CN112287554B - Method for predicting contact fatigue damage of gear surface - Google Patents
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Abstract
A method of predicting gear surface contact fatigue damage comprising the steps of: 1) Calculating the comprehensive curvature radius, the half width of a meshing surface and the maximum meshing stress of the gear pair according to the Hertz contact theory and the gear meshing principle; 2) Establishing a dynamic model of the gear meshing process at any selected meshing point on the meshing line of the gear pair; 3) Dispersing the dynamic model into a plurality of analysis steps with equal intervals; 4) Calculating and recording the contact pressure and the surface stress of each analysis step; 5) Obtaining a time sequence table based on contact pressure and surface stress in the gear meshing process; 6) Establishing a stress sequence only containing surface stress peak value-valley value; 7) Extracting stress cycles by using a rain flow counting method to obtain a stress amplitude value and an average stress of each stress cycle; 8) Calculating the contact fatigue life corresponding to the stress cycle in the step 7); 9) Accumulating the contact fatigue life corresponding to each stress cycle obtained in the step 8) to obtain the tooth surface fatigue damage at the contact point.
Description
Technical Field
The invention relates to the field of mechanical parts, in particular to a method for predicting the surface contact fatigue damage of a gear.
Background
The gear has a rough surface and is in a motion state of coexistence of sliding and rolling, gear contact fatigue is a common failure mode of mechanical parts, and with the further improvement of the power density and service performance requirements of modern mechanical equipment, the problem of gear contact fatigue becomes one of bottlenecks which restrict the development of high-end equipment.
The existing gear contact fatigue life prediction method does not consider the influence of sliding and rolling contact, so that the failure problem caused by inaccurate gear contact fatigue life prediction often occurs in engineering practice, the safety and the reliability of mechanical equipment are seriously influenced, and the technical problem in the engineering practice in the field is that the relative sliding between tooth surfaces is considered and the contact fatigue damage prediction of the tooth surfaces at different positions is considered.
Disclosure of Invention
The invention aims to provide a method for predicting the contact fatigue damage of the surface of a gear aiming at the corresponding defects of the prior art, and the influence of sliding and rolling contact is considered, so that the problem of failure caused by inaccurate prediction of the contact fatigue life of the gear in engineering practice can be solved, and the safety and the reliability of mechanical equipment are greatly improved.
The purpose of the invention is realized by adopting the following scheme: a method of predicting gear surface contact fatigue damage comprising the steps of:
1) According to the Hertz contact theory and the gear meshing theory, the comprehensive curvature radius r, the half width b of the meshing surface and the maximum meshing stress p of the gear pair are calculated according to the following formulas H ：
r＝r 1 r 2 /(r 1 +r 2 )
Wherein F is the engaging force, E is the equivalent elastic modulus, r 1 Is the equivalent radius of curvature, r, of gear A in the gear pair 2 Is the equivalent curvature radius of the gear B in the gear pair, and pi is the circumferential ratio;
because a straight gear is often a single-pair gear meshing area near a node point, the stress of gear teeth is large, and therefore pitting corrosion firstly occurs near the node point. Therefore, the contact fatigue strength of the node is generally calculated.
2) Establishing a dynamic model of the gear meshing process at any selected meshing point on a meshing line of the gear pair, wherein the tooth surface speed of a gear A in the gear pair is u in the dynamic model 1 Tooth surface speed of gear B in gear pair is u 2 ；
The rolling speed u between the gear A and the gear B in the gear pair is calculated according to the following formula r And sliding velocity u s ：
u r ＝(u 1 +u 2 )/2
u s ＝u 1 -u 2
In the formula u 1 Tooth flank speed, u, of gear A in geared transmission for gear pairs 2 The tooth surface speed of the gear B in the meshing transmission for the gear pair;
the influence of the rolling contact on the fatigue damage of the tooth surface in the gear meshing process is considered by utilizing the rolling speed and the sliding speed of the gears in the gear pair.
