PARTICULAR RELIABILITY CHARACTERISTICS OF TWO ELEMENT PARALLEL TECHNICAL (MECHATRONIC) SYSTEMS

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in

Production Engineering

2012, No 3 (7), pp 3‐8

Abstract:

The paper characterizes the basic distribu ons of failure of elements that cons tute the technical (mechatronical) sys‐ tems: exponen al, Weibull, normal, log‐normal distribu on. The descrip on of two‐element parallel technical systems with reliability characteris cs has been made. Specific cases are studied where up me of the technical system elements have exponen al, Weibull, normal, log‐normal distribu ons and where the system consists of two parts with parallel reliability structure and of diﬀerent types of distribu ons of elements up mes. The order of elements in the analysis does not ma er. The relevant characteris cs of reliability for a system with two parallel elements are presented: up me distribu on func on of the system, system reliability, up me probability density of the system, the system failure intensity.

PARTICULAR

RELIABILITY

CHARACTERISTICS

OF

TWO

‐

ELEMENT

PARALLEL

TECHNICAL

(MECHATRONIC)

SYSTEMS

INTRODUCTION

The combina on of mechanical, electrical, electronic, pneuma c elements into one opera ng technical system has been called in recent years a mechatronic device (system). Each of these elements has a specific character of durability, damage suscep bility and reliability. Reliability of devices and mechatronic device components is described by mathema cal models – distribu ons of random varia‐ bles, and par cularly, the characteris cs of reliability, which directly aﬀect the reliability characteris cs of mechatronic devices, into which they are included.

The most commonly used mathema cal models of stu‐ dying the reliability of technical devices are distribu ons of random variables: exponen al, Weibull, normal, logarithmo ‐normal, normal truncated at zero, gamma, binomial (Bernoulli), Poisson, hypergeometric, geometric, and pro‐ cesses: Poisson, normal, Markov and semi‐Markov. Distri‐ bu ons are probabilis c models, and processes are stocha‐ s c models [1, 2, 3, 4, 5].

In complex, responsible technical and mechatronic sys‐ tems (a combina on of mechanical, electrical, electronic, pneuma c opera ng components into a coherent technical system) reserving components are o en used. The simplest reserving consists in a parallel inclusion of the same compo‐ nent into the system, which replaces the damaged compo‐

nent at the me of the primary failure. In more complex technical systems, the func ons of the defec ve part may be taken by a different system component, with different opera ng characteris cs, and thus with different reliability characteris cs. Each of the components may have a diffe‐ rent durability, damage and reliability character, which is described by the mathema cal models ‐ sta s cal distribu‐

ons, and par cularly the characteris cs of reliability, which directly aﬀect the reliability characteris cs of a tech‐ nical system they cons tute.

The most commonly used mathema cal models of stu‐ dying the reliability of technical devices are distribu ons of random variables: exponen al, Weibull, normal, logarithmo ‐normal, normal truncated at zero, gamma, binomial (Bernoulli), Poisson, hypergeometric, geometric, and pro‐ cesses: Poisson, normal , Markov and semi‐Markov. Distri‐ bu ons are probabilis c models, and processes are stocha‐ s c models [1, 2, 3, 4, 5]. In further analysis the simplest and the most common models are assumed: exponen al, Weibull, normal, or logarytmo‐normal.

Many years of field tests and databases available on failure of components and technical equipment indicate that specific distribu ons for their reliability characteris cs can be a ributed to specific components and devices as well as to the typical types of damage [6, 7, 8, 9] (Table 1). Zbigniew MATUSZAK

Mari me University of Szczecin

Key words: technical (mechatronical) system, distribu ons of elements failure, distribu ons of failures of two‐element

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Z. MATUSZAK — Szczególne charakterystyki niezwodnościowe dwuelementowych równoległych systemów technicznych

Up me of the i‐th element is a random variable with a distribu on defined by the following characteris cs:

reliability of the component

probability density of the component up me

failure intensity of the component

the expected up me of the component

In the next part the analysis of four distribu ons is given as an example, and then selected characteris cs of reliability composi ons of elements of mechatronic devices are presented. In the given examples, the presenta on is limited to composi ons of distribu ons for two elements.

