Method for calculating meshing stiffness of variable position planetary gear based on tooth contact analysis

Abstract

The application discloses a kind of based on the meshing stiffness calculation method of modification planetary gear based on tooth surface contact analysis, comprising the following steps: S100. Based on the engagement of the cutter and the modification gear when the cutter processes modification gear, the full tooth surface equation of the modification gear is established;S200. Based on the engagement of two modification gears, the contact equation of two modification gears is established, and the contact trajectory when two modification gears engage is calculated according to the contact equation;S300. According to the full tooth surface equation and the contact trajectory, the tooth surface point information on the contact trajectory is obtained, and the meshing stiffness is calculated using potential energy method according to the tooth surface point information.The meshing stiffness calculation method of modification planetary gear based on tooth surface contact analysis of the application can quickly and accurately calculate the time-varying meshing stiffness of planetary gear after modification, which is beneficial to the subsequent analysis of planetary gear system and suitable for popularization and use.

Description

TECHNICAL FIELD

[0001] The present application relates to gear mesh stiffness calculation technology, in particular to a kind of meshing stiffness calculation method of modification planetary gear based on tooth surface contact analysis. BACKGROUND

[0002] Time-varying mesh stiffness is one of the most important internal excitations in planetary gear system, and the modification of planetary gear can significantly improve the time-varying mesh stiffness of planetary gear, and then have a great influence on the dynamic characteristics of planetary gear. Therefore, it is of great significance to quickly and accurately calculate the time-varying mesh stiffness of planetary gear after modification for the subsequent dynamic analysis of planetary gear system.

[0003] In the prior art, the finite element method is generally used to analyze and predict the time-varying mesh stiffness of planetary gear after modification, but the finite element method is complex and time-consuming, which is not conducive to the subsequent analysis of planetary gear system. SUMMARY

[0004] The present application aims to solve at least one of the technical problems existing in the prior art. To this end, the present application provides a meshing stiffness calculation method of modification planetary gear based on tooth surface contact analysis, which can more quickly and accurately calculate the time-varying mesh stiffness of planetary gear after modification, is conducive to the subsequent analysis of planetary gear system, and is suitable for popularization and use.

[0005] The meshing stiffness calculation method of modification planetary gear based on tooth surface contact analysis according to the first aspect of the present application comprises the following steps:

[0006] S100. Based on the engagement of the tool and the modification gear during the machining of the modification gear by the tool, a full tooth surface equation of the modification gear is established;

[0007] S200. Based on the engagement of two modification gears, a contact equation of the two modification gears is established, and the contact trajectory of the two modification gears during engagement is calculated according to the contact equation;

[0008] S300. According to the full tooth surface equation and the contact trajectory, tooth surface point information on the contact trajectory is obtained, and the potential energy method is used to calculate the meshing stiffness according to the tooth surface point information.

[0009] The meshing stiffness calculation method of modification planetary gear based on tooth surface contact analysis according to the embodiment of the present application has at least the following beneficial effects:

[0010] In the present application, firstly, based on the engagement of the cutter and the modified gear when the cutter processes the modified gear, the full tooth surface equation of the modified gear can be established, the full tooth surface equation can be used to calculate the tooth surface point information of the modified gear, then based on the engagement of two modified gears, the contact equation is established, according to the contact equation, the contact trajectory when the two modified gears engage, that is, the engagement area, can be calculated, and then combined with the full tooth surface equation and the contact trajectory, the tooth surface point information at the engagement area can be quickly obtained, and according to the tooth surface point information at the engagement area, the potential energy method can be used to calculate the engagement stiffness. According to the modified planetary gear engagement stiffness calculation method based on tooth surface contact analysis in the embodiment of the present application, the time-varying engagement stiffness of the planetary gear after modification can be quickly and accurately calculated, which is beneficial to the subsequent analysis of the planetary gear system and is suitable for popularization and use.

[0011] According to some embodiments of the present application, in step S100, the cutter processes the modified gear by a gear hobbing cutter or a gear shaping cutter.

[0012] According to some embodiments of the present application, in step S100, a first coordinate system is established with any point on the center line of the modified gear as the origin, the plane where the first coordinate system is located is perpendicular to the center line of the modified gear, and the first coordinate system is fixedly connected with the modified gear;

[0013] A second coordinate system is established with any point on the center line of the gear hobbing cutter as the origin, the plane where the second coordinate system is located is perpendicular to the center line of the gear hobbing cutter, and the second coordinate system is fixedly connected with the gear hobbing cutter;

[0014] The modified gear and the gear hobbing cutter are relatively rotated, and the rotation angle of the modified gear is defined as φ ;

[0015] The distance between the center line of the gear hobbing cutter and the center line of the modified gear is calculated as follows:

[0016]

[0017] In the formula, r pg is the pitch circle radius of the modified gear, x is the modification coefficient of the modified gear, m is the module of the modified gear;

[0018] The full tooth surface equation is expressed as:

[0019]

[0020] In the formula, ( )is the coordinate transformation matrix between the first coordinate system and the second coordinate system, is a tooth surface equation of the hobbing cutter in the second coordinate system, a profile of a tooth top to a tooth root of a gear tooth of the hobbing cutter is composed of a CD segment, a DM segment and an MN segment in sequence, and the CD segment, the DM segment and the MN segment have different tooth surface equations.

