CN109271688B - Accurate calculation method for high-speed gear meshing transmission temperature field - Google Patents
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Abstract
The invention relates to a method for accurately calculating a high-speed gear meshing transmission temperature field, which comprises the following steps of: s1, calculating the actual meshing line length of gear meshing; s2, calculating the distance from the dividing point of the single meshing area and the double meshing area to the center of the gear; s3, calculating a functional relation between the heat flux density of the meshing point in the non-meshing area and the single-tooth and double-tooth meshing areas and the distance between the meshing point and the center of the gear; s4, calculating in finite element software to obtain the distance from each point to the center of the gear; s5, calculating to obtain the heat flux density of each point; s6, selecting corresponding nodes to finish the application of the heat flux density of each point; s7, calculating the convection heat transfer coefficients of different surfaces of the gear; s8, selecting corresponding surfaces to complete the application of the convection heat transfer coefficient; and S9, calculating to obtain a temperature field of high-speed gear transmission. The gear meshing surface is dispersed into a plurality of points based on the discretization idea, the temperature fields of the points are calculated, and then the subsequent analysis and processing are carried out, so that the gear meshing surface temperature calculation method has the advantages of high calculation precision, simplicity and convenience in method, high efficiency and the like.
Description
Technical Field
The invention relates to the technical field of high-speed transmission gears, in particular to an accurate calculation method for a high-speed gear meshing transmission temperature field.
Background
With the continuous innovation of technology and the gradual decrease of petroleum resources, new energy automobiles become a research hotspot. The high-speed low-temperature gear-raising technology is a leading-edge technology of new energy automobile transmission system research, and how to accurately calculate a high-speed gear meshing transmission temperature field is a key for regulating and controlling the temperature rise of the new energy automobile transmission system. However, the heat flux density distribution on the meshing surface of the high-speed gear is very complex, the heat flux density span is large, the heat flux density distribution on the meshing surface is not continuously changed due to the existence of the single meshing area and the double meshing area, and an obvious step of the heat flux density exists at the point of conversion of the single meshing area and the double meshing area, which brings great difficulty to the calculation of the temperature field of the high-speed transmission gear.
At present, the calculation method of the high-speed gear transmission temperature field mainly comprises two methods. One is a simple algorithm, namely, one tooth of the gear is intercepted, and the average value of the heat flux density of the meshing surface is directly taken and added to the meshing surface. The other method is to establish a gear set model in finite element software and define various constraints and boundary conditions so as to calculate the heat generated by the sliding friction between the gear sets.
Disclosure of Invention
The invention aims to provide an accurate calculation method for a high-speed gear meshing transmission temperature field, and the method has the advantages of high calculation accuracy, simplicity and convenience, high efficiency and the like.
The technical scheme adopted by the invention for solving the technical problem is as follows: a method for accurately calculating a high-speed gear meshing transmission temperature field is constructed, and comprises the following steps:
s1, according to various parameters and transmission requirements of a driving gear and a driven gear, a gear set meshing model is made, and the actual meshing line length of gear meshing is calculated;
s2, determining the lengths of the single meshing area and the double meshing area by combining the contact ratio of the meshing of the gear pair with the S1 as a basis, and calculating the distance from a boundary point of the single meshing area and the double meshing area to the center of the gear according to a geometric relationship;
s3, respectively calculating a functional relation between the heat flow density of an engagement point in the non-engagement area and the single-tooth and double-tooth engagement areas and the distance between the engagement point and the center of the gear according to a Hertz contact theory;
s4, establishing a three-dimensional model in finite element software, giving gear material attributes and various physical parameters, dispersing a meshing surface into a plurality of points based on a discretization idea, and calculating the distance from each point to the center of the gear in the finite element software;
s5, comparing S2 to determine the area of each point according to the distance between each point and the center of the gear, and substituting the area into the corresponding relational expression in S3 to calculate the heat flow density of each point;
s6, adding a layer of surface effect unit with nodes overlapped with the discrete points on the meshing surface in finite element software, and selecting corresponding nodes by using APDL command flow to finish the application of heat flow density of each point;
s7, calculating the convection heat transfer coefficients of different surfaces of the gear;
s8, selecting a corresponding surface by using an APDL command stream in finite element software to complete the application of the convection heat transfer coefficient;
and S9, in finite element software, combining boundary conditions in S4, S6 and S8, and finally calculating to obtain the temperature field of the high-speed gear transmission according to a boundary condition formula.