CHARACTERISTICS OF THE ANALYZED RELIABILITY DISTRI‐

BUTIONS

In the work below [10], par cular cases of distribu on composi ons are analysed.

Exponentialdistribution

Exponen al distribu on is useful for tes ng the reliabili‐ ty of such devices, which are the result of impact of shock loads (so‐called discrete s muli). Exponen al distribu on can be used to test the reliability of equipment and compo‐ nents if:

 changes to the technical condi on and the resul ng damage is irreversible,

 the level of resistance (wear resistance) is constant, which means no damage caused by aging (derived from cumula ve extor on),

 damage is the result of external or internal random shock interac ons (discrete s muli).

Up me characteris cs (i=1,2,...,n) of a component are as follows:

 reliability of the component

 failure intensity of the component

Weibulldistribution

Weibull distribu on describes the me of normal opera‐ on of such devices, in which damage is independent, any damage causes loss of equipment up me, each unit con‐ sists of a suﬃciently large number of homogeneous compo‐ nents.

The element has a Weibull distribu on with the param‐ eters (αi,βi)(i=1,2,…,n), when its characteris cs take the

form:

A special case of Weibull distribu on is a Rayleigh distri‐ bu on in which the parameter αi=2.

Normaldistribution

The normal distribu on is a model of reliability of any technical object in which there are damages resul ng from the aging process, including wear. This is to be used if a described random variable depends on a number of phe‐ nomena and causes, none of which can be considered dom‐ inant.

The component τi up me has a normal distribu on

when the probability density has the form

Table 1.

Func ons of failure intensity of selected components and devices as well as typical types of damage

Component, device Distribu on of func ons

of failure intensity Type of damage

Distribu on of func ons of

failure intensity

small rubber parts such as seals, diaphragms Weibull catastrophic exponen al

components and equipment damaged by external

factors exponen al ageing Weibull, gamma

electronic elements exponen al very slow wear exponen al

devices with a dominant number of moving parts Weibull rapid wear normal, logarithmo‐normal

corrosive wear gamma



t



F(t),t ,i , ,...,n, P ) t ( Ri  i 1 i 0 12 (1) , n ,..., , i , t , dt ) t ( dF ) t ( f i i  0 12 (2)





,t ,i , ,...,n, ) t ( R ) t ( f ) t ( F ) t ( f ) t ( R ln dt d ) t ( i i i i i i 0 12 1        ₍₃₎

 

R(t)dt,i , ,...,n. E T_i _i _i 12 0   



  ₍₄₎ , t, e ) t ( R t i i ₀   (5) , t, e ) t ( f t i i i ₀ _  (6) . const ) t ( i i    (7)

 

. i E i T i    1  ₍₈₎



t



,t , exp ) t ( R i i i   0   (9)



t



,t , exp t ) t ( f i i i i i i 0 1 _ _ __  _  (10) , t , t ) t ( i i i i 0 1  __   (11)

 

,t . E T _i i i i i 0 1 1 _ 1                ₍₁₂₎

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Z. MATUSZAK — Szczególne charakterystyki niezwodnościowe dwuelementowych równoległych systemów technicznych

and its distribu on func on

where Ti is the expected up me of a component and σi2 is

its variance.

The normal distribu on is defined for all tR, while the random variable τi being the up me of the element takes

only nonnega ve values. One can put up with such an inad‐ equacy of the model, where the probabili es P{τi<0} are

negligibly small, no larger than measurement errors. In order to write simpler and more convenient up me characteris cs of the element with normal distribu on the characteris cs of the normal distribu on N (0,1) probability density has been used:

and the distribu on func on

One can then write the characteris cs of the system components in the form:

In case one cannot accept the assump on that the probabili es P{τi<0} are negligibly small, one should use the

truncated normal distribu on, at which me the fault‐free opera on of the component takes only nonnega ve values τi≥0

In this case, the reliability characteris cs of elements has the form:

Logarithmo‐normaldistribution

Logarithmo‐normal distribu on in reliability theory on the basis of the empirical research characterizes metal components safe fa gue life, the strength of metals sub‐ jected to prolonged opera on stress, as well as electronic components up me.