3) Dispersing the dynamic model of the gear meshing process in the step 2) into a plurality of analysis steps with equal intervals;
4) Performing contact analysis on each analysis step, calculating the contact pressure and the surface stress of each analysis step according to the following method, and recording:
(1) the contact pressure is found by the unified reynolds equation, which can be expressed as:
in the formula eta * For equivalent viscosity, to describe the effect of a non-Newtonian fluid, p is the fluid pressure, h is the oil film thickness, ρ is the fluid density, u r The rolling speed between the gear A and the gear B in the gear pair in the step 2), namely the entrainment speed;
(2) calculation of surface stress sigma from pressure-stress influence coefficient in contact mechanics ij (x,z)：
In the formula (I), the compound is shown in the specification,
for surface pressure-stress influence coefficient, in the document "contact mechanics" k.l. johnson; the specific expression of the coefficient is described in the Xutake industry equivalent translation, higher education Press, page 21.
The calculation of surface and subsurface stresses is described in The literature "Lubrication and contact surface models for roller and gear contacts" Li s., phD, the Ohio State University, USA,2009. ("model of Lubrication and contact of rollers and gears", li s., doctrine, ohio State University, USA, 2009), pages 95-101.
5) Arranging the contact pressure and the surface stress of each analysis step obtained in the step 4) according to a time sequence to obtain a time sequence table based on the contact pressure and the surface stress in the gear meshing process;
6) Extracting surface stress peak values and valley values in the surface stress in the time sequence table in the step 5), and establishing a stress sequence only containing the surface stress peak values-valley values;
7) Extracting stress cycles by using a rain flow counting method according to the stress sequence in the step 6) to obtain a stress amplitude and an average stress of each stress cycle;
the literature "Standard, ASTM E1049-85 for Cycle Counting in Fatigue Analysis", west Consho ken, USA,2011, ("American society for testing and materials Standard E1049-85: standard practice of Cycle Counting method in Fatigue Analysis" West Consheuchen, USA, 2011) describes specific implementations of the rain flow Counting method.
8) Calculating the fatigue life corresponding to each stress cycle in the step 7) according to the following Basquin formula:
(Δσ e ) i ＝σ f ′(2N i ) c
in the formula, σ f ' and c are the axial fatigue coefficient and fatigue strength index, respectively, Δ σ e Is the equivalent stress amplitude, N i Is the contact fatigue life corresponding to the ith stress cycle;
9) Accumulating the fatigue life corresponding to each stress cycle obtained in the step 8) according to a Palmgren-Miner linear damage accumulation criterion to obtain the tooth surface fatigue damage at the contact point as shown in the following formula:
wherein D is the surface contact fatigue damage of the gear at the current position, N is the total number of stress cycles, and N is i Is the contact fatigue life corresponding to the ith stress cycle.
The product of the number of the analysis steps and the node distance in the step 3) is equal to the total number of the nodes in the target area.
The equivalent stress amplitude Delta sigma in the step 8) e Is obtained by correcting the mean stress contribution according to the Goodman criterion, as shown in the following equation:
in the formula, σ a Is the stress amplitude, σ m Is the mean stress, Δ σ e Is equivalent stressAmplitude, σ b Is the material bending fatigue limit.
The method has the advantages that because the influence of the contact of the sliding roller is considered, the fatigue damage distribution of the rough tooth surface in the motion state of the sliding roller can be predicted, the influence of the relative sliding between the tooth surfaces on the contact fatigue is analyzed, a basis is provided for the manufacture and the use of the gear, the service performance of the gear is improved, the failure problem caused by inaccurate prediction of the contact fatigue life of the gear in the engineering practice can not occur, and the safety and the reliability of mechanical equipment are greatly improved.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a simplified model diagram of gear tooth engagement;
FIG. 3 is a schematic view of a dynamic model of a gear meshing process taking into account the rolling contact;
FIG. 4 is a schematic flow diagram of a rain flow technique;
FIG. 5 is a stress sequence chart of an embodiment;
FIG. 6 is a stress cycle profile of the embodiment;
FIG. 7 is a profile of tooth flank damage for an embodiment.
