CN106355039A - Method for calculating fatigue life and reliability of rolling bearing based on independent contact pair - Google Patents

Abstract

The invention relates to a method for calculating fatigue life and reliability of a rolling bearing based on an independent contact pair. In a calculation process, the stress distributions in different contact states between a rolling element and different roller paths are firstly calculated according to a Lundberg-Palmgren formula and based on the clearance and internal geometric parameters of the rolling bearing. The fatigue characteristics of the different roller paths are considered based on contact stresses of the different roller paths, the fatigue lives of the different roller paths under needed reliability are calculated, the life of the roller path having the shorter life is further selected as the life of a whole set of bearing, and the new reliability of the roller path having longer life is calculated by returning under the condition of the shorter life, so that the reliability of the whole set of bearing is determined under the condition of the life, and the fatigue life under the needed reliability is obtained by means of further conversion.

Description

Based on the secondary rolling bearing fatigue life of independent contact and reliability degree calculation method

Technical field

The invention belongs to rolling bearing fatigue life and the technical field of reliability calculating, it is related to the tired longevity of rolling bearing Life calculating and the assessment computational methods of fatigue life reliability.

Background technology

Rolling bearing is important mechanical basic part, and its life-span is for main frame important.The life-span of rolling bearing Refer to the material of Bearing inner arbitrary part fatigue equivalent occurs first before total revolution (or under a certain given constant rotational speed Operating hours number).The fatigue life discreteness of rolling bearing is more notable, a collection of identical structure, same size, identical material The rolling bearing of material, identical heat treatment condition and identical processing method, its life-span difference is also very huge, thus reality should Rated life time with middle general provision rolling bearing.The rated life time of rolling bearing refers to a collection of model identical bearing, identical Operating condition under, the wherein 90% total revolution (l that can operate before fatigue equivalent₁₀) or can operate under given rotating speed total Work hours (l_h10), 90% reliability referring to this batch of bearing here.

Different application scenarios, the reliability demand of rolling bearing is different.Some are related to great personal safety as well as the property safety Occasion, such as aerospace field, manned vehicle etc., the reliability of rolling bearing has high demands；And some simple application fields Close, such as curtain bearing, toy bearing etc., be then not required to too high reliability.Reliability requires different, the then fatigue life that it calculates Also there were significant differences, and reliability requirement is higher, then its mathematic(al) expectation is shorter.Therefore, it is possible to think the fatigue reliability of rolling bearing Degree refers to the reliability under certain fatigue life determining, the fatigue life of rolling bearing then refers to the Fatigue Reliability premise determining Under fatigue life, thus, both are closely coupled, mutual dependence for existence.Current engineering practice is all shown with the result of scientific research, The fatigue life of rolling bearing obeys Weibull (w.weibull) distribution.

In existing international standard and scientific research, rolling bearing calculation methods of fatigue reliability has two big types: 1, Based on bearing dynamic load rating computational methods.The rolling bearing fatigue that this method is carried out according to the dynamic load rating information of product The simplification algorithm of Life Calculation, its advantage is that calculating process is relatively simple, but accuracy is not enough, in order to ensure reliability, often Sacrifice the residual life of rolling bearing in a large number；2nd, the computational methods based on lundberg-palmgren formula.This method root According to external applied load size, connecing between each rolling element and raceway is calculated by the clearance and interior geometry parameter of rolling bearing Tactile stress, calculates the l of every raceway respectively₁₀In the life-span, then adopt the method for statistical disposition to trade off again and obtain a whole set of bearing reliability Spend for life-span when 90%, i.e. l₁₀.This method is more accurate with respect to first method, but its fussy degree also accordingly increases Plus.

Content of the invention

It is an object of the invention to provide based on the secondary rolling bearing fatigue life of independent contact and reliability degree calculation method.Meter During calculation, first by clearance and the inner geometry parameter of rolling bearing, calculate according to lundberg-palmgren formula and roll The stress distribution under different contact conditions between body and different raceways.Different rollings are considered by the contact stress between different raceways The fatigue characteristic in road, calculates the fatigue life of the different raceways under demand reliability, further selection shorter raceway longevity in life-span Order the life-span for complete bearing, return again to new reliability under the shorter life-span for the mathematic(al) expectation longer raceway, thus really Fixed reliability under this life-span for the complete bearing, is scaled the fatigue life under demand reliability further.