[0021] According to some embodiments of the present application, in step S100, a third coordinate system is established with any point on a center line of the variable gear as an origin, a plane where the third coordinate system is located is perpendicular to the center line of the variable gear, and the third coordinate system is fixed with the variable gear; a fourth coordinate system is established with any point on a center line of the gear shaping cutter as an origin, a plane where the fourth coordinate system is located is perpendicular to the center line of the gear shaping cutter, and the fourth coordinate system is fixed with the gear shaping cutter; the variable gear and the gear shaping cutter are relatively rotated, a rotation angle of the variable gear is defined as φ g , and a rotation angle of the gear shaping cutter is defined as φ s ;

[0022] The full tooth surface equation is expressed as:

[0023]

[0024] In the formula, , is a coordinate transformation matrix between the third coordinate system and the fourth coordinate system, is a tooth surface equation of the gear shaping cutter in the fourth coordinate system, a profile of a tooth top to a tooth root of a gear tooth of the gear shaping cutter is composed of an EF segment, an FG segment and a GH segment in sequence, and the EF segment, the FG segment and the GH segment have different tooth surface equations.

[0025] According to some embodiments of the present application, in step S200, based on external meshing of two variable gears, the process of establishing the contact equation of the two variable gears includes: establishing a working tooth surface equation and a tooth surface unit external normal vector equation of a driving gear;

[0026] In the formula,

[0027]

[0028] In the formula, r ap is a tooth top radius of the driving gear, r bp is a base circle radius of the driving gear, θ 0p =π / 2Z ​​​p - inv α 1, Z p is the number of teeth of the driving gear, α 1 is the pressure angle of the driving gear;

[0029] The tooth surface unit outer normal vector equation of the driving gear is expressed as:

[0030] .

[0031] According to some embodiments of the present application, the process of establishing the contact equations of the two variable position gears further comprises:

[0032] establishing the working tooth surface equation and the tooth surface unit outer normal vector equation of the driven gear;

[0033] wherein the working tooth surface equation of the driven gear is expressed as:

[0034]

[0035] wherein, r ag is the addendum radius of the driven gear, r bg is the base circle radius of the driven gear, θ 0g =π / 2Z g - inv α 2 ， Z g is the number of teeth of the driven gear, α 2 is the pressure angle of the driven gear;

[0036] The tooth surface unit outer normal vector equation of the driven gear is expressed as:

[0037] .

[0038] According to some embodiments of the present application, the process of establishing the contact equations of the two variable position gears further comprises:

[0039] establishing a fifth coordinate system and a sixth coordinate system with any point on the center line of the driving gear as the origin, the plane where the fifth coordinate system and the sixth coordinate system are located is perpendicular to the center line of the driving gear, and the fifth coordinate system and the sixth coordinate system are located on the same plane;

[0040] wherein the fifth coordinate system is fixedly arranged, and the sixth coordinate system is fixedly arranged with the driving gear;

[0041] A seventh coordinate system and an eighth coordinate system are established with any point on the center line of the driven gear as the origin, and the planes of the seventh coordinate system and the eighth coordinate system are both perpendicular to the center line of the driven gear, wherein the seventh coordinate system is fixedly arranged, and the eighth coordinate system is fixedly arranged with the driven gear;

[0042] The driving gear and the driven gear are relatively rotated, and the rotation angle of the driving gear is defined as φ 1, and the rotation angle of the driven gear is defined as φ 2.

[0043] According to some embodiments of the present application, the process of establishing the contact equation of the two variable-gear gears further comprises:

[0044] The working tooth surface equation and the tooth surface unit outer normal vector equation of the driving gear are transformed into the fifth coordinate system:

[0045]

[0046] In the formula, ( φ 1) is the coordinate transformation matrix between the fifth coordinate system and the sixth coordinate system;

[0047] The working tooth surface equation and the tooth surface unit outer normal vector equation of the driven gear are transformed into the fifth coordinate system:

[0048]

[0049] In the formula, is the coordinate transformation matrix between the eighth coordinate system and the fifth coordinate system, ( φ 2) is the coordinate transformation matrix between the seventh coordinate system and the eighth coordinate system;

[0050] According to the local contact principle of gears, the contact equation of the driving gear and the driven gear in the fifth coordinate system is established:

[0051]

[0052]

[0053] The contact trajectory of the driving gear and the driven gear when meshing is obtained by solving the contact equation.

