CN114048653A - Herringbone gear pair meshing dislocation amount calculation method considering system flexibility - Google Patents

Herringbone gear pair meshing dislocation amount calculation method considering system flexibility Download PDF

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CN114048653A
CN114048653A CN202111331660.4A CN202111331660A CN114048653A CN 114048653 A CN114048653 A CN 114048653A CN 202111331660 A CN202111331660 A CN 202111331660A CN 114048653 A CN114048653 A CN 114048653A
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meshing
unit
gear pair
flexibility
matrix
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栾圣罡
王鑫
尹逊民
刘更
韩冰
袁冰
杨小辉
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Northwestern Polytechnical University
703th Research Institute of CSIC
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703th Research Institute of CSIC
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H57/00—General details of gearing
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H57/00—General details of gearing
    • F16H2057/0087—Computer aided design [CAD] specially adapted for gearing features ; Analysis of gear systems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

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Abstract

The invention aims to provide a herringbone gear pair meshing dislocation amount calculation method considering system flexibility. The method can calculate the meshing dislocation quantity of the herringbone gear pair under different system structure parameters and various layout forms, and provides an effective means for tooth surface meshing quality control and subsequent dynamics analysis and calculation.

Description

Herringbone gear pair meshing dislocation amount calculation method considering system flexibility
Technical Field
The invention relates to a gear pair calculation method, in particular to a gear pair dislocation amount calculation method.
Background
The herringbone gear pair consists of two pairs of large-tooth-width bevel gears with equal helical angles and opposite rotation directions, and the single-side tooth width is generally more than 10 times of the modulus. Because the herringbone gear is generally high in contact ratio and relatively large in the number of meshed gears, compared with a conventional straight/helical gear pair, the influence of manufacturing/assembling errors on the actual three-dimensional contact state of the tooth surfaces of the herringbone gear is more complicated. Due to the large radial dimension, the structure is unique and complex, and the manufacturing/assembling error control, the tooth surface modification design, the vibration noise control and the like are more difficult. In addition, the basic characteristic of large tooth width enables the actual three-dimensional contact state of the tooth surface to be obviously influenced by system deformation (shafting deformation, bearing deformation and box body deformation), and further causes the actual meshing state of the left and right meshing tooth surfaces to have larger difference.
As a key parameter for measuring the influence of system flexibility on the actual meshing state of the tooth surface, the calculation method of the meshing dislocation quantity is extremely few, the calculation precision and the solving efficiency are difficult to be considered, and the calculation method of the meshing dislocation quantity of the herringbone gear pair considering the system flexibility is not reported.
Disclosure of Invention
The invention aims to provide a herringbone gear pair meshing dislocation amount calculation method considering system flexibility, which organically combines a beam unit theory and a finite element substructure technology, has higher calculation precision and higher solving efficiency.
The purpose of the invention is realized as follows:
the invention relates to a method for calculating meshing dislocation quantity of a herringbone gear pair considering system flexibility, which is characterized by comprising the following steps of:
(1) discretizing each shaft into a shaft system unit, and establishing a rigidity matrix of each shaft system unit by adopting an Euler-Bernoull beam theory;
(2) respectively dispersing left and right meshing pairs of the herringbone gears into a thin-plate gear meshing unit along the axial direction, and establishing a rigidity matrix of the thin-plate gear meshing unit according to basic parameters and a meshing principle of the gears;
(3) discretizing the four bearings respectively along the axial direction, establishing a bearing-box unit, and simulating the rigidity of the bearing-box unit by adopting a group of parallel springs respectively;
(4) regarding the box body as a basic unit, establishing a three-dimensional finite element model of the box body, respectively coupling inner holes of four bearing seats into four concentrated mass points, and extracting a rigidity matrix of the four concentrated mass points by adopting a finite element substructure technology;
(5) based on a structure finite element method, a system stiffness matrix is obtained by assembling element stiffness matrixes, and a system statics balance equation set is further established;
(6) and obtaining the meshing dislocation quantity of the herringbone gear pair considering the flexibility of the system according to the relative displacement of each slicing gear meshing unit by solving a system static equilibrium equation set.
The present invention may further comprise:
1. the rigidity of the sheet gear meshing unit is calculated by adopting a flexibility matrix method, and the calculation formula of the flexibility matrix method is
Figure BDA0003349043920000021
In the formula, [ lambda ] is an elastic deformation compliance matrix of the contact points, { F } is a load borne by each contact point, C is a deformation amount of the contact point, and P is a total load.
2. The static equilibrium equation of the bearing-box unit is
Figure BDA0003349043920000022
In the formula, KbIs a bearing stiffness matrix, qbiIs the displacement vector of the bearing node, qhiAnd the displacement vector of the equivalent node of the box body.
3. The statics balance equation of the box body unit is
Khqh＝0
