CN115952653A - Method for calculating meshing stiffness of modified planet gear based on tooth surface contact analysis - Google Patents
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Abstract
The invention discloses a method for calculating meshing rigidity of a modified planet gear based on tooth surface contact analysis, which comprises the following steps of: s100, establishing a full tooth surface equation of the modified gear based on the engagement of the cutter and the modified gear when the modified gear is machined by the cutter; s200, establishing a contact equation of the two modified gears based on the engagement of the two modified gears, and calculating a contact track when the two modified gears are engaged according to the contact equation; s300, obtaining tooth surface point information on the contact track according to the full tooth surface equation and the contact track, and calculating the meshing stiffness by using a potential energy method according to the tooth surface point information. The method for calculating the meshing stiffness of the modified planetary gear based on the tooth surface contact analysis can quickly and accurately calculate the time-varying meshing stiffness of the modified planetary gear, is favorable for subsequent analysis of a planetary gear system, and is suitable for popularization and application.
Description
Technical Field
The invention relates to a gear meshing rigidity calculation technology, in particular to a method for calculating the meshing rigidity of a modified planet gear based on tooth surface contact analysis.
Background
The time-varying meshing stiffness is one of the most important internal excitations in a planetary gear system, and the deflection of the planetary gear can obviously improve the time-varying meshing stiffness of the planetary gear, thereby having great influence on the dynamic characteristics of the planetary gear. Therefore, the time-varying meshing stiffness of the planetary gear after deflection can be quickly and accurately calculated, and the time-varying meshing stiffness of the planetary gear has important significance for subsequent dynamic analysis of the planetary gear system.
In the prior art, the time-varying meshing stiffness of the planetary gear after deflection is generally analyzed and predicted by a finite element method, but the finite element method is complex in calculation, long in calculation time and not beneficial to subsequent analysis of the planetary gear system.
Disclosure of Invention
The present invention is directed to solving at least one of the problems of the prior art. Therefore, the method for calculating the meshing stiffness of the modified planet gear based on the tooth surface contact analysis can calculate the time-varying meshing stiffness of the planet gear after modification more quickly and accurately, is favorable for subsequent analysis of a planet gear system, and is suitable for popularization and application.
According to the method for calculating the meshing stiffness of the modified planetary gear based on the tooth surface contact analysis, the method comprises the following steps:
s100, establishing a full tooth surface equation of the modified gear based on the engagement of the cutter and the modified gear when the modified gear is machined by the cutter;
s200, establishing a contact equation of the two modified gears based on the engagement of the two modified gears, and calculating a contact track when the two modified gears are engaged according to the contact equation;
s300, obtaining tooth surface point information on the contact track according to the full tooth surface equation and the contact track, and calculating the meshing stiffness by using a potential energy method according to the tooth surface point information.
The method for calculating the meshing stiffness of the modified planetary gear based on the tooth surface contact analysis has the following beneficial effects:
according to the method, firstly, based on the meshing of the cutter and the modified gear when the cutter is used for machining the modified gear, a full tooth surface equation of the modified gear can be established, the full tooth surface equation can be used for calculating the tooth surface point information of the modified gear, then, based on the meshing of the two modified gears, a contact equation is established, according to the contact equation, the contact track when the two modified gears are meshed, namely, a meshing area can be calculated, further, the full tooth surface equation and the contact track are combined, the tooth surface point information at the meshing area can be quickly obtained, and according to the tooth surface point information at the meshing area, the potential energy method is utilized to calculate the meshing rigidity. According to the method for calculating the meshing stiffness of the modified planetary gear based on the tooth surface contact analysis, the time-varying meshing stiffness of the modified planetary gear can be quickly and accurately calculated, the subsequent analysis of a planetary gear system is facilitated, and the method is suitable for popularization and application.
According to some embodiments of the invention, in step S100, the tool machines the modified gear by: and machining the modified gear by a hobbing cutter or a slotting cutter.
According to some embodiments of the present invention, in step S100, a first coordinate system is established by using any point on a center line of the modified gear as an origin, a plane of the first coordinate system is perpendicular to the center line of the modified gear, and the first coordinate system is fixedly connected to the modified gear;
establishing a second coordinate system by taking any point on the central line of the hobbing cutter as an origin, wherein the plane of the second coordinate system is vertical to the central line of the hobbing cutter, and the second coordinate system is fixedly connected with the hobbing cutter;
enabling the modified gear and the hobbing cutter to rotate relatively, wherein the rotation angle of the modified gear is defined as phi;
calculating the distance between the center line of the hobbing cutter and the center line of the modified gear:
r mg ＝r pg +xm
in the formula, r pg The radius of a reference circle of the modified gear is defined as x, the modification coefficient of the modified gear is defined as m, and the modulus of the modified gear is defined as m;
the full tooth surface equation is expressed as:
in the formula, M gc (φ j ) Is a coordinate transformation matrix between the first coordinate system and the second coordinate system,
and the contour from the tooth top to the tooth bottom of the gear tooth of the hobbing cutter sequentially consists of a CD section, a DM section and an MN section, and the CD section, the DM section and the MN section have different tooth surface equations.