For avoiding repeating and tediously long, only choose centripetal point contact ball bearing in the present invention and centripetal linear contact lay roller bearing is Example explanation, thrust point contact ball bearing can foundation with the fatigue life of thrust line contact roller bearing and the computational methods of reliability Method proposed by the present invention is analogized.

For achieving the above object, be the technical scheme is that based on the secondary rolling bearing fatigue life of independent contact with Reliability degree calculation method, comprises the following steps:

(1) theoretical according to lundberg-palmgren, the contact fatigue life between rolling element and raceway can be estimated according to following formula:

In formula (1), q_cFor contact rating load, q is applied load；

(2) during one's term of military service, the speed between two lassos has larger difference to rolling bearing, and one is swivel ferrule, and another is low Fast lasso or fixing lasso are non-rotating rings；

For spot contact bearing,

In formula (2), q_cμFor swivel ferrule contact rating load, q_eμFor swivel ferrule applied load；q_cvFor fixing lasso contact volume Constant load, q_evFor fixing lasso applied load；Subscript μ refers to swivel ferrule, and subscript ν refers to non-rotating rings；

For line contact bearing, it is similar to and is expressed as:

The implication of symbology in formula (3) is identical with formula (2)；

(3) contact rating load is only relevant with contact type, material and heat treatment situation, with rolling element and inside/outside circle indirect Touch unrelated, thus the inside and outside circle contact rating load of point contact be represented by:

The inside and outside circle contact rating load of linear contact lay is represented by:

In formula (4) and (5), using upper symbol during inner ring Calculation of Contact Stress, it is "-" number on molecule, denominator is "+"；Outward Using upper symbol during circle Calculation of Contact Stress, on molecule be "+" number, denominator is "-", f is ditch curvature number；α is contact angle； D is rolling element diameter；Z is rolling element number；γ is represented by:

$\mathrm{\γ}=\frac{dcos\mathrm{\α}}{d_m}---\left(6\right)$

In formula (6), d_mFor bearing pitch diameter；

(4) because applied load acts on, between rolling element and inside and outside circle, produce contact stress, due to the raceway diameter of Internal and external cycle, The impact of curvature direction, contact stress produced by applied load；

For spot contact bearing,

$\left\{\begin{array}{c}{q}_{e\mathrm{\μ}}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^3\right)}^{\frac{1}{3}}\\ q_{ev}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^{\frac{10}{3}}\right)}^{0.3}\end{array}\right.---\left(7\right)$

For line contact bearing,

$\left\{\begin{array}{c}{q}_{e\mathrm{\μ}}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^4\right)}^{\frac{1}{4}}\\ q_{ev}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^{4.5}\right)}^{\frac{1}{4.5}}\end{array}\right.---\left(8\right)$

In formula (7) and formula (8), j refers to j-th rolling element, q_jRefer to the contact stress of j-th rolling element, z is total rolling element number；

(5) formula (4) and formula (7) are substituted into formula (2), formula (5) and formula (8) substitute into formula (3), can try to achieve inside and outside raceway contact pair respectively Life-span when reliability is for r；It is assumed that between outer ring and rolling element fatigue life during Contact Pair reliability r be l_ro, inner ring with Between rolling element, fatigue life during Contact Pair reliability r is l_ri, then can determine that a whole set of axle because a Contact Pair occurs to lose efficacy Hold inefficacy, so that choosing the life-span as complete bearing for the shorter mathematic(al) expectation, you can require as complete bearing during r by degree Life-span be:

l_r=min [l_ri,l_ro] (9)

(6) for the raceway of longer life, because its demand life-span reduces, then its reliability also will accordingly increase, that is, at one Under shorter requirements for life, its reliability will increase；

It is assumed that l_ri>l_ro, then the reliability lifting of interior raceway, that is, its reliability is changed into r from r_i；In the same manner it is assumed that l_ro>l_ri, then outer rolling The reliability lifting in road, that is, its reliability is changed into r from r_o；That is:

$r_{i/o}=1-s_{i/o}=1-e^{0.1053{\left(\frac{l_s}{l_{10}}\right)}^e}---\left(10\right)$

In formula (10), s is probability of escaping by luck, and subscript i/o in formula refers to inner ring or outer ring；l_sIt is calculating during s for probability of escaping by luck In the life-span, now reliability r is 1-s；

