CN113868801B - Gear tooth contact analysis method based on full tooth surface search - Google Patents
Gear tooth contact analysis method based on full tooth surface search Download PDFInfo
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- CN113868801B CN113868801B CN202111167325.5A CN202111167325A CN113868801B CN 113868801 B CN113868801 B CN 113868801B CN 202111167325 A CN202111167325 A CN 202111167325A CN 113868801 B CN113868801 B CN 113868801B
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Abstract
The invention relates to a gear tooth contact analysis method based on full tooth surface search, and belongs to the technical field of gear contact analysis. On the basis of establishing the real tooth surface of the gear, the real tooth surface of the gear is subjected to grid division, and the distance between the tooth surface point of the pinion gear and the tooth surface point of the bull gear at the grid point is calculated. And (3) giving a pinion rotation angle, adjusting a bull gear rotation angle, finding a minimum value between a pinion tooth surface and a bull gear tooth surface, and taking a tooth surface point at the moment as an instantaneous contact point. Given different pinion rotation angles, different instantaneous contact points are obtained, forming a contact trajectory. And calculating transmission error according to the minimum distance between the gear corner and the tooth surface, wherein the region where the distance value between the two tooth surfaces is smaller than the thickness of the rolling inspection experimental coating is the contact patch. The method is not limited by a specific tooth surface, and the tooth surface error and the condition that a plurality of meshing points exist on a single tooth surface pair at the same time can be considered when the method is used for carrying out the gear tooth contact analysis, and all the meshing points on the single tooth surface pair at the same time can be solved.
Description
Technical Field
The invention belongs to the technical field of gear contact analysis, and particularly relates to a gear tooth contact analysis method based on full tooth surface search.
Background
The gear Tooth Contact Analysis method (Tooth Contact Analysis) is used for calculating and solving the Contact patch and the static transmission error of a gear pair, and is an important means for evaluating the design quality, the machining quality and the meshing performance of the gear Tooth surface.
The contact analysis algorithm based on the same phase vector and normal vector of two rigid smooth tooth surfaces, which is proposed by the professor of Litvin, is the most widely applied traditional analysis method at present and is suitable for a point contact gear pair formed by two smooth tooth surfaces. The core of the method is that five nonlinear equations are constructed according to the common normal vector and the common bit vector at the tooth surface points of the two gears (wherein the bit vectors are the same to determine 3 equations, and the normal vectors are the same to determine 2 equations), and simultaneous solution is carried out to determine the position of a meshing point and the rotation angle of the gears. When the method is used for solving, the position and the rotating angle of the pinion can be given, only one instantaneous contact point or contact line (line contact condition) can be solved, the condition that the tooth surface locally has interference cannot be considered, and the condition that the tooth surface contact is discontinuous and the like caused by machining errors cannot be considered.
The Fangzongde professor obtains a normal deviation curved surface through cubic B-spline fitting based on the tooth surface deviation measured by three coordinates, constructs the actual tooth surface of the helical gear by superposing a theoretical tooth surface and the deviation curved surface, deduces the normal vector of the tooth surface, the contact condition of the tooth surface and the contact condition of the edge, and realizes the contact analysis simulation of various actual tooth surfaces. The method can be used for tooth surface contact analysis simulation under various machining errors and modification conditions, and provides a method for the actual tooth surface contact analysis of the cylindrical gear. However, the method still solves the problem based on the principle that normal vectors and bit vectors of two tooth surfaces are equal, and full tooth surface search cannot be carried out.
The Sun palace column teaches and considers the gear tooth thermal deformation and the tooth surface processing error, and alternately fuses a curved surface modeling technology, a CAGD theory and a classic gear meshing principle, and provides two solving algorithms of real tooth surface instantaneous meshing points for a spiral cylindrical gear: the zero-gap method and the tangent method. The tangent method has high calculation accuracy and high speed, but is sensitive to the selection of the initial value of the nonlinear equation set. The zero clearance method has the advantages of low solving speed, no requirement on initial value conditions and capability of checking whether intersection interference exists outside the meshing point. The two methods can be used complementarily.
Disclosure of Invention
Technical problem to be solved
In order to avoid the defects of the prior art, the invention provides a gear tooth contact analysis method based on full tooth surface search, so as to solve the problem that the existing traditional gear tooth contact analysis method cannot consider that multiple meshing points exist on a single tooth surface.