[0054] According to some embodiments of the present application, the potential energy method comprises: the teeth of the variable-gear gears are equivalent

[0055] The cantilever beam is equivalent to two variable-gear gears, and the meshing stiffness is solved based on the force balance and the energy conservation principle during the meshing process of the two variable-gear gears.

[0056] According to some embodiments of the present application, the bending stiffness of the shifted gear is calculated according to the potential energy method k b , the shear stiffness k s , the axial compression stiffness k a , the elastic matrix stiffness k f and the Hertz contact stiffness k h ;

[0057] The single tooth meshing stiffness of the shifted gear is expressed as:

[0058]

[0059] wherein, i denotes the i-th gear in the pair; i The double tooth meshing stiffness of the shifted gear is expressed as:

[0060]

[0061] .

[0062] Other features and advantages of the present application will be set forth in the description that follows, and in part will be apparent from the description, or can be learned by practice of the application. BRIEF DESCRIPTION OF DRAWINGS

[0063] The above and / or additional aspects and advantages of the present application will become apparent and be readily understood from the following description, taken in conjunction with the accompanying drawings, in which:

[0064] Figure 1 Flow chart for solving the meshing stiffness of the present application;

[0065] Figure 2 Schematic diagram of the gear tooth of the hobbing cutter;

[0066] Figure 3 Schematic diagram of the shifted gear machined by the hobbing cutter;

[0067] Figure 4 Schematic diagram of the gear tooth of the gear shaping cutter;

[0068] Figure 5 Schematic diagram of the shifted gear machined by the gear shaping cutter;

[0069] Figure 6 Schematic diagram of the external meshing of two shifted gears;

[0070] Figure 7 Schematic diagram of the internal meshing of two shifted gears; ​

[0071] Figure 8 Schematic diagram of tooth of external gear in variable pitch gear;

[0072] Figure 9 Schematic diagram of tooth of internal gear in variable pitch gear;

[0073] Figure 10 Comparison diagram of finite element method and calculation result of the present application;

[0074] Figure 11 Schematic diagram of meshing stiffness of external meshing gear pair under different variable pitch coefficients;

[0075] Figure 12 Schematic diagram of meshing stiffness of internal meshing gear pair under different variable pitch coefficients. DETAILED DESCRIPTION

[0076] The embodiments of the present application are described in detail below, examples of which are shown in the accompanying drawings. The embodiments described below by referring to the accompanying drawings are exemplary, only for explaining the present application, and cannot be understood as a limitation of the present application.

[0077] In the description of the present application, it is to be understood that, in relation to the description of the orientation, for example, the orientation or position relationship indicated by up, down, front, back, inner, outer, top, bottom, etc. is based on the orientation or position relationship shown in the drawings, only for the convenience of describing the present application and simplifying the description, and is not intended to indicate or imply that the device or element referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as a limitation of the present application.

[0078] In the description of the present application, multiple refers to two or more than two. If there is a description of first, second, only for the purpose of distinguishing technical features, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated or implicitly indicating the sequence of technical features indicated.

[0079] In the description of the present application, unless otherwise explicitly limited, the words such as setting, establishing, etc. should be broadly understood, and the person skilled in the art can reasonably determine the specific meaning of the above words in the present application in combination with the specific content of the technical scheme.

[0080] The following refers to Figures 1 to 12 The variable pitch planetary gear meshing stiffness calculation method based on tooth surface contact analysis according to the embodiments of the present application is described.

[0081] The method for calculating the meshing stiffness of modified planetary gears based on tooth surface contact analysis according to an embodiment of the present invention includes the following steps: S100. Based on the meshing between the cutting tool and the modified gear during machining, establish the full tooth surface equation of the modified gear; S200. Based on the meshing of two modified gears, establish the contact equation of the two modified gears, and calculate the contact trajectory of the two modified gears during meshing according to the contact equation; S300. Based on the full tooth surface equation and the contact trajectory, obtain the tooth surface point information on the contact trajectory, and calculate the meshing stiffness using the potential energy method based on the tooth surface point information.

[0082] The method for calculating the meshing stiffness of modified planetary gears based on tooth surface contact analysis according to embodiments of the present invention first establishes the full tooth surface equation of the modified gear based on the meshing between the cutting tool and the modified gear during machining. This full tooth surface equation can be used to calculate the tooth surface point information of the modified gear. Then, based on the meshing of two modified gears, a contact equation is established. According to the contact equation, the contact trajectory, i.e., the meshing region, when the two modified gears mesh can be calculated. Furthermore, by combining the full tooth surface equation and the contact trajectory, the tooth surface point information at the meshing region can be quickly obtained. Based on this tooth surface point information, the meshing stiffness can be calculated using the potential energy method. The method for calculating the meshing stiffness of modified planetary gears based on tooth surface contact analysis according to embodiments of the present invention can quickly and accurately calculate the time-varying meshing stiffness of the modified planetary gear, which is beneficial for subsequent analysis of planetary gear systems and is suitable for widespread use.