In the above scheme, in the step S1, the actual meshing line length g of the gear meshing s The calculation formula of (a) is as follows:
g s ＝r b1 (tanα a1 -tanα′)+r b2 (tanα a2 -tanα’) (1)
wherein: r is a radical of hydrogen b1 、r b2 The base circle radius of the driving gear and the driven gear are respectively; alpha is alpha a1 、α a2 The gear tooth top pressure angles of the driving gear and the driven gear; α' is the engagement angle.
In the above scheme, in the step S2, the meshing areas when the gears mesh are distributed as follows: the gear is characterized in that a non-meshing area is arranged outside the meshing line, two ends of the meshing line are respectively provided with a section of double-tooth meshing area with the same length, the middle section sandwiched by the two double-tooth meshing areas is a single-tooth meshing area, the number of dividing points of the single-tooth meshing area, the double-tooth meshing area and the non-meshing area is 4, and the distance l from the 4 dividing points to the center of the gear is calculated according to the geometric relationship 1 、l 2 、l 3 、l 4 ：
Wherein: e is the contact ratio of the engaging gears, g s1 、g s2 Respectively, the lengths of the single and double meshing areas are respectively
In the above scheme, in the step S3, the functional relation q (l) between the heat flux density q of the meshing point in the single-tooth meshing zone and the non-meshing zone and the distance l from the meshing point to the gear center is as follows:
wherein: k is a radical of p For heat flow densityA degree distribution coefficient having a value of
Gamma is a heat energy conversion coefficient; t is 0 The meshing period of the driving gear is set; f is the friction coefficient; f n1 The load force of the meshing point when the single tooth is meshed is equal to
F n2 A load force of a mesh point at the time of meshing two teeth, which is set to
k is the interdental load distribution coefficient; p is input power; n is the input rotation speed; v. of 1 、v 2 The sliding speed of the meshing point of the driving gear and the driven gear is respectively the value
And B is the gear tooth width.
In the above scheme, in the step S5, the distances l from each boundary point to the gear center obtained in S2 are compared according to the distances l from each discrete point to the gear center in S4 1 、l 2 、l 3 、l 4 When l is 1 <l<l 2 Or l 3 <l<l 4 Substituting l into equation (7) to calculate q (l); when l is 2 <l<l 3 Then, substituting l into equation (6) to calculate q (l); when l is<l 1 Or l>l 4 Then, q (l) is calculated by substituting l into equation (8).
In the above scheme, in the step S6, in the finite element software, according to the points scattered on the meshing surface, a layer of surface effect units in which nodes coincide with the scattered points is added on the meshing surface, and then each node in the surface effect units is numbered; because the distances l from the meshing points axially distributed on the meshing surface to the center of the gear are equal, the heat flux densities q of the meshing points are also equal, the meshing points axially distributed are selected layer by layer from the addendum circle to the dedendum circle and are compiled into a series of point groups, and then the point groups are numbered; and selecting a point group needing to apply the heat flux q and point groups adjacent to the two sides of the point group by using the APDL command stream through numbering, loading the heat flux density calculated in the S5 to all the selected point groups by using the APDL command stream, and selecting the point groups needing to apply the heat flux density layer by layer from the addendum circle to the dedendum circle through numbering by using the same method to load the heat flux density until all the point groups are loaded with the heat flux density.
In the above scheme, in the step S7, the heat convection coefficients h of different surfaces of the gear are as follows:
wherein: lambda is the heat conductivity coefficient of the lubricating oil; p is r Is lubricating oil prandtl number; r is e Is the lubricating oil Reynolds number; l is a characteristic length; m is an empirical correction coefficient; and omega is the gear rotation angular speed.
In the above scheme, in step S8, the tooth crest, the meshing surface, and the tooth end surface of each gear are respectively numbered in the finite element software, the APDL command stream is used to select the corresponding surface according to the number, and then the corresponding convective heat transfer coefficient h in S4 is loaded on the selected surface.
In the above scheme, in the step S9, the temperature field of the high-speed gear transmission is calculated according to the boundary condition formula (10) by combining the boundary conditions in S4, S6 and S8;
wherein: λ' is the thermal conductivity of the gear; t is 0 Is ambient temperature; t is the thermal equilibrium temperature.
