The theory of anthropogenic catastrophes: a new approach to the safety problems

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THE THEORY OF ANTHROPOGENIC CATASTROPHES: A NEW APPROACH TO THE SAFETY PROBLEMS

Stanislav Korshakovsky, Professor, MSTU MIREA, Electrical Engineering Department, 78, Vernadskji, 119454, Moscow, Russia, E-mail: korshakovsky.mirea@gmail.com

Miсhael Krasnenkov, Professor, MSTU MIREA, Physics Department, E-mail: kma1942@yandex.ru

Nikolai Chekalkin, Professor, MSTU MIREA, Higher Mathematics Department, E-mail: chekalkin@rambler.ru

 

Moscow State Technical University of Radio Engineering, Electronics and Automation (MSTU MIREA), Moscow, Russia

On the basis of the statistical regularity the anthropogenic catastrophe risk evaluation criterion for units and components failure under operation has been developed. The paper contains both the analysis of possible failure reasons on the basis of an abstract unit as an example and the appropriate calculations of the criteria obtained. The investigations conducted allow to predict equipment failures of technical objects and to improve safety in their operation.

Nomenclature

Aij – the events wherein conjugate the components parameters deviations from the average are maximum

Aj –the events in which the combination of j- group parameters is unfavorable

clip_image008[4]the probability of an opposing event

clip_image010[4]matrices whose elements are parameters of conjugate elements clip_image012[4]respectively

i, j–numbers of conjugate elements parameters and parameter group numbers

k–the degree of the weight function clip_image014[12]

clip_image016[4]mode

m–maximum number of the object element groups

clip_image018[4]– a set of natural numbers

nj –maximum number of parameters in the j-group

clip_image020[4] the event probability of Aij

P – limiting probability: the outcome probability of at least one critical state for an object or unit

clip_image022[4] limiting probability: the probability of the event of just as many q critical states as possible from m

clip_image024[8],clip_image026[4]– the event probability of Aj

clip_image028[4]– total risk probability: the outcome probability of at least one critical state for an object or unit

clip_image030[4]–total risk probability: the probability of the event of just as many q critical states as possible from m

clip_image014[13]–weighting function

x – continuous random variable

clip_image032[4]–matrices whose elements clip_image034[4] are the upper limits of the integrals at the definition of clip_image036[4]

clip_image038[4]–matrices whose elements clip_image040[4]are respectively mathematical expectations, average measurement errors and root-mean-square deviations

clip_image042[10]–accuracy rating

clip_image044[4]–the probability integral

clip_image046[4]–probability density

clip_image048[4]–diameter, mm

q– the number of critical states among m groups

Introduction

When speaking about the problem of safety for power-consuming objects and particularly for the airspace ones one can’t but admit that the successful solving of it depends largely on the objects reliability that is defined by the extent to which internal and external factors may have the influence on.

Any object consists of a set of components operating under appropriate technical parameters with each of them having its own range of deviation from the standard states. This range also called the tolerance zone is of great importance for seeking new criteria to evaluate the level of anthropogenic rick for both the object’s individual components and the object itself as a whole. At the same time the creation of the emergency protection system for the objects involves certain difficulties covering nondestructive control methods design and diagnostic means provision as well. The first thing to do is to reach a compromise between external devices (sensors, communication, infrastructure) mating and the object itself under control [1–3].

But the present well-known methods of control can’t provide complete safety. In the absence of extreme conditions the failure relating to the possible standard state parameter deviation is purely internal: it is built into the object components during their manufacturing, mounting and assembling and the operating conditions only add to the failure. The investigation of the statistical regularities mechanism for these processes is especially urgent for both the safety problem and the problem of keeping power-consuming and other objects safe from anthropogenic catastrophes during their usage.

Risk area. Setting up a problem

In any technical object unit one can specify a group or groups of interconnected and interdependent components whose parameters keep the object operating properly. Generally these components may not make the immediate contact with each other. Let’s name these or those components of an abstract unit conjugate for the sake of definiteness and as an example refer to the air engine’s two units: the body of the turbine and its rotor’s blades.

