CN113987716A - Dynamic three-dimensional contact stress calculation method for tooth surface of multistage gear pair - Google Patents

Dynamic three-dimensional contact stress calculation method for tooth surface of multistage gear pair Download PDF

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CN113987716A
CN113987716A CN202111332364.6A CN202111332364A CN113987716A CN 113987716 A CN113987716 A CN 113987716A CN 202111332364 A CN202111332364 A CN 202111332364A CN 113987716 A CN113987716 A CN 113987716A
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tooth surface
dimensional contact
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栾圣罡
尹逊民
杨龙
袁冰
刘更
韩冰
杨小辉
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Northwestern Polytechnical University
703th Research Institute of CSIC
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Abstract

The invention aims to provide a method for calculating dynamic three-dimensional contact stress of a tooth surface of a multistage gear pair, which deeply fuses a tooth surface three-dimensional contact analysis method and a gear rotor system dynamics solving technology and constructs a gear rotor system three-dimensional contact-dynamics coupling model. By combining a Fourier series fast solving method and a tooth surface three-dimensional contact stress calculating method, tooth surface three-dimensional contact stress distribution of each stage of gear pair at different rotating speeds can be obtained. The invention realizes the establishment of the tooth surface dynamic three-dimensional contact stress calculation method on the basis of tooth surface quasi-static three-dimensional contact stress calculation, realizes the real-time tracking of the tooth surface instantaneous three-dimensional contact state under the dynamic three-dimensional contact condition, and lays a more scientific and reasonable foundation for the dynamic strength design of gear parameters.

