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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 22. Выпуск 1.
УДК 517 DOI 10.22405/2226-8383-2021-22-1-92-104
Стохастический анализ механической системы на предмет ее надежности с различными услугами по ремонту
Дж. Бхатти, М. К. Каккар, М. Каур, Дипика, П. Кханна
Джасдев Бхатти — доцент, Институт инженерии и технологий Университета Читкары, Университет Читкары (Пенджаб, Индия). e-mail: jasdev. bhattiMchitkara. edu. in
Мохит Кумар Каккар — профессор, Институт инженерии и технологий Университета Читкары, Университет Читкары (Пенджаб, Индия). e-mail: mohit.kakkar@chitkara.edu.in
Манприт Каур — доцент, Институт инженерии и технологий Университета Читкары, Университет Читкары (Пенджаб, Индия). e-mail: manpreet.kaur@chitkara.edu. in
Дипика — доцент, Институт инженерии и технологий Университета Читкары, Университет Читкары (Пенджаб, Индия). e-mail: deepika.goyaWchitkara.edu. in
Панкадж Кханна — Институт инженерии и технологий Университета Читкары, Университет Читкары (Пенджаб, Индия). e-mail: pankaj.khanna@chitkara.edu.in
Аннотация
Надежность увеличивает свою ценность в развитии механического и промышленного мира за счет включения механизма ремонта, доступности и возможности изготовления машин с различной рабочей мощностью в любых условиях. Настоящая статья представляет собой инициативу, предпринятую с механической системой, работающей с единым сервером ремонта для различного характера отказов и услуг. Стратегия пассивной резервной машины используется для поддержания надежности системы на удовлетворительном уровне. Процесс проверки включен для фильтрации машин в зависимости от их неисправности или уровня ремонтных услуг. Вычисленные числовые и графические данные оказались полезными для выяснения поведения прибыли и надежности при увеличении / уменьшении скорости механизма ремонта и интенсивности отказов. Политика предпочтений была инициирована для регулярных сбоев или сбоев, требующих обычных затрат на обслуживание и периода времени, чем основные, чтобы избежать времени ожидания для обычного клиента.
Ключевые слова: Моделирование надежности, геометрическое распределение, доступность системы, отказ, проверка, стоимость обслуживания и функция прибыли.
Библиография: 30 названий. Для цитирования:
Дж. Бхатти, М.К. Каккар, М. Каур, Дипика, П. Кханна. Стохастический анализ механической системы на предмет ее надежности с различными услугами по ремонту // Чебышевский сборник, 2021, т. 22, вып. 1, с. 92-104.
CHEBYSHEVSKII SBORNIK Vol. 22. No. 1.
UDC 517 DOI 10.22405/2226-8383-2021-22-1-92-104
Stochastic analysis to mechanical system to its reliability with
varrying repairing services
J. Bhatti, M. K. Kakkar, M. Kaur, Deepika, P. Khanna
Jasdev Bhatti — Associate Professor, Chitkara University Institute of Engineering and Technology, Chitkara University (Punjab, India). e-mail: jasdev. bhatti®chitkara. edu. in
Mohit Kumar Kakkar — Professor, Chitkara University Institute of Engineering and Technology, Chitkara University (Punjab, India). e-mail: mohit.kakkar@chitkara.edu.in
Manpreet Kaur — Assistant Professor, Chitkara University Institute of Engineering and Technology, Chitkara University (Punjab, India). e-mail: manpreet.kaur@chitkara.edu. in
Deepika — Assistant Professor, Chitkara University Institute of Engineering and Technology, Chitkara University (Punjab, India). e-mail: deepika.goyal@chitkara.edu. in
Pankaj Khanna — Chitkara University Institute of Engineering and Technology, Chitkara University (Punjab, India). e-mail: pankaj.khanna@chitkara.edu.in
Abstract
Reliability is enhancing its value in the advancement of mechanical and industrial world by incorporating the repair mechanism, availability and manufacturing possibility of machines with varying working capacity in all conditions. The present paper is an initiative taken with a mechanical system operating with single repair server facility for varying nature of failures and services. Passive standby machine strategy is adopted for maintaining reliability at a gratified level in the system. The inspection process is included for filtering the machines according to its failure or to the level of repair services. The computed numerical and graphical data is proved to be beneficial for clarifying the profit and reliability behaviour with increasing/decreasing rate of repair mechanism and failure rate. The preference policy has been initiated for regular failure or to the failure requiring normal servicing charges and time period than major ones to avoid the waiting time for normal customer.