(7) as it was previously stated, the then complete bearing failure of arbitrary Contact Pair element failure, then for the reliability of complete bearing, adopt Series connection reliability description is more accurate, and a whole set of bearing is in life-span l_rUnder reliability r ' can be described as:

R '=r × [r_i×r_o] (11)

Formula (11) represents one or computing, i.e. new reliability r of complete bearing ' it is demand reliability r and the interior r lasting a long time_i Or outer r_oCalculating reliability product, interior raceway mathematic(al) expectation longer then select r_i, outer raceway last a long time then selection r_o, r_i Or r_oValue by formula (11) determine；

(8) reliability r being obtained after calculating ' different from former demand reliability r, then need to return under calculating demand reliability Mathematic(al) expectation, this calculating process can solve Different Reliability according to the Slope relationship of Weibull distribution, and that is, difference is escaped by luck generally In the different fatigue life-span under rate, its relation can be described as:

$ln\frac{1}{s}={\left(\frac{l_s}{l_{s^{,}}}\right)}^{e}ln\frac{1}{s^{,}}---\left(12\right)$

In formula (12), s is the corresponding probability of escaping by luck of demand reliability r, and s ' is then actual reliability r calculating under the corresponding life-span ' Corresponding probability of escaping by luck, l_sIt is mathematic(al) expectation during s for probability of escaping by luck, l_s' it is mathematic(al) expectation during s ' for probability of escaping by luck, typically In the case of, Calculation of Fatigue Life its l of many demands of rolling bearing₁₀Life-span, then formula (12) can be reduced to:

$l_{10}=\frac{l_{s^{,}}}{{\left(\frac{1}{0.1053}ln\frac{1}{s^{,}}\right)}^{\frac{1}{e}}}---\left(13\right).$

The medicine have the advantages that in the present invention, rolling bearing fatigue life and reliability degree calculation method exist Expand on the basis of lundberg-palmgren formula, calculate the distribution of rolling bearing internal load first, using need The reliability asked calculates the life-span of inside and outside raceway respectively, then just carries out the new method proposing in this patent.

In the present invention, rolling bearing is not re-used as a whole removing consideration, but is split as interior Gun Dao rolling element contact Secondary, outer raceway two different units of rolling element Contact Pair go to calculate its fatigue life and reliability, any cell lost efficacy then a whole set of Bearing failure.

According to the rolling bearing fatigue life secondary based on independent contact proposed by the present invention and reliability degree calculation method, with not Short life with Contact Pair is basic as the calculating of complete bearing life, can ensure the reliability of calculating.

According to the rolling bearing fatigue life secondary based on independent contact proposed by the present invention and reliability degree calculation method, calculate The fatigue life going out is noticeably greater than original method, which solves actual life in current engineering practice and is much larger than mathematic(al) expectation, but Situation about being theoretically unsound.

Brief description

Fig. 1 is the Contact Pair of radial ball bearing；

Fig. 2 is the Contact Pair of radial roller bearing；

Fig. 3 is the supporting region load distribution of radial ball bearing；

Fig. 4 is the supporting region load distribution of radial roller bearing.

Specific embodiment

The rolling bearing fatigue life secondary based on independent contact proposed by the present invention and reliability degree calculation method, existing pass Still can use as current methods in material, lubrication, working environment clean conditions equivalent life correction factor, in this patent Do not repeat again.

As follows with the step of reliability degree calculation method based on the secondary rolling bearing fatigue life of independent contact:

(1) theoretical according to lundberg-palmgren, the contact fatigue life between rolling element and raceway can be estimated according to following formula:

In formula (1), q_cFor contact rating load, q is applied load.Due to the contact between different types of rolling element and raceway State is different, and its fatigue life is also variant, thus separately shown.

(2) rolling bearing during one's term of military service, fix by a general lasso, and another one lasso rotates.Also there is a small amount of Internal and external cycle The operating mode of equal rotations, but the speed between two swivel ferrules has larger difference, thus low speed circle is assumed to non-rotating rings.

For spot contact bearing,

In formula (2), q_cμFor swivel ferrule contact rating load, q_eμFor swivel ferrule applied load；q_cvFor fixing lasso contact volume Constant load, q_evFor fixing lasso applied load；Subscript μ refers to swivel ferrule, and subscript ν refers to non-rotating rings；Spot contact bearing In, the contact stress state between the contact condition of steel ball and Internal and external cycle and single steel ball and inside and outside raceway is shown in accompanying drawing 1.