The accurate calculation method for the high-speed gear meshing transmission temperature field has the following beneficial effects:
1. the invention solves the problems that the distribution of the heat flux density on the meshing surface of the high-speed transmission gear is complex, the distribution span of the heat flux density is large, and the heat flux density has step due to the existence of a single meshing area and a double meshing area, so that the accurate calculation is difficult, and has the advantages of high calculation precision, simple method, high efficiency and the like.
2. The gear meshing surface is dispersed into a plurality of points based on the discretization idea, the temperature fields of the points are calculated, and then the subsequent analysis and processing are carried out, so that the gear meshing surface temperature-measuring method has the advantages of high calculation precision, simplicity and convenience in method, high efficiency and the like.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2 is a meshing pattern of a high speed drive gear set in an embodiment of the present invention;
FIG. 3 is a heat flux density distribution function graph of the meshing surface of the high-speed transmission gear in the embodiment of the invention;
FIG. 4 is a three-dimensional model of a high speed drive gear in an embodiment of the invention;
FIG. 5 is a dot matrix after the meshing surface of the high-speed transmission gear is dispersed in the embodiment of the invention;
FIG. 6 is a surface effect sheet added to the meshing surface of a high speed drive gear in an embodiment of the present invention;
FIG. 7 is an analysis of the temperature field of the high speed drive gear set in an embodiment of the present invention.
Detailed Description
For a more clear understanding of the technical features, objects and effects of the present invention, embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
The invention discloses a method for accurately calculating a high-speed gear meshing transmission temperature field, which takes a high-speed transmission gear set with known parameters as an example, and analyzes the temperature field according to the method, wherein the basic parameters of the high-speed transmission gear set are shown in table 1 and are shown in figures 1-7, and the method comprises the following steps:
TABLE 1 high-speed drive gear set parameters
S1, as shown in figure 2, determining the central position O of two gears according to the central distance 1 、O 2 Drawing base circle and addendum circle at the center of each of the two gears, making the common tangent line on the inner sides of the two base circles, intersecting with the addendum circle at B 1 、B 2 Line segment B 1 B 2 I.e. the actual meshing line.
Engagement angle:
tooth tip pressure angle: alpha (alpha) ("alpha") a ＝arccos(r b /r a ) (2)
Therefore alpha a1 ＝35.8966°，α a2 ＝24.2092°。
Wherein: in the lower corner marks, the numeral 1 represents a driving gear, 2 represents a driven gear, and the same applies below, unless otherwise specified.
The contact ratio and the actual meshing line length can be calculated according to the meshing angle and the tooth crest pressure angle:
g s ＝r b1 (tanα a1 -tanα’)+r b2 (tanα a2 -tanα’)＝5.5811mm (3)
s2, determining the lengths of the single and double meshing areas according to the contact ratio epsilon of the meshing of the combined gear pair by taking the S1 as a basis:
when the gears are meshed, the meshing area is distributed as follows: the non-meshing zone is arranged outside the meshing line, two ends of the meshing line are respectively provided with a double-tooth meshing zone with the same length, and the middle section between the two double-tooth meshing zones is a single-tooth meshing zone, so as shown in figure 2, E 1 、E 2 Is a dividing point of a single-tooth meshing zone and a double-tooth meshing zone, i.e. B 1 E 1 、B 2 E 2 Being a double toothed zone, E 1 E 2 A single tooth meshing zone, B 1 、B 2 The outer is the non-engagement zone.
The regions of the double-tooth meshing region and the single-tooth meshing region can be obtained according to the contact ratio:
double-tooth meshing zone:
therefore, it is
Single-tooth meshing zone:
therefore, it is
Respectively calculating B according to the geometric relationship 1 、E 1 、E 2 、B 2 Distance l from the gear center at these 4 dividing points 1 、l 2 、l 3 、l 4 I.e. O in FIG. 2 1 B 1 、O 1 E 1 、O 1 B 2 、O 1 E 2 ：
Therefore, when the distance l between the meshing point and the gear center is between 11.4344-11.990 mm and 12.7585-13.775 mm, the gear is in a double-tooth meshing area; when the diameter is between 11.990 and 12.7585mm, the gear is in a single-tooth meshing area; otherwise in the non-engagement zone.