According to some embodiments of the present invention, in step S100, a third coordinate system is established with any point on the center line of the modified gear as an origin, a plane of the third coordinate system is perpendicular to the center line of the modified gear, and the third coordinate system is fixedly connected to the modified gear; establishing a fourth coordinate system by taking any point on the central line of the pinion cutter as an origin, wherein the plane of the fourth coordinate system is vertical to the central line of the pinion cutter, and the fourth coordinate system is fixedly connected with the pinion cutter; the modified gear and the gear shaping cutter are relatively rotated, and the rotation angle of the modified gear is defined as phi g The rotation angle of the pinion cutter is defined as phi s ；
The full tooth surface equation is expressed as:
in the formula (I), the compound is shown in the specification,
for a coordinate transformation matrix between the third coordinate system and the fourth coordinate system, ->
For the gear shaper cutter is in the flank of tooth equation under the fourth coordinate system, the profile of the tooth top to the tooth root of the teeth of the gear shaper cutter comprises EF section, FG section and GH section in proper order, and EF section, FG section and GH section have different flank of tooth equations.
According to some embodiments of the present invention, the process of establishing the contact equation for two of the modified gears based on the external engagement of the two modified gears in step S200 includes: establishing a working tooth surface equation and a tooth surface unit external normal vector equation of the driving wheel;
the working tooth surface equation of the driving wheel is expressed as follows:
in the formula, r ap The tooth crest radius of the driving wheel, r bp Is the base radius of the driving wheel, theta 0p ＝π/2Z p -invα 1 ，Z p Is the number of teeth of the driving wheel, alpha 1 Is the pressure angle of the driving wheel;
the tooth surface unit external normal vector equation of the driving wheel is expressed as follows:
according to some embodiments of the invention, establishing the contact equation for two of the modified gears further comprises:
establishing a working tooth surface equation and a tooth surface unit external normal vector equation of the driven wheel;
wherein the working tooth surface equation of the driven wheel is expressed as:
in the formula, r ag Is the tooth crest radius of the driven wheel, r bg Is the base radius of the driven wheel, theta 0g ＝π/2Z g -invα 2 ，Z g Is the number of teeth of the driven wheel, α 2 Is the pressure angle of the driven wheel;
the tooth surface unit external normal vector equation of the driven wheel is expressed as follows:
according to some embodiments of the invention, establishing the contact equation for two of the modified gears further comprises:
establishing a fifth coordinate system and a sixth coordinate system by taking any point on the center line of the driving wheel as an origin, wherein the planes of the fifth coordinate system and the sixth coordinate system are vertical to the center line of the driving wheel, the fifth coordinate system is fixedly arranged, and the sixth coordinate system is fixedly connected with the driving wheel;
establishing a seventh coordinate system and an eighth coordinate system by taking any point on the central line of the driven wheel as an origin, wherein the planes of the seventh coordinate system and the eighth coordinate system are vertical to the central line of the driven wheel, the seventh coordinate system is fixedly arranged, and the eighth coordinate system is fixedly connected with the driven wheel;
the driving wheel and the driven wheel are relatively rotated, and the rotation angle of the driving wheel is defined as phi 1 The rotation angle of the driven wheel is defined as phi 2 。
According to some embodiments of the invention, establishing the contact equation for two of the modified gears further comprises:
transforming the working tooth surface equation and the tooth surface unit external normal vector equation of the driving wheel to the fifth coordinate system:
in the formula, M fp (φ 1 ) A coordinate transformation matrix between the fifth coordinate system and the sixth coordinate system;
transforming the working tooth surface equation and the tooth surface unit external normal vector equation of the driven wheel to the fifth coordinate system:
in the formula, M fc Is a coordinate transformation matrix between the eighth coordinate system and the fifth coordinate system, M cg (φ 2 ) A coordinate transformation matrix between the seventh coordinate system and the eighth coordinate system;
according to the gear local contact principle, establishing a contact equation of the driving wheel and the driven wheel in the fifth coordinate system:
and solving the contact equation to obtain a contact track when the driving wheel is meshed with the driven wheel.