Description

Dynamic three-dimensional contact stress calculation method for tooth surface of multistage gear pair
Technical Field
The invention relates to a dynamics analysis method, in particular to a gear contact stress calculation method.
Background
In the actual operation process of the gear transmission system, due to the influence of system vibration displacement, meshing dynamic loads different from rated loads are borne between meshing tooth surfaces under different rotating speed working conditions, and at the moment, the three-dimensional contact state of the tooth surfaces is greatly different from that under a quasi-static condition.
Manufacturing/assembly errors are inevitable for gear transmission systems. Under quasi-static conditions, the introduction of tooth flank errors will cause the actual contact state of the tooth flanks to differ from an error-free gear pair. When the gear transmission system further introduces system vibration displacement in the operation process, the actual three-dimensional contact state of the tooth surface is more complicated and is different under different rotating speed working conditions. Tooth surface errors and system vibration displacement will have a coupling effect on the actual three-dimensional contact state of the tooth surface.
At present, the method for calculating the three-dimensional contact stress of the tooth surface mainly aims at the calculation of the three-dimensional contact stress under a static condition, namely a rated load condition, and the method for efficiently calculating the dynamic three-dimensional contact stress of the tooth surface by comprehensively considering the error distribution of the tooth surface and the influence of system vibration is not reported.
Disclosure of Invention
The invention aims to provide a dynamic three-dimensional contact stress calculation method for a tooth surface of a multistage gear pair, which has higher calculation precision and higher solving efficiency.
The purpose of the invention is realized as follows:
the invention discloses a dynamic three-dimensional contact stress calculation method for a tooth surface of a multistage gear pair, which is characterized by comprising the following steps of:
(1) calculating time-varying meshing rigidity, tooth surface three-dimensional contact stress and no-load transmission error under a quasi-static condition by adopting a tooth surface bearing contact analysis method;
(2) inputting shafting parameters, bearing parameters, gear basic parameters and power input/output parameters, and establishing a rigidity matrix and a damping matrix of a shafting unit, a rigidity matrix and a damping matrix of a bearing unit and a rigidity matrix and a damping matrix of a gear meshing unit;
(3) establishing a box body finite element model, coupling nodes of inner holes of all bolts and constraining all degrees of freedom, coupling nodes of inner holes of all bearing seats, and extracting a rigidity matrix of the bearing seats by adopting a finite element substructure technology;
(4) assembling a system stiffness matrix and a damping matrix according to a structure finite element method, and establishing a gear-shafting-bearing-box body system dynamic model;
(5) performing approximate transformation on a system dynamic model to convert a parametric differential equation set into a stationary differential equation set, solving system vibration displacement response by adopting a Fourier series method, and calculating dynamic meshing force and dynamic transmission error of each stage of gear pair;
(6) introducing the dynamic meshing force into a tooth surface bearing contact equation, and solving dynamic contact performance parameters including time-varying meshing rigidity and tooth surface three-dimensional contact stress again;
(7) and (5) substituting the solved dynamic meshing excitation into a system dynamic model, calculating a dynamic transfer error, judging whether the dynamic transfer error reaches a convergence condition, returning to the step (6) to perform iterative calculation if the dynamic transfer error does not meet the convergence condition, and inputting the dynamic three-dimensional contact stress of each stage of gear pair until the convergence condition is met.
The present invention may further comprise:
1. the tooth surface bearing contact equation is
Figure BDA0003349258110000021
In the formula, [ lambda ]G]Is the bending deformation compliance matrix of the contact points, { uLHertz contact deformation of contact points, { d } residual gap between contact points after loading, { epsilon } residual gap between contact points, { F } tooth transfer error, gear pair transfer error, and tooth transfer erroriLoad at contact point i{ I } is an n-dimensional unit vector, { F } is a tooth surface load distribution vector, and P is a normal meshing force;
the time-varying meshing stiffness calculation formula is:
Figure BDA0003349258110000022
2. the calculation formula of the three-dimensional contact stress of the tooth surface is
Figure BDA0003349258110000023
Where rho1And ρ2Radius of curvature of contact points on driving and driven wheels, E1And E2Young's modulus, v, of driving and driven wheels1V and v2Poisson's ratio of driving wheel and driven wheel, FiThe load carried by the contact point.
3. The original form of the system dynamics equation is
Figure BDA0003349258110000031
In the formula MGIs a system quality matrix, CGIs a system damping matrix, KGIs a system stiffness matrix, xGIs a generalized coordinate vector of a system node, eGFor integration of the meshing error vectors, FGIs the system external load vector;
the differential equation of motion of the gear-shaft-bearing-box body system is rewritten into
Figure BDA0003349258110000032
Removing an inertia term and a damping term in the equation to obtain a matrix form of a gear-shaft-bearing system statics equilibrium equation
KG(t)[xGs(t)-eG(t)]＝FG
Wherein x isGs(t) is the system static displacement vector;
time-varying stiffness matrix KG(t) writing as a superposition of mean and undulation values
KG(t)＝KG0+ΔKG(t)
Can obtain
Figure BDA0003349258110000033
X at the right end of the above formulaG(t) with xGs(t) when the substitution is made, the above formula can be approximately rewritten as
Figure BDA0003349258110000034
4. For the system dynamic equation after the normalization, the solving process of the Fourier series method is as follows:
the system dynamic displacement vector xG(t) written in the form of a Fourier series:
Figure BDA0003349258110000041
Figure BDA0003349258110000042
Figure BDA0003349258110000043
wherein, ω ismThe gear pair meshing frequency; n is the order of Fourier series;
