip condition, which is of great significance for the prediction of STBS (starting torque of ball screw).

[0128] This invention comprehensively considers load distribution, surface roughness, and the microscopic sliding mechanics of the ball-raceway contact interface. The conjugate gradient method (CGM) is used to solve for the tangential pressure distribution and to delineate the adhesion / sliding regions. Experiments verified the predicted results, and the influence of preload and material properties on the starting torque was systematically analyzed, leading to the following conclusions:

[0129] (1) The model shows high prediction accuracy for starting torque, with a relative error of less than 3% compared to the existing NSK empirical formula. In addition, the 95% confidence interval of the model always covers all experimental measurements under different test conditions, which confirms its reliability in engineering applications.

[0130] (2) As the tangential load increases, the contact interface exhibits a significant shift from localized slippage to overall slippage. This characteristic manifests as a redistribution of tangential traction force from a distribution dominated by adhesion and concentrated at the center to a distribution dominated by slippage. This shift is accompanied by the gradual expansion of the slippage area, eventually leading to the disappearance of the central adhesion zone and triggering complete sliding.

[0131] The ball screw pair starting torque calculation method according to embodiments of the present invention achieves accurate prediction of the starting torque of the ball screw pair without relying on a large number of physical tests, and is suitable for rapid evaluation of starting torque under multiple working conditions and multiple parameters. In addition, the present invention integrates the critical slip state of each ball and establishes a distributed prediction calculation model for the starting torque, which can truly reflect the influence mechanism of the cooperative sliding of multiple balls in the ball screw pair on the starting torque, making the calculation results more accurate and more physically meaningful.

[0132] Although various embodiments of the invention have been described above, it should be understood that they are presented by way of example only and not as limitations. It will be apparent to those skilled in the art that various combinations, modifications, and alterations can be made without departing from the spirit and scope of the invention. Therefore, the breadth and scope of the invention disclosed herein should not be limited by the exemplary embodiments disclosed above, but should be defined solely by the appended claims and their equivalents.

Claims

1. A method for calculating the starting torque of a ball screw pair, characterized in that, Includes the following steps: Calculate the force on each ball in the ball screw pair under load; Calculate the tangential and normal displacements at any contact point between the ball and the raceway; The boundary element method is used to discretize the contact area between the ball and the raceway, and the Boussinesq-Cerruti equation is used to construct the discrete convolution relationship between tangential and normal displacements and shear stress. The discrete convolution relationship between tangential and normal displacements and shear stress is solved using the fast Fourier transform and the conjugate gradient method to obtain the shear stress distribution; The adhesion and slippage regions of the balls are determined based on the friction coefficient and shear stress distribution, and the evolution process of the stick-slip interface is tracked. When all the balls enter the slip state, the corresponding critical tangential load is recorded, and the starting torque of the screw is calculated through the distributed prediction calculation model of the starting torque. The force on each ball is calculated based on the deformation coordination relationship between the inner screw, nut, and ball of the ball screw pair after deformation. The calculation formula is as follows: ， In the formula, This represents the preload force on the leadscrew, which is equal to the sum of the forces acting on each ball in the axial direction. Indicates the first Each ball is subjected to normal force on the nut side; Indicates the first The contact angle of each ball; This indicates the normal force on the i-th ball on the nut side; Indicates each The contact angle of each ball; Indicates the helix angle; and Indicates material parameters; and These represent the stiffness coefficients of the nut and the lead screw, respectively. Any contact point between the ball and the raceway The calculation methods for the tangential and normal displacements at the location are as follows: Constructing the Green function The normal and tangential displacements can be expressed as the convolution of shear stress and normal pressure with the Green's function: ， in, ， ， ， ， In the formula, Represent the Green's function; and These represent the elastic moduli of the balls and raceways, respectively. and These represent the Poisson's ratios of the balls and raceways, respectively. , , These are intermediate parameters used to simplify expressions; It represents the reciprocal of the equivalent modulus of elasticity; The expression for the Boussinesq-Cerruti equation is: ， In the formula, Indicates continuous convolution; Indicates in Tangential displacement in the direction; Indicates in Tangential displacement in the direction; Indicates in Normal displacement in the direction; Represent the Green's function; , They represent in and Shear stress in the direction, Indicates normal force; The expression for shear stress distribution is as follows: ， In the formula, , and Representing the rectangular unit on , and Contact displacement in the direction; Indicates the influence coefficient; , They represent in and Shear stress in the direction; Indicates normal force; The normal and tangential displacements satisfy the following relationship: ， In the formula, , and Represents the rigid body displacement components. and Indicates tangential and normal displacement. This indicates the contact gap under normal loading conditions. Indicates the initial gap; , and Representing the rectangular unit on , , Contact displacement in the direction.

2. The method for calculating the starting torque of a ball screw pair according to claim 1, characterized in that, The contact area between the ball and the raceway is divided into contact areas. Non-contact area And satisfy the following conditions: The contact area meets the requirements of positive pressure and zero clearance; the non-contact area meets the requirements of zero pressure and has a clearance. The condition equations it satisfies are as follows: ， In the formula, Indicates the subscript of a rectangular cell; Indicates normal force; Indicates the contact gap; This indicates the normal force acting on the ball.

3. The method for calculating the starting torque of a ball screw pair according to claim 2, characterized in that, Contact area Divided into adhesion zones and slip zone And satisfy the following conditions: In the adhesion zone Within the slip zone, the shear stress is less than the static friction limit, and no relative sliding occurs; Internally, the direction of the tangential friction force is opposite to the direction of local slip, and the shear stress is equal to the friction limit; the condition equations it satisfies are as follows: ， In the formula, , They represent in and Shear stress in the direction; Indicates the subscript of a rectangular cell; Indicates normal force; Indicates the static friction coefficient; and This indicates the tangential displacement.

4. The method for calculating the starting torque of a ball screw pair according to claim 3, characterized in that, The criteria for determining the starting torque are as follows: With increasing tangential load, the adhesion zone... Gradually shrinking, slip zone Gradually expand until the entire contact area is reached. Full slip occurs; define critical tangential load. For the first The tangential load that enables the ball to slip as a whole, and the starting torque is the sum of the torques when all the balls simultaneously enter the fully slipped state; Among them, the critical tangential load The expression is: ， In the formula, Indicates the critical tangential load; express Shear stress in the direction; Indicates the size of the rectangular unit; The formula for the distributed prediction calculation model of the starting torque is: ， In the formula, Indicates the starting torque; Indicates the lead screw radius; Indicates the radius of the ball bearing; This indicates the contact angle of each ball; Indicates the helix angle; This indicates the total number of balls.

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