The analysis and choice of models of durability at probes of reliability of drive lines

From practice of design of cars it is known that their demanded reliability is put in the course of design, and mistakes at this stage are wrapped in huge losses in production and operation. On the example of internal combustion engines it is possible to note that if to take mistake cost at a stage of research work for unit, at a stage of developmental works its price will increase in 10, at a stage of pilot production and tests - in 100, and serial - in 1000 of times [1]. In this regard questions of a reasonable choice and use of mathematical models of durability of transport and technological cars and the equipment hubs for the purpose of working off of experimental actions of increase of their reliability are very actual.

Bases of the theory of definition of durability of rolling bearings are stated in A.G.Komissara, L.Ya.Perel, I.Ya.Birger's works, etc. In relation to сardan bearing mount assemblies A.I.Grishkevich, E.P.Flika, M.S. Vysotsky's works, etc. are known. Life cycle of transport and technological cars and the equipment drive lines is defined by durability of the bearing mount assemblies formed by joint "a crosspiece thorn – a needle bearer". Considering that drive lines possess the smallest durability among units of mechanical transmissions, it is possible to claim that the durability of cardan joints limits durability of all transmission [2].

MATERIALS AND RESERCH METHODS

On condition of a right choice of a standard size of cardan joints and an exception of consideration of the phenomena of corrosion, damage of assembly surfaces and the breakages caused by rough defects of production and operation, main types of collapse for cardan bearing mount assemblies are the fatigue spalling of surfaces of swing, abrasive wear, education longitudinal, often inclined, the indents called by effect of a false brinelling and the tease on butts [2]. Specified phenomena define primary fact of a choice of model of durability.

With big amplitude of swing, in cases when the angle of rotation of a glass of the bearing concerning a thorn from a midposition in extreme is more, than a corner between the next bodies of swing, calculation of durability carry out by criterion of contact fatigue on the basis of Lundberg-Palmgrena theory on a formula [3].

LY

= a 1 a ₂ a ₃

60 - 10⁶

n

where L γ – durability of the bearing, h; а 1 , а 2 , а 3 – coefficients of reliability, a material, lubricant and a condition of surfaces of bodies of swing; C – the dynamic loading capacity of the bearing, N; Р – equivalent applied bearing load, N; m – empirical constant; n – working rotary speed of a shaft, min^-1.

For the purpose of approach of results of calculation for classical dependence to data of operation of bearings in field environment the new theory of durability on the basis of probes of the center of SKF firm (Sweden) was offered, thus the formula of durability considers a condition of a greasing layer, extent of pollution and a load limit on fatigue [4].

L

haa

= a 1 a SKF

Г C ^ P I P J

where a SKF - coefficient considers a condition of a greasing layer, extent of pollution and rhee's load limit Р и on the fatigue, defined depending on parameter η С Р и /Р ; Р и - a maximum load of the bearing on fatigue; η С - the coefficient characterizing extent of pollution. On the basis of this theory it is possible to predict surely durability on fatigue in the conditions of rather small loadings and intensive pollution (transport and agricultural).

In relation to durability of needle bearers of сardan joints of L ho cars, h, the mathematical model generalizing experience of the Minsk automobile plant and some foreign firms on design of drive lines in which modes of loading and operation of units of the car [5] are considered is known:

10/3

1,5 - 10 6 , C ( H - L w ) n ^{- в} _     ^Тк;ш    .

where β - a corner of a break of the joint, grade; H – a dimensional factor of a crosspiece, mm; L W – length of a needle roller, mm; T KШ – the given torque transferred by shaft, Nm. This model allows to count durability of the cardan bearing under conditions that the total interneedle spacing doesn't exceed 0,5 mm, the different dimensions of rollers is equal to ±1 microns, the hardness of paths of swing within 60 … 64 HRC, at a working temperature in knot less than 100°C and 90% level of reliability of bearings.

