A calculation method of single tooth meshing stiffness based on spline tooth surface fretting friction

Abstract

The present invention discloses a method for calculating the meshing stiffness of a single tooth based on the micro-friction of a spline tooth surface, comprising: establishing a calculation model for the micro-friction of the spline tooth surface; establishing a stiffness characterization model for a single tooth of an internal spline and a single tooth of an external spline; establishing a contact stiffness model for the spline; and establishing a spline single-tooth meshing stiffness model for the spline according to the coupling relationship between the calculation model for micro-friction, the stiffness characterization model and the contact stiffness model. The present invention establishes a method for calculating the meshing stiffness of a single tooth taking into account the micro-friction of a spline tooth surface. The method can more accurately describe the meshing stiffness of a spline, which is of great significance for the evaluation and prediction of the dynamic performance of a shaft system.

Description

Technical Field

[0001] The invention relates to the technical field of data analysis, and in particular to a single-tooth meshing stiffness calculation method based on spline tooth surface micro-friction. Background Art

[0002] As a key component of the mechanical transmission system, the shaft system is usually connected by splines. Due to the space limitation of the shaft system, splines can usually only work under non-lubricated conditions. In addition, there is micro-friction between the tooth surfaces of the spline pair, and the friction coefficient is much higher than the sliding friction coefficient. Therefore, a large friction force will be generated between the tooth surfaces of the spline pair, seriously affecting the spline meshing stiffness. With the rapid development of high-end equipment, the requirements for the dynamic performance of the shaft system are increasing. As an important source of spline excitation, the spline meshing stiffness has a significant impact on the dynamic characteristics of the shaft system. More importantly, the direction of the spline tooth surface friction force is constantly changing during operation, and the effects of friction forces in different directions on the spline stiffness are different. For example, Figure 2 As shown in the figure, the continuous change of friction force causes the spline meshing stiffness to also change continuously, which seriously affects the dynamic characteristics of the shaft system.

[0003] At present, most of the research on spline meshing stiffness calculation ignores the influence of micro-friction between spline pairs, which makes it difficult to accurately calculate the spline meshing stiffness. In addition, although some calculation methods for gear meshing stiffness consider the influence of friction on gear meshing stiffness, due to the sliding friction between gear tooth surfaces, the classic Coulomb friction model is used to calculate tooth surface friction, while the spline tooth surface is micro-sliding, which does not meet the applicable conditions of Coulomb friction, so it cannot be used for spline meshing force calculation. Summary of the invention

[0004] An embodiment of the present invention provides a method for calculating the single-tooth meshing stiffness based on the micro-friction of the spline tooth surface, so as to solve the problem in the prior art that due to the sliding friction between the gear tooth surfaces, the classic Coulomb friction model is used to calculate the tooth surface friction, while the spline tooth surface is micro-sliding and does not meet the applicable conditions of Coulomb friction, so it cannot be used for spline meshing force calculation.

[0005] On the one hand, an embodiment of the present invention provides a method for calculating the single tooth meshing stiffness based on the spline tooth surface micro-friction, comprising:

[0006] A calculation model of the fretting friction of the tooth surface of the spline is established according to the shear stress in the sliding area and the shear stress in the adhesion area of ​​the spline;

[0007] Establish the stiffness characterization model of single tooth of internal spline and single tooth of external spline;

[0008] Establishing a contact stiffness model of the spline;

[0009] Establishing a spline single-tooth meshing stiffness model of the spline according to the coupling relationship between the micro-friction calculation model, the stiffness characterization model and the contact stiffness model;

[0010] The single tooth meshing stiffness of the spline is calculated according to the single tooth meshing stiffness model of the spline.

[0011] In a possible implementation, the stiffness characterization models of the internal spline single tooth and the external spline single tooth include a bending stiffness model, a shear stiffness model, and an axial compression stiffness model.

[0012] In a possible implementation, the calculation model of the fretting friction of the tooth surface of the spline is established based on the shear stress in the sliding area and the shear stress in the adhesion area of ​​the spline by Mindlin fretting friction theory.