Technical scheme
A gear tooth contact analysis method based on full tooth surface search is characterized by comprising the following steps:
step 1: establishing a follow-up coordinate system S 1 And S 2 Said coordinate system S 1 Has coordinate axis of x 1 ，y 1 ，z 1 Said coordinate system S 2 Coordinates of (2)Axis x 2 ，y 2 ，z 2 ，；S 1 Rigidly connected to pinion gear, z 1 The shaft coincides with the pinion axis; likewise, S 2 Rigidly connected to the gear wheel, z 2 The shaft is superposed with the large gear shaft;
gear tooth surface deviation data is obtained through three-coordinate measurement, and a normal deviation curved surface is obtained through cubic B spline fitting
The theoretical tooth surface and the normal deviation curved surface are superposed to construct the actual tooth surface of the gear;
pinion tooth surface Σ (1) In a following coordinate system S 1 The following tooth surface equation is:
wherein
Is pinion tooth surface Σ (1) Position vector of (1), p (1) And q is (1) Is a surface parameter;
likewise, the tooth flanks Σ of the gearwheel (2) In a following coordinate system S 2 The tooth surface equation below is:
wherein
Is tooth surface Σ of large gear (2) Position vector of (a), p (2) And q is (2) Is a surface parameter;
step 2: selecting an initial position: setting the initial engagement position to O a And O b Position at time of coincidence, wherein the initial point O a And O b The tooth width middle points of the small gear and the big gear are respectively positioned at the positions which are 1 time of the module away from the tooth top surface; pinion and bullAngle of rotation of gear
And
from the initial position, the initial rotation angle of the pinion is taken as
The initial turning angle of the large gear is
The two gears are defaulted to solve the contact point from the initial position;
and step 3: grid division: with pinion tooth surfaces Σ in the initial position (1) Upper initial point O a As an origin, a following coordinate system S rigidly connected with the pinion is established a Coordinate axis being x a ，y a ，z a Plane x of coordinate system a y a With pinion tooth surface Σ (1) Tangent to the origin O a Axis x a The direction of the gear is the same as the tooth direction of the pinion; in the coordinate system plane x a y a Grid T for dividing m rows and n columns (1) Grid transverse and axis x a Parallel, longitudinal and axis y a Parallel connection; for the same reason, with O b As an origin, a follow-up coordinate system S rigidly connected with the gearwheel is established b Coordinate axis being x b ，y b ，z b Dividing the grid T (2) (ii) a Definition of T (1) Or T (2) Points which are positioned at the geometric center of each grid are grid points, and the rest points are non-grid points; points obtained by projecting grid points or non-grid points on the gear tooth surface are projection points; will be at T in the following distance search (1) The grid points on the grid are indexes;
to facilitate coordinate transformation in subsequent distance searches, at an initial position, i.e. at
Then, a coordinate system S is established a Coincident fixed coordinate system S p Coordinate axis being x p ，y p ，z p Establishing and coordinating system S in the same way b Coincident fixed coordinate system S g Coordinate axis being x g ，y g ，z g (ii) a Due to the initial position O a And O b Coincidence, from a following coordinate system S a 、S b Coordinate system S known by relation of each axis p About axis x p Rotating 180 degrees to obtain a coordinate system S g ；
And 4, step 4: calculating Z coordinates of tooth surface points
Calculating pinion tooth flank ∑ (1) In a coordinate system S a The tooth surface equation below is:
wherein M is a1 As a coordinate system S 1 To a coordinate system S a The coordinate transformation matrix of (2);
as a coordinate system S 1 Origin O of 1 To the initial point M (1) A vector of (a);
solving in a coordinate system S a Lower grid point
The coordinates of (a):
left side of equation
And
as a grid point
The horizontal and vertical coordinates of (1); to the right of the equation is in the coordinate system S a The following vector equation;
according to pinion tooth surface Σ (1) In a coordinate system S a Tooth surface equation under the condition that grid points can be obtained
Along z a Towards pinion tooth flank ∑ (1) Projection point
Z coordinate of (2), as
In the same way, the tooth surface sigma of the large gear can be solved (2) Upper point of
In a coordinate system S b Z coordinate of lower, noted