According to some embodiments of the invention, the potential energy method comprises: and (3) equivalent the gear teeth of the modified gears to be cantilever beams, and solving the meshing stiffness based on the stress balance and energy conservation principle in the meshing process of the two modified gears.
According to some embodiments of the invention, the bending stiffness k of the modified gear is calculated according to the potential energy method b Shear stiffness k s Axial compression stiffness k a Elastic matrix stiffness k f And Hertz contact stiffness k h ；
The single-tooth meshing stiffness of the modified gear is expressed as follows during the whole meshing process:
wherein i represents the i-th gear in the meshing pair;
the double-tooth meshing rigidity of the modified gear is expressed as follows:
additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a flow chart for solving for meshing stiffness of the present invention;
FIG. 2 is a schematic gear tooth view of a hobbing cutter;
FIG. 3 is a schematic view of a hobbing cutter machining a modified gear;
FIG. 4 is a gear tooth schematic of the pinion cutter;
FIG. 5 is a schematic view of a pinion cutter machining a modified gear;
FIG. 6 is a schematic view of two modified gears being externally engaged;
FIG. 7 is a schematic view of the meshing of two profile gears;
FIG. 8 is a schematic view of the teeth of an outer gear of the modified gear;
FIG. 9 is a schematic illustration of the teeth of the inner gear in the modified gear;
FIG. 10 is a comparison of the results of finite element method and calculations of the present invention;
FIG. 11 is a schematic diagram of meshing stiffness of an external gear pair under different deflection coefficients;
fig. 12 is a schematic view of mesh stiffness of the ring gear pair under different deflection coefficients.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
In the description of the present invention, it should be understood that the positional descriptions referred to, for example, the upper, lower, front, rear, inner, outer, top, bottom, etc. are based on the positional or positional relationships shown in the drawings, which are for convenience of description and simplicity of description, and do not 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, are not to be considered as limiting.
In the description of the present invention, a plurality means two or more. If there is a description of first and second for the purpose of distinguishing technical features only, this is not to be understood as indicating or implying a relative importance or implicitly indicating the number of technical features indicated or implicitly indicating the precedence of technical features indicated.
In the description of the present invention, unless otherwise explicitly defined, terms such as setup, establishment, etc. should be understood in a broad sense, and those skilled in the art can reasonably determine the specific meanings of the above terms in the present invention by combining the specific contents of the technical solutions.
A method of calculating the meshing stiffness of a modified planetary gear based on the tooth surface contact analysis according to an embodiment of the present invention will be described below with reference to fig. 1 to 12.
The method for calculating the meshing rigidity of the modified planetary gear based on the tooth surface contact analysis comprises the following steps: s100, establishing a full tooth surface equation of the modified gear based on the engagement of the cutter and the modified gear when the cutter processes the modified gear; s200, establishing a contact equation of the two modified gears based on the engagement of the two modified gears, and calculating a contact track when the two modified gears are engaged according to the contact equation; s300, obtaining tooth surface point information on the contact track according to the full tooth surface equation and the contact track, and calculating the meshing stiffness by using a potential energy method according to the tooth surface point information.
According to the method for calculating the meshing stiffness of the modified planet gear based on the tooth surface contact analysis, firstly, a full tooth surface equation of the modified gear is established based on the meshing of a cutter and the modified gear when the cutter processes the modified gear, the full tooth surface equation can be used for calculating the tooth surface point information of the modified gear, then, a contact equation is established based on the meshing of two modified gears, according to the contact equation, the contact track when the two modified gears are meshed, namely the meshing area, can be calculated, further, the full tooth surface equation and the contact track are combined, the tooth surface point information at the meshing area can be quickly obtained, and according to the tooth surface point information at the meshing area, the potential energy method is utilized to calculate the meshing stiffness. According to the method for calculating the meshing stiffness of the modified planetary gear based on the tooth surface contact analysis, the time-varying meshing stiffness of the modified planetary gear can be quickly and accurately calculated, the subsequent analysis of a planetary gear system is facilitated, and the method is suitable for popularization and application.
The following describes steps S100 to S300 of the method for calculating the meshing stiffness of a modified planetary gear based on the tooth surface contact analysis according to the embodiment of the present invention in more detail.
In some embodiments of the present invention, step S100 is: based on the meshing of the cutter and the modified gear when the cutter processes the modified gear, the full tooth surface equation of the modified gear is established according to the envelope principle. The cutter can process the modified gear by processing the gear teeth of an external gear in the modified gear through a hobbing cutter; or the gear teeth of the internal gear in the modified gear are processed through the gear shaping cutter.