Detailed Description
As shown in fig. 1, a method for predicting the contact fatigue damage of the gear surface comprises the following steps:
1) According to the Hertz contact theory and the gear meshing theory, the comprehensive curvature radius r, the half width b of a meshing surface and the maximum meshing stress p of the gear pair are calculated according to the following formulas H ：
r＝r 1 r 2 /(r 1 +r 2 )
Wherein F is the engaging force, E is the equivalent elasticityModulus of elasticity, r 1 Is the equivalent radius of curvature, r, of gear A in the gear pair 2 Is the equivalent curvature radius of the gear B in the gear pair, and pi is the circumferential ratio;
because a straight gear is often a single-pair gear meshing area near a node point, the stress of gear teeth is large, and therefore pitting corrosion firstly occurs near the node point. Therefore, the contact fatigue strength of the node is generally calculated.
2) Establishing a dynamic model of the gear meshing process at any selected meshing point on a meshing line of a gear pair, wherein in the dynamic model, the tooth surface speed of a gear A in the gear pair is u 1 Tooth surface speed of gear B in gear pair is u 2 ；
The rolling speed u between the gear A and the gear B in the gear pair is calculated according to the following formula r And sliding velocity u s ：
u r ＝(u 1 +u 2 )/2
u s ＝u 1 -u 2
In the formula u 1 Tooth flank speed, u, of gear A in meshing transmission for gear pairs 2 The tooth surface speed of the gear B in the meshing transmission for the gear pair;
3) Dispersing the dynamic model of the gear meshing process in the step 2) into a plurality of analysis steps with equal intervals;
the product of the number of the analysis steps and the node distance in the step 3) is equal to the total number of the nodes in the target area.
4) Performing contact analysis on each analysis step, calculating the contact pressure and the surface stress of each analysis step according to the following method, and recording:
(1) the contact pressure is determined by a unified reynolds equation, which can be expressed as:
in the formula eta * For equivalent viscosity, to describe the effect of a non-Newtonian fluid, p is the fluid pressure, h is the oil film thickness, ρ is the fluid density, u r The middle teeth of the gear pair in the step 2)The rolling speed between wheel a and gear B, i.e. the entrainment speed;
(2) calculation of surface stress sigma from pressure-stress influence coefficients in contact mechanics ij (x,z)：
In the formula (I), the compound is shown in the specification,
for surface pressure-stress influence coefficient, in the document "contact mechanics" k.l. johnson; the specific expression of the coefficient is described in the Xutake industry equivalent translation, higher education Press, page 21.
The calculation of surface and subsurface stresses is described in The literature "Lubrication and contact surface models for roller and gear contacts" Li s., phD, the Ohio State University, USA,2009. ("model of Lubrication and contact of rollers and gears", li s., doctrine, ohio State University, USA, 2009), pages 95-101.
5) Arranging the contact pressure and the surface stress of each analysis step obtained in the step 4) according to a time sequence to obtain a time sequence table based on the contact pressure and the surface stress in the gear meshing process;
6) Extracting surface stress peak values and valley values in the surface stress in the time sequence table in the step 5), and establishing a stress sequence only containing the surface stress peak values-valley values;
7) Extracting stress cycles by using a rain flow counting method according to the stress sequence in the step 6) to obtain a stress amplitude and an average stress of each stress cycle;
the literature "Standard, ASTM E1049-85 for Cycle Counting in Fatigue Analysis", west Consho ken, USA,2011, ("American society for testing and materials Standard E1049-85: standard practice of Cycle Counting method in Fatigue Analysis" West Consheuchen, USA, 2011) describes specific implementations of the rain flow Counting method.