[0083] The steps S100 to S300 of the method for calculating the meshing stiffness of modified planetary gears based on tooth surface contact analysis according to the embodiments of the present invention will be described in more detail below.

[0084] In some embodiments of the present invention, step S100 is: based on the meshing between the cutting tool and the modified gear during machining, and according to the envelope principle, establishing the full tooth surface equation of the modified gear. The cutting tool can machine the modified gear by machining the teeth of the external gear in the modified gear with a hobbing cutter; or by machining the teeth of the internal gear in the modified gear with a gear shaper.

[0085] Specifically, in some embodiments of the present invention, when the teeth of an external gear are machined by a hobbing cutter, the hobbing cutter meshes with the external gear in the modified gear, such as... Figure 2 and Figure 3 As shown, a first coordinate system is established with any point on the centerline of the modified gear as the origin. First coordinate system The plane in which the gear is located is perpendicular to the centerline of the gear, in the first coordinate system. Fixed to the modified gear, i.e., the first coordinate system It can rotate in sync with the rotation of the modified gear;

[0086] A second coordinate system is established with any point on the center line of the hobbing cutter as the origin , the plane of the second coordinate system is perpendicular to the center line of the hobbing cutter, the second coordinate system is fixedly connected with the hobbing cutter, that is, the second coordinate system can move along with the rotation of the hobbing cutter;

[0087] A ninth coordinate system is established with any point on the center line of the modified gear as the origin , the plane of the ninth coordinate system is perpendicular to the center line of the modified gear, the ninth coordinate system is fixedly arranged and does not rotate along with the modified gear;

[0088] The modified gear is relatively rotated with the hobbing cutter, and the rotation angle of the modified gear is defined as φ , that is, the angle of rotation of the first coordinate system relative to the ninth coordinate system is φ .

[0089] Then the distance between the center line of the hobbing cutter and the center line of the modified gear is calculated r mg :

[0090]

[0091] In the formula, r pg is the pitch circle radius of the modified gear, x is the modification coefficient of the modified gear, and m is the modulus of the modified gear;

[0092] Then, the full tooth surface equation of the modified gear can be expressed as:

[0093]

[0094] In the formula, ( )is the coordinate transformation matrix between the first coordinate system and the second coordinate system, ( )is the tooth surface equation of the hobbing cutter in the second coordinate system, which can be obtained by referring to the data, the tooth top to the tooth root profile of the hobbing cutter is composed of CD segment, DM segment and MN segment in turn, and the CD segment, DM segment and MN segment have different tooth surface equations;

[0095] Among them, ( )can be expressed as:

[0096]

[0097] It should be noted that the profile of the gear tooth is symmetrically divided into two halves, and the CD segment, the DM segment and the MN segment constitute one of the halves. The CD segment, the DM segment and the MN segment have different shapes, for example, as shown in Figure 2 , the CD segment is approximately a straight segment, the DM segment is approximately a circular arc segment, and the MN segment is approximately a straight segment, and further, the CD segment, the DM segment and the MN segment have different tooth surface equations.

[0098] In some embodiments of the present application, when the gear tooth of the internal gear is processed by the gear shaping cutter, as shown in Figure 4 and Figure 5 , a third coordinate system is established with any point on the center line of the shifted gear as the origin, the plane of the third coordinate system is perpendicular to the center line of the shifted gear, the third coordinate system is fixedly connected with the shifted gear, that is, the third coordinate system can rotate with the rotation of the shifted gear;

[0099] a fourth coordinate system is established with any point on the center line of the gear shaping cutter as the origin, the plane of the fourth coordinate system is perpendicular to the center line of the gear shaping cutter, the fourth coordinate system is fixedly connected with the gear shaping cutter, that is, the fourth coordinate system can rotate with the rotation of the gear shaping cutter;

[0100] a tenth coordinate system is established with any point on the center line of the shifted gear as the origin, the plane of the tenth coordinate system is perpendicular to the center line of the shifted gear, the tenth coordinate system is fixedly arranged and does not rotate with the shifted gear; and an eleventh coordinate system is established with any point on the center line of the gear shaping cutter as the origin, the plane of the eleventh coordinate system is perpendicular to the center line of the gear shaping cutter, the eleventh coordinate system is fixedly arranged and does not rotate with the gear shaping cutter;

[0101] the shifted gear and the gear shaping cutter are relatively rotated, the rotation angle of the shifted gear is defined as φ g , that is, the rotation angle of the third coordinate system relative to the tenth coordinate system is φ g the rotation angle of the gear shaping cutter is defined as φ s , that is, the rotation angle of the fourth coordinate system relative to the eleventh coordinate system isφ s .