Life cycle of drive lines of cars and the equipment, applied in agro-industrial complex determine by the following model of calculation of durability of cardan bearings [6].

L

ho

1,5 - 10⁶ n - в

C ( H - L w ) ^Кd^TКШ

- а _х а 2 а 3 ,

where K d - the coefficient of dynamism determined by tensometric data of loading; а 1 – the coefficient considering demanded level of reliability; а 2 – the coefficient considering structure of a material, purity concerning nonmetallic inclusions and hardness, greasing conditions; а 3 – the coefficient considering service conditions of agricultural cars. In case the propeller shaft works at break coal equal to zero, in calculation the deviation to 3 ° is allowed, otherwise the durability will be equal to infinity.

During the work with a small amplitude of swing, there are specific conditions of lubricant of the loaded contact piece of bodies and swing paths which lead to formation of prints of bodies of swing in the form of flutes from rollers that testifies to process of the fretting corrosion promoting formation of defect of work faces of cardan bearing mount assemblies in a look of false brinelling therefore calculation of durability of L10h, h, make at constant rotary speed under the actual working conditions [3].

^L 10 h

= 500( f_h ) P = 500 -

2,6 CR

у maxV dJn P у

where f h - the coefficient of durability established by requirements of operation, h; R -distance from a pivot shaft to the average section of the bearing, m; p - an exponent; T max -the maximum torque, Nm; f d - the dynamic coefficient considering existence of pushes and blows during the work; f nβ - coefficient of work of rotary speed on a break corner in the nβ joint (at β ≥ 3 °).

Assessment of durability of rolling bearings by criterion of the wear determined by intensity of wear of I h , carry out on the basis of mathematical model of process of wear of bearings in the assumption of proportionality of wear to specific work of friction forces of slippage on the empirical expression approximating results of bench tests [7].

mn pq

¹ h ^^a ^см^у ^ш ,

where Ф a , Ф см , Ф у , Ф ш – the dimensionless complexes considering respectively a tension of contact piece, relative thickness of a lubricant layer, fatigue strength of contacting surfaces and their roughness; m , n , p , q - the empirical parameters determined by results of tests; k – proportionality coefficient.

Result of synthesis of theories of contact fatigue and wear for the purpose of an assessment of durability of drive lines is the mathematical model considering additional influence of a radial play in needle bearers on a limit operating time of сardan joints at bench resource tests [8].

_T 2,2 - 10 6

^Lh = „ ГТ” "

C ( H - L w )

T _КШ

3,165

,

where Δ Н - initial spacing in cardan bearings, micron.

The analysis of the given mathematical models of durability of cardan joints (a formula (1)…(7)) shows that the major factors having impact on durability of drive lines are constructive (a dimensional factor of a crosspiece of H ( R ), length of a needle roller of L W ), technological (an initial radial play Δ н , dynamic loading capacity C , the hardness of work faces of HRC ) and operational ( T KШ torque, coefficient of dynamism of K d , rotary speed of n , a break corner in the hinge β and their variability). Taking note of the factors determining initial and used potential of durability of cardan joints, in particular, factors, considering level of reliability, quality of a material, hardness of work faces, conditions of greasing and operation of agricultural cars isn't less important.

On the base above the specified we suggest a choice of mathematical models of durability of drive lines to define proceeding from the following provisions: 1) accounting of a goal; 2) main types of refusals; 3) roadability factors; 4) reproducibility of factors in the conditions of bench or operational trials; 5) susceptibility of actions of improvement and modernization; 6) completeness of the accounting of efficiency, technology and operational factors.

RESULTS AND DISCUSSION

Let's review an example of a choice of mathematical model of durability of needle bearers of drive lines on the basis of the offered provisions.

The major factor of constructive and technological character defining durability of cardan joints, the dynamic loading capacity of their bearing mount assemblies as this parameter is included into the durability equations in p=10/3 or m=3,165 (1)-(7) degree is.