8) Calculating the fatigue life corresponding to each stress cycle in the step 7) according to the following Basquin formula:
(Δσ e ) i ＝σ f ′(2N i ) c
in the formula, σ f ' and c are the axial fatigue coefficient and fatigue strength index, respectively, Δ σ e Is the equivalent stress amplitude, N i Is the contact fatigue life corresponding to the ith stress cycle;
the equivalent stress amplitude Delta sigma in the step 8) e Is derived by correcting the mean stress contribution according to the Goodman criterion, as shown in the following equation:
in the formula, σ a Is the stress amplitude, σ m Is mean stress, Δ σ e For equivalent stress amplitude, σ b Is the material bending fatigue limit.
9) Accumulating the fatigue life corresponding to each stress cycle obtained in the step 8) according to a Palmgren-Miner linear damage accumulation criterion to obtain the tooth surface fatigue damage at the contact point as shown in the following formula:
wherein D is the surface contact fatigue damage of the gear at the current position, N is the total number of stress cycles, and N is i Is the contact fatigue life corresponding to the ith stress cycle.
In the embodiment, the sample gear is selected from a middle parallel-stage gear pair of a certain 2MW wind power gear box, the failure frequency of a pinion of the gear pair is obviously higher than that of other gears in engineering practice, and the main geometric and working condition parameters of the gears are shown in table 1:
TABLE 1
In this embodiment, the contact fatigue damage of the gear pair with the above parameters is estimated, and the steps are as follows:
1) According to the Hertz contact theory and the gear meshing theory, as shown in figure 2, the contact of the gear at the meshing point can be simplified into a line contact model of a rigid cylinder and an elastic semi-infinite plane, and the comprehensive curvature radius r, the half width b of the meshing surface and the maximum meshing stress p of the gear pair are calculated according to the following formulas H ：
r＝r 1 r 2 /(r 1 +r 2 )
Wherein F is the engaging force, E is the equivalent elastic modulus, r 1 Is the equivalent radius of curvature, r, of gear A in the gear pair 2 Is the equivalent curvature radius of the gear B in the gear pair, and pi is the circumferential ratio;
because a straight gear is often a single-pair gear meshing area near a node point, the stress of gear teeth is large, and therefore pitting corrosion firstly occurs near the node point. Therefore, the contact fatigue strength of the node is generally calculated.
Calculating to obtain a comprehensive curvature radius r =38.84mm and a unit tooth width load F =1530N/mm, and obtaining the Hertz contact pressure as p according to the formula (1) H =1.2GPa, hz contact half-width b =0.8mm.
2) At any meshing point selected on the meshing line of the gear pair, as shown in fig. 3, a dynamic model of the gear meshing process is established, in which the tooth surface speed of the gear a in the gear pair is u 1 Tooth surface speed of gear B in gear pair is u 2 ；
The rolling speed u between the gear A and the gear B in the gear pair is calculated according to the following formula r And sliding velocity u s ：
u r ＝(u 1 +u 2 )/2
u s ＝u 1 -u 2
In the formula u 1 Tooth flank speed, u, of gear A in geared transmission for gear pairs 2 The tooth surface speed of the gear B in the meshing transmission for the gear pair;
the surface speed of the gear A is calculated to be u 1 =4.6m/s, surface speed of gear B is u 2 =3.6m/s, rolling speed u r =4m/s, sliding speed u r ＝1m/s。
3) Dispersing the dynamic model of the gear meshing process in the step 2) into 128 equally spaced analysis steps;
the product of the number of the analysis steps and the node distance in the step 3) is equal to the total number of the nodes in the target area.
4) Performing contact analysis on each analysis step, calculating the contact pressure and the surface stress of each analysis step according to the following method, and recording:
(1) the contact pressure is determined by a unified reynolds equation, which can be expressed as:
in the formula eta * For equivalent viscosity, to describe the effect of a non-Newtonian fluid, p is the fluid pressure, h is the oil film thickness, ρ is the fluid density, u r The rolling speed between the gear A and the gear B in the gear pair in the step 2), namely the entrainment speed;
(2) calculation of surface stress sigma from pressure-stress influence coefficient in contact mechanics ij (x,z)：
In the formula (I), the compound is shown in the specification,
for surface pressure-stress influence coefficient, in the document "contact mechanics" K.LJohnson; the specific expression of the coefficient is described in the Xutake industry equivalent translation, higher education Press, page 21.