Specifically, in some embodiments of the present invention, when the teeth of the external gear are machined by the hobbing cutter, the hobbing cutter meshes with the external gear in the modified gear, and as shown in fig. 2 and 3, the first coordinate system x is established with any point on the center line of the modified gear as the origin g O g y g First coordinate system x g O g y g The plane is perpendicular to the central line of the modified gear, and the first coordinate system x g O g y g Fixedly connected to the modified gear, i.e. the first coordinate system x g O g y g Can rotate along with the rotation of the modified gear;
establishing a second coordinate system x by taking any point on the central line of the hobbing cutter as an origin c O c y c Second coordinate system x c O c y c The plane is perpendicular to the central line of the hobbing cutter, and a second coordinate system x c O c y c Attached to the hobbing cutter, i.e. the second coordinate system x c O c y c Can move along with the rotation of the hobbing cutter;
establishing a ninth coordinate system x with any point on the center line of the modified gear as the origin f O f y f Ninth coordinate system x f O f y f The plane is perpendicular to the central line of the modified gear, and the ninth coordinate system x f O f y f The gear is fixedly arranged and does not rotate along with the modified gear;
the modified gear and the hobbing cutter are relatively rotated, and the rotation angle of the modified gear is defined as phi, namely a first coordinate system x g O g y g Relative to a ninth coordinate system x f O f y f The angle of rotation is phi.
Then calculating the distance r between the center line of the hobbing cutter and the center line of the modified gear mg ：
r mg ＝r pg +xm
In the formula, r pg The radius of a reference circle of the modified gear is shown, x is the modification coefficient of the modified gear, and m is the modulus of the modified gear;
the full flank equation for a modified gear can then be expressed as:
in the formula, M gc (φ j ) Is a coordinate transformation matrix between a first coordinate system and a second coordinate system,
the tooth surface equation of the hobbing cutter under the second coordinate system can be obtained by looking up data, the contour from the tooth top to the tooth bottom of the gear tooth of the hobbing cutter sequentially consists of a CD section, a DM section and an MN section, and the CD section, the DM section and the MN section have different tooth surface equations;
wherein, M gc (φ j ) Can be expressed as:
it should be noted that the profile of the gear tooth is symmetrically divided into two halves, and the CD section, the DM section, and the MN section constitute one half of the profile. The CD section, the DM section, and the MN section have different shapes, for example, as shown in fig. 2, the CD section is a substantially straight section, the DM section is a substantially circular arc section, and the MN section is a substantially straight section, and thus the CD section, the DM section, and the MN section have different tooth surface equations.
In some embodiments of the present invention, when the teeth of the internal gear are machined by the slotting cutter, as shown in fig. 4 and 5, a third coordinate system x is established with any point on the center line of the modified gear as an origin g O g y g Third coordinate System x g O g y g The plane is perpendicular to the central line of the modified gear, and a third coordinate system x g O g y g Fixedly connected with modified gears, i.e. a third coordinate system x g O g y g Can rotate along with the rotation of the modified gear;
establishing a fourth coordinate system x by taking any point on the central line of the pinion cutter as an origin s O s y s Fourth coordinate system x s O s y s The plane is vertical to the central line of the pinion cutter, and a fourth coordinate system x s O s y s Fixedly connected pinion cutters, i.e. fourth coordinate system x s O s y s The gear shaper cutter can rotate along with the rotation of the gear shaper cutter;
establishing a tenth coordinate system x by taking any point on the center line of the modified gear as an origin p O p y p The tenth coordinate system x p O p y p The plane is perpendicular to the central line of the modified gear, and the tenth coordinate system x p O p y p The gear is fixedly arranged and does not rotate along with the modified gear; establishing an eleventh coordinate system x with any point on the centerline of the pinion cutter as an origin f O f y f Eleventh coordinate system x f O f y f The plane is perpendicular to the central line of the pinion cutter, and the eleventh coordinate system x f O f y f The gear shaping cutter is fixedly arranged and does not rotate along with the gear shaping cutter;
make the modified gearThe rotation angle of the modified gear is defined as phi when the modified gear rotates relative to the gear shaper cutter g I.e. a third coordinate system x g O g y g Relative to a tenth coordinate system x p O p y p The angle of rotation is phi g The rotation angle of the slotting cutter is defined as phi s I.e. the fourth coordinate system x s O s y s Relative to an eleventh coordinate system x f O f y f The angle of rotation is phi s 。
The full flank equation for a modified gear can then be expressed as:
in the formula (I), the compound is shown in the specification,
for a coordinate transformation matrix between the third coordinate system and the fourth coordinate system, ->
The tooth surface equation of the gear shaper cutter under the fourth coordinate system can be obtained by looking up data, the contour from the tooth top to the tooth bottom of the gear tooth of the gear shaper cutter sequentially consists of an EF section, an FG section and a GH section, and the EF section, the FG section and the GH section have different tooth surface equations.