[0013] In a possible implementation, the bending stiffness model, the shear stiffness model, and the axial compression stiffness model are established based on the calculation model of the micro-friction.

[0014] In a possible implementation, the contact stiffness model of the spline is established by Hertz contact theory.

[0015] In a possible implementation, the calculation model of the micro-friction of the tooth surface of the spline is used to obtain the axial force component, the bending force component and the bending moment generated by the axial force component of the spline teeth of the spline.

[0016] In a possible implementation, the bending stiffness model, the shear stiffness model, and the axial compression stiffness model are established based on a potential energy method.

[0017] The single tooth meshing stiffness calculation method based on spline tooth surface fretting friction in the present invention has the following advantages:

[0018] A single tooth meshing stiffness calculation method considering the spline tooth surface fretting friction is established. It can describe the spline meshing stiffness more accurately, which is of great significance for the evaluation and prediction of the dynamic performance of the shaft system. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0020] Figure 1 A schematic flow chart of a method for calculating single tooth meshing stiffness based on spline tooth surface fretting friction provided by an embodiment of the present invention;

[0021] Figure 2 A schematic diagram of the spline force process of a method for calculating the single-tooth meshing stiffness based on spline tooth surface fretting friction provided by an embodiment of the present invention;

[0022] Figure 3 A schematic diagram of the contact area of ​​two contact bodies of a single tooth meshing stiffness calculation method based on spline tooth surface fretting friction provided by an embodiment of the present invention;

[0023] Figure 4 A schematic diagram of force analysis of a single tooth on an external spline according to a method for calculating the single tooth meshing stiffness based on spline tooth surface fretting friction provided by an embodiment of the present invention;

[0024] Figure 5 A schematic diagram of a cantilever beam model of a single tooth meshing stiffness calculation method based on spline tooth surface fretting friction provided by an embodiment of the present invention;

[0025] Figure 6 A schematic diagram of the geometry of the external spline tooth root circle of a single tooth meshing stiffness calculation method based on spline tooth surface fretting friction provided by an embodiment of the present invention;

[0026] Figure 7 A schematic diagram of an internal spline equivalent cantilever beam according to a single-tooth meshing stiffness calculation method based on spline tooth surface fretting friction provided by an embodiment of the present invention;

[0027] Figure 8 A schematic diagram of the internal spline tooth root circle geometry of a single tooth meshing stiffness calculation method based on spline tooth surface fretting friction provided in an embodiment of the present invention;

[0028] Fig. 9 A schematic diagram of the rotation angle of a spline under different torques with and without considering fretting friction, according to a method for calculating the single-tooth meshing stiffness based on fretting friction of a spline tooth surface provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0029] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0030] Figure 1 A flow chart of a method for calculating the meshing stiffness of a single tooth based on the fretting friction of a spline tooth surface provided in an embodiment of the present invention; an embodiment of the present invention provides a method for calculating the meshing stiffness of a single tooth based on the fretting friction of a spline tooth surface, comprising:

[0031] A calculation model of the fretting friction of the tooth surface of the spline is established according to the shear stress in the sliding area and the shear stress in the adhesion area of ​​the spline;

[0032] Establish the stiffness characterization model of single tooth of internal spline and single tooth of external spline;

[0033] Establishing a contact stiffness model of the spline;

[0034] Establishing a spline single-tooth meshing stiffness model of the spline according to the coupling relationship between the micro-friction calculation model, the stiffness characterization model and the contact stiffness model;

[0035] The single tooth meshing stiffness of the spline is calculated according to the single tooth meshing stiffness model of the spline.

[0036] Exemplarily, this application establishes a calculation model for spline tooth surface micro-friction force based on Mindlin micro-friction theory; obtains the axial force component, bending force component and bending moment generated by the axial force component of the spline tooth based on the meshing force and micro-friction force; derives the expressions for the bending stiffness, shear stiffness and axial compression stiffness of the inner and outer spline single teeth based on the potential energy method; establishes a spline meshing Hertz contact stiffness characterization model based on the Hertz contact theory; considers the coupling relationship of each stiffness, and finally establishes a single tooth meshing stiffness calculation method considering the micro-friction of the spline tooth surface. Being able to more accurately describe the spline meshing stiffness is of great significance to the evaluation and prediction of the dynamic performance of the shaft system.