From this, the two-gear tooth surface sigma can be calculated (1) Or sigma (2) Upper arbitrary point P in corresponding following coordinate system S a Or S b The lower Z coordinate can be obtained by the interpolation of the Z coordinate of the projection point of any point P around the point P; therefore, during distance searching, only the projection point of the grid point on the tooth surface is subjected to Z coordinate calculation through a tooth surface equation, and the Z coordinate of the projection point of the non-grid point on the tooth surface is obtained through a two-dimensional interpolation method;
and 5: tooth surface distance calculation
Let T (1) A certain grid point on
On pinion tooth surface ∑ (1) Projected points on are points
Dot
To T (2) The projection point of the plane is marked as M, and the point M is on the grid plane T at the moment (2) The upper is usually a non-grid point, so the point M is found to the flank of the bull gear (2) Projected point on
The interpolation described above is needed; to calculate points
To point
The two tooth surface equations need to be placed under the same coordinate system; dot
In a coordinate system S b The following coordinate vectors can be expressed as:
wherein
As a coordinate system S a The following vector equation; m ba Is S a To S b The coordinate transformation matrix of (2) is in a transformation mode of: s a →S p →S g →S b ；H a Is the pinion initial point M (1) To the initial point M of the gearwheel (2) A vector of (a); i.e. i b ，j b ，k b Are respectively S b Unit vectors of the next three axes;
setting point
In a coordinate system S b Z coordinate of
Dot
In a coordinate system S b Z coordinate of
Then point is reached
To point
Is recorded as
Assuming that the pinion initial rotation angle is
Calculating the initial rotation angle of the large gear according to the transmission ratio
Calculating grid T of m rows and n columns (1) Upper all grid points along z a Towards pinion tooth flank ∑ (1) All projected points of (a); according to the tooth surface distance calculation method, the small wheel tooth surface sigma is calculated (1) All projected points on the large gear tooth surface sigma (2) The distance of the upper corresponding point; finding the minimum distance, noted as Δ z ba,min ；
If Δ z is ba，min If the contact point is more than epsilon, the tooth surfaces of the two gears are not contacted with each other, and the instantaneous contact point needs to be solved iteratively; if Δ z is ba，min If the tooth surfaces of the two gears are in mutual contact or even embedded, the condition is indicated; epsilon is a given tolerance and is used for judging the states of the two tooth surfaces;
and 6: iterative solution of instantaneous contact points
If Δ z is ba，min If the value is more than epsilon, the following adjustment is needed, otherwise, the transmission error is directly calculated; small gear tooth surface sigma at minimum distance (1) The upper point is K, the big gear tooth surface sigma (2) The upper point is Q, the point K and the point Q are tooth surface jointsA near point;
Δz ba,min at Q point line velocity V Q Is set as V Then, then
In the formula, QC is the shortest distance from a point Q to a bull gear shaft;
thereafter, the pinion rotation angle is maintained
The rotation angle of the large gear is unchanged
Continuing to search the approaching point of the tooth surface and calculating new
Such continuous iterative adjustment of the rotation angle of the large gear
Solving the minimum distance of the tooth surface; when the minimum distance Δ z ba,min When the tolerance is less than or equal to the given tolerance epsilon, the tooth surfaces are considered to be contacted at the moment, and the corresponding tooth surface approach point is a contact point;
when the tooth surfaces contact, calculating the transmission error according to the definition of the transmission error;
find all instantaneous contact points: to simulate the rolling inspection of coating thickness, T (1) Upper each grid point on pinion tooth face Σ (1) The projection point on the projection point moves a distance xi along the direction of the external normal; using new projection point as index to search the distance delta z between two tooth surfaces again ba The rotation angle of the bull gear is not adjusted iteratively in the search; tooth flank distance Δ z ba All the tooth surface points of which the thickness xi is less than or equal to the rolling inspection experiment coating are regarded as contact points;
and 7: in increments of
Changing the angle of rotation of the pinion in positive and negative directions, respectively
Calculating the rotation angle of the large gear according to the transmission ratio
Calculating and solving a tooth surface contact point at the next meshing position according to the tooth surface distance calculation method and the instantaneous contact point solution method in the steps 5 and 6, and calculating a transmission error, an instantaneous contact line or a contact area; when the pinion is rotated
Is subject to [ -2 π/N 1 ，2π/N 1 ]Then stop the calculation, where N 1 Is the pinion tooth number.