as above, the excitation term on the right-hand side of the equation is written in the form of a Fourier series:
Figure BDA0003349258110000044
making the harmonic coefficients at the left and right ends of the equation equal, then
Figure BDA0003349258110000045
Wherein, ω isk＝iωm(i ═ 1, 2.., N), the harmonic coefficient a is obtained by solving the linear equation system of the above formulaiAnd BiFurther obtain the dynamic displacement vector x of the systemG(t)。
The invention has the advantages that: the invention deeply fuses a tooth surface three-dimensional contact analysis method and a gear rotor system dynamics solving technology, and constructs a gear rotor system three-dimensional contact-dynamics coupling model. By combining a Fourier series fast solving method and a tooth surface three-dimensional contact stress calculating method, tooth surface three-dimensional contact stress distribution of each stage of gear pair at different rotating speeds can be obtained. The invention realizes the establishment of the tooth surface dynamic three-dimensional contact stress calculation method on the basis of tooth surface quasi-static three-dimensional contact stress calculation, realizes the real-time tracking of the tooth surface instantaneous three-dimensional contact state under the dynamic three-dimensional contact condition, and lays a more scientific and reasonable foundation for the dynamic strength design of gear parameters.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic view of the meshing action surface of the gear pair;
FIG. 3 is a schematic view of a finite element model of a tank;
FIG. 4 is a two-stage gerotor bearing system dynamics model diagram.
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 method for calculating the dynamic three-dimensional contact stress of the tooth surface of the multistage gear pair provided by the invention comprises the following specific steps:
(1) calculating quasi-static meshing excitation of each stage of gear pair according to gear basic parameters, gear error parameters and working condition parameters;
(2) establishing a gear-shafting-bearing-box body system dynamic model, and converting a system dynamic equation into a steady dynamic equation;
(3) solving the dynamic engaging force and the dynamic transmission error of each gear pair by adopting a Fourier series method;
(4) bringing the dynamic meshing force into a tooth surface bearing contact equation to solve meshing excitation again;
(5) and substituting the meshing excitation into a system dynamic model to solve dynamic meshing force and dynamic transmission errors, and judging whether convergence precision is met. If not, returning to the step 4; and if so, outputting the dynamic three-dimensional contact stress of the tooth surface.
The quasi-static meshing excitation calculation method in the step (1) is as follows:
the dynamic meshing process of the helical gear pair is shown in fig. 2. Rectangular region B1B2B3B4Is the meshing action surface of the gear pair. N is a radical of1N2And B1B2Respectively a theoretical end surface meshing line and an actual end surface meshing line. r isb1And rb2The base circle radius of the driving wheel and the driven wheel are respectively. O isM-XMYMZMIs a calculated coordinate system on the engagement reaction surface. O isi-XiYiZiIs a geometric coordinate system of the driving wheel and the driven wheel. Wherein, when i equals 1, it represents the driving wheel, and when i equals 2, it represents the driven wheel.
The tooth surface bearing contact equation is
Figure BDA0003349258110000051
In the formula, [ lambda ]G]A bending deformation flexibility matrix of the contact point; { uLHertz' contact deformation at the point of contact; TE is gear pair transfer error, { d } is residual gap between contact points after loading, { ε } is residual gap between contact points, FiFor the load at contact point I, { I } is the n-dimensional unit vector, { F } is the tooth flank load distribution vector, and P is the normal meshing force.
The time-varying meshing stiffness calculation formula is:
Figure BDA0003349258110000061
the calculation formula of the three-dimensional contact stress of the tooth surface is
Figure BDA0003349258110000062
Where rho1And ρ2The curvature radius of the contact point on the driving wheel and the driven wheel; e1And E2The Young modulus of the driving wheel and the driven wheel; v is1V and v2The Poisson ratio of the driving wheel and the driven wheel is obtained; fiThe load carried by the contact point.
When the gear-shafting-bearing-box body system dynamic model is established in the step (2), the flexibility of the box body is considered by adopting a three-dimensional finite element method and a substructure technology, as shown in figure 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. The original form of the system dynamics equation is
Figure BDA0003349258110000063
In the formula MGIs a system quality matrix; cGIs a system damping matrix; kGIs a system stiffness matrix; x is the number ofGGeneralized coordinate vectors of system nodes are taken as the generalized coordinate vectors; e.g. of the typeGIs a comprehensive meshing error vector; fGIs the system external load vector.
The differential equation of motion of the gear-shaft-bearing-box body system can be rewritten as
Figure BDA0003349258110000064
The matrix form of the gear-shaft-bearing system static equilibrium equation can be obtained by removing the inertia term and the damping term in the equation
KG(t)[xGs(t)-eG(t)]＝FG
Wherein x isGsAnd (t) is a system static displacement vector.
Time-varying stiffness matrix KG(t) writable as a superposition of mean and ripple values
KG(t)＝KG0+ΔKG(t)
Can obtain
Figure BDA0003349258110000071
X at the right end of the above formulaG(t) with xGs(t) when the substitution is made, the above formula can be approximately rewritten as
Figure BDA0003349258110000072
When the dynamic engaging force and the dynamic transmission error of each stage of gear pair are solved in the step (3), the dynamic displacement vector x of the system is obtainedG(t) writing in the form of a Fourier series
Figure BDA0003349258110000073
Figure BDA0003349258110000074
Figure BDA0003349258110000075
Wherein, ω ismThe gear pair meshing frequency; n is the order of Fourier series.
Similarly, the excitation term at the right end of the equation is written in the form of Fourier series
Figure BDA0003349258110000076
Making the harmonic coefficients at the left and right ends of the equation equal, then
Figure BDA0003349258110000077
Wherein, ω isk＝iωm(i ═ 1, 2.., N). The harmonic coefficient A can be obtained by solving the linear equation system of the above formulaiAnd BiFurther obtain the dynamic displacement vector x of the systemG(t)。