For line contact bearing, can be similar to and be expressed as:

The implication of symbology in formula (3) is identical with formula (2)；In line contact bearing, the contact condition of roller and Internal and external cycle with And the contact stress state between single roller and inside and outside raceway is shown in accompanying drawing 2.

(3) contact rating load is only relevant with initial factors such as contact type, material and heat treatment situations, with rolling element with Contact unrelated between inside/outside circle, thus the inside and outside circle contact rating load of point contact be represented by:

The inside and outside circle contact rating load of linear contact lay is represented by:

In formula (4) and (5), using upper symbol during inner ring Calculation of Contact Stress, it is "-" number on molecule, denominator is "+"；Outward Using upper symbol during circle Calculation of Contact Stress, on molecule be "+" number, denominator is "-".F is ditch curvature number；α is contact angle； D is rolling element diameter；Z is rolling element number；γ is represented by:

$\mathrm{\γ}=\frac{dcos\mathrm{\α}}{d_m}---\left(6\right)$

In formula (6), d_mFor bearing pitch diameter.

(4) because applied load acts on, between rolling element and inside and outside circle, produce contact stress.Raceway due to Internal and external cycle Diameter, the impact of the factor such as curvature direction, contact stress produced by applied load.

For spot contact bearing,

In spot contact bearing, the internal load distribution of complete bearing and supporting region are shown in accompanying drawing 3.

For line contact bearing,

In line contact bearing, the internal load distribution of complete bearing and supporting region are shown in accompanying drawing 4.In formula (7) and formula (8), j refers to J rolling element, q_jRefer to the contact stress of j-th rolling element；

Above 4 steps consistent with the existing computational methods based on lundberg-palmgren formula it is therefore intended that first obtaining The load distribution of Bearing inner.Following steps are new method proposed by the present invention.

(5) formula (4) and formula (7) are substituted into formula (2), formula (5) and formula (8) substitute into formula (3), can try to achieve inside and outside raceway respectively and connect Touch the secondary life-span when reliability is for r.It is assumed that between outer ring and rolling element fatigue life during Contact Pair reliability r be l_ro, interior Between circle and rolling element, fatigue life during Contact Pair reliability r is l_ri, due to Contact Pair occur to lose efficacy then can determine that whole Set bearing failure, so that choose the life-span as complete bearing shorter mathematic(al) expectation, you can requires as complete during r by degree The life-span of bearing is:

l_r=min [l_ri,l_ro] (9)

(6) for the raceway of longer life, because its demand life-span reduces, then its reliability also will accordingly increase, that is, at one Under shorter requirements for life, its reliability will increase.

It is assumed that l_ri>l_ro, then the reliability lifting of interior raceway, that is, its reliability is changed into r from r_i；Same reason is it is assumed that l_ro >l_ri, then the reliability lifting of outer raceway, that is, its reliability is changed into r from r_o；That is:

$r_{i/o}=1-s_{i/o}=1-e^{0.1053{\left(\frac{l_s}{l_{10}}\right)}^e}---\left(10\right)$

In formula (10), s is probability of escaping by luck, and subscript i/o in formula refers to inner ring or outer ring；l_sIt is calculating during s for probability of escaping by luck In the life-span, now reliability r is 1-s；Formula (10) derives from Weibull distribution expression formula.

(7) as it was previously stated, the then complete bearing failure of arbitrary Contact Pair element failure, then for the reliability of complete bearing, More accurate using series connection reliability description, a whole set of bearing is in life-span l_rUnder reliability r ' can be described as:

R '=r × [r_i×r_o] (11)

Formula (11) represents one or computing, i.e. new reliability r of complete bearing ' it is demand reliability r and certain lasting a long time Raceway (interior r_iOr outer r_o) calculating reliability product, interior raceway mathematic(al) expectation longer then select r_i, outer raceway lasts a long time then Select r_o.r_iOr r_oValue by formula (11) determine.