[0102] Then, the full tooth surface equation of the shifted gear can be expressed as:

[0103]

[0104] In the formula, ( , ) is the coordinate transformation matrix between the third coordinate system and the fourth coordinate system, ( ) is the tooth surface equation of the gear shaping cutter in the fourth coordinate system, which can be obtained by consulting the data. The tooth profile of the gear shaping cutter is composed of EF segment, FG segment and GH segment in turn, and the EF segment, FG segment and GH segment have different tooth surface equations.

[0105] Among them, ( , ) can be expressed as:

[0106]

[0107] In the formula, E is the center distance of the two gears without shifting, E+ E is the center distance of the two shifted gears, E is the difference between the center distance before and after the shifting of the two gears.

[0108] It should be noted that the tooth profile is symmetrically divided into two halves, and the EF segment, FG segment and GH segment constitute one half of the profile. The EF segment, FG segment and GH segment have different shapes, and thus the EF segment, FG segment and GH segment have different tooth surface equations.

[0109] φ g and φ s The ratio of

[0110]

[0111] In the formula, Z s is the number of teeth of the gear shaping cutter, Z g is the number of teeth of the shifted gear, therefore, only one of the values of φ g and φ s is needed to calculate the other value, which is more convenient for calculation.

[0112] In some embodiments of the present application, step S200 is: based on the meshing of the two profile shifted gears, a contact equation of the two profile shifted gears is established by using a tooth surface contact analysis method, and a contact trajectory when the two profile shifted gears are meshing is calculated according to the contact equation.

[0113] Specifically, in some embodiments of the present application, as shown in FIG. 2, the external meshing of the two profile shifted gears is simulated, and the contact equation of the two profile shifted gears is established, and the process includes: Figure 6

[0114] Firstly, a working tooth surface equation and a tooth surface unit outer normal vector equation of the driving gear are established.

[0115] The working tooth surface equation of the driving gear is expressed as:

[0116]

[0117] In the formula, r is the base circle radius of the driving gear, r ap r is the addendum radius of the driving gear, r bp r is the base circle radius of the driving gear, θ 0p =π / 2Z p - inv a 1, Z p z is the number of teeth of the driving gear, α 1 is the pressure angle of the driving gear;

[0118] The tooth surface unit outer normal vector equation of the driving gear is expressed as:

[0119] .

[0120] Secondly, a working tooth surface equation and a tooth surface unit outer normal vector equation of the driven gear are established.

[0121] The working tooth surface equation of the driven gear is expressed as:

[0122]

[0123] In the formula, r is the base circle radius of the driving gear, r ag r is the addendum radius of the driving gear, r bg r is the base circle radius of the driving gear, θ 0g =π / 2Z g - inv a 2 ， Z g z is the number of teeth of the driving gear, α ​2 represents the pressure angle of the driven wheel;

[0124] The equation of the unit external normal vector of the tooth surface of the driven gear is expressed as:

[0125] .

[0126] In addition, a fifth coordinate system is established with any point on the center line of the drive wheel as the origin. and the sixth coordinate system Fifth coordinate system and the sixth coordinate system The planes in which they lie are all perpendicular to the centerline of the drive wheel, where the fifth coordinate system... Fixed setting, sixth coordinate system Fixed to the driving wheel, which is the sixth coordinate system It can rotate in sync with the rotation of the drive wheel;

[0127] Establish a seventh coordinate system with any point on the center line of the driven wheel as the origin. and the eighth coordinate system The seventh coordinate system and the eighth coordinate system The planes in which they lie are all perpendicular to the centerline of the driven wheel, where the seventh coordinate system... Fixed settings, eighth coordinate system Fixed to the driven wheel, which is the eighth coordinate system. It can rotate in accordance with the rotation of the driven wheel;

[0128] The driving wheel and the driven wheel rotate relative to each other; the rotation angle of the driving wheel is defined as... φ 1, which is the sixth coordinate system Relative to the fifth coordinate system The angle of rotation is φ 1. The rotation angle of the driven wheel is defined as follows: φ 2, which is the eighth coordinate system Relative to the seventh coordinate system The angle of rotation is φ 2.

[0129] Finally, the equations of the working tooth surface and the unit external normal vector of the tooth surface of the driving gear are transformed to the fifth coordinate system.