Between the blades and the body of the turbine under operation there must be a process clearance to keep the turbine trouble free running. However the turbine units made with the certain precision has natural dispersion (spread) in sizes of both the body inner diameter and the blades length. These deviations may cause the minimum clearance between the moving and stationary parts of the turbine when the two events coincide simultaneously–the body inner diameter reaches the lower line of the tolerance and the blade length does the upper one. Let us mark that these deviations do not exceed the accepted allowable ones. However the probability of the turbine failure may increase under external factors influence leading to temperature, pressure and loads rise. This means that the object is in the area of risk.

The performance of such analysis for the conjugate components with regard to their interconnection will allow defining the probability for destruction development potential risk and comparing it with the necessary (specified) level of trouble-free operating as well as develop an algorithm for the appropriate actions to provide safety. These probabilities, worked out for separate units (components) or the object as a whole are anthropogenic catastrophes criteria. A certain complexity of such approach lies in the relative uncertainty of unit (component) size deviation distribution from the average meaning in the tolerance field. Besides the standard normal distribution it makes sense to look at other possible probability distribution in this field. The uncertainty relating to the choice of limiting deviations range for the object mating components and the probability distribution density region width needed for its calculation may cause some difficulties.

By analogy with the presented analysis of mechanical units operation one can consider electronic and other object units subject to failures whose parameters and characteristics also have statistical distribution. Let us next consider the most «problematic» mating unit components and evaluate the described situation leading to a failure probability increase.

Probabilistic models. Critical states

Let us consider the problem of an abstract object safety in the context of statistical regularities taking into consideration definitions and simplifications accepted. Let us also specify in this object (or separate unit) the conjugate components with the specified parameters for the operation in the standard state (each such component may have several parameters). The component parameters must be grouped as follows 1, 2,…, j,…, m (m clip_image050[6]1) in each of them the conjugate components have numbers 1, 2,…, i,…, n (nj clip_image050[7]2). For each group the events are represented by A1j, A2j,…, Aij,…, clip_image052[4]wherein conjugate components parameters deviations from the average ones are maximum and these events probabilities are p1j, p2j,…, pij,…, clip_image054[4] respectively.

clip_image056[4] . (1)

Fig. 1 To the definition of the critical state for two conjugate elements.

clip_image058[4]

Suppose that in some j-group conjugate components with the probabilityclip_image060[6]the event Aj may happen resulting in unfavorable combination of nj parameters further leading to the unit possible failure under external factors influence. Let’s name the unit’s conjugate components state relevant to this event a critical state (fig. 1). In this case the unit appears to be in the risk area (nj =2) or highrisk area (nj ≥2). The group unfavorable parameters set are given through the probabilistic model with the associated outcomeclip_image060[7].The simplest model of these sets is the agreement of maximum parameter deviations of all mating components from the average at which the Aj probability of j-group critical state is defined as the product of probabilities for independent events [4, 5]:

For the dependent events the conditional probability must be used:

clip_image062[4] (2)

When defining the clip_image024[9] probability one should take into account only those components (with the relevant probabilities pij) the events with which will cause the situation change for the worse (close to anthropogenic catastrophe). At the same time a consideration of the interrelation between several mating components at a time will demand far more complicated probabilistic models.

Having entered the probability of an opposing event for the set of independent m-groups clip_image065[4], the outcome probability of at least one critical state for an object or unit may be defined as

clip_image067[4]. (3)

This simple equation shows the most common nature of critical states for the unit or object independent parameters group. For the simple probabilistic model (see equations (1) and (2)) this value depends both on maximal probability deviation pij, maximum number of parameters for conjugate components nj and m-groups number.