(8) reliability r being obtained after calculating ' different from former demand reliability r, then need to return calculating demand reliability Mathematic(al) expectation under degree.This calculating process can solve Different Reliability (i.e. different good fortunes according to the Slope relationship of Weibull distribution Exempt from probability) under the different fatigue life-span, its relation can be described as:

$ln\frac{1}{s}={\left(\frac{l_s}{l_{s^{,}}}\right)}^{e}ln\frac{1}{s^{,}}---\left(12\right)$

In formula (12), s is the corresponding probability of escaping by luck of demand reliability r, and s ' is then actual reliability r calculating under the corresponding life-span ' Corresponding probability of escaping by luck.l_sIt is mathematic(al) expectation during s for probability of escaping by luck, l_s' it is mathematic(al) expectation during s ' for probability of escaping by luck.Typically In the case of, Calculation of Fatigue Life its l of many demands of rolling bearing₁₀Life-span, then formula (12) can be reduced to:

$l_{10}=\frac{l_{s^{,}}}{{\left(\frac{1}{0.1053}ln\frac{1}{s^{,}}\right)}^{\frac{1}{e}}}---\left(13\right)$

Describe the rolling bearing fatigue life secondary based on independent contact and reliability degree calculation method below in conjunction with instantiation in detail Specific implementation step, it should be noted that this embodiment be exemplary type example, rather than as a example limit the scope of the present invention Extremely apply.

In order to preferably realize contrasting, this example select in t.a.harris works " rolling bearing analysis " two with regard to The fatigue life of deep groove ball bearing and cylinder roller bearing and the calculated examples of reliability, selection and this example identical bearing, Identical internal structure parameter and identical application conditions, more clearly to show method proposed by the present invention, more The clearly difference between contrast new method and existing computational methods；In order to simplify calculating, become apparent from shows new calculating Method, is not introduced into life adjustment factor in following example, as existing method, with regard to material, lubrication, working environment cleaning shape State equivalent life correction factor also can introduce the computational methods of the present invention.

Embodiment 1:

6209 deep groove ball bearings, steel ball number z is 9, and steel ball size d is 12.7mm, bearing pitch diameter d_mFor 65mm, Internal and external cycle Ditch coefficient of curvature f_i=f_o=0.52, end-play is 0.015mm.Under the Radial Loads of 8900n, its inner ring with The speed rotation of 1800rpm, calculates the l of two raceways₁₀Life-span, and calculate the actual reliability of this life conditions.

(1) from formula (6):

$\mathrm{\γ}=\frac{dcos\mathrm{\α}}{d_m}=\frac{12.7}{65}=0.1954---\left(14\right)$

(2) inside and outside circle contact rating load is tried to achieve respectively by formula (4)

$q_{ci}=98.1{\left(\frac{2f}{2f-1}\right)}^{0.41}\frac{{(1-\mathrm{\γ})}^{1.39}}{{(1+\mathrm{\γ})}^{\frac{1}{3}}}{\left(\frac{\mathrm{\γ}}{cos\mathrm{\α}}\right)}^{0.3}{d}^{1.8}{z}^{-\frac{1}{3}}=7058n---\left(15\right)$

$q_{co}=98.1{\left(\frac{2f}{2f-1}\right)}^{0.41}\frac{{(1+\mathrm{\γ})}^{1.39}}{{(1-\mathrm{\γ})}^{\frac{1}{3}}}{\left(\frac{\mathrm{\γ}}{cos\mathrm{\α}}\right)}^{0.3}{d}^{1.8}{z}^{-\frac{1}{3}}=13970n---\left(16\right)$

In formula (15) and formula (16), because clear and definite swivel ferrule is inner ring, non-rotating rings are outer ring, thus adopt q_ciGeneration For q_cμ, q_coReplace q_cv.

(3) according to the end-play of this bearing, interior geometry parameter and the external applied load born, using Approach by inchmeal The load such as table 1 that method or the different rolling elements of solution by iterative method Bearing inner 8 bear:

Table 1 6209 Bearing inner load is distributed

φ(°)	cosφ	qφ(n)
			0	1	4536
±40	0.766	2842
			±80	0.1737	61
±120	-0.5000	0
			±160	-0.9737	0

(4) because swivel ferrule is inner ring, therefore the data in table 1 is substituted into formula (7) and can obtain:

$q_{ei}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^3\right)}^{\frac{1}{3}}=2475n---\left(17\right)$

In formula (17), q_eiWith q in formula (7)_eμSymbol implication is identical, and being also due to clear and definite swivel ferrule is inner ring, thus more Change subscript.