A computer system, comprising: one or more processors, a computer readable storage medium, for storing one or more programs, which when executed by the one or more processors, cause the one or more processors to implement the above-described method.
Advantageous effects
The invention provides a gear tooth contact analysis method based on full tooth surface search. And (3) giving a pinion rotation angle, adjusting a bull gear rotation angle, finding a minimum value between a pinion tooth surface and a bull gear tooth surface, and taking a tooth surface point at the moment as an instantaneous contact point. Given different pinion rotation angles, different instantaneous contact points are obtained, forming a contact trajectory. And calculating a transmission error according to the minimum distance between the gear corner and the tooth surface, wherein the region with the distance value between the two tooth surfaces being smaller than the thickness of the rolling inspection experimental coating is the contact patch.
The method can be suitable for the contact analysis of different types of gear tooth surfaces, and the algorithm of the method is not limited by specific tooth surfaces. The tooth surface mentioned in the method is constructed by superposing a tooth surface error generated in gear machining with a theoretical tooth surface after spline fitting. Therefore, the method can take the tooth surface error into consideration when the tooth contact analysis is carried out, and has a result closer to the actual result. In addition, the method can consider the situation that a plurality of meshing points exist on a single tooth surface pair at the same time, can solve all the meshing points of the single tooth surface pair at the same time, and overcomes the defect that only one meshing point can be calculated by the traditional gear tooth contact analysis method.
Drawings
The drawings are only for purposes of illustrating particular embodiments and are not to be construed as limiting the invention, wherein like reference numerals are used to designate like parts throughout.
FIG. 1 is a coordinate system S of the present invention based on a full-flank search contact analysis method 1 、S 2 And an initial point O a And O b The distance between two gear teeth is increased for clarity;
FIG. 2 is a schematic view of pinion meshing based on the full tooth search contact analysis method of the present invention;
FIG. 3 is a following coordinate system S based on the full-tooth-surface search contact analysis method of the present invention a 、S b And a fixed coordinate system S p 、S g A schematic diagram of the relationship of each axis;
FIG. 4 is a schematic diagram of a Z coordinate calculation method for a projection point on a pinion tooth surface based on a full tooth surface search contact analysis method according to the present invention, for convenience of representing the pinion tooth surface Σ (1) The curvature of (a) increases;
FIG. 5 is a graph of the full tooth surface search based contact analysis method of the present invention calculating Δ from tooth surface approach points V A schematic diagram of (a);
FIG. 6 is a schematic of a two tooth surface distance search based on a full tooth surface search contact analysis method of the present invention;
FIG. 7 is a flow chart of the full flank search based contact analysis method of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
As shown in fig. 7, which is a flowchart of the present invention, the method for analyzing tooth contact based on full tooth surface search mainly includes the following steps: (1) establishing a local coordinate system at reference points of two tooth surfaces, setting an initial position of a gear, performing mesh division on the tooth surfaces of the gear, and giving rotation angles of the two gears as
And
(2) calculating the Z coordinate of a projection point of the grid point on the tooth surface of the gear, wherein the projection point on the tooth surface of the gear at the non-grid point can quickly obtain the Z coordinate in a two-dimensional interpolation mode; (3) along direction Z b Solving any tooth surface point of pinion
Shortest distance to tooth surface of large gear
And searching the tooth surface approaching point; (4) iteratively adjusting the turning angle of the large gear according to the shortest distance and the tooth surface approach point
Making the shortest distance between the tooth surfaces smaller than a given tolerance epsilon, and then considering the two tooth surfaces to be tangent and obtaining an instantaneous contact point; (5) and solving a transmission error according to the instantaneous contact point and the rotation angles of the two gears. And considering the thickness of the coating layer during rolling impression inspection of the gear pair, and performing distance search on the gear tooth surface grid points once again to find a contact area.
The steps are as follows:
1. establishing a follow-up coordinate system S 1 (coordinate axis is x) 1 ，y 1 ，z 1 ) And S 2 (coordinate axis is x) 2 ，y 2 ，z 2 )。S 1 Rigidly connected to pinion gear, z 1 The shaft coincides with the pinion axis. Likewise, S 2 Rigidly connected to the gear wheel, z 2 The shaft coincides with the large gear shaft.