If probabilities pij differ little from each other and nj are the same, one can evaluate the critical state probability when among the m-groups there are just as many q critical states (q ≤ m). In this case pij is necessary to substitute for its average meaning clip_image069[4].This is the event of just as many q critical states as possible from m are realized and falls into so many incompatible variants as many ways q may be chosen from m-cases (the number of combinations from m by q). With the well-known equation of the repeated tests and the theorems [4] the desired probability is defined as

clip_image071[4]. (4)

The probability obtained will be the most at clip_image073[4]. At sufficiently high values of m and q Pq,m is approximately defined according to Sterling equation. At small clip_image075[4]the Poisson equation [5] gives more accurate value.

The approach based on average meanings pij allows evaluating the frequency of interest for the event q/m occurrence. With the increase of m the most real frequency of such event occurrence (the greatest statistical probability) will approach Pq,m In this case the probability that this frequency lying between clip_image077[4]andclip_image079[4]will approach the probability integral clip_image081[4](Laplace theorem). But all looks much more complex if pij averaging is done not in a proper way.

Limiting probabilities

When considering the common case of Aij probability definition it is necessary to use the probability dense clip_image083[4]of a continuous random value. Let’s believe that x values are limited at the top and bottom by the elements aij and bij parameters (fig. 2), i.e.

clip_image085[4] (5)

where clip_image087[14]is an absolute mean error of the parameter definition x (positive value).

For a mechanical unit for example aij and bij are boundary tolerance fields, and clip_image087[15]–the mean value of measurements. The elements, whose parameters don’t satisfy the condition of the (5) are rejected. If inequalities (5) are satisfied, then the probability of x appearance in the limits of clip_image090[4] is the conditional probability. Here cij is the j-group element i-parameter defining the entry of x in the tolerance field boundaries to the accuracy of clip_image087[16] (in the common case to an accuracy of clip_image087[17]aij cij

clip_image106[4] (6)

 

clip_image100[4]
bij). Its value may be defined as a ratio of areas on the plotclip_image102[4]:

Fig. 2 Probability dense clip_image104[4]for the even accidental value.

It’s well to bear in mind that error considerationclip_image087[18]broadens the range of limiting parameters by the value of 2clip_image087[19]. At the same time for the units with the specified standard tolerable errors for parameter measurements at acceptance check out the limiting parameter boundaries according to the standard include these errors (see, for example, [6]). In these cases the limiting boundaries of the relevant parameters are fixed and must not be changed (fig. 3) and that is especially important for highly accurate unit components.

With regard to the equations (1) and (3) the probability P can be written as

clip_image110[4], (7)

where clip_image112[4]– are the matrix elements

clip_image114[4]. (8)

The integration limits are the relevant matrix elements derived from addition and subtraction of A, B, C with clip_image116[4]:

clip_image124[4]
clip_image126[4]. (9)

Fig.3 Probability dense for the fixed tolerance field boundaries:

1–even distribution; 2– normal distribution law.

The equation derived for P calculation in which the simplest probabilistic model is used (1) is the special case of the more common equation (3) for which the equation is done:

clip_image128[4]. (10)

To calculate the event probability Pq,m wherein just as many q critical states from m are realized it is necessary to consider a more complicated probabilistic model where this event is presented by the sum of incompatible variants the number of which is equal to the number of combinations from m by q (clip_image130[4]):

clip_image132[4] (11)

Performing the inequalityclip_image134[4]the numerator of the fractionclip_image136[4] (function clip_image138[4] is continuous within the limits considered). A new equation forclip_image024[10]

clip_image141[4] (12)

allows to simplify the probability calculations P and Pq,m .

In critical events evaluation both even distribution of probability density (ed) and distribution normal law (nd) are of practical interest, which can be written as

clip_image143[4] (13)

clip_image145[4] , (14)

where clip_image147[4]–is the root-mean-square deviation,clip_image149[8]–mathematical expectation all being the matrix elements

clip_image151[4]. (15)

Believing that the distributions clip_image153[6]are symmetric relative to the mathematical expectation at least for even and normal distributions when the Mij mode and the mathematical expectationclip_image149[9] (fig. 3) are equal the values clip_image149[10] can be specified as

clip_image157[4]. (16)

While using the probability nonsymmetrical probability density clip_image153[7] or in particular at Gauss distribution when the field tolerance boundaries are nonsymmetrical relative to Mij mode, the elements of the matrix clip_image160[4] should be determined by the method described in [5].