Wherein the content of the first and second substances,
can be expressed as:
in the formula, E is the center distance between two gears which are not displaced, E + Delta E is the center distance between two gears which are displaced, and Delta E is the difference value of the center distances between the two gears before and after the displacement.
It should be noted that the profile of the gear tooth is symmetrically divided into two halves, and the EF section, the FG section, and the GH section constitute one half of the profile. The EF section, FG section, and GH section have different shapes, and thus the EF section, FG section, and GH section have different tooth surface equations.
φ g And phi s The ratio of (A) has the following relationship:
in the formula, Z s Number of teeth of the slotting cutter, Z g For changing the number of teeth of the gear, so that only phi needs to be determined g And phi s The other value can be calculated by one value, and the calculation is more convenient.
In some embodiments of the present invention, step S200 is: based on the meshing of the two modified gears, a contact equation of the two modified gears is established by utilizing a tooth surface contact analysis method, and a contact track when the two modified gears are meshed is calculated according to the contact equation.
Specifically, in some embodiments of the present invention, as shown in fig. 6, the external engagement of two modified gears is simulated, and a contact equation of the two modified gears is established, which includes the following steps:
firstly, establishing a working tooth surface equation and a tooth surface unit external normal vector equation of a driving wheel;
the working tooth surface equation of the driving wheel is expressed as follows:
in the formula, r ap The tooth crest radius of the driving wheel, r bp Is the base radius of the driving wheel, theta 0p ＝π/2Z p -invα 1 ，Z p Number of teeth of the driving wheel, α 1 Is the pressure angle of the driving wheel;
the tooth surface unit external normal vector equation of the driving wheel is expressed as:
secondly, establishing a working tooth surface equation and a tooth surface unit external normal vector equation of the driven wheel;
wherein the working tooth surface equation of the driven wheel is expressed as:
in the formula, r ag Is the tooth crest radius of the driven wheel, r bg Is the base radius of the driven wheel, theta 0g ＝π/2Z g -invα 2 ，Z g Number of teeth of driven wheel, α 2 Is the pressure angle of the driven wheel;
the tooth surface unit external normal vector equation of the driven wheel is expressed as:
in addition, a fifth coordinate system x is established by taking any point on the center line of the driving wheel as an origin f O f y f And a sixth coordinate system x p O p y p The fifth coordinate system x f O f y f And a sixth coordinate system x p O p y p The planes are all vertical to the central line of the driving wheel, wherein, a fifth coordinate system x f O f y f Fixed set up, sixth coordinate system x p O p y p Fixedly connected to the driving wheel, i.e. in a sixth coordinate system x p O p y p The driving wheel can rotate along with the rotation of the driving wheel;
establishing a seventh coordinate system x by taking any point on the central line of the driven wheel as an origin g O g y g And an eighth coordinate system x c O c y c The seventh coordinate system x g O g y g And an eighth coordinate system x c O c y c All lying in a plane perpendicular to the centre line of the driven wheel, wherein a seventh coordinate system x g O g y g Fixed in the eighth coordinate system x c O c y c Attached to the driven wheel, i.e. the eighth coordinate system x c O c y c The driven wheel can rotate along with the rotation of the driven wheel;
the driving wheel and the driven wheel are rotated relatively, and the rotation angle of the driving wheel is defined as phi 1 I.e. a sixth coordinate system x p O p y p Relative to a fifth coordinate system x f O f y f The angle of rotation is phi 1 . The angle of rotation of the driven wheel is defined as phi 2 I.e. an eighth coordinate system x c O c y c Relative to a seventh coordinate system x g O g y g The angle of rotation is phi 2 。
And finally, transforming the working tooth surface equation and the tooth surface unit external normal vector equation of the driving wheel to a fifth coordinate system:
in the formula, M fp (φ 1 ) A coordinate transformation matrix between the fifth coordinate system and the sixth coordinate system;
wherein M is fp (φ 1 ) The expression of (c) is:
and transforming the working tooth surface equation and the tooth surface unit external normal vector equation of the driven wheel to a fifth coordinate system:
in the formula, M fc Is a coordinate transformation matrix between the fifth coordinate system and the eighth coordinate system, M cg (φ 2 ) A coordinate transformation matrix between the seventh coordinate system and the eighth coordinate system;
wherein M is fc The expression of (a) is:
M cg (φ 2 ) The expression of (a) is:
according to the gear local contact principle, establishing a contact equation of the driving wheel and the driven wheel in a fifth coordinate system:
the contact equation is expanded to obtain three independent nonlinear equation sets, and the three independent nonlinear equation sets contain four unknowns in phi 1 For inputting parameters, three nonlinear equation sets are solved, and then the contact track when the driving wheel and the driven wheel are meshed can be obtained.