[0130] The following is a list of related terms:

[0131]

[0132] In the formula, ( φ 1) is the coordinate transformation matrix between the fifth and sixth coordinate systems;

[0133] in, (φ The expression for 1) is:

[0134]

[0135] Transform the equations of the working tooth surface and the unit external normal vector of the tooth surface of the driven gear to the fifth coordinate system:

[0136]

[0137] In the formula, This is the coordinate transformation matrix between the fifth and eighth coordinate systems. ( φ 2) This is the coordinate transformation matrix between the seventh and eighth coordinate systems;

[0138] in, The expression is:

[0139]

[0140] ( φ The expression for 2) is:

[0141]

[0142] Based on the principle of local contact in gears, the contact equations between the driving gear and the driven gear in the fifth coordinate system are established as follows:

[0143]

[0144]

[0145] Expanding the contact equations above yields three independent sets of nonlinear equations, each containing four unknowns. φ With 1 as the input parameter, solving three nonlinear equations yields the contact trajectory when the driving wheel and driven wheel mesh.

[0146] like Figure 7 As shown, when the two modified gears are internally meshed, a similar coordinate system can be established according to the process of the two modified gears being externally meshed. Similarly, the contact equations of the driving gear and the driven gear can be established, and the contact trajectory of the driving gear and the driven gear when meshing can be obtained according to the contact equations. The solution principle and process are the same as when the two modified gears are externally meshed, so they will not be described again.

[0147] In some embodiments of the present invention, step S300 is as follows: based on the full tooth surface equation and the meshing trajectory, tooth surface point information on the meshing trajectory is obtained, and based on the tooth surface point information, the meshing stiffness is calculated using the potential energy method.

[0148] Specifically, after obtaining the contact trajectory of the driving and driven gears during meshing, the starting point, ending point, and range of the contact trajectory can be determined. Within the range of the contact trajectory, the tooth surface information within the contact trajectory range can be solved using the full tooth surface equation established in step S100, which can solve for tooth surface point information. Tooth surface point information includes various aspects, such as the moment of inertia, width, area, and direction of the meshing force at the corresponding position. Based on the tooth surface point information within the contact trajectory range, the meshing stiffness can be calculated using the potential energy method.

[0149] In some embodiments of the present invention, the principle of the potential energy method is as follows: the teeth of the modified gear are equivalent to a cantilever beam, and the meshing stiffness is solved based on the principle of force balance and energy conservation during the meshing process of the two modified gears.

[0150] like Figure 8 and Figure 9 As shown, a stiffness calculation coordinate system is established at the tooth root of the gear. coordinate system The origin is located on the root circle and on the line of symmetry of the tooth profile. The line of symmetry of the tooth profile is taken as... x In the axial direction, the tangential direction of the root circle is y Axial direction. Wherein F i To be at a distance from the tooth root d Dynamic meshing force at the point, F a For dynamic meshing force in x Components in the axial direction, F b For dynamic meshing force in y Components in the axial direction, for F i and F b The included angle, h To be at a distance from the tooth root d Half the width of the cross-section at that point, which is the distance between the tooth profile and the tooth root. d The distance h between the point at the location and the line of symmetry of the tooth profile. x To be at a distance from the tooth root x Half the width of the cross-section at that point, which is the distance between the tooth profile and the tooth root. x The distance between the point and the line of symmetry of the tooth profile, the cross section mentioned in this embodiment and x The axis is perpendicular, that is, perpendicular to the line of symmetry of the tooth profile.

[0151] In some embodiments of the present invention, the bending stiffness of the modified gear is determined according to the potential energy method. k b Shear stiffness ks Axial compressive stiffness k a Elastic matrix stiffness k f and Hertzian contact stiffness k h They can be represented as:

[0152]

[0153] in:

[0154]

[0155] In the formula, I x , representing the distance from the tooth root x Moment of inertia of the cross section at that point A x Represents the distance from the tooth root x The area of ​​the cross section at that point, E Represents Young's modulus, G Represents shear modulus, v Representative material: Poisson B Represents tooth width, This represents the distance from the intersection of the line of action and the line of symmetry of the gear teeth to the root circle. It is the arc length on the root circle between the two intersection points of the tooth profile curve and the root circle of a single gear tooth. It is a coefficient related to gear design parameters.

[0156] Furthermore, during the entire meshing process, the single-tooth meshing stiffness of the modified gear can be expressed as:

[0157]

[0158] In the formula, i Indicates the first in the meshing pair i One gear;

[0159] The double-tooth meshing stiffness of a modified gear can be expressed as:

[0160] .

[0161] In some embodiments of the present invention, step S400 is further included: verifying and analyzing the calculation method of the present invention.