The calculation of surface and subsurface stresses is described in The literature "Lubrication and contact surface models for roller and gear contacts" Li s., phD, the Ohio State University, USA,2009. ("model of Lubrication and contact of rollers and gears", li s., doctrine, ohio State University, USA, 2009), pages 95-101.
The resulting stress sequence is shown in FIG. 5: the surface experiences a plurality of stress peak-to-valley fluctuations during one movement, which can be considered as a superposition of a plurality of stress amplitudes and an average stress, i.e. a material point on the surface is subjected to a plurality of stress cycles.
5) Arranging the contact pressure and the surface stress of each analysis step obtained in the step 4) according to a time sequence to obtain a time sequence table based on the contact pressure and the surface stress in the gear meshing process;
6) Extracting surface stress peak values and valley values in the surface stress in the time sequence table in the step 5), and establishing a stress sequence only containing the surface stress peak values-valley values;
7) Extracting stress cycles by using a rain flow counting method according to the stress sequence in the step 6) to obtain a stress amplitude and an average stress of each stress cycle;
the document "Standard, ASTM E1049-85 for Cycle Counting in facial Analysis", west conshohon, USA,2011 ("american society for testing and materials Standard E1049-85: standard practice of Cycle Counting method in Fatigue Analysis" West conschhouken, USA, 2011) describes a specific implementation of the rain flow Counting method, as shown in fig. 4;
stress cycles were extracted by rain flow counting and the stress cycles obtained are shown in fig. 6: the stress sequence contained 14 stress cycles, with two stress cycles having a magnitude in excess of 1GPa.
During one movement, the material points on the surface will undergo 14 stress cycles, and the 14 stress cycles have different stress amplitudes and different damages.
8) Equivalent stress amplitude Δ σ e Is derived by correcting the mean stress contribution according to the Goodman criterion, as shown in the following equation:
in the formula, σ a To the stress amplitude, σ m Is mean stress, Δ σ e To an equivalent stress amplitude, σ b Is the material bending fatigue limit.
In the present embodiment, the gear material bending fatigue limit σ b And (8) =1200MPa, and obtaining an equivalent stress amplitude. The equivalent stress magnitude is the equivalent stress magnitude for each stress cycle.
Calculating the fatigue life corresponding to each stress cycle in the step 7) according to the following Basquin formula:
(Δσ e ) i ＝σ f ′(2N i ) c
in the formula, σ f ' is the axial fatigue coefficient, c is the fatigue strength index, delta sigma e Is the equivalent stress amplitude, N i Is the contact fatigue life corresponding to the ith stress cycle;
according to the document "Materials Data for Cyclic Loading: low-Alloy columns" Boller, C.; seeger, T.T., elsevier, amsterdam, the Netherlands,2013, ("Material parameters under cyclic load: low carbon alloy steels", boller, C.; seeger, T.T., isville Press, amsterdam, netherlands, 2013), it can be seen that in this embodiment, the axial fatigue coefficient of The selected gear is σ f ′＝1.5σ b Fatigue strength index c = -0.087.
As shown in fig. 7, the tooth surface damage risk distribution at the selected meshing point indicates the position of the fatigue damage occurring on the gear surface, the target region length is the moving direction, and the target region width is the tooth width direction.
9) Accumulating the fatigue life corresponding to each stress cycle obtained in the step 8) according to a Palmgren-Miner linear damage accumulation criterion to obtain the tooth surface fatigue damage at the contact point as shown in the following formula:
wherein D is the surface contact fatigue damage of the gear at the current position, N is the total number of stress cycles, and N is i Is the contact fatigue life corresponding to the ith stress cycle.