As shown in fig. 7, when the two modified gears are in inner engagement, a similar coordinate system is established according to the process that the two modified gears are in outer engagement, a contact equation set of the driving wheel and the driven wheel can also be established, and a contact track when the driving wheel and the driven wheel are engaged can be obtained according to the contact equation set, and the solving principle and the solving process are consistent with those when the two modified gears are in outer engagement, so that the description is omitted.
In some embodiments of the present invention, step S300 is: and obtaining tooth surface point information on the meshing track according to the full tooth surface equation and the meshing track, and calculating the meshing stiffness by using a potential energy method according to the tooth surface point information.
Specifically, after the contact trajectory when the driving wheel and the driven wheel are engaged is obtained, the starting point, the ending point and the range of the contact trajectory can be determined, and in the range of the contact trajectory, the tooth surface point information in the range of the contact trajectory can be solved by using the full tooth surface equation which is established in the step S100 and can be used for solving the tooth surface point information, wherein the tooth surface point information includes various types, for example, including the section inertia moment, the section width, the section area of the corresponding position, the direction of the engagement force when the driving wheel and the driven wheel are engaged with each other, and the like. And calculating the meshing stiffness by using a potential energy method according to the tooth surface point information in the contact track range.
In some embodiments of the present invention, the principle of the potential energy method is: and (3) equivalent the gear teeth of the modified gears to be cantilever beams, and solving the meshing stiffness based on the stress balance and energy conservation principle in the meshing process of the two modified gears.
As shown in fig. 8 and 9, a stiffness calculation coordinate system xOy is established at the tooth root of the gear, the origin of the coordinate system xOy is located on the dedendum circle and on the symmetry line of the tooth profile of the gear, the symmetry line of the tooth profile of the gear is the x-axis direction, and the tangential direction of the tooth root circle is the y-axis direction. Wherein F i For dynamic engaging forces at a distance d from the root, F a Component of dynamic engagement force in the x-axis direction, F b The component of the dynamic engagement force in the y-axis direction, alpha being F i And F b H is half of the width of the cross section at a distance d from the dedendum, i.e. the distance between the point on the tooth profile at a distance d from the dedendum and the line of symmetry of the tooth profile of the wheel tooth, h x The cross-section referred to in this embodiment is perpendicular to the x-axis direction, i.e. perpendicular to the symmetry line of the gear tooth profile, for half the width of the cross-section at x from the dedendum, i.e. the distance between the point on the tooth profile at x from the dedendum and the symmetry line of the tooth profile of the gear tooth.
In some embodiments of the invention, the bending stiffness k of the modified gear is determined according to the potential energy method b Shear stiffness k s Axial compression stiffness k a Elastic matrix stiffness k f And Hertz contact stiffness k h Can be respectively expressed as:
wherein:
in the formula I x Representing the area moment of inertia at a distance x from the root, A x Represents the area of the cross section at a distance x from the root of the tooth, E represents the Young's modulus, G represents the shear modulus, v represents the material Poisson, B represents the width of the tooth, u represents the thickness of the tooth f Indicating the distance from the intersection of the meshing line and the tooth symmetry line to the root circle, S f Is the arc length L on the root circle between two intersection points of the tooth profile curve of a single gear tooth and the root circle * 、M * 、P * 、Q * Is a coefficient related to the gear design parameter.
Further, the single-tooth meshing stiffness of the modified gear can be expressed as:
wherein i represents the i-th gear in the meshing pair;
the double tooth meshing stiffness of a modified gear can be expressed as:
in some embodiments of the present invention, the method further comprises step S400: the calculation method of the present invention is verified and analyzed.
First, in order to verify the correctness of the calculation method of the present invention, the calculation results of the calculation method of the present invention and the calculation results of the finite elements are compared, and specific calculation parameters of the meshing pair are shown in table 1. The calculation result is shown in fig. 10, the abscissa is a standardized time variable, the ordinate is the meshing stiffness, the maximum error of the calculation method and the finite element method in the double-tooth meshing region is 2.45%, the maximum error of the calculation method and the finite element method in the single-tooth meshing region is only 0.59%, and the errors of the calculation method and the finite element method are both below 5%, so that the accuracy of the calculation method can be verified.