The component τi up me has logarithmo‐normal distri‐

bu on when the random variable Y=lnτi is normally distrib‐

uted withthe parameters N(Ti,σi) . Using the probability

density and distribu on func on of the normal distribu on N(0,1) component reliability characteris cs of logaithmo‐ normal distribu on can be wri en as:

CHARACTERISTICS OF THE DISTRIBUTIONS OF TWO‐

ELEMENT PARALLEL TECHNICAL SYSTEMS

Here the analysis of n‐element system with a parallel reliability structure is presented. The same designa ons, which have been defined in the previous sec on, will be used. Since the elements are independent, and the system





, t , T t exp ) t ( f i i i i              2 2 2 2 1    (13) , t , dx ) T x ( exp ) t ( F t i i i i           

_

  2 2 2 2 1    (14) , R t , t exp ) t ( f _         2 2 1 2 0  (15)



           t exp x dx,t R. ) t ( F 2 2 1 2 0  (16) , R t , T t f ) t ( f i i i i         



0 1 (17)          i i i T t F ) t ( R  0 1 ₍₁₈₎                         i i i i i i T t F T t f ) t (     0 0 1 (19) , t , T F T t f T F T t f ) t ( f i i i i i i i i i i i 0 1 1 0 0 0 0                                      (20)                                                       i i i i i i i i i i i i i i i T F t T F T F T t F T F T F T t F ) t ( R        0 0 0 0 0 0 0 1 1 1 ₍₂₁₎ . t , t T F T t f T t F T t f ) t ( i i i i i i i i i i i 0 1 0 0 0 0                                                ₍₂₂₎





, t , T t ln f t T t ln exp t ) t ( f i i i i i i i 0 1 2 2 1 0 2 2                 _        (23) , t ln T F T t ln F ) t ( R i i i i i                    0 0 1 ₍₂₄₎

 

, F f ) t ( i t ln i T i i T t ln i t i     _   0 0 2 1 (25)

 



2



2 i i i expT E   (26)

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Z. MATUSZAK — Szczególne charakterystyki niezwodnościowe dwuelementowych równoległych systemów technicznych is working flawlessly un l the damage of all the compo‐

nents:

 the up me distribu on is

 the system reliability

 probability density of the system up me

Failure intensity of the system has the form

The expected up me of the system with parallel reliability structure is

The equa on (31) shows that it is rarely possible to cal‐ culate the expected up me of the system in the explicit form. Even for simple distribu ons the expected up me of the system is quite complicated. In a par cular case, all components of the system have the same up me distribu‐

ons. This happens usually when several components per‐ form one and the same func on.

To perform it one component is suﬃcient, therefore, the remaining elements are hot reserving and in that case o en Fi(t)=F1(t) for i=1,2,…,n. . Then for a system with a

parallel structure consis ng of iden cal components relia‐ bility characteris cs have the form:

the distribu on func on of the system up me

the system reliability

probability density of the system up me

failure intensity of the system

the expected up me of the system

The following reliability characteris cs of systems with a parallel two‐element structure and diﬀerent types of com‐ ponents up me distribu ons have been determined. The order of components is arbitrary, it does not ma er for the dependence.