Keywords: Reliability modelling, geometric distribution, system availability, failure, inspection, maintenance cost and profit function.
Bibliography: 30 titles. For citation:
J. Bhatti, M. K. Kakkar, M. Kaur, Deepika, P. Khanna, 2021, "Stochastic analysis to mechanical system to its reliability with varrying repairing services", Chebyshevskii sbornik, vol. 22, no. 1, pp. 92-104.
1. Introduction
In the real world full of industrial advancement and variety of machines, reliability is adding its major importance for their evaluation and analysis such as manufacturing and availability of machines and system, their repairing mechanism and working capacity in all environmental situations and many more. For keeping the technology more reliable there is always a new initiative or substitution policy be kept ready for balancing the possibilities of damage to technology. Along with this keeping standby units in active or passive form is also an alternative method that usually been adopted by many industries for maintaining reliability of system in satisfied level. It has been observed in several systems that rather than placing direct repair server to the failed unit, the inspection process is included for filtering the machines according to its failure or to the level of repair services. The present paper is initially contributed to analysis of the system based on above strategies.
From past many years there has been lots of initiative been taken in reliability evaluation of industrial models for its improvement. In 2008, Bhardwaj et al. fl, 2] had stochastically examined distinct repair and failure mechanism under discrete distribution for the redundant system. In 2009 evaluation of industrial system with linear first order differential equations was initiated by Haggag et al. [3, 4]. Rizwan et al. [5] also investigated hot standby PLC system for enhancing its reliability. An computer based working system with replacement preference to S/W over H/W replacement subject to MOT and MRT was examined by Kumar, A. [6]. In 2014, Singh et al. [7] had stochastically studied a gas turbine plant processing with one gas and steam turbine to an industrial system i.e. power generating system. Malhotra. R et. al. [8, 9], had also initiated with examining system with varying repair demands. Bhatti and Kakkar et al [10, 11, 12, 13, 14] had also initiated together in evaluating different real life models following correlation relation concept and using geometric distributions.
In 2016, Hua et al. [15, 16] had spatially distributed units in his research with major challenge of assessing systems with involving unit degradation paths. Pervaiz et al [17] used Boolean function for assessment of paper plant industry and S.Z. Taj, et al [18] assessed the cable plant subsystem by framing probabilistic modelling. In 2017 N. Adlakha [19] had investigated mechanical system having assembling and activation time for cold standby unit before being into an operative state. Cui et al [20, 21] and Endharta et al. [22] in 2018 had applied F and G balanced systems under Markov processes for k-out-of-n systems reliability. Chen W-L. et al. [23] enhanced his study for retrial machine repair systems with operating units in 'M' numbers out of which 'W are to be in warm standby mode and a single recovery policy for server breakdown. In 2019, S. Bhardwaj [24] studied neural network prediction model and Dong Q.L. et al. [25] used Bivariate Wiener processes by emerging stochastic degradation system for two-stage failure process. Recently, new balanced mechanisms for examining balanced systems and common cause failures had been initiated by Wu H. et al. [26], Jia H.P. [27] and Fang et al. [28]. Bhatti and Kakkar et al. [29, 30] also enhance his study under reliability with active or passive standby systems with common failure.