With the equation (3) keeping in mind the probabilities Ped,nd for these kinds of distributions are written in rather simple forms:

clip_image162[4],clip_image164[4], (17)

where the clip_image166[4] functions entered for the case (1) are determined as

clip_image168[4]clip_image170[4] . (18)

The event probability Pq,m may accordingly be calculated based on the above given relations (4) and (11).

For the limiting parameter boundaries fixed (in the tolerance field) it is essential to enter the integration limits correction into the equations (6), (7), (10), (12) and (18) obtained for the left (cij=aij) and right (cij=bij) tolerable boundaries:

clip_image172[4]. (19)

In this case the probability density at the even distribution is determined as

clip_image174[4]. (20)

At the normal distribution the equation (14) is used. The equations (19) may be integrated considering the integration low limit cij, while the upper one is clip_image176[4] (for the left and right tolerance field boundaries, respectively). In order not to change the integration limits for these two cases the fraction numerators in the equation (19) should be taken over the model:

clip_image178[4] , (21)

where the upper limit in the integral numerator is the matrix elements

clip_image180[4]. (22)

Similar transformations are done for the rest of equations.

The greater the probability values in these equations the higher the failure risk for the unit or the whole object considered. The critical values obtained for these probabilities appropriate to the extremes of the allowable interval parameters are named limiting probabilities.

The conjugate component parameter number increase for the nj unit (group) causes clip_image182[4]to decrease sharply (the events at pij<<1 are considered). Performing the addition of the probabilities the contribution of this component might be insignificant. That is why it makes sense to specify two groups with the same number of mating component parameters nj (nj = 2, 3,…) to which critical state probabilities clip_image184[6] (2), clip_image184[7] (3),…, are appropriate and compare them in between. Limiting probabilities under these conditions are critical states levels – the first, the second and so on respectively. The same is also relevant to the total probability risks notion introduced below.

Solving the problem of evaluation of complex system high degree failure risk one can face the difficulties when building separate group parameter probabilistic models. It is worthwhile making up these groups (subsets) into bigger ones (sets) with the appropriate probabilistic models.

Total risk probability

clip_image187[4]

Fig. 4 To the weight function definition:

1 – probability dense clip_image195[4]; 2 – power function clip_image197[6]; 3 – product clip_image199[4]clip_image197[7]

The events described at the object operation occur undeniably and are of interest, however are very seldom. In the real life all is more complicated. However besides the above mentioned critical states one should refer to the cases when the unfavorable coincidences of the conjugate element parameters deviation from the average including the parameter vicinity cij of the length considered (aij; bij) and as far as to the point of its middle (mathematical expectationclip_image189[4]ij) take place. So, consideration must be given to the introduction of a weigh function clip_image191[4]under the integral sign multiplied by the probability densityclip_image193[4](x), which allows for the degree of approximation to the tolerable parameter intervals extremes (limiting probabilities), fig. 4.

Thus at the calculation of the pij probability in the fraction numerator (6), the low integration limit should be replaced by clip_image202[4]ij and the equation clip_image204[4] be considered the upper limit for the left and right tolerance field boundaries, respectively. The equation

clip_image206[4] (23)

may be written by analogy with equation (22) and the upper limit of the integral numerator is the matrix elements for this case:

clip_image208[4]. (24)

The introduced weigh function must have the following features:

clip_image210[4] (25)

Besides clip_image014[14]must be continuous and monotonic within this interval and its first derivative clip_image213[4] must approach 0 when xclip_image215[6]0 and go up when clip_image217[4].The power function clip_image219[4]satisfies these requirements where kclip_image221[4]. Using the condition (25) to find the coefficient clip_image223[4] the weigh function may be written as (clip_image225[4]):

clip_image227[4] . (26)

With consideration for the equations clip_image014[15]and clip_image230[4](x) we have

clip_image232[4], (27)

clip_image234[4]. (28)

These equations are used further for the definition of the object (unit) critical state probabilities by analogy with (3), (4) and (11) which we name total risk probabilities:

clip_image236[4], (29)

clip_image238[4], (30)

clip_image240[4] (31)

where for a simple probabilistic model (1) clip_image242[4].