[0037] In a possible embodiment, the stiffness characterization model of the single tooth of the internal spline and the single tooth of the external spline includes a bending stiffness model, a shear stiffness model and an axial compression stiffness model; the calculation model of the micro-friction of the tooth surface of the spline established according to the shear stress in the sliding area and the shear stress in the adhesion area of ​​the spline is established through the Mindlin micro-friction theory; the bending stiffness model, the shear stiffness model and the axial compression stiffness model are established based on the calculation model of the micro-friction; the contact stiffness model of the spline is established through the Hertz contact theory; the calculation model of the micro-friction of the tooth surface of the spline is used to obtain the axial component of the spline tooth, the bending component and the bending moment generated by the axial component of the axial force; the bending stiffness model, the shear stiffness model and the axial compression stiffness model are established based on the potential energy method.

[0038] For example, a calculation model for spline tooth surface micro-friction force is first established, and the calculation formulas for bending stiffness, shear stiffness and axial compression stiffness considering tooth surface micro-friction are derived based on the potential energy method. Finally, the influence of spline bending stiffness, shear stiffness, axial compression stiffness and Hertz contact stiffness is integrated, and according to the coupling relationship between each stiffness, a calculation method for spline meshing stiffness considering tooth surface micro-friction is established. The overall calculation process is as follows: Figure 2 As shown;

[0039] 1. Fretting friction model of spline tooth surface

[0040] According to Mindlin's fretting friction theory, the contact area of ​​two contact bodies in fretting friction is divided into sliding area and adhesion area, such as Figure 3 As shown in the figure, p is the contact pressure distribution, τ x is the shear stress distribution, b is the contact area contact half-width, c is the half width of the adhesion area,

[0041] (1)

[0042] (2)

[0043] In the formula, L is the spline tooth width; is the curvature and of the contact body; E 1 / 2 and ξ 1 / 2 is the Young's modulus and Poisson's ratio of contact body 1 and contact body 2; is the tangential traction coefficient, defined as f / μF .

[0044] The shear stress in the sliding area is

[0045] (3)

[0046] The shear stress in the adhesion area is

[0047] (4)

[0048] The micro-friction force is

[0049]

[0050] In the formula, p 0 is the maximum contact stress, and its calculation formula is

[0051] (6)

[0052] Substituting formula (6) into formula (5), we have

[0053] (7)

[0054] Substituting equation (2) into equation (7), we have

[0055] (8)

[0056] 2. External spline bending, shear and axial compression stiffness model considering fretting friction

[0057] The bending, shear and axial compression potential energies of the external spline are:

[0058] (9)

[0059] In the formula, A x and I x is the vertical distance from the intersection of the involute and the minor diameter circle of the external spline x The area and moment of inertia of the section at Figure 4 As shown; M Axial force F a Bending moment; E and G are Young's modulus and shear modulus, respectively; F b is the shear component; U b is the bending potential energy; U s is the shear potential energy; U a is the axial compression potential energy, according to Figure 5 Knowable

[0060] (10)

[0061] In the formula, ζ is the directional coefficient of the fretting friction force. When the fretting friction force is positive, its value is 1, and when the fretting friction force is negative, its value is -1; h is the half tooth thickness at the meshing point; α 1 is the angle between the meshing line and the tooth thickness direction.

[0062] The geometric diagram of the root circle of the external spline is as follows Figure 6 As shown, substituting formula (10) into formula (9), we have

[0063]

[0064] The expressions of related variables in formula (11-13) are:

[0065] (14)

[0066] (15)

[0067] (16)

[0068] (17)

[0069] (18)

[0070] (19)

[0071] (20)

[0072] (twenty one)

[0073] (twenty two)

[0074] (twenty three)

[0075] (twenty four)