The tooth surface damage Prediction result shown in fig. 7 is consistent with the distribution trend of the sliding contact surface fatigue damage observed through experiments in the documents "Prediction of microscopic texture in gear tooth contacts constraints of surface texture and ground wind" templates-Espejel GE, rycez P, kadiric a, wear,2018. ("tooth surface contact micro pitting damage Prediction considering the common action of surface fatigue and slight Wear", morales-Espejel GE, rycez P, kadiric a., wear, 2018), so as to verify the reliability of the present invention.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention, and modifications of the present invention by those skilled in the art can be made without departing from the spirit of the present invention.
Claims (3)
1. A method of predicting gear surface contact fatigue damage, comprising the steps of:
1) According to the Hertz contact theory and the gear meshing theory, the comprehensive curvature radius r, the half width b of a meshing surface and the maximum meshing stress p of the gear pair are calculated according to the following formulas H ：
r＝r 1 r 2 /(r 1 +r 2 )
Wherein F is the engaging force, E is the equivalent elastic modulus, r 1 Is the equivalent radius of curvature, r, of gear A in the gear pair 2 The equivalent curvature radius of a gear B in the gear pair, and pi is the circumferential rate;
2) Establishing a dynamic model of the gear meshing process at any selected meshing point on a meshing line of a gear pair, wherein in the dynamic model, the tooth surface speed of a gear A in the gear pair is u 1 Tooth surface speed of gear B in gear pair is u 2 ；
The rolling speed u between the gear A and the gear B in the gear pair is calculated according to the following formula r And sliding velocity u s ：
u r ＝(u 1 +u 2 )/2
u s ＝u 1 -u 2
In the formula u 1 Tooth flank speed, u, of gear A in meshing transmission for gear pairs 2 The tooth surface speed of the gear B in meshing transmission for the gear pair;
3) Dispersing the dynamic model of the gear meshing process in the step 2) into a plurality of analysis steps with equal intervals;
4) And (3) performing contact analysis on each analysis step, calculating the contact pressure and the surface stress of each analysis step according to the following methods, and recording:
(1) the contact pressure is determined by a unified reynolds equation, which can be expressed as:
in the formula eta * Equivalent viscosity, p is fluid pressure, h is oil film thickness, ρ is fluid density, u r The rolling speed between the gear A and the gear B in the gear pair in the step 2);
(2) calculation of surface stress sigma from pressure-stress influence coefficients in contact mechanics ij (x,z)：
In the formula (I), the compound is shown in the specification,
surface pressure-stress influence coefficient;
5) Arranging the contact pressure and the surface stress of each analysis step obtained in the step 4) according to a time sequence to obtain a time sequence table based on the contact pressure and the surface stress in the gear meshing process;
6) Extracting surface stress peak values and valley values in the surface stress in the time sequence table in the step 5), and establishing a stress sequence only containing the surface stress peak values-valley values;
7) Extracting stress cycles by using a rain flow counting method according to the stress sequence in the step 6) to obtain a stress amplitude and an average stress of each stress cycle;
8) Calculating the fatigue life corresponding to each stress cycle in the step 7) according to the following Basquin formula:
(Δσ e ) i ＝σ f ′(2N i ) c
in the formula, σ f ' and c are the axial fatigue coefficient and fatigue strength index, respectively, Δ σ e Is the equivalent stress amplitude, N i Is the contact fatigue life corresponding to the ith stress cycle;
9) Accumulating the fatigue life corresponding to each stress cycle obtained in the step 8) according to a Palmgren-Miner linear damage accumulation criterion to obtain the tooth surface fatigue damage at the contact point as shown in the following formula:
wherein D is the surface contact fatigue damage of the gear at the current position, N is the total number of stress cycles, and N is i Is the contact fatigue life corresponding to the ith stress cycle.
2. The method of claim 1, wherein: the product of the number of the analysis steps and the node distance in the step 3) is equal to the total number of the nodes in the target area.
3. The method of claim 1, wherein: the equivalent stress amplitude Delta sigma in the step 8) e Is obtained by correcting the mean stress contribution according to the Goodman criterion, as shown in the following equation:
in the formula, σ a To the stress amplitude, σ m Is mean stress, Δ σ e For equivalent stress amplitude, σ b Is the material bending fatigue limit.
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