(5) outer ring is ring for fixing, therefore the data in table 1 is substituted into formula (7) and can obtain:

$q_{eo}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^3\right)}^{\frac{1}{3}}=2475n---\left(18\right)$

In formula (18), q_eoWith q in formula (7)_evSymbol implication is identical, and being also due to clearly fix lasso is outer ring, thus more Change subscript.

(6) by formula (3), the l of inside and outside circle can be obtained₁₀It is respectively as follows:

$\left\{\begin{array}{c}{l}_{10i}={\left(\frac{q_{ci}}{q_{ei}}\right)}^3=23.2\×{10}^6\left(r\right)\\ l_{10o}={\left(\frac{q_{co}}{q_{eo}}\right)}^3=154.4\×{10}^6\left(r\right)\end{array}\right.---\left(19\right)$

In formula (19), l_10iRepresent that inner ring raceway/rolling element Contact Pair reliability is fatigue life when 90%；l_10oRepresent outer ring Raceway/rolling element Contact Pair reliability is fatigue life when 90%.

(7) by formula (19), the l of inside and outside circle₁₀Difference, and have a long way to go, weak link is inner ring, and inner ring/rolling element connects After touching secondary inefficacy, this set bearing had lost efficacy that is to say, that the l of this set bearing_10oMore than l_10iPart be insignificant, this meaning Taste the reliability of outer ring under this operating mode more than 90%.

(8) by formula (19), under this application conditions, the outer ring life-span only needs to reach 23.2 × 10⁶, by formula (10), can obtain Under this requirements for life, outer ring reliability is:

$r_{i/o}=1-e^{0.1053{\left(\frac{l_s}{l_{10}}\right)}^e}=0.987---\left(20\right)$

(9) by formula (11), a whole set of bearing life is 23.2 × 10⁶Actual reliability during r can be described as:

R=0.9 × 0.987=0.89 (21)

(10) by formula (12), the calculating fatigue life that reverse goes out that reliability is when 90% is:

l₁₀=23.0 × 10⁶(22)

(11), in existing computational methods, the calculating process of front 4 steps is consistent with the calculating process of the present invention.But it calculates overall axle During the life-span held, the directly probability of equivalent each element, the life-span Integrative expression of complete bearing is:

$l={(l_i^{-1.11}+l_o^{-1.11})}^{-0.9}---\left(23\right)$

(12) each Contact Pair life-span is substituted into formula 16, the complete bearing reliability that can try to achieve existing method calculating is tired when 90% The labor life-span is:

l₁₀=20.9 × 10⁶(24)

Contrast (19) and formula (20) understand, the race bearing fatigue life being calculated using the computational methods of the present invention is significantly carried High by 10%, in practical application the calculating fatigue life of deep groove ball bearing often be substantially less than mathematic(al) expectation, the present invention closer to Practical application.

Embodiment 2:

N209 cylinder roller bearing, roller number z is 14, and roller diameter d is 10mm, bearing pitch diameter d_mFor 65mm, roller Effective length 29.6mm, under the Radial Loads of 4450n, bearing is to revise linear contact lay and inner ring rotation, calculates two raceways L₁₀Life-span, and calculate the actual reliability of this life conditions.

(1) from formula (6)

$\mathrm{\γ}=\frac{dcos\mathrm{\α}}{d_m}=0.1538---\left(25\right)$

(2) inside and outside circle contact rating load is tried to achieve respectively by formula (4)

$q_{ci}=552\frac{{(1-\mathrm{\γ})}^{\frac{29}{27}}}{{(1+\mathrm{\γ})}^{\frac{1}{4}}}{\left(\frac{\mathrm{\γ}}{cos\mathrm{\α}}\right)}^{\frac{2}{9}}{d}^{\frac{29}{27}}{z}^{-\frac{1}{4}}=6381n---\left(26\right)$

$q_{co}=552\frac{{(1+\mathrm{\γ})}^{\frac{29}{27}}}{{(1-\mathrm{\γ})}^{\frac{1}{4}}}{\left(\frac{\mathrm{\γ}}{cos\mathrm{\α}}\right)}^{\frac{2}{9}}{d}^{\frac{29}{27}}{z}^{-\frac{1}{4}}=9621n---\left(27\right)$