[0076] In the formula, h Half tooth thickness; d It is the distance from the meshing point to the intersection of the involute and the minor diameter circle along the direction of the tooth profile symmetry line; d 2 is the distance from the starting point of the involute tooth profile to the intersection of the involute and the minor diameter circle along the direction of the tooth profile symmetry line; d 1 is the distance from the starting point of the involute tooth profile to the end point of the root fillet along the direction of the tooth profile symmetry line; h x It is the vertical distance from any point on the involute tooth profile to the tooth symmetry line; x It is the distance from any point on the involute tooth profile to the intersection of the involute and the minor diameter circle along the direction of the tooth profile symmetry line; x 1 is the distance from any point of the tooth root fillet to the starting point of the involute tooth profile along the direction of the tooth profile symmetry line; h x1 It is the vertical distance from any point on the tooth root fillet to the tooth profile symmetry line; d x for x The integration unit of d x1 for x 1 integral unit; I x and A x The distance on the involute line from the starting point of the involute tooth profile x The moment of inertia and cross-sectional area of ​​the gear tooth section at ; I x1 and A x1 The distance from the starting point of the involute tooth profile on the tooth root fillet x 1. Moment of inertia and cross-sectional area of ​​the gear tooth section; R b is the base circle radius; r e is the tooth root fillet radius; α 2 is the half tooth angle;α 2′ is the angle between the meshing line and the intersection of the base circle at the starting point of the involute tooth profile in the tooth thickness direction; α is the distance between the meshing point and the tooth profile line in the vertical direction; β Distance from the starting point of the involute tooth profile on the tooth root fillet x The angle between the radius of the tooth root circle at 1 and the perpendicular line of the tooth profile symmetry line passing through the center of the tooth root fillet circle; β 0 is the angle between the root fillet radius at the starting point of the involute tooth profile and the perpendicular line of the tooth profile symmetry line passing through the center of the root fillet circle.

[0077] In addition, there are

[0078] (25)

[0079] In the formula, K b , K s and K a It is the bending stiffness, shear stiffness and axial compression stiffness of a single spline tooth.

[0080] Combining equation (25) and equation (11-13), we have

[0081]

[0082] In the formula, σ 1 is an intermediate variable for convenient expression, and its expression is

[0083] (29)

[0084] α 1, α 2 and α The expression of 2′ is

[0085] (30)

[0086] (31)

[0087] (32)

[0088] The geometric diagram of the root circle of the external spline is as follows Figure 6 , β 1, β 2 and β The expression for 0 is

[0089] (32)

[0090] (33)

[0091] (34)

[0092] 3. Internal spline bending, shear and axial compression stiffness models considering fretting friction

[0093] The schematic diagram of the internal spline equivalent cantilever beam model is as follows Figure 7 As shown, the geometric diagram of the internal spline tooth root circle is as follows Figure 8 , the bending potential energy, shear potential energy and axial compression potential energy of the internal spline are

[0094]

[0095] In the formula, γ 1 is the angle between the meshing line of the intersection of the pitch circle and the involute tooth profile and the tangent point of the base circle and the symmetry line of the tooth profile; d i It is the distance from the intersection of the pitch circle and the involute tooth profile to the end point of the involute tooth profile along the symmetry line of the tooth profile; x i It is the distance from any point on the internal spline involute tooth profile to the end point of the internal spline involute tooth profile along the tooth profile symmetry line; d i It is the distance from the intersection of the pitch circle and the involute tooth profile to the end point of the internal spline involute tooth profile along the symmetry line of the tooth profile; h i It is the vertical distance from the intersection of the pitch circle and the involute tooth profile to the symmetry line of the tooth profile; h ix It is the vertical distance from any point on the involute tooth profile of the internal spline to the symmetry line of the tooth profile; x i1 It is the distance from any point on the root circle of the internal spline to the end point of the involute tooth profile along the symmetry line of the tooth profile; I ix and A ix The distance on the involute line from the end point of the involute tooth profile x i The moment of inertia and cross-sectional area of ​​the gear tooth section at ; I ix1 and A ix1 The distance from the end point of the involute tooth profile on the root circle of the inner gear ring x i1 The moment of inertia and cross-sectional area of ​​the gear tooth section at ; its expression is

[0096] (38)

[0097] (39)

[0098] (40)

[0099] (41)

[0100] (42)

[0101] (43)