At the fixed parameter boundaries clip_image244[4] with the upper limit in the equation numerators (27) and (28) is replaced by cij while the integration limits in the denominator are replaced by aij and bij, respectively.

The main drawback of the limiting probabilities considered is that they are generally defined by the error the value of which is chosen conventionally enough: the measuring instrument error may change in the wide range of limits. Another approach to the problem of total risk probability gives the opportunity to predict the discussed unit or object failure independently from the error values if the tolerance boundaries are fixed. The complexity is that the weight function degree (26) is unknown. With clip_image246[4] clip_image014[16] is a linear function not showing the true situation. If clip_image249[4]the total risk probability approaches clip_image215[7] 0. Complex investigation of a big amount of statistical data may give more or less exact situation, unfortunately the authors of the paper don’t know about them. At the present stage another thing is important. Speaking about the comparison of an anthropogenic catastrophe criteria obtained for different units of one and the same object/objects that will be enough to consider k as the same value for these cases and make it equal 3 (cubic parabola) that sketches true situation. When conducting further investigations this value may be adjusted and the results refined.

Potential risk probabilities calculation even for the simplest case of independent events is a rather difficult problem. Let us specify these values for a simple specific case where the events have pure mechanical nature, a bearing, for example, that is the unit of an abstract object and research the influence of different factors on it related to its manufacturing, accuracy measurement, accuracy rating and probability density type.

Anthropogenic catastrophe risk evaluation for a mechanical unit

To evaluate potential probability risk let us consider a radial roller bearing having a short cylindrical 2000 type roller, fig. 5 [7]. Let’s also assume the following system of mating for the bearing conjugate components: transition mating into the body opening and stressed one on the shaft. At last let’s specify 3 parameter groups for the conjugate components (m = 3): the opening diameter 1 – the outer diameter of the outer ring 2 (j =1); the outer rolling way diameter 2 – the roller diameter 3 – the inner ring rolling way diameter 4 (j =2); the inner ring inner diameter 4 – the shaft diameter 5 (j =3). You can find the bearing components, opening and shaft dimension tolerances [8] in table 1 and the data necessary for doing calculations in accordance to the above said their presentation structure – in table 2. The definition accuracy of geometric parameters clip_image252[4],in general is determined by accuracy rating clip_image042[11] specified for this unit. Errors presented by the matrix clip_image255[4] are accepted as equal to the allowable ones under the State Standard [6] at the acceptance test.

clip_image259[4]clip_image260[4]clip_image261[4]clip_image262[4]clip_image263[4]clip_image264[4]

Fig. 5 The sketch of the cylindrical

bearing:

1– the body; 2 – the outer ring;

3 – the cylindrical roller; 4 – the inner ring; 5 – the shaft.

 

In accordance to the expected events in critical situations we choose the components field tolerances boundaries (table 2): cij = aij or cij = bij. It is necessary further to specify both a probabilistic event model and interrelation between conjugate components keeping in mind for the simplicity that the elements when critical states of the unit components occur are independent. Critical states in the first and third groups are defined in accordance to the equation (1), when n1,3 = 2 and the product ofclip_image266[4]are equalclip_image268[4] andclip_image270[4], respectively. If for these groups the critical states are defined by the simple probabilistic model for the conjugate components 2, 3 and 4 (fig. 5) of the second parameter group it is necessary to consider the event Аj = А2 at least one critical state for three components having in total three pairs of geometric parameters. This event may be presented as the sum of four incompatible variants:

To find the limiting probability for the whole unit it is necessary to add all the probabilities using the probabilities of the opposite events:

clip_image272[4] (33)

clip_image274[4]. (32)

Table 1 The bearing conjugate components tolerance field.