Gear tooth surface deviation data is obtained through three-coordinate measurement, and a normal deviation curved surface s is obtained through cubic B spline fitting 1 (1)(u (1) ,v (1) ) And the theoretical tooth surface and the normal deviation curved surface are superposed to construct the actual tooth surface of the gear.
Pinion tooth surface Σ (1) In a following coordinate system S 1 The following tooth surface equation is:
wherein
Is pinion gear flank ∑ (1) Position vector of (1), p (1) And q is (1) Is a surface parameter.
Likewise, the tooth flanks Σ of the gearwheel (2) In a following coordinate system S 2 The following tooth surface equation is:
wherein
Is the tooth surface sigma of the gearwheel (2) Position vector of (1), p (2) And q is (2) Is a surface parameter.
2. And selecting an initial position. Setting the initial engagement position to O a And O b Position at time of coincidence, wherein the initial point O a And O b On the pinion and bull gear teeth, respectivelyWide midpoint and 1 modulus from the addendum face. Corner of small gear and big gear
And
from the initial position, the initial rotation angle of the pinion is taken as
The initial turning angle of the large gear is
I.e. by default both gears start from the initial position to solve for the contact point. Coordinate system S 1 、S 2 And an initial point O a And O b As shown in fig. 1, the space between two gear teeth is increased for clarity.
3. And (5) dividing the grids. With pinion tooth surfaces Σ in the initial position (1) Upper initial point O a As an origin, a following coordinate system S rigidly connected with the pinion is established a (coordinate axis is x) a ，y a ，z a ) Plane x of coordinate system a y a With pinion tooth surface Σ (1) Tangent to the origin O a Axis x a In the same direction as the pinion tooth direction. In the coordinate system plane x a y a Grid T for dividing m rows and n columns (1) Grid transverse and axis x a Parallel, longitudinal and axis y a Parallel. By the same token, with O b As an origin, a follow-up coordinate system S rigidly connected with the gearwheel is established b (coordinate axis is x) b ，y b ，z b ) Dividing the grid T (2) . Definition of T (1) Or T (2) Points which are positioned at the geometric center of each grid are grid points, and the rest points are non-grid points; the points obtained by projecting the grid points or the non-grid points on the tooth surface of the gear are projected points. Will be at T in the following distance search (1) The grid points above are the indices. A schematic of pinion meshing is shown in fig. 2.
To facilitate sitting in subsequent distance searchesAt the initial position, i.e. at the mark transfer
Then, a coordinate system S is established a Coincident fixed coordinate system S p (coordinate axis is x) p ，y p ，z p ) Establishing and coordinating system S in the same way b Coincident fixed coordinate system S g (coordinate axis is x) g ，y g ，z g ). Due to the initial position O a And O b Coincidence, from a following coordinate system S a 、S b Coordinate system S known by relation of each axis p About axis x p Rotating 180 degrees to obtain a coordinate system S g As shown in fig. 3.
4. And calculating a Z coordinate of a tooth surface point. Calculating pinion tooth flank ∑ (1) In a coordinate system S a The following tooth surface equation is:
wherein M is a1 As a coordinate system S 1 To a coordinate system S a The coordinate transformation matrix of (2);
as a coordinate system S 1 Origin O of 1 To the initial point M (1) The vector of (2).
Solving in a coordinate system S a Lower grid point
The coordinates of (a):
left side of equation
And
as a grid point
The horizontal and vertical coordinates of (1); to the right of the equation is in the coordinate system S a The following vector equation.
According to pinion tooth surface Σ (1) In a coordinate system S a Tooth surface equation under the condition that grid points can be obtained
Along z a Towards pinion tooth flank ∑ (1) Projection point
Z coordinate of (2), as
In the same way, the tooth surface sigma of the large gear can be solved (2) Upper point of
In a coordinate system S b Z coordinate of lower, noted
From this, the two-gear tooth surface sigma can be calculated (1) Or sigma (2) Upper arbitrary point P in corresponding following coordinate system S a Or S b The lower Z coordinate may be obtained by interpolating the Z coordinates of the projected points of the arbitrary point P around it. Therefore, in the distance search, only the projection points of the grid points on the tooth surface are subjected to Z coordinate calculation through the tooth surface equation, and the Z coordinate of the projection points of the non-grid points on the tooth surface is obtained through a two-dimensional interpolation method. The method of calculating the Z coordinate of the projected point on the pinion tooth surface is shown in FIG. 4, and the pinion tooth surface Σ is shown for convenience (1) The curvature of (a) increases.