The first component has the up me of the exponen al distribu on with the parameter λ, and the second – of the Weibull distribu on with the parameters (α,β). Reliability characteris cs of the system take the form:

The first component has the up me of the exponen al distribu on with the parameter λ, and the second – of the Weibull distribu on with the parameters (T,σ). Reliability characteris cs of the system take the form:

the distribu on func on of the system up me the system reliability

The first component has the up me of the exponen al distribu on with the parameter λ, and the second – of the logarithmo‐normal distribu on with the parameters (T,σ). Reliability characteris cs of the system take the form:

  



_

        n i i n t F(t), ,..., t , t P t P ) t ( F 1 2 1     ₍₂₇₎



   n i i(t), F ) t ( R 1 1 (28)

 

   n i i j j j i(t) F(t), f ) t ( f 1 1 (29)







      _n i i n i i j j j i ) t ( F ) t ( F ) t ( f ) t ( 1 1 1 1  (30)

 

_





        0 1 1 F(t) dt E T n i i  (31) , ) t ( F ) t ( F  1n (32)



R(t)



, ) t ( F ) t ( R 1 1n 11 1 n (33) , ) t ( F ) t ( nf ) t ( f 1 1n1   (34) , ) t ( F ) t ( F ) t ( nf ) t ( _n n 1 1 1 1 1    ₍₃₅₎

 



_







 0 1 1 F (t)dt E T  n (36)



e





e



t, , ) t ( F t t 0 1 1      ₍₃₇₎



e





e



e e e , ) t ( R    t  t  t  t  (tt) 1 1 1 (38)











t



e , e t e e e t e e ) t ( f ) t t ( t t t t t t                                          1 1 1 1 1 (39)





_, e e e e t e t e ) t ( ) t t ( t t ) t t ( t t               _ _ _                   1 1 ₍₄₀₎





 

 t t T F e ) t ( F 1  ₀  (41)



e



F( ), ) t ( R t t T       1 1 ₀ (42)



e



f ( ), ) ( F e ) t ( f t tT t tT     



  _ _    0 1 0 1 (43)







e



F ( ) ) ( f e ) ( F e ) t ( T t t T t t T t t            _ _       0 0 1 0 1 1 1 (44)



e



F( )t, , ) t ( F t lntT 0 1 ₀       (45)

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Z. MATUSZAK — Szczególne charakterystyki niezwodnościowe dwuelementowych równoległych systemów technicznych the system reliability

The first component has the up me of the Weibull dis‐ tribu on with the parameter (α,β), and the second – of the normal distribu on with the parameters (T,σ). Reliability characteris cs of the system take the form:

The first component has the up me of the Weibull dis‐ tribu on with the parameter (α,β), and the second – of the logarithmo‐normal distribu on with the parameters (T,σ). Reliability characteris cs of the system take the form:

The first component has the up me of the normal dis‐ tribu on with the parameter (T1,σ1), and the second – of

the logarithmo‐normal distribu on with the parameters (T2,σ2). Reliability characteris cs of the system take the

form:

CONCLUDING REMARKS

In technical systems, par cularly in mechatronic sys‐ tems, reserving the system components is popular. In the presented material specific examples of components re‐ serving are shown, when the components are not reserved by the objects of the same type, with the same characteris‐

cs of reliability, but when the func onal reserving by a diﬀerent component with a diﬀerent reliability characteris‐

cs is used. In case of reserving by the same components with the same failure distribu ons, the analysis of reliability characteris cs is rela vely simple and defined by the de‐ pendences (32)¸(35). In other cases it is complex.

REFERENCES

[1] Prażewska M. (ed.): Niezawodność urządzeń elektro‐ nicznych. WKiŁ. Warszawa 1987.

[2] Rausand M., Høyland A.: System Reliability Theory: Models, Sta s cal Methods, and Applica ons. Second edi on. Interscience. New Jersey: Wiley 2004.

[3] Saleh J. H., Marais K.: Reliability: How much is it worth? Beyond its es ma on or predic on, the (net) present value of reliability. Reliability Engineering and System Safety. Vol. 91, 2006, pp. 665–673.

[4] Sotskow B.S.: Niezawodność elementów i urządzeń automatyki. WNT. Warszawa 1973.

[5] Ważyńska‐Fiok K., Jaźwiński J.: Niezawodność syste‐ mów technicznych. PWN. Warszawa 1990.