Conjugate elements

Nominal size, mm

Tolerance field for accuracy rating,

clip_image276[4]

4

5

6

7

8

body (position 1

in the fig. 5)

opening diameter 90

+15

0

+22

0

+35

0

+54

0

outer ring

(position 2)

outer diameter 90

0

10

0

15

0

22

0

35

0

54

rolling way diameter 80

+2,5

2,5

+4

4

+5,5

5,5

+9

9

+13,5

13,5

roller

(position 3)

diameter 10

0

5

0

8

0

11

0

18

0

27

inner ring

(position 4)

rolling way diameter 59,92

+4

4

+6,5

6,5

+9,5

9,5

+15

15

+23

23

inner diameter 50

+7

0

+11

0

+16

0

+25

0

+39

0

shaft

(position 5)

diameter 50

+24

+17

+37

+26

+42

+26

+68

+43

+109

+70

Conjugate elements and groups

Component geometric parameters

The event expected in the critical state

component position in the figure 5

number of element

i

number

of group

j

low boundary aij

upper boundary bij

mathematical expectations μij

maximum deviation from the average, cij

tolerance field boundary

parameter value

1

1

1

0

15

7,5

b11

15

outer ring slippage

2

2

–15

0

–7,5

a21

–15

2

1

2

–4

4

0

a12

–4

roller jam between the outer and inner rings

3

2

–8

0

–4

b22

0

4

3

–6,5

6,5

0

b32

6,5

4

1

3

0

11

5,5

b13

11

inner ring slippage

5

2

26

37

31,5

a23

26

Table 2 Data for calculation, clip_image278[6]=5 (clip_image280[4]).

clip_image282[4]

As it is evident from tables 1 and 2 the probabilities pij differ greatly from each other and the equations (4) and (30) are unacceptable. Using the equation (31) one can determine the probability of the unit critical state when among the m-groups there is as many q critical states as possible restricting to two cases for the simplicity at m=3 : q=2 and q=3 (the worst case). Limiting probabilities and probabilities of total risks may be determined as

clip_image284[4], clip_image286[4], (34)

clip_image288[4], clip_image290[4]. (35)

Probabilistic model for the mechanical unit critical state

Even distribution

Normal distribution law

At least one critical state

limiting

probability P

0,525

0,484

total risk probability clip_image292[4]

0,073

0,036

Among three groups there are just as many as two in the critical state

limiting

probability P2,3

0,081

0,072

total risk probability clip_image294[4]

0,00156

0,000401

All three groups in the critical state

limiting

probability P3,3

0,00565

0,00332

total risk probability clip_image296[4]

1,49clip_image298[6]

1,418clip_image298[7]

Using the equations obtained the necessary calculations have been done for limiting and total risk probabilities for both even and normal distributions. The calculation algorithm developed for the fixed limiting tolerance field boundaries for the discussed mechanical unit was realized in Mathcad medium. Some results of these calculations are given in table 3 and figures 6 – 9. The discussion of the results gained is carried out below.

Table 3 Some calculation results for the mechanical unit (clip_image278[7]=5, k = 3).

Le Chatelier approach

As one can see from the results dispersion σ2 , accuracy rating clip_image300[8], accepted at the unit manufacturing, index k of the power function clip_image302[4], the type of dense probability distribution have the main influence on the potential risk probability. Root-mean-square σ depends not only on accuracy rating clip_image300[9] , but on the component dimensions (fig. 6). At dispersion increase the total risk probability

under the normal distribution rises approaching its limiting value i.e., the probability under the even distribution (fig. 7). In fig. 8 the influence on potential risk probability index k, which is the most essential at k < 2÷4 has been shown.

clip_image306[4]

clip_image312[6]

Fig. 7 Total risk probability dependence of the root- mean- square deviation (clip_image042[12]= 5, k=3).