5. And calculating the tooth surface distance. Let T (1) A certain grid point on
On pinion tooth surface ∑ (1) Projected points on are points
Dot
To T (2) The projection point of the plane is marked as M, and the point M is on the grid plane T at the moment (2) The upper is usually a non-grid point, so the point M is found to the flank of the bull gear (2) Projected point on
The aforementioned interpolation is needed. To calculate points
To point
The two tooth surface equations need to be placed under the same coordinate system. Dot
In a coordinate system S b The following coordinate vectors can be expressed as:
wherein
As a coordinate system S a The following vector equation; m ba Is S a To S b The coordinate transformation matrix of (2) is in a transformation mode of: s a →S p →S g →S b 。H a Is the pinion initial point M (1) To the initial point M of the gearwheel (2) The vector of (2). i.e. i b ，j b ，k b Are respectively S b Unit vector of the next three axes.
Setting point
In a coordinate system S b Z coordinate of
Dot
In a coordinate system S b Z coordinate of
Then point is reached
To point
Is recorded as
Assuming that the pinion initial rotation angle is
Calculating the initial rotation angle of the large gear according to the transmission ratio
Calculating grid T of m rows and n columns (1) Upper all grid points along z a Towards pinion tooth flank ∑ (1) All proxels of (a). According to the tooth surface distance calculation method, the small wheel tooth surface sigma is calculated (1) Go all projected points to large gear tooth surface sigma (2) Distance of the upper corresponding point. Find the minimum distance, noted as Δ z ba,min 。
If Δ z is ba，min If the tooth surfaces of the two gears are not in contact with each other, the instantaneous contact point needs to be solved iteratively; if Δ z is ba，min If the tooth surfaces of the two gears are contacted with each other or even embedded into each other, the condition is indicated. Epsilon is a given tolerance for judging the state of both tooth surfaces.
6. And (5) iteratively solving the instantaneous contact point. If Δ z is ba，min If the transmission error is larger than epsilon, the following adjustment needs to be made, otherwise, the transmission error is directly calculated. Small gear tooth surface sigma at minimum distance (1) The upper point is K, the big gear tooth surface sigma (2) The upper point is Q, and the point K and the point Q are the points near the tooth surface, as shown in FIG. 5.
Δz ba,min At Q point line velocity V Q Is set as Δ V Then, then
Where QC is the shortest distance from point Q to the bull gear axis.
Thereafter, the pinion rotation angle is maintained
The rotation angle of the large gear is unchanged
Continuing to search the approaching point of the tooth surface and calculating new
Such continuous iterative adjustment of the rotation angle of the large gear
And solving the minimum distance of the tooth surface. When the minimum distance Δ z ba,min And less than or equal to a given tolerance epsilon, at which point the tooth flanks are considered to have contacted, and the corresponding tooth flank approach point is the contact point.
When the tooth surfaces are in contact, the transmission error is calculated according to the definition of the transmission error.
Find all instantaneous contact points: to simulate the rolling inspection of coating thickness, T (1) On the pinion tooth surfaceΣ (1) The projected point on the screen is moved in the direction of the external normal by a distance xi. Using new projection point as index to search the distance delta z between two tooth surfaces again ba And the rotation angle of the large gear is not iteratively adjusted in the search. Tooth flank distance Δ z ba All the tooth surface points with the thickness xi smaller than or equal to the rolling inspection experimental coating are regarded as contact points.
7. In increments of
Changing the angle of rotation of the pinion in positive and negative directions, respectively
Calculating the rotation angle of the large gear according to the transmission ratio
According to the tooth flank distance calculation method and the instantaneous contact point solution method described in steps 5) and 6), the tooth flank contact point at the next meshing position is calculated and solved, and the transmission error, the instantaneous contact line, or the contact area is calculated therefrom. When the pinion is rotated
Is subject to [ -2 π/N 1 ，2π/N 1 ]Then stop the calculation, where N 1 Is the pinion tooth count.
While the invention has been described with reference to specific embodiments, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention.