TABLE 1 calculation parameters of the meshing pairs
And secondly, verifying by calculating the meshing rigidity of the external meshing modified gear pair and the internal meshing modified gear pair under different modification coefficients. The external meshing modified gear pair comprises a sun gear and a planet gear, the internal meshing modified gear pair comprises a planet gear and an internal gear, and parameters for calculation are shown in the table 2.
In table 2, the mesh angle at an angle of 22.4 is the mesh angle between the sun gear and the planetary gear, the mesh angle at an angle of 20 is the mesh angle between the planetary gear and the internal gear, the center distance is 93mm, which means that the center distance between the sun gear and the planetary gear is 93mm, the center distance between the planetary gear and the internal gear is also 93mm, the young's moduli of the sun gear, the planetary gear, and the internal gear are the same, and the poisson ratios of the sun gear, the planetary gear, and the internal gear are the same.
TABLE 2 calculated parameters for planetary gear systems
According to the parameters in table 2, software programming is adopted to calculate, so as to obtain the meshing stiffness of the external meshing modified gear pair under different modification coefficients as shown in fig. 11 and the meshing stiffness of the internal meshing modified gear pair under different modification coefficients as shown in fig. 12, wherein the abscissa is a standardized time variable, the ordinate is the meshing stiffness, and different types of line segments represent different modification coefficients.
With the increase of the deflection coefficient, as shown in fig. 11, the contact ratio and the average meshing stiffness of the s-p deflection gear pair, that is, the external meshing deflection gear pair, do not change much, as shown in fig. 12, the contact ratio and the average meshing stiffness of the r-p deflection gear pair, that is, the internal meshing deflection gear pair, are continuously increased and change obviously, which is consistent with the reality, that is, the correctness of the present invention is also verified from the contact ratio and the average meshing stiffness.
The embodiments of the present invention have been described in detail with reference to the accompanying drawings, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the gist of the present invention.
Claims (10)
1. A method for calculating the meshing stiffness of a modified planetary gear based on tooth surface contact analysis is characterized by comprising the following steps:
s100, establishing a full tooth surface equation of the modified gear based on the engagement of the cutter and the modified gear when the modified gear is machined by the cutter;
s200, establishing a contact equation of the two modified gears based on the engagement of the two modified gears, and calculating a contact track when the two modified gears are engaged according to the contact equation;
s300, obtaining tooth surface point information on the contact track according to the full tooth surface equation and the contact track, and calculating the meshing stiffness by using a potential energy method according to the tooth surface point information.
2. The method for calculating the meshing stiffness of the modified planetary gear based on the tooth surface contact analysis according to claim 1, wherein in step S100, the manner of machining the modified gear by the tool is as follows: and processing the modified gear by a hobbing cutter or a gear shaping cutter.
3. The method of calculating meshing stiffness of a modified planetary gear according to claim 2,
in the step S100, a first coordinate system is established by taking any point on the center line of the modified gear as an origin, the plane of the first coordinate system is vertical to the center line of the modified gear, and the first coordinate system is fixedly connected with the modified gear;
establishing a second coordinate system by taking any point on the central line of the hobbing cutter as an origin, wherein the plane of the second coordinate system is vertical to the central line of the hobbing cutter, and the second coordinate system is fixedly connected with the hobbing cutter;
enabling the modified gear and the hobbing cutter to rotate relatively, wherein the rotation angle of the modified gear is defined as phi;
calculating the distance between the center line of the hobbing cutter and the center line of the modified gear:
r mg ＝r pg +xm
in the formula, r pg The radius of a reference circle of the modified gear is defined as x, the modification coefficient of the modified gear is defined as m, and the modulus of the modified gear is defined as m;
the full tooth surface equation is expressed as:
in the formula, M gc (φ j ) Is a coordinate transformation matrix between the first coordinate system and the second coordinate system,
and the contour from the tooth top to the tooth bottom of the gear tooth of the hobbing cutter sequentially consists of a CD section, a DM section and an MN section, and the CD section, the DM section and the MN section have different tooth surface equations.
4. The method of calculating meshing stiffness of a modified planetary gear according to claim 2,
in the step S100, a third coordinate system is established by taking any point on the center line of the modified gear as an origin, the plane of the third coordinate system is vertical to the center line of the modified gear, and the third coordinate system is fixedly connected with the modified gear;
establishing a fourth coordinate system by taking any point on the central line of the pinion cutter as an origin, wherein the plane of the fourth coordinate system is vertical to the central line of the pinion cutter, and the fourth coordinate system is fixedly connected with the pinion cutter;
the modified gear and the gear shaping cutter are relatively rotated, and the rotation angle of the modified gear is defined as phi g The rotation angle of the gear shaper cutter is defined as phi s ；
The full tooth surface equation is expressed as:
in the formula (I), the compound is shown in the specification,
is a coordinate transformation matrix between the third coordinate system and the fourth coordinate system,
for the gear shaper cutter is in the flank of tooth equation under the fourth coordinate system, the profile of the tooth top to the tooth root of the teeth of the gear shaper cutter comprises EF section, FG section and GH section in proper order, and EF section, FG section and GH section have different flank of tooth equations.