[0162] First, to verify the correctness of the calculation method of this invention, the calculation results of the calculation method of this invention are compared with the calculation results of the finite element method. The specific calculation parameters of the meshing pair are shown in Table 1. The calculation results are as follows: Figure 10As shown in the figure, the abscissa is the normalized time variable, and the ordinate is the engagement stiffness. In the double-tooth engagement area, the maximum error of the calculation method of the application and the finite element method is 2.45%, and in the single-tooth engagement area, the maximum error of the calculation method of the application and the finite element method is only 0.59%, both of which are below 5%, which can verify the correctness of the calculation method of the application.

[0163] Table 1 Calculation parameters of the engagement pair

[0164]

[0165] Secondly, the engagement stiffness of the external engagement variable pitch gear pair and the internal engagement variable pitch gear pair under different variable pitch coefficients is calculated and verified. The external engagement variable pitch gear pair includes the sun gear and the planet gear, and the internal engagement variable pitch gear pair includes the planet gear and the ring gear. The parameters used for calculation are shown in Table 2.

[0166] It should be noted that in Table 2, the engagement angle of 22.4 degrees is the engagement angle between the sun gear and the planet gear, the engagement angle of 20 degrees is the engagement angle between the planet gear and the ring gear, the center distance is 93 mm, indicating that the center distance between the sun gear and the planet gear is 93 mm, and the center distance between the planet gear and the ring gear is also 93 mm. In addition, the Young's modulus of the sun gear, the planet gear and the ring gear is consistent, and the Poisson's ratio of the sun gear, the planet gear and the ring gear is consistent.

[0167] Table 2 Calculation parameters of the planetary gear system

[0168]

[0169] According to the parameters in Table 2, the engagement stiffness of the external engagement variable pitch gear pair under different variable pitch coefficients is calculated by programming, as shown in the figure Figure 11 , and the engagement stiffness of the internal engagement variable pitch gear pair under different variable pitch coefficients is calculated by programming, as shown in the figure Figure 12 . The abscissa is the normalized time variable, and the ordinate is the engagement stiffness. Different types of line segments represent different variable pitch coefficients.

[0170] As the variable pitch coefficient increases, as shown in the figure Figure 11 , the coincidence degree and the average engagement stiffness of the s-p variable pitch gear pair, that is, the external engagement variable pitch gear pair, do not change much, as shown in the figure Figure 12 , the coincidence degree and the average engagement stiffness of the r-p variable pitch gear pair, that is, the internal engagement variable pitch gear pair, are constantly increasing and changing more obviously, which is consistent with the actual situation, that is, the correctness of the application is verified from the coincidence degree and the average engagement stiffness.

[0171] The embodiments of the present application are described in detail above with reference to the drawings, but the present application is not limited to the above-described embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the spirit of the present application.

Claims

1. A method for calculating meshing stiffness of a variable position planetary gear based on tooth contact analysis, characterized by, The method comprises the following steps: S100. Based on the engagement of the cutter and the profile shifted gear during the machining of the profile shifted gear by the cutter, a full tooth surface equation of the profile shifted gear is established; S200. Based on the engagement of the two profile shifted gears, a contact equation of the two profile shifted gears is established, and according to the contact equation, a contact trajectory of the engagement of the two profile shifted gears is calculated; S300. According to the full tooth surface equation and the contact trajectory, tooth surface point information on the contact trajectory is obtained, and the potential energy method is used for the calculation of the engagement stiffness according to the tooth surface point information; In step S200, the process of establishing the contact equation based on the engagement of the two profile shifted gears comprises: a working tooth surface equation and a tooth surface unit outer normal vector equation of the driving gear are established; wherein the working tooth surface equation of the driving gear is expressed as: wherein r ap is the addendum radius of the driving wheel, r bp is the base radius of the driving wheel, θ 0p =π / 2Z p - invα 1, Z p is the number of teeth of the driving wheel, α 1 is the pressure angle of the driving wheel; The tooth surface unit outer normal vector equation of the driving gear is expressed as: ； The potential energy method comprises: the gear teeth of the profile shifted gear are equivalent to a cantilever beam, and the engagement stiffness is solved based on the force balance and the energy conservation principle during the engagement of the two profile shifted gears; According to the potential energy method, the bending stiffness of the variable pitch gear is calculated k b , the shear stiffness k s , the axial compression stiffness k a , the elastic matrix stiffness k f and the Hertz contact stiffness k h ; During the whole engagement process, the single-tooth engagement stiffness of the profile shifted gear is expressed as: In the formula, i Indicates the first in the meshing pair i One gear; The double-tooth engagement stiffness of the profile shifted gear is expressed as: 。 2. The meshing stiffness calculation method of a modified planetary gear based on a tooth contact analysis according to claim 1, characterized in that, In step S100, the profile shifted gear is machined by the gear hobbing cutter or the gear shaping cutter.