[0102] (44)

[0103] (45)

[0104] (46)

[0105] (47)

[0106] (48)

[0107] In the formula, R Fi and r i is the radius of the termination circle of the internal spline involute and the radius of the tooth root fillet; γ2 is the angle between the intersection of the involute and the base circle and the symmetry line of the internal spline tooth profile; γ is the angle between any point of the involute and the symmetry line of the internal spline tooth profile; Φ is the angle between the termination point of the internal spline involute and the tangent point of the base circle and the symmetry line of the internal spline tooth profile; β Distance from the end point of the involute tooth profile on the tooth root fillet x The angle between the radius of the tooth root fillet at 1 and the perpendicular line of the tooth profile symmetry line passing through the center of the tooth root fillet; β i0 It is the angle between the root fillet radius at the starting point of the involute tooth profile and the perpendicular line of the tooth profile symmetry line passing through the center of the root fillet circle; h ix1 It is the vertical distance from any point on the tooth root fillet to the tooth profile symmetry line.

[0108] Similar to the external spline, the expressions for the bending stiffness, shear stiffness and axial compression stiffness of the internal spline can be derived as follows:

[0109]

[0110] γ and Φ The expression is

[0111] (52)

[0112] (53)

[0113] β i1 and βi0 The expression is

[0114] (54)

[0115] (55)

[0116] (56)

[0117] (57)

[0118] 4. Spline single tooth meshing stiffness considering micro-friction

[0119] The expression of spline single tooth meshing stiffness is:

[0120] (58)

[0121] In the formula, K h is the Hertzian contact stiffness, which is expressed as

[0122] (59)

[0123] In the formula, ν is the Poisson's ratio of the material.

[0124] The results of the spline rotation angle under different torques when considering and not considering the fretting friction are as follows: Fig. 9 The results show that the single tooth meshing stiffness of the timing spline is 1.0608×10 9 N / m, when the micro-friction is negative, the single tooth meshing stiffness of the spline is 1.0092×10 9 N / m, without considering the micro-friction, the spline single tooth meshing stiffness is 1.0459×10 9 N / m.

[0125] Although the preferred embodiments of the present invention have been described, those skilled in the art may make additional changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention.

[0126] Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A method for calculating the meshing stiffness of a single tooth based on the micro-friction of the spline tooth surface, characterized in that: include: A calculation model of the fretting friction of the tooth surface of the spline is established according to the shear stress in the sliding area and the shear stress in the adhesion area of ​​the spline; The calculation model of the fretting friction of the spline tooth surface established according to the shear stress in the sliding area and the shear stress in the adhesion area of ​​the spline is established by Mindlin fretting friction theory; Establish the stiffness characterization model of single tooth of internal spline and single tooth of external spline; The stiffness characterization models of the internal spline single tooth and the external spline single tooth include a bending stiffness model, a shear stiffness model and an axial compression stiffness model; Establishing a contact stiffness model of the spline; Establishing a spline single-tooth meshing stiffness model of the spline according to the coupling relationship among the micro-friction calculation model, the stiffness characterization model and the contact stiffness model; The single tooth meshing stiffness of the spline is calculated according to the single tooth meshing stiffness model of the spline.

2. The method for calculating the single tooth meshing stiffness based on the spline tooth surface fretting friction according to claim 1, characterized in that: The bending stiffness model, the shear stiffness model and the axial compression stiffness model are established based on the calculation model of the micro-friction.

3. The method for calculating the single tooth meshing stiffness based on the spline tooth surface fretting friction according to claim 1, characterized in that: The contact stiffness model of the spline is established by Hertz contact theory.

4. The method for calculating the single tooth meshing stiffness based on the spline tooth surface fretting friction according to claim 1, characterized in that: The calculation model of the micro-friction of the tooth surface of the spline is used to obtain the axial force component, bending force component and bending moment generated by the axial force component of the spline teeth of the spline.

5. The method for calculating the single tooth meshing stiffness based on the spline tooth surface fretting friction according to claim 2, characterized in that: The bending stiffness model, the shear stiffness model and the axial compression stiffness model are established based on the potential energy method.

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