(3) load that can be born using successive approximation approach or the different rolling element (referring to Fig. 2) of solution by iterative method Bearing inner 3 Lotus such as table 2:

Table 1 6209 Bearing inner load is distributed

φ(°)	cosφ	qφ(n)
			0	1	1915n
25.17	0.905	1348n
			51.42	0.624	0
…	…	…
			180	-1	0

(4) because swivel ferrule is inner ring, therefore the data in table 2 is substituted into formula (7) and can obtain:

$q_{ei}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^4\right)}^{\frac{1}{4}}=1095n---\left(28\right)$

(5) outer ring is ring for fixing, therefore the data in table 2 is substituted into formula (7) and can obtain:

$q_{eo}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^{4.5}\right)}^{\frac{1}{4.5}}=1148n---\left(29\right)$

(6) by formula (3), the l of inside and outside circle can be obtained₁₀It is respectively as follows:

$\left\{\begin{array}{c}{l}_i={\left(\frac{q_{ci}}{q_{ei}}\right)}^4=1155\×{10}^6\left(r\right)\\ l_o={\left(\frac{q_{co}}{q_{eo}}\right)}^4=4937\×{10}^6\left(r\right)\end{array}\right.---\left(30\right)$

(7) by formula (9), the l of inside and outside circle₁₀Difference, and have a long way to go, weak link is inner ring, i.e. the l of this set bearing₁₀Super Cross l_iPart be insignificant, also imply that the reliability of outer ring under this operating mode exceeds well over 90%.

(8) by formula (10), under this application conditions, outer ring requirements for life is 1155 × 10⁶, then its reliability be:

r_o=0.98 (31)

(9) by formula (11), a whole set of bearing life is 1155 × 10⁶Actual reliability during r can be described as:

R=0.9 × 0.98=0.882 (32)

(10) by formula (12), the calculating fatigue life that reverse goes out that reliability is when 90% is:

l₁₀=9.88 × 10⁸(33)

(11), in existing computational methods, the calculating process of front 4 steps is consistent with the calculating process of the present invention.But it calculates overall axle During the life-span held, the directly probability of equivalent each element, the life-span Integrative expression of complete bearing is:

$l={(l_i^{-1.11}+l_o^{-1.11})}^{-0.9}---\left(34\right)$

(12) each Contact Pair life-span is substituted into formula 16, the complete bearing reliability that can try to achieve existing method calculating is tired when 90% The labor life-span is:

l₁₀=9.85 × 10⁸(35)

Contrast (33) and formula (35) understand, the race bearing fatigue life being calculated using the computational methods of the present invention is improved 0.03%, in practical application, the calculating fatigue life of cylindrical bearing is higher with mathematic(al) expectation degree of closeness, and this also reflects from side The correctness of the computational methods of the present invention.

Claims (1)

1. based on the secondary rolling bearing fatigue life of independent contact with reliability degree calculation method it is characterised in that: include following walking Rapid:

(1) theoretical according to lundberg-palmgren, the contact fatigue life between rolling element and raceway can be estimated according to following formula:

In formula (1), q_cFor contact rating load, q is applied load；

(2) during one's term of military service, the speed between two lassos has larger difference to rolling bearing, and one is swivel ferrule, and another is low Fast lasso or fixing lasso are non-rotating rings；

For spot contact bearing,

In formula (2), q_cμFor swivel ferrule contact rating load, q_eμFor swivel ferrule applied load；q_cvFor fixing lasso contact volume Constant load, q_evFor fixing lasso applied load；Subscript μ refers to swivel ferrule, and subscript ν refers to non-rotating rings；

For line contact bearing, it is similar to and is expressed as:

The implication of symbology in formula (3) is identical with formula (2)；

(3) contact rating load is only relevant with contact type, material and heat treatment situation, with rolling element and inside/outside circle indirect Touch unrelated, thus the inside and outside circle contact rating load of point contact be represented by:

The inside and outside circle contact rating load of linear contact lay is represented by:

In formula (4) and (5), using upper symbol during inner ring Calculation of Contact Stress, it is "-" number on molecule, denominator is "+"；Outward Using upper symbol during circle Calculation of Contact Stress, on molecule be "+" number, denominator is "-", f is ditch curvature number；α is contact angle； D is rolling element diameter；Z is rolling element number；γ is represented by:

$\mathrm{\γ}=\frac{dcos\mathrm{\α}}{d_m}---\left(6\right)$

In formula (6), d_mFor bearing pitch diameter；

(4) because applied load acts on, between rolling element and inside and outside circle, produce contact stress, due to the raceway diameter of Internal and external cycle, The impact of curvature direction, contact stress produced by applied load；

For spot contact bearing,

$\left\{\begin{array}{c}{q}_{e\mathrm{\μ}}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^3\right)}^{\frac{1}{3}}\\ q_{ev}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^{\frac{10}{3}}\right)}^{0.3}\end{array}\right.---\left(7\right)$

For line contact bearing,

$\left\{\begin{array}{c}{q}_{e\mathrm{\μ}}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^4\right)}^{\frac{1}{4}}\\ q_{ev}={\left(\frac{1}{z}\underset{j=1}{\overset{j=z}{\σ}}{q}_j^{4.5}\right)}^{\frac{1}{4.5}}\end{array}\right.---\left(8\right)$

In formula (7) and formula (8), j refers to j-th rolling element, q_jRefer to the contact stress of j-th rolling element, z is total rolling element number；

(5) formula (4) and formula (7) are substituted into formula (2), formula (5) and formula (8) substitute into formula (3), can try to achieve inside and outside raceway contact pair respectively Life-span when reliability is for r；It is assumed that between outer ring and rolling element fatigue life during Contact Pair reliability r be l_ro, inner ring with Between rolling element, fatigue life during Contact Pair reliability r is l_ri, then can determine that a whole set of axle because a Contact Pair occurs to lose efficacy Hold inefficacy, so that choosing the life-span as complete bearing for the shorter mathematic(al) expectation, you can require as complete bearing during r by degree Life-span be:

l_r=min [l_ri,l_ro] (9)

(6) for the raceway of longer life, because its demand life-span reduces, then its reliability also will accordingly increase, that is, at one Under shorter requirements for life, its reliability will increase；

It is assumed that l_ri>l_ro, then the reliability lifting of interior raceway, that is, its reliability is changed into r from r_i；In the same manner it is assumed that l_ro>l_ri, then outer rolling The reliability lifting in road, that is, its reliability is changed into r from r_o；That is:

$r_{i/o}=1-s_{i/o}=1-e^{0.1053{\left(\frac{l_s}{l_{10}}\right)}^e}---\left(10\right)$

In formula (10), s is probability of escaping by luck, and subscript i/o in formula refers to inner ring or outer ring；l_sIt it is calculating longevity during s for probability of escaping by luck Life, now reliability r is 1-s；

(7) as it was previously stated, the then complete bearing failure of arbitrary Contact Pair element failure, then for the reliability of complete bearing, adopt Series connection reliability description is more accurate, and a whole set of bearing is in life-span l_rUnder reliability r ' can be described as:

R '=r × [r_i×r_o] (11)

Formula (11) represents one or computing, i.e. new reliability r of complete bearing ' it is demand reliability r and the interior r lasting a long time_iOr Outer r_oCalculating reliability product, interior raceway mathematic(al) expectation longer then select r_i, outer raceway last a long time then selection r_o, r_iOr r_o Value by formula (11) determine；

(8) reliability r being obtained after calculating ' different from former demand reliability r, then need to return under calculating demand reliability Mathematic(al) expectation, this calculating process can solve Different Reliability according to the Slope relationship of Weibull distribution, and that is, difference is escaped by luck generally In the different fatigue life-span under rate, its relation can be described as:

$ln\frac{1}{s}={\left(\frac{l_s}{l_{s^{,}}}\right)}^{e}ln\frac{1}{s^{,}}---\left(12\right)$

In formula (12), s is the corresponding probability of escaping by luck of demand reliability r, and s ' is then actual reliability r calculating under the corresponding life-span ' Corresponding probability of escaping by luck, l_sIt it is mathematic(al) expectation during s for probability of escaping by luck, ls ' is the mathematic(al) expectation that probability of escaping by luck is during s ', typically In the case of, Calculation of Fatigue Life its l of many demands of rolling bearing₁₀Life-span, then formula (12) can be reduced to:

$l_{10}=\frac{l_{s^{,}}}{{\left(\frac{1}{0.1053}ln\frac{1}{s^{,}}\right)}^{\frac{1}{e}}}---\left(13\right).$