S3, respectively calculating a functional relation q (l) of the heat flow density q of an engagement point in the single-tooth engagement zone and the double-tooth engagement zone and the distance l from the engagement point to the gear center according to a Hertz contact theory:
assume a mesh point of C, i.e. O 1 C has a length of l and P is O 1 O 2 The intersection with the meshing line is O 1 The length of P is:
the calculated length of PC is:
wherein: the upper operator being adapted to mesh point at PB 1 Between segments, i.e. l is between 11.4344 and 12.0147 mm; the lower layer operator being adapted for mesh point at PB 2 The same applies to the sections l between 12.0147 and 13.775 mm.
Sliding speed of meshing points of driving gears:
driven gear mesh point slip speed:
single tooth meshing zone meshing point load force:
double-tooth meshing zone meshing point load force:
wherein: k is the interdental load distribution coefficient, typically 0.5.
Comprehensive curvature radius:
wherein: rho c1 、ρ c2 Radius of curvature for meshing gear pairs, value thereof
Comprehensive elastic modulus:
wherein: e c1 、E c2 Is the modulus of elasticity, epsilon, of the material of the meshing gear pair 1 、ε 2 Poisson's ratio as material for meshing gear pair
Contact half width of the two gears:
maximum hertzian contact stress:
coefficient of friction:
wherein: v is the kinematic viscosity of the lubricating oil; r a1 、R a2 The roughness of the meshing tooth surface of the two gears; x 1 To correct the coefficient for the lubricating oil, the value thereof
Heat flux density:
wherein: k is a radical of p Distribution coefficient of heat flux density, value thereof
Gamma is the heat energy conversion coefficient which is generally 0.9-0.98; t is t 0 Is Hertz contact half-width in the time domain, the value of which
T 0 For the meshing period of the driving gear, its value
Combining the formulas (8) to (19), finally simplifying:
and S4, as shown in figure 4, establishing a three-dimensional model in finite element software, giving gear material attributes and various physical parameters, dispersing the meshing surface into a plurality of points based on a discretization idea, and calculating in the finite element software to obtain the distance l from each discrete point to the center of the gear.
S5, comparing the distances l from the boundary points to the gear center obtained in the S2 according to the distances l from the discrete points to the gear center in the S4 1 、l 2 、l 3 、l 4 When l is 1 <l<l 2 Or l 3 <l<l 4 Substituting l into equation (24) to calculate q (l); when l is 2 <l<l 3 When, substituting l into equation (23) calculates q (l); when l is<l 1 Or l>l 4 Then, q (l) is calculated by substituting l into equation (25), and the final results are shown in Table 2.
TABLE 2 Heat flow Density at discrete points
S6, as shown in FIG. 6, in the finite element software, according to the dispersed points on the meshing surface, adding a layer of surface effect units with nodes superposed with the dispersed points on the meshing surface, and numbering each node in the surface effect units; the distances l from the meshing points distributed on the meshing surface along the axial direction to the center of the gear are equal, so the heat flux density q is also equal, the meshing points distributed along the axial direction are selected layer by layer from the addendum circle to the dedendum circle and are compiled into a series of point groups, and then the point groups are numbered; and selecting a point group needing to apply the heat flux q and point groups adjacent to the two sides of the point group by using the APDL command stream through numbering, loading the heat flux density calculated in the S5 to all the selected point groups by using the APDL command stream, and selecting the point groups needing to apply the heat flux density layer by layer from the addendum circle to the dedendum circle through numbering by using the same method to load the heat flux density until all the point groups are loaded with the heat flux density.
S7, calculating the convective heat transfer coefficients h of different surfaces of the gear:
wherein: lambda is the heat conductivity coefficient of the lubricating oil; p is r Is lubricating oil prandtl number; r e Is the lubricating oil Reynolds number; l is the characteristic length, here the gear reference circle diameter; m is an empirical correction parameter, and is generally 2; and omega is the gear rotation angular speed.