5. The method for calculating meshing stiffness of a modified planetary gear according to any one of claims 1 to 4, wherein the step S200 of establishing the contact equation based on the external meshing of the two modified gears includes:
establishing a working tooth surface equation and a tooth surface unit external normal vector equation of the driving wheel;
the working tooth surface equation of the driving wheel is expressed as follows:
in the formula, r ap The tooth crest radius of the driving wheel, r bp Is the base radius of the driving wheel, theta 0p ＝π/2Z p -invα 1 ，Z p Is the number of teeth of the driving wheel, alpha 1 Is the pressure angle of the driving wheel;
the tooth surface unit external normal vector equation of the driving wheel is expressed as follows:
6. the method of calculating tooth surface contact analysis-based meshing stiffness of a modified planetary gear according to claim 5, wherein the process of establishing the contact equation further comprises:
establishing a working tooth surface equation and a tooth surface unit external normal vector equation of the driven wheel;
wherein the working tooth surface equation of the driven wheel is expressed as:
in the formula, r ag Is the tooth crest radius of the driven wheel, r bg Is the base radius of the driven wheel, theta 0g ＝π/2Z g -invα 2 ，Z g Is the number of teeth of the driven wheel, α 2 Is the pressure angle of the driven wheel;
the tooth surface unit external normal vector equation of the driven wheel is expressed as follows:
7. the method of calculating tooth surface contact analysis-based meshing stiffness of a modified planetary gear according to claim 6, wherein the process of establishing the contact equation further comprises:
establishing a fifth coordinate system and a sixth coordinate system by taking any point on the center line of the driving wheel as an origin, wherein the planes of the fifth coordinate system and the sixth coordinate system are vertical to the center line of the driving wheel, the fifth coordinate system is fixedly arranged, and the sixth coordinate system is fixedly connected with the driving wheel;
establishing a seventh coordinate system and an eighth coordinate system by taking any point on the central line of the driven wheel as an origin, wherein the planes of the seventh coordinate system and the eighth coordinate system are vertical to the central line of the driven wheel, the seventh coordinate system is fixedly arranged, and the eighth coordinate system is fixedly connected with the driven wheel;
the driving wheel and the driven wheel are relatively rotated, and the rotation angle of the driving wheel is defined as phi 1 The angle of rotation of the driven wheel is defined as phi 2 。
8. The method of calculating tooth surface contact analysis-based meshing stiffness of a modified planetary gear according to claim 7, wherein the process of establishing the contact equation further comprises:
and transforming the working tooth surface equation and the tooth surface unit external normal vector equation of the driving wheel to a fifth coordinate system:
in the formula, M fp (φ 1 ) A coordinate transformation matrix between the fifth coordinate system and the sixth coordinate system;
transforming the working tooth surface equation and the tooth surface unit external normal vector equation of the driven wheel to the fifth coordinate system:
in the formula, M fc Is a coordinate transformation matrix between the fifth coordinate system and the eighth coordinate system, M cg (φ 2 ) Transforming a matrix for coordinates between the seventh coordinate system and the eighth coordinate system；
According to the gear local contact principle, establishing a contact equation of the driving wheel and the driven wheel in the fifth coordinate system:
and solving the contact equation to obtain a contact track when the driving wheel is meshed with the driven wheel.
9. The method for calculating meshing stiffness of a modified planetary gear according to any one of claims 1 to 4, wherein the potential energy method includes: and (3) equivalent the gear teeth of the modified gears to be cantilever beams, and solving the meshing stiffness based on the stress balance and energy conservation principle in the meshing process of the two modified gears.
10. The method for calculating the meshing stiffness of a modified planet gear based on the tooth surface contact analysis according to any one of claims 1 to 4, wherein the bending stiffness k of the modified gear is calculated according to the potential energy method b Shear stiffness k s Axial compression stiffness k a Elastic matrix stiffness k f And Hertz contact stiffness k h ；
The single-tooth meshing stiffness of the modified gear is expressed as follows during the whole meshing process:
wherein i represents the ith gear in the meshing pair;
the double-tooth meshing rigidity of the modified gear is expressed as follows:
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