Claims (2)
1. A gear tooth contact analysis method based on full tooth surface search is characterized by comprising the following steps:
step 1: establishing a follow-up coordinate system S 1 And S 2 Said coordinate system S 1 Has coordinate axis of x 1 ，y 1 ，z 1 Said coordinate system S 2 Has coordinate axis of x 2 ，y 2 ，z 2 ，；S 1 Rigidly connected to pinion gear, z 1 The shaft coincides with the pinion axis; likewise, S 2 Rigidly connected to the gear wheel, z 2 The shaft is superposed with the large gear shaft;
gear tooth surface deviation data is obtained through three-coordinate measurement, and a normal deviation curved surface is obtained through cubic B spline fitting
The theoretical tooth surface and the normal deviation curved surface are superposed to construct the actual tooth surface of the gear;
pinion tooth flank Σ (1) In a following coordinate system S 1 The following tooth surface equation is:
wherein
Is the pinion tooth surface ∑ (1) Position vector of (1), p (1) And q is (1) Is a surface parameter;
likewise, the tooth flanks of the bull gear (2) In a following coordinate system S 2 The following tooth surface equation is:
wherein
Is the tooth surface sigma of the gearwheel (2) Position vector of (1), p (2) And q is (2) Is a surface parameter;
step 2: selecting an initial position: setting the initial engagement position to O a And O b Position at time of coincidence, wherein the initial point O a And O b Are respectively positioned at the middle points of the tooth widths of the small gear and the big gearThe tooth crest is at the position of 1 time of modulus; angle of rotation of pinion and gearwheel
And
from the initial position, the initial rotation angle of the pinion is taken as
The initial turning angle of the large gear is
The two gears are defaulted to solve the contact point from the initial position;
and step 3: grid division: with pinion tooth flank sigma in the home position (1) Upper initial point O a As an origin, a following coordinate system S rigidly connected with the pinion is established a Coordinate axis being x a ，y a ，z a Plane x of coordinate system a y a Tooth surface sigma of pinion (1) Tangent to the origin O a Axis x a The direction of the gear is the same as the tooth direction of the pinion; in the coordinate system plane x a y a Grid T for dividing m rows and n columns (1) Grid transverse and axis x a Parallel, longitudinal and axis y a Parallel connection; for the same reason, with O b As an origin, a follow-up coordinate system S rigidly connected with the gearwheel is established b Coordinate axis of x b ，y b ，z b Dividing the grid T (2) (ii) a Definition of T (1) Or T (2) Points which are positioned at the geometric center of each grid are grid points, and the rest points are non-grid points; points obtained by projecting grid points or non-grid points on the gear tooth surface are projection points; will be at T in the following distance search (1) The grid points on the grid are indexes;
to facilitate coordinate transformation in subsequent distance searches, at an initial position, i.e. at
Then, a coordinate system S is established a Coincident fixed coordinate systems S p Coordinate axis being x p ，y p ，z p Establishing and coordinating system S in the same way b Coincident fixed coordinate system S g Coordinate axis of x g ，y g ，z g (ii) a Due to the initial position O a And O b Coincidence, from a following coordinate system S a 、S b Relation of axes to a coordinate system S p About axis x p Rotating 180 degrees to obtain a coordinate system S g ；
And 4, step 4: calculating Z coordinates of tooth surface points
Calculating pinion tooth flank ∑ (1) In a coordinate system S a The following tooth surface equation is:
wherein M is a1 As a coordinate system S 1 To a coordinate system S a The coordinate transformation matrix of (2);
as a coordinate system S 1 Origin O of 1 To the initial point M (1) The vector of (a);
solving in a coordinate system S a Lower grid point
The coordinates of (a):
left side of equation
And
as a grid point
The horizontal and vertical coordinates of (2); to the right of the equation is in the coordinate system S a The following vector equation;
according to the pinion tooth flank ∑ (1) In a coordinate system S a Tooth surface equation under the condition that grid points can be obtained
Along z a Towards the pinion flank surface ∑ (1) Projection point
Z coordinate of (2), as
In the same way, the tooth surface sigma of the large gear can be solved (2) Upper point of
In a coordinate system S b Z coordinate of lower, noted
From this, the tooth surfaces sigma of the two gears can be calculated (1) Or Σ (2) Upper arbitrary point P in corresponding following coordinate system S a Or S b The lower Z coordinate can be obtained by the interpolation of the Z coordinate of the projection point of any point P around the point P; therefore, during distance searching, only the projection point of the grid point on the tooth surface is subjected to Z coordinate calculation through a tooth surface equation, and the Z coordinate of the projection point of the non-grid point on the tooth surface is obtained through a two-dimensional interpolation method;