3. The profile shifted planetary gear engagement stiffness calculation method based on tooth surface contact analysis according to claim 2, characterized in that, In step S100, a first coordinate system is established with any point on the center line of the profile shifted gear as the origin, the plane of the first coordinate system is perpendicular to the center line of the profile shifted gear, and the first coordinate system is fixed with the profile shifted gear; a second coordinate system is established with any point on the center line of the gear hobbing cutter as the origin, the plane of the second coordinate system is perpendicular to the center line of the gear hobbing cutter, and the second coordinate system is fixed with the gear hobbing cutter; rotating the modified gear and the hobbing cutter relative to each other, the rotation angle of the modified gear being defined as φ ; the distance between the center line of the gear hobbing cutter and the center line of the profile shifted gear is calculated: wherein r pg is the reference circle radius of the gear, x is the modification coefficient of the gear, m is the module of the gear; the full tooth surface equation is expressed as: wherein is a coordinate transformation matrix between the first coordinate system and the second coordinate system, is a tooth surface equation of the hob in the second coordinate system, a profile of a tooth tip to a tooth root of a gear tooth of the hob is composed of a CD segment, a DM segment and an MN segment in sequence, the CD segment, the DM segment and the MN segment have different tooth surface equations.​​ 4. The profile shifted planetary gear engagement stiffness calculation method based on tooth surface contact analysis according to claim 2, characterized in that, In step S100, a third coordinate system is established with any point on the center line of the profile shifted gear as the origin, the plane of the third coordinate system is perpendicular to the center line of the profile shifted gear, and the third coordinate system is fixed with the profile shifted gear; a fourth coordinate system is established with any point on the center line of the gear shaping cutter as the origin, the plane of the fourth coordinate system is perpendicular to the center line of the gear shaping cutter, and the fourth coordinate system is fixed with the gear shaping cutter; rotating the profile gear and the gear shaper cutter relative to each other, the rotation angle of the profile gear being defined as φ g , and the rotation angle of the gear shaper cutter being defined as φ s ; the full tooth surface equation is expressed as: wherein , , is a coordinate transformation matrix between the third coordinate system and the fourth coordinate system, , is a tooth surface equation of the gear shaping cutter in the fourth coordinate system, the addendum-to-dedendum profile of the gear tooth of the gear shaping cutter is composed of the EF segment, the FG segment and the GH segment in turn, and the EF segment, the FG segment and the GH segment have different tooth surface equations.

5. The meshing stiffness calculation method of the modified planetary gear based on the tooth contact analysis according to any one of claims 1 to 4, characterized in that, The process of establishing the contact equation further comprises: a working tooth surface equation and a tooth surface unit outer normal vector equation of the driven gear are established; wherein the working tooth surface equation of the driven gear is expressed as: wherein r ag is the addendum radius of the driven gear, r bg is the base radius of the driven gear, θ 0g =π / 2Z g - invα 2 ， Z g is the number of teeth of the driven gear, α 2 is the pressure angle of the driven gear; The tooth surface unit outer normal vector equation of the driven gear is expressed as: 。 6. The meshing stiffness calculation method of a modified planetary gear based on a tooth contact analysis according to claim 5, characterized in that, The process of establishing the contact equation further comprises: A fifth coordinate system and a sixth coordinate system are established with any point on the center line of the driving gear as the origin, and the planes of the fifth coordinate system and the sixth coordinate system are perpendicular to the center line of the driving gear, wherein the fifth coordinate system is fixedly arranged, and the sixth coordinate system is fixedly arranged with the driving gear; A seventh coordinate system and an eighth coordinate system are established with any point on the center line of the driven gear as the origin, and the planes of the seventh coordinate system and the eighth coordinate system are perpendicular to the center line of the driven gear, wherein the seventh coordinate system is fixedly arranged, and the eighth coordinate system is fixedly arranged with the driven gear; rotating the driving wheel and the driven wheel relative to each other, the rotation angle of the driving wheel being defined as φ 1, the rotation angle of the driven wheel being defined as The process of establishing the contact equation further comprises:

2.

7. The meshing stiffness calculation method of a modified planetary gear based on a tooth contact analysis according to claim 6, characterized in that, The working tooth surface equation and the tooth surface unit outer normal vector equation of the driving gear are transformed into the fifth coordinate system: φ In the formula, The working tooth surface equation and the tooth surface unit outer normal vector equation of the driven gear are transformed into the fifth coordinate system: 1) is a coordinate transformation matrix between the fifth coordinate system and the sixth coordinate system;​ φ wherein is a coordinate transformation matrix between the fifth coordinate system and the eighth coordinate system, According to the local contact principle of gears, the contact equation of the driving gear and the driven gear in the fifth coordinate system is established: 2) is a coordinate transformation matrix between the seventh coordinate system and the eighth coordinate system;​ The contact equation is solved to obtain the contact trajectory when the driving gear and the driven gear mesh. ​

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