In the formula, KhBox stiffness matrix, q, obtained for finite element substructurehAnd the displacement vector of the equivalent node of the box body.
4. The meshing dislocation quantity of the herringbone gear pair is as follows:
Figure BDA0003349043920000023
in the formula (I), the compound is shown in the specification,
Figure BDA0003349043920000024
indicating the amount of meshing misalignment of the ith sheet gear unit of the left meshing pair,
Figure BDA0003349043920000025
indicating the amount of meshing misalignment of the ith sheet gear unit of the right meshing pair,
Figure BDA0003349043920000026
showing the relative displacement of the ith sheet gear pair of the left meshing pair along the normal meshing line direction,
Figure BDA0003349043920000031
showing the relative displacement of the ith plate gear pair of the right meshing pair in the direction of the normal line of engagement.
5. The stiffness calculation for the sheet engagement unit is written as:
Figure BDA0003349043920000032
wherein N is the number of contact points of the sheet engagement unit.
The invention has the advantages that: the method can accurately predict the tooth surface meshing quality under the flexibility of the system, thereby effectively guiding the structural parameter design and the model selection of the gear transmission system and further laying a foundation for considering the tooth surface modification design of the flexibility of the system.
Drawings
FIG. 1 is a schematic axial discretization of a gear pair;
FIG. 2 is a schematic view of a wafer gear engagement unit;
FIG. 3 is a schematic view of a finite element model of a case;
FIG. 4 is a schematic view of the meshing misalignment of the herringbone gear pair.
Detailed Description
The invention will now be described in more detail by way of example with reference to the accompanying drawings in which:
with reference to fig. 1-4, the invention provides a herringbone gear pair meshing misalignment amount calculation method considering system flexibility, which comprises the following steps:
(1) according to the structure and layout characteristics of the gear-shafting-bearing-box body, the system is discretized, and a Euler-Bernoull beam unit is adopted to simulate a shafting unit.
(2) The left side and the right side of the herringbone helical gear pair are dispersed into a series of thin sheets along the tooth width, and the meshing process of the left meshing pair after the herringbone helical gear pair is dispersed along the axial direction is shown in figure 1, wherein rpAnd rgIs the radius of the base circle of the driving and driven gears, omegapAnd ωgIs the rotational speed of the driving and driven gears, OpAnd OgIs the rotation center of the driving and driven gears. N is a radical of1N2Is a theoretical line of mesh, B1B2B3B4Is a plane of engagement, betabIs the helix angle of the base circle. A model of the i-th nonlinear contact element is shown in fig. 2, where ψ and a are a mounting angle and an engagement angle,
Figure BDA0003349043920000033
and
Figure BDA0003349043920000034
is the rotation center of the i-th nonlinear contact unit.
Figure BDA0003349043920000035
Is the meshing stiffness of the ith non-linear contact unit of the left gear pair,
Figure BDA0003349043920000036
indicating the meshing gap.
Rigidity of ith sheet gear engagement unit of left and right side engagement pair
Figure BDA0003349043920000041
And
Figure BDA0003349043920000042
are respectively calculated as
Figure BDA0003349043920000043
Where M and N are the numbers of contact points on the ith sheet gear engagement units of the left and right engagement pairs, respectively. Is the stiffness of the ith contact point.
The generalized coordinates of the ith chip gear engagement unit may be defined as
Figure BDA0003349043920000044
In the formula
Figure BDA0003349043920000045
And
Figure BDA0003349043920000046
for the lateral displacement about each axis,
Figure BDA0003349043920000047
and
Figure BDA0003349043920000048
is the angle of rotation about each axis.
The ith sheet gear meshing unit is relatively displaced in the direction of the normal meshing line by
Figure BDA0003349043920000049
The expression for the projection vector T can be written as
Figure BDA00033490439200000410
In the formula
Figure BDA00033490439200000411
The + and sign in the "+" and "+" signs
Figure BDA00033490439200000412
The "-" in the symbol represents that the driving wheel rotates anticlockwise, and the "-" and the symbol in the "+/-" symbol
Figure BDA00033490439200000413
The "+" in (1) indicates that the driving wheel rotates clockwise.
(3) Three-dimensional finite element methods and substructure techniques are used to take into account the flexibility of the tank, as shown in fig. 3. For each bolt hole, six degrees of freedom are constrained for all nodes on its inner surface. For each bearing block, all nodes on its inner surface are coupled and then the six-order stiffness matrix of the four bearing holes is compressed using the substructure method. The stiffness matrix thus obtained reflects the flexibility of the tank substantially. After models of the shafting unit, the bearing unit, the sheet gear meshing unit and the box body unit are established, the rigidity matrix of the system can be assembled according to a finite element method. The matrix form of the system statics equilibrium equation can be written as
KGS(t)XGS(t)＝PG
In the formula KGS(t) is a system stiffness matrix; xGS(t) is the system static displacement vector; pGIs an external load vector.
(4) Solving the system statics equilibrium equation can obtain the relative displacement and the meshing dislocation quantity of the left and right side meshing pairs along the normal meshing line direction, as shown in fig. 4. The calculation formula of the engagement displacement of the left and right engagement pairs is
Figure BDA0003349043920000051
In the formula
Figure BDA0003349043920000052
The engagement misalignment amount of the ith sheet gear engagement unit on the left side,
Figure BDA0003349043920000053
is the meshing misalignment amount of the ith right-hand gear meshing unit.
In a word, the method calculates the meshing dislocation quantity of the herringbone gear pair considering the flexibility of the system, and has higher calculation efficiency and more accurate calculation precision.