[6] Matuszak Z.: Charakterystyki niezawodnościowe wie‐ loelementowych struktur mieszanych i odnawialnych. Collec on of research papers of the Bal c Associa on of Mechanical Engineering Experts No 4, Mechanical Engineering of the Bal c Region. Kaliningrad State Technical University. Kaliningrad 2004, pp. 146‐148. [7] Matuszak Z.: Charakterystyki niezawodnościowe kilku‐

elementowych systemów technicznych o strukturze szeregowej i równoległej o różnych rozkładach czasów zdatności. Collec on of research papers of the Bal c Associa on of Mechanical Engineering Experts No 4, Mechanical Engineering of the Bal c Region. Kalinin‐ grad State Technical University. Kaliningrad 2004, pp. 149‐153.



e



F( ), ) t ( R t lntT       1 1 0 (46)



e



f ( ), ) ( F e ) t ( f t lntT t T t ln t     



  _ _    0 1 0 1 (47)

















       T t ln t T t ln t t T t ln t F e ) ( f e ) ( F e ) t ( _ _          0 0 1 0 1 1 1 (48)



e



F( ),t , ) t ( F t t T 0 1 ₀        (49)



e



F ( ), ) t ( R t t T        1 1 0 (50)



e



f ( ), ) ( F e t ) t ( f t t T t t T             _ _    1 ₀ 0 1 1 (51)









 

             T t t T t t T t t F e ) ( f e ) ( F e t ) t (             0 0 1 0 1 1 1 1 (52)



e



F ( ),t , ) t ( F t lnt T 0 1 0        ₍₅₃₎



e



F ( ),t , ) t ( R t lnt T 0 1 1  0        ₍₅₄₎



e



f ( ), ) ( F e t ) t ( f t lnt T t T t ln t             _ _    0 1 0 1 1 (55)















            T t ln t T t ln t t T t ln t F e ) ( f e ) ( F e t ) t (             0 0 1 0 1 1 1 1 (56)

   

F

,

t

,

F

)

t

(

F

t T lnt₂T2

0

0 1 1 0





    (57)

   

F

,

t

,

F

)

t

(

R

1

t T lnt T

0

2 2 0 1 1 0







    (58)

   

₂2 0 1 1 0 2 1 2 2 0 1 1 0 1 1       t t T lnt T T t ln T t

f

F

f

)

t

(

f



 



  (59)

   

2 2 0 1 1 0 2 2 0 1 1 0 2 1 2 2 0 1 1 0 1 1 1 _ _      



_t_T _ln_t_T T t ln T t t T t ln T t F F f F F f ) t ( _ _        ₍₆₀₎

(6)

Z. MATUSZAK — Szczególne charakterystyki niezwodnościowe dwuelementowych równoległych systemów technicznych [8] Matuszak Z.: Selected safety models of elements and

systems in the engine room. Interna onal Scien fic Journal „Problems Of Applied Mechanics”. Georgian Commi ee of Interna onal Federa on for the Ma‐ chines and Mechanics. Tbilisi 2004, No 1(14), 2004, pp. 30‐39.

[9] Nowakowski T.: Bazy wiedzy w badaniach niezawod‐ ności maszyn. Materiały Konferencji Naukowej

"Metody doświadczalne w budowie i eksploatacji ma‐ szyn roboczych, technologicznych oraz środków trans‐ portu". Wydawnictwo Politechniki Wrocławskiej. Wro‐ cław‐Szklarska Poręba 1993, s. 219‐225.

[10] Matuszak Z.: Szczególne charakterystyki niezawodno‐ ściowe szeregowych systemów mechatronicznych. Studia i Materiały Polskiego Stowarzyszenia Zarządza‐ nia Wiedzą. Nr 45, 2011, s. 176‐190.

dr hab. inż. Zbigniew Matuszak, prof AM Mari me University of Szczecin

Faculty of Mechanical Engineering Ins tute of Ship Power Plant Opera on

ul. Wały Chrobrego 1‐2, 70‐500 Szczecin, POLAND e‐mail: [email protected]