In domestic practice the dynamic loading capacity of the roller radial bearing C , H, determine by a formula [3].

3/4  29/27               7/9

С f C Z D W  ⁽ L W i ^{cos a )}   ,            (8)

where , _/ d w cos a 4 - the coefficient depending on geometry of details of the bearing, ^fc = f I

V D о )

accuracy of their production and material; D W - diameter of a needle roller, mm; D 0 - diameter of a circle of the centers of a set of rollers, mm; i =1 - number of ranks of rollers in the bearing; L W - roller length for calculation of loading capacity, mm; α=0° - a nominal corner of contact piece of the bearing, grade; Z - number of rollers in the bearing.

On the basis of methods of calculation of loading capacity and durability of the rolling bearings accepted by the international organization of standardization of ISO, the basic dynamic radial loading capacity of C r , N, a radial roller bearer is equal [3].

C = b m f c ( iL we cos a) 79 Z ^3/4 D WWe ^/27,      (9)

where b m =1,1 - the coefficient considering properties of steel taking into account a mode of its production; D WE - diameter of a roller for calculation of loading capacity, mm; D pW -diameter of a circle of the centers of a set of rollers, mm; L We - roller length for calculation of loading capacity, mm.

On the basis of numerous probes of the drive lines which have been carried out by GWB firm (Germany), dynamic loading capacity of the cardan bearing C , H, it can be determined by a formula [5].

C = 4 • V Z ² • D W

. T

^LW ■

For establishment of the fact of applicability of the given dependences in calculations of durability of cardan bearings we will carry out comparison of results of calculation and data of the specifications and technical documentation on standard sizes (table 1).

The basis of calculation of dynamic loading capacity of needle bearers is made by formulas (8), (9) and (10). Results of calculations it is representable in table 2 taking into account help data on coefficients.

Table 1 – Parameters of cardan bearings

Type size	Symbol	Bearing	Z	d W , mm	L W , mm	C , кN
I	КШ 50	904902	22	2,4	10,0	8,0
II	КШ 160	704902	22	2,5	12,5	7,2
III	КШ 250	704702К2	29	2,0	13,8	7,7
III	КШ 250	704702КУ2	20	3,0	13,5	7,7
IV	КШ 400	804704	26	3,0	18,0	13,7
V	КШ 630	804805	29	3,0	18,1	14,5
VI	КШ 1000	804907	38	3,0	18,0	16,5
VII	КШ 1600	804707	38	3,0	24,0	21,0
VIII	КШ 1600	804709	50	3,0	24,0	31,5

The dispersion of the received results is explained by absence in settlement formulas of the parameters considering influence of different dimensions of rollers, the hardness of surfaces of friction pairs, a radial play, etc. Thus, the rocking mode of behavior and limitation of a zone of loading of a сardan needle bearer isn't accepted in the account. In this regard smaller values of dynamic loading capacity are preferable to further calculations of durability (formula (10).

Table 2 – Results of calculation

Type size	Symbol	Bearing	C , kN (8)	C r , kN (9)	C , kN (10)
I	КШ 50	904902	12,1	15,2	7,54
II	КШ 160	704902	15,1	18,9	9,82
III	КШ 250	704702К2	15,5	19,4	10,4
III	КШ 250	704702КУ2	18,3	22,9	12,0
IV	КШ 400	804704	27,3	34,1	19,0
V	КШ 630	804805	29,5	36,8	20,5
VI	КШ 1000	804907	34,3	42,9	24,4
VII	КШ 1600	804707	42,9	53,7	32,6
VIII	КШ 1600	804709	50,2	62,6	39,1

On the basis of the chosen model of calculation of dynamic loading capacity we will consider calculation of durability of cardan bearing mount assemblies in modes of normal and extreme loading. Results of calculations are presented in tables 3 and 4.