It makes also sense to investigate the influence of the measurement instrumental error clip_image315[4]on the results of the calculations in the limits of accuracy rating chosen: clip_image317[4]. According to the calculations if clip_image319[4] decrease up to 0,5…1 µm the relative probabilities P and clip_image321[4] differ little from each other.

clip_image328[4]

Fig. 8 Total risk probability

dependence of the k parameter (clip_image042[13]=5).

clip_image334[4]

Fig. 9 Total risk probability dependence of accuracy rating (k = 3).

The dependence of the obtained potential risk probabilities of the unit accuracy rating is shown in fig. 9. Going to higher accuracy rating (clip_image300[10] decrease) followed by the tolerance field narrowing does not at all mean that the potential risk probability must go down: the unit conjugate components critical states probabilities at least for the fixed tolerance field boundaries go up (the denominators of the appropriate equation fractions decrease). The Le Chatelier – Brown (1884) approach may be arranged for the problem specified. That means that the unit accuracy rating increase involving the limiting parameters boundaries narrowing and thus providing its reliability in service causes the system state under which the stimulated processes lead to the system reliability decrease.

However the result gained does not actually mean that the accuracy rating increase necessarily lead to the object reliability decrease which is determined not only by the considered criteria expressed through the laws of probability. It may be explained by pure inner conflicts, unique to each technical object: for the units with high accuracy rating one can expect conjugate components critical states events more often than for the units with wide tolerance fields.

The example considered shows that the offered approach to the problem of technical safety can be used for the evaluation of the anthropogenic catastrophe risk for the objects of different purposes. At the same time, it is quiet clear that in real conditions the concept of independent events does not fully cover the processes reviewed. In many modern overloaded technical objects together with mechanical processes other processes such as thermodynamic, electro dynamic, chemical and other ones are taking place. Their interrelated running makes the existing private evaluation strength methods doubtful and especially if it concerns the life of such objects. One of the examples of such objects as the least reliable from the point of view of reliability are bearing between two rotors for gas-turbine engines operating between two quickly rotatable vibrating rotors surrounded by close open flame separated by a rather thin current-conducting low-magnetic sheaths and air-gas films. Under these conditions the bearing destruction event is as a rule random, sudden and quick. The physics of the process is not clear and the methods for the complex evaluation of the influence of different by its nature factors do not exist (fig. 10).

Conclusion

clip_image337[8] It has been shown that at the statistical parameters dispersion defining a technical object operation within the limits not exceeding the tolerance field of the relevant accuracy rating under normal operating conditions and manufacturing and staff error absence, the object as a whole with the certain event probability appear to be in the group of risk without regard to operating qualities, its state and environment. The reasons for the event indicated are referred to the inner nature and related to only final components or units manufacturing accuracy, their assembling and mounting as well as natural parameters and characteristic dispersion. During the object operation these parameters may go far beyond the tolerance field that will add to failure probability. To avoid these events it makes sense to carry out parameter control of the components and units that are less reliable in number of failures at regular intervals.

clip_image337[9]A change to higher accuracy rating involving tolerance field narrowing at least for their fixed boundaries under normal probability density distribution may lead to unit probability potential risk rise, the reason of which is the increase of the unit conjugate component critical state probabilities.

clip_image337[10] The offered statistical approach to the consideration of technical object units and components safe operation has made it possible to specify two criteria for anthropogenic risk severity evaluation for the object under operation:

-limiting probability dependent on parameter measurement and evaluation errors;

-total risk probability independent of these errors at least for the fixed tolerance field boundaries.

Fig. 10 To the definition of the air engine bearing destruction mechanisms.

image

The calculation of these values allows evaluating the potential risk probability for both separate units and the object as a whole. The failure forecast will make it possible to provide certain safety measures and particularly to create emergency protection means based on innovative technologies to provide the unit with global technical safety. It makes sense further to consider a more detailed description of object components and units interconnection based on hypotheses concept, multivariate probability distributions, mathematical statistics modern methods as well as other approaches.

References

[1] Лозовский В.Н., Бондал Г.В., Каксис А.О., Колтунов А.Е. Диагностика авиационных двигателей.clip_image341[20]Москва: Машиностроение, 1988.

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