Claims (5)

1. A dynamic three-dimensional contact stress calculation method for a tooth surface of a multistage gear pair is characterized by comprising the following steps:
(1) calculating time-varying meshing rigidity, tooth surface three-dimensional contact stress and no-load transmission error under a quasi-static condition by adopting a tooth surface bearing contact analysis method;
(2) inputting shafting parameters, bearing parameters, gear basic parameters and power input/output parameters, and establishing a rigidity matrix and a damping matrix of a shafting unit, a rigidity matrix and a damping matrix of a bearing unit and a rigidity matrix and a damping matrix of a gear meshing unit;
(3) establishing a box body finite element model, coupling nodes of inner holes of all bolts and constraining all degrees of freedom, coupling nodes of inner holes of all bearing seats, and extracting a rigidity matrix of the bearing seats by adopting a finite element substructure technology;
(4) assembling a system stiffness matrix and a damping matrix according to a structure finite element method, and establishing a gear-shafting-bearing-box body system dynamic model;
(5) performing approximate transformation on a system dynamic model to convert a parametric differential equation set into a stationary differential equation set, solving system vibration displacement response by adopting a Fourier series method, and calculating dynamic meshing force and dynamic transmission error of each stage of gear pair;
(6) introducing the dynamic meshing force into a tooth surface bearing contact equation, and solving dynamic contact performance parameters including time-varying meshing rigidity and tooth surface three-dimensional contact stress again;
(7) and (5) substituting the solved dynamic meshing excitation into a system dynamic model, calculating a dynamic transfer error, judging whether the dynamic transfer error reaches a convergence condition, returning to the step (6) to perform iterative calculation if the dynamic transfer error does not meet the convergence condition, and inputting the dynamic three-dimensional contact stress of each stage of gear pair until the convergence condition is met.
2. The method for calculating the dynamic three-dimensional contact stress of the tooth surface of the multistage gear pair as claimed in claim 1, wherein the method comprises the following steps: the tooth surface bearing contact equation is
Figure FDA0003349258100000011
In the formula, [ lambda ]G]Is the bending deformation compliance matrix of the contact points, { uLHertz contact deformation of contact points, { d } residual gap between contact points after loading, { epsilon } residual gap between contact points, { F } tooth transfer error, gear pair transfer error, and tooth transfer erroriLoad of a contact point I, { I } is an n-dimensional unit vector, { F } is a tooth surface load distribution vector, and P is a normal meshing force;
the time-varying meshing stiffness calculation formula is:
Figure FDA0003349258100000021
3. the method for calculating the dynamic three-dimensional contact stress of the tooth surface of the multistage gear pair as claimed in claim 1, wherein the method comprises the following steps: the calculation formula of the three-dimensional contact stress of the tooth surface is
Figure FDA0003349258100000022
Where rho1And ρ2Radius of curvature of contact points on driving and driven wheels, E1And E2Young's modulus, v, of driving and driven wheels1V and v2Poisson's ratio of driving wheel and driven wheel, FiThe load carried by the contact point.
4. The method for calculating the dynamic three-dimensional contact stress of the tooth surface of the multistage gear pair as claimed in claim 1, wherein the method comprises the following steps: the original form of the system dynamics equation is
Figure FDA0003349258100000023
In the formula MGIs a system quality matrix, CGIs a system damping matrix, KGIs a system stiffness matrix, xGIs a generalized coordinate vector of a system node, eGFor integration of the meshing error vectors, FGIs the system external load vector;
the differential equation of motion of the gear-shaft-bearing-box body system is rewritten into
Figure FDA0003349258100000024
Removing an inertia term and a damping term in the equation to obtain a matrix form of a gear-shaft-bearing system statics equilibrium equation
KG(t)[xGs(t)-eG(t)]＝FG
Wherein x isGs(t) is the system static displacement vector;
time-varying stiffness matrix KG(t) writing as a superposition of mean and undulation values
KG(t)＝KG0+ΔKG(t)
Can obtain
Figure FDA0003349258100000031
X at the right end of the above formulaG(t) By xGs(t) when the substitution is made, the above formula can be approximately rewritten as
Figure FDA0003349258100000032
5. The method for calculating the dynamic three-dimensional contact stress of the tooth surface of the multistage gear pair as claimed in claim 1, wherein the method comprises the following steps: for the system dynamic equation after the normalization, the solving process of the Fourier series method is as follows:
the system dynamic displacement vector xG(t) written in the form of a Fourier series:
Figure FDA0003349258100000033
Figure FDA0003349258100000034
Figure FDA0003349258100000035
wherein, ω ismThe gear pair meshing frequency; n is the order of Fourier series;
as above, the excitation term on the right-hand side of the equation is written in the form of a Fourier series:
Figure FDA0003349258100000036
making the harmonic coefficients at the left and right ends of the equation equal, then
Figure FDA0003349258100000037
Wherein, ω isk＝iωm(i ═ 1, 2.., N), the harmonic coefficient a is obtained by solving the linear equation system of the above formulaiAnd BiFurther obtain the dynamic displacement vector x of the systemG(t)。
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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108052760A (en) * 2017-12-25 2018-05-18 长安大学 A kind of gear pair nonlinear kinetics computational methods
CN110321655A (en) * 2019-07-11 2019-10-11 西北工业大学 Flank of tooth dynamic load distribution calculation method
CN110334460A (en) * 2019-07-11 2019-10-15 西北工业大学 Roller gear mesh stiffness calculation method

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108052760A (en) * 2017-12-25 2018-05-18 长安大学 A kind of gear pair nonlinear kinetics computational methods
CN110321655A (en) * 2019-07-11 2019-10-11 西北工业大学 Flank of tooth dynamic load distribution calculation method
CN110334460A (en) * 2019-07-11 2019-10-15 西北工业大学 Roller gear mesh stiffness calculation method

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
袁冰等: "大齿宽齿轮耦合转子系统准静态接触分析及动态特性", 《机械工程学报》 *
袁冰等: "对角修形对斜齿轮系统准静态及动态特性的影响研究", 《西北工业大学学报》 *

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