Claims (6)

1. A herringbone gear pair meshing dislocation amount calculation method considering system flexibility is characterized by comprising the following steps:
(1) discretizing each shaft into a shaft system unit, and establishing a rigidity matrix of each shaft system unit by adopting an Euler-Bernoull beam theory;
(2) respectively dispersing left and right meshing pairs of the herringbone gears into a thin-plate gear meshing unit along the axial direction, and establishing a rigidity matrix of the thin-plate gear meshing unit according to basic parameters and a meshing principle of the gears;
(3) discretizing the four bearings respectively along the axial direction, establishing a bearing-box unit, and simulating the rigidity of the bearing-box unit by adopting a group of parallel springs respectively;
(4) regarding the box body as a basic unit, establishing a three-dimensional finite element model of the box body, respectively coupling inner holes of four bearing seats into four concentrated mass points, and extracting a rigidity matrix of the four concentrated mass points by adopting a finite element substructure technology;
(5) based on a structure finite element method, a system stiffness matrix is obtained by assembling element stiffness matrixes, and a system statics balance equation set is further established;
(6) and obtaining the meshing dislocation quantity of the herringbone gear pair considering the flexibility of the system according to the relative displacement of each slicing gear meshing unit by solving a system static equilibrium equation set.
2. The method for calculating the meshing misalignment of the herringbone gear pair considering the flexibility of the system as claimed in claim 1, wherein the method comprises the following steps: the rigidity of the sheet gear meshing unit is calculated by adopting a flexibility matrix method, and the calculation formula of the flexibility matrix method is
Figure FDA0003349043910000011
In the formula, [ lambda ] is an elastic deformation compliance matrix of the contact points, { F } is a load borne by each contact point, C is a deformation amount of the contact point, and P is a total load.
3. The method for calculating the meshing misalignment of the herringbone gear pair considering the flexibility of the system as claimed in claim 1, wherein the method comprises the following steps: the static equilibrium equation of the bearing-box unit is
Figure FDA0003349043910000012
In the formula, KbIs a bearing stiffness matrix, qbiIs the displacement vector of the bearing node, qhiAnd the displacement vector of the equivalent node of the box body.
4. The method for calculating the meshing misalignment of the herringbone gear pair considering the flexibility of the system as claimed in claim 1, wherein the method comprises the following steps: the statics balance equation of the box body unit is
Khqh＝0
In the formula, KhBox stiffness matrix, q, obtained for finite element substructurehAnd the displacement vector of the equivalent node of the box body.
5. The method for calculating the meshing misalignment of the herringbone gear pair considering the flexibility of the system as claimed in claim 1, wherein the method comprises the following steps: the meshing dislocation quantity of the herringbone gear pair is as follows:
Figure FDA0003349043910000021
in the formula (I), the compound is shown in the specification,
Figure FDA0003349043910000022
indicating the amount of meshing misalignment of the ith sheet gear unit of the left meshing pair,
Figure FDA0003349043910000023
indicating the amount of meshing misalignment of the ith sheet gear unit of the right meshing pair,
Figure FDA0003349043910000024
showing the relative displacement of the ith sheet gear pair of the left meshing pair along the normal meshing line direction,
Figure FDA0003349043910000025
showing the relative displacement of the ith plate gear pair of the right meshing pair in the direction of the normal line of engagement.
6. The method for calculating the meshing misalignment of the herringbone gear pair considering the flexibility of the system as claimed in claim 2, wherein the method comprises the following steps: the stiffness calculation for the sheet engagement unit is written as:
Figure FDA0003349043910000026
wherein N is the number of contact points of the sheet engagement unit.
CN202111331660.4A 2021-11-11 2021-11-11 Herringbone gear pair meshing dislocation amount calculation method considering system flexibility Pending CN114048653A (en)

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