The calculated values of the convective heat transfer coefficient h of each face of the gear are shown in Table 3
TABLE 3 convective heat transfer coefficient of each face of gear
| Item | (symbol) | Numerical value (W/(m) 2 K)) |
| Tooth crest convection heat transfer coefficient | h d | 2446 |
| Convective heat transfer coefficient of tooth surface | h m | 4438.5 |
| End face convective heat transfer coefficient | h e | 5428.1 |
And S8, as shown in the figure 4, numbering each tooth top surface, each meshing surface and each tooth end surface of the gear in finite element software respectively, selecting a corresponding surface according to the numbering by using an APDL (advanced persistent programming language) command stream, and then loading the corresponding convective heat exchange coefficient h in the table 2 onto the selected surface.
And S9, in finite element software, combining boundary conditions in S4, S6 and S8, and calculating to obtain a temperature field of the high-speed gear transmission according to a boundary condition formula (27), as shown in FIG. 7.
Wherein: λ' is the heat conductivity of the gear; t is 0 Is ambient temperature; t is the thermal equilibrium temperature.
While the present invention has been described with reference to the particular illustrative embodiments, it is to be understood that the invention is not limited to the disclosed embodiments, but is intended to cover various modifications, equivalent arrangements, and equivalents thereof, which may be made by those skilled in the art without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (9)
1. A method for accurately calculating a high-speed gear meshing transmission temperature field is characterized by comprising the following steps of:
s1, according to various parameters and transmission requirements of a driving gear and a driven gear, a gear set meshing model is made, and the actual meshing line length of gear meshing is calculated;
s2, determining the lengths of the single meshing area and the double meshing area by combining the contact ratio of the meshing of the gear pair with the S1 as a basis, and calculating the distance from a boundary point of the single meshing area and the double meshing area to the center of the gear according to a geometric relationship;
s3, respectively calculating a functional relation between the heat flow density of an engagement point in the non-engagement area and the single-tooth and double-tooth engagement areas and the distance between the engagement point and the center of the gear according to a Hertz contact theory;
s4, establishing a three-dimensional model in finite element software, giving gear material properties and various physical parameters, dispersing a meshing surface into a plurality of points based on a discretization thought, and calculating in the finite element software to obtain the distance from each point to the center of the gear;
s5, comparing S2 to determine the area of each point according to the distance between each point and the center of the gear, and substituting the area into the corresponding relational expression in S3 to calculate the heat flow density of each point;
s6, adding a layer of surface effect unit with nodes overlapped with the discrete points on the meshing surface in finite element software, and selecting corresponding nodes by using APDL command flow to finish the application of heat flow density of each point;
s7, calculating the convection heat transfer coefficients of different surfaces of the gear;
s8, selecting a corresponding surface by using an APDL command stream in finite element software to complete the application of the convection heat transfer coefficient;
and S9, in finite element software, combining boundary conditions in S4, S6 and S8, and finally calculating to obtain the temperature field of the high-speed gear transmission according to a boundary condition formula.
2. The method for accurately calculating the temperature field of a high-speed gear meshing transmission according to claim 1, wherein in step S1, the actual meshing line length g of the gear meshing is calculated s The calculation formula of (a) is as follows:
g s ＝r b1 (tanα a1 -tanα′)+r b2 (tanα a2 -tanα’) (1)
wherein: r is b1 、r b2 The base circle radius of the driving gear and the driven gear are respectively; alpha is alpha a1 、α a2 The gear tooth top pressure angles of the driving gear and the driven gear; α' is the engagement angle.
3. The method for accurately calculating the temperature field of high-speed gear meshing transmission according to claim 2, wherein in the step S2, the distribution of the meshing area when the gears are meshed is as follows: the non-meshing area is arranged outside the meshing line, two ends of the meshing line are respectively provided with a double-tooth meshing area with equal length, the middle section between the two double-tooth meshing areas is a single-tooth meshing area, the single-tooth meshing area and the double-tooth meshing area are non-meshing areasThe total number of the dividing points of the combined area is 4, and the distance l from the 4 dividing points to the center of the gear is calculated according to the geometric relationship 1 、l 2 、l 3 、l 4 ：
Wherein: e is the contact ratio of the engaging gears, g s1 、g s2 Respectively, the lengths of the single and double meshing areas are respectively
4. The method for accurately calculating the temperature field of a high-speed gear meshing transmission according to claim 3, wherein in the step S3, the heat flux density q of the meshing point in the single-tooth meshing zone and the non-meshing zone and the distance l from the meshing point to the gear center have the functional relation q (l) as follows:
wherein: k is a radical of formula p A coefficient of heat flux density is assigned a value of
Gamma is a heat energy conversion coefficient; t is 0 The meshing period of the driving gear is shown; f is the friction coefficient; f n1 The load force of the meshing point when the single tooth is meshed is equal to
F n2 A load force of a mesh point at the time of meshing two teeth, which is set to
k is an interdental load distribution coefficient; p is input power; n is the input rotation speed; v. of 1 、v 2 The sliding speeds of the meshing points of the driving gear and the driven gear are respectively the value
And B is the gear tooth width.