Table 3 - Results of calculation of durability of cardan bearings (MNL)

Type size	Symbol	Bearing	Durability, h
			L γ (1)	L haa (2)	L ho (3)	L ho (4)	L 10h (5)	L h (7)
I	КШ50	904902	3,19·106	4839	190·103	12,9·103	15872	34733
II	КШ160	704902	18,8·106	28537	1,10·106	75,2·103	93837	186788
III	КШ250	704702К2	0,25·106	370	14,6·103	999	1233	3055
III	КШ250	704702КУ2	0,40·106	611	23,9·103	1631	1985	4818
IV	КШ400	804704	0,50·106	757	30,0·103	2053	2538	5352
V	КШ630	804805	0,31·106	427	17,8·103	1220	1514	3160
VI	КШ1000	804907	1,02·106	1550	60,5·103	4139	5117	9169
VII	КШ1600	804707	0,87·106	1319	0,10·103	3487	4308	7785
VIII	КШ1600	804709	2,47·106	3518	147·103	10,1·103	12440	20525

Table 4 - Results of calculation of durability of cardan bearings (MEL)

Type size	Symbol	Bearing	Durability, h
			L γ (1)	L haa (2)	L ho (3)	L ho (4)	L 10h (5)	L h (7)
I	КШ50	904902	9672	14,7	102	6,98	8,16	25,0
II	КШ160	704902	31167	47,2	336	23,0	27,0	77,4
III	КШ250	704702К2	15497	23,5	168	11,5	13,4	40,0
III	КШ250	704702КУ2	11662	17,7	126	8,62	10,0	30,4
IV	КШ400	804704	2590	3,92	27,9	1,91	2,22	6,39
V	КШ630	804805	1441	2,18	15,4	1,05	1,24	3,54
VI	КШ1000	804907	1836	2,78	19,6	1,34	1,57	4,05
VII	КШ1600	804707	17122	25,9	184	12,6	14,6	33,7
VIII	КШ1600	804709	3546	5,37	38,2	2,61	3,04	7,30

The mode of normal loading (MNL) is characterized: in nominal rates of a torque according to a standard size, rotary speed and a break corner ( β min =3°), causing refusal in the form of a false brinelling, and the mode of extreme loading (MEL) is characterized by the greatest operational values of a torque according to a standard size and a break corner in hinges ( β max =20°), causing refusal in the form of fatigue damage [2].

The comparative analysis of results of calculation of durability at MNL shows that the greatest values are received on model (1), and the smallest - on models (2), thus excess of sizes makes approximately by 700 times. For MEL this comparison shows that the greatest values are received also on model (1), and the smallest - on models (4), thus excess - in 1400

of times. Thus, with increase in values of parameters of a mode of loading value of durability has essential dispersion, thus models (1), (4), (5) and (7) yield the closest results.

Proceeding from the carried-out reasonings, we will note that by the provisions given above the mathematical models (1), (4), (5) and (7) mentioned above possess the following characteristics: 1) realize the research purposes; 2) consider nature of emergence and a type of refusal; 3) allow to steer studied factors; 4) reproducibility of studied factors as in the conditions of bench, and operational trials is satisfactory; 5) the accounting of actions of improvement of processes and modernization of a design assumes introduction of the additional parameters established by results of probes that is reflected in the accounting of increase of loading capacity of needle bearers by their self-installation, dynamic change of a corner of a break of hinges, carrying out maintenance operation by replacement of work faces of bearings, fitness to carrying out repair by use of additional repair barrels and movable spigot joints, etc.; 6) these models consider the greatest number of efficiency, technology and operational factors.

CONCLUSIONS

• 1.    Review of mathematical models for calculating longevity propeller bearings to show the diversity of the existing theories, allowing to explore different phenomena occurring in them and determine their longevity.

• 2.    Proposed provisions of informed choice models of durability depending on the objectives and expected results of scientific research.

• 3.    Our studies confirm the effectiveness of this approach, which suggests the need to create virtual models of simulation study of factors and their effects - durability.