and 5: tooth surface distance calculation
Let T (1) A certain grid point on
On the pinion tooth flank ∑ (1) Projected points on are points
Dot
To T (2) The projection point of the plane is marked as M, and the point M is on the grid plane T at the moment (2) The upper is usually a non-grid point, so the point M is found to the bull gear flank ∑ (2) Projection point on
The interpolation described above is needed; to calculate points
To point
The two tooth surface equations need to be placed under the same coordinate system; dot
In a coordinate system S b The following coordinate vectors may be expressed as:
wherein
As a coordinate system S a The following vector equation; m ba Is S a To S b The coordinate transformation matrix of (2) is in a transformation mode of: s a →S p →S g →S b ；H a Is the pinion initial point M (1) To the initial point M of the gearwheel (2) A vector of (a); i.e. i b ，j b ，k b Are respectively S b Unit vectors of the next three axes;
setting point
In a coordinate system S b Z coordinate of
Dot
In a coordinate system S b Z coordinate of
Then point is reached
To point
Is recorded as
Assuming that the pinion initial rotation angle is
Calculating the initial rotation angle of the large gear according to the transmission ratio
Calculating grid T of m rows and n columns (1) Upper all grid points along z a Towards the pinion flank surface ∑ (1) All projected points of (a); according to the tooth surface distance calculation method, the small wheel tooth surface sigma is calculated (1) All projection points are projected to the gear surface of the bull wheel (2) The distance of the upper corresponding point; find the minimum distance, noted as Δ z ba,min ；
If Δ z is ba，min If the tooth surfaces of the two gears are not contacted with each other, the instantaneous connection needs to be solved iterativelyA contact; if Δ z is ba，min If the tooth surfaces of the two gears are in mutual contact or even embedded, the condition is indicated; epsilon is a given tolerance and is used for judging the states of the two tooth surfaces;
step 6: iterative solution of instantaneous contact points
If Δ z is ba，min If the transmission error is larger than epsilon, the following adjustment is needed, otherwise, the transmission error is directly calculated; flank of small wheel sigma at minimum distance (1) The upper point is K, the big wheel tooth surface is sigma (2) The upper point is Q, and the point K and the point Q are tooth surface approach points;
Δz ba，min at Q point line velocity V Q Is set as Δ V Then, then
In the formula, QC is the shortest distance from a point Q to a bull gear shaft;
thereafter, the pinion rotation angle is maintained
The rotation angle of the big gear is unchanged
Continuously searching the approaching point of the tooth surface and calculating new
Such continuous iterative adjustment of the rotation angle of the large gear
Solving the minimum distance of the tooth surface; when the minimum distance Δ z ba，min Less than or equal to a given tolerance epsilon, at which point the flanks are considered to have contacted, their corresponding teethThe surface approach point is a contact point;
when the tooth surfaces contact, calculating the transmission error according to the definition of the transmission error;
find all instantaneous contact points: to simulate the rolling inspection of coating thickness, T (1) Each grid point is on the pinion tooth surface ∑ (1) The projection point on the upper plate moves a distance xi along the direction of the external normal; using new projection point as index to search the distance delta z between two tooth surfaces again ba The rotation angle of the large gear is not iteratively adjusted in the search; tooth flank distance Δ z ba All the tooth surface points of which the thickness xi is less than or equal to the rolling inspection experiment coating are regarded as contact points;
and 7: in increments of
Changing the angle of rotation of the pinion in positive and negative directions, respectively
Calculating the rotation angle of the large gear according to the transmission ratio
Calculating and solving a tooth surface contact point at the next meshing position according to the tooth surface distance calculation method and the instantaneous contact point solution method in the steps 5 and 6, and calculating a transmission error, an instantaneous contact line or a contact area; when the pinion is rotated
Is subject to [ -2 π/N 1 ，2π/N 1 ]Then stop the calculation, where N 1 Is the pinion tooth number.
2. A computer system, comprising: one or more processors, a computer readable storage medium, for storing one or more programs, wherein the one or more programs, when executed by the one or more processors, cause the one or more processors to implement the method of claim 1.
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| 刚支时弧齿锥齿轮齿面啮合迹的确定;刘光磊等;《机械科学与技术》;20010530(第03期);全文 * |
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