5. The method for accurately calculating the temperature field of a high-speed gear meshing transmission according to claim 4, wherein in the step S5, the distances l from each boundary point to the gear center obtained in the step S2 are compared according to the distances l from each discrete point to the gear center in the step S4 1 、l 2 、l 3 、l 4 When l is 1 <l<l 2 Or l 3 <l<l 4 Substituting l into equation (7) to calculate q (l); when l is 2 <l<l 3 When, substituting l into equation (6) calculates q (l); when l is<l 1 Or l>l 4 Then, q (l) is calculated by substituting l into equation (8).
6. The method for accurately calculating the temperature field of the high-speed gear meshing transmission according to claim 5, wherein in the step S6, in finite element software, according to points scattered on a meshing surface, a layer of surface effect units with nodes coincident with the scattered points is added on the meshing surface, and then each node in the surface effect units is numbered; the distances l from the meshing points distributed on the meshing surface along the axial direction to the center of the gear are equal, so the heat flux density q is also equal, the meshing points distributed along the axial direction are selected layer by layer from the addendum circle to the dedendum circle and are compiled into a series of point groups, and then the point groups are numbered; and selecting a point group needing to apply the heat flux q and point groups adjacent to the two sides of the point group by using the APDL command stream through numbering, loading the heat flux density calculated in the S5 to all the selected point groups by using the APDL command stream, and selecting the point groups needing to apply the heat flux density layer by layer from the addendum circle to the dedendum circle through numbering by using the same method to load the heat flux density until all the point groups are loaded with the heat flux density.
7. The method for accurately calculating the temperature field of the high-speed gear meshing transmission according to claim 6, wherein in the step S7, the convective heat transfer coefficients h of different surfaces of the gear are as follows:
wherein: lambda is the heat conductivity coefficient of the lubricating oil; p r Is lubricating oil prandtl number; r is e Is the lubricating oil Reynolds number; l is a characteristic length; m is an empirical correction coefficient; and omega is the gear rotation angular speed.
8. The method for accurately calculating the temperature field in meshing transmission of a high-speed gear according to claim 7, wherein in the step S8, the tooth crest surface, the meshing surface and the tooth end surface of the gear are respectively numbered in finite element software, APDL command streams are used for selecting corresponding surfaces according to the numbers, and then the corresponding convective heat transfer coefficient h in the step S4 is loaded on the selected surfaces.
9. The method for accurately calculating the temperature field of the high-speed gear meshing transmission according to claim 8, wherein in the step S9, the temperature field of the high-speed gear transmission is calculated according to a boundary condition formula (10) by combining boundary conditions in S4, S6 and S8;
wherein: λ' is the thermal conductivity of the gear; t is a unit of 0 Is ambient temperature; t is the thermal equilibrium temperature.
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| CN104268343A (en) * | 2014-09-28 | 2015-01-07 | 北京理工大学 | Method of cutting force prediction and temperature prediction for end-milling cutting |
| CN107763173A (en) * | 2017-11-22 | 2018-03-06 | 电子科技大学 | A kind of helical gear time-variant mesh stiffness computational methods based on finite element analysis |
| WO2018086160A1 (en) * | 2016-11-09 | 2018-05-17 | 北京工业大学 | Rough surface-based three-dimensional contact stiffness calculation method for spur gear |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN104268343A (en) * | 2014-09-28 | 2015-01-07 | 北京理工大学 | Method of cutting force prediction and temperature prediction for end-milling cutting |
| WO2018086160A1 (en) * | 2016-11-09 | 2018-05-17 | 北京工业大学 | Rough surface-based three-dimensional contact stiffness calculation method for spur gear |
| CN107763173A (en) * | 2017-11-22 | 2018-03-06 | 电子科技大学 | A kind of helical gear time-variant mesh stiffness computational methods based on finite element analysis |
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