In all, all researchers had contributed to their best for enhancing the reliability of mechanical system and machines. The present paper is also one step forward been taken with initiative as an inspection process for filtering the machines according to its failure or to the level of repair service but with some preferences. The preference is been given to regular or to the failure required normal servicing charges and time period to avoid the waiting time for normal customer. The two machines or customer 'X^d 'Y' having their individual mechanical problems. But due to the possibility of only normal/regular failure to machine 'Y' it has been given preference to machine 'X' for being repaired and free from inspection procedure. Possible states of the system under operative and failed states are reflected through transition model Figure 1 and Table 1.
Up States
So = (Хо ,Yo ), St = (XT ,Yo ), S2 = (Xri ,Yo ), S3 = (ХГ2 ,Yo ), S7 = (X0 ,Yr2 ).
Phc. 1: Transition Model
Таблица 1: Nomenclature
^0 ,Yo Operative behaviour of unit 'X' and 'Y' .
Xi Inspection behaviour of failed unit 'X'.
Xri /Xriw Special failure repaired service or waiting rate to unit 'X'.
Xr2/Xr2W / Yr2 Regular services or waiting rate to unit 'X' and fYf.
Pi Probability value for 'X' to fall in failure mode
p\ /P2 Probabilityc value for 'X' to get inspected.
P3 Probability value to fYf for falling in failure mode with regular repair
n/r2 Probability repair value to C'X' and 'Y' for being repaired (special/regular) successfully.
Down States
S4 = (Xriw ,YT2), S5 = (Xr2w ,YT2), Sq = (X/ ,YT2). 2. Transition Probabilities
Using the transition diagram shown in Figure 1, the steady state transition probabilities from state Si to Sjcan be calculated by applying:
Pii = lim Qij
J t^^ J
where Qij depicts the 'cumulative density function' from first regenerative state 'i' to second state 'j'. The evaluated transition probabilities are as follows:
Poi P06
P07
Pi2
Pi3 Pi4 Pi5
Pi6
PiQ3 1 - qiQ3
PiP3 1 - qiQ3
qiP3 1 - qiQ3
P2Q3 1 - Q2 Q3
P2Q3 1 - Q2 Q3
P2P3 1 - Q2 Q3
2
P2P3 1 - Q2 Q3
Q2P3
1 - Q2 Q3
P20 =
P24 =
P27 =
P30 =
P35 =
P37 =
riQ3
1 - Siq3
SiP3
1 - SiQ3
riP3
1 - SiQ3
T2l3
1 - S2Q3
S2P3 1 - S2Q3
T2P3
1 - S2q3
Pt
62
P42 = P5 3(t) =
T2
1 - S2
Pei =
Q2^2
Pe3 =
Pt
64 =
Pes
Pro Pji Pre
P2r2 1 - Q2S2
p2t2
1 - Q2S2
P2S2
1 - Q2S2 2
P2s2
1 - Q2S2
T2li 1 - S2Qi
T2Pi 1 - S2Qi
S2Pi 1 - S2Qi
1 - Q2S2
2.1. Mean Sojourn Times
By mentioning sojourn time in state Si(i = 0 - 9) by svmbol the value of mean sojourn time for state Si is calculated as:
Vo =
1
1 - QiQ3 ' №4 = № =
Vi =
1
V2 =
1
V3 =
1
1 - Q2Q3 1 - SiQ3 1 - S2Q3
1 1 1
1 - S2
Ve =
1 - Q2S2 '
V7 =
1 - S2Qi '
3. Mean Time to System Failure (MTSF)
To calculate MTSF of the proposed system, the absorbing states is taken to be the failure ones. Then, reliability analysis Ri at time't' is obtained by solving the following equation:
Ro = Zo + qoi© Ri + qor© Rr
Ri = Zi + qi2© R2 + qi3© R3 R2 = Z2 + q2o© Ro + q27© Ri R3 = Z3 + q3o© Ro + q37© Ri
Rr = Z7 + qro© Ro + qri© Ri
By solving above equations, we obtain
MTSF = - 1 = £
h^i Di(h) Di
where
Ni = /lo [1 - Pri(Pi2 + Pi3)] + »i[Poi + Por Pri] + IJ.2(Pi2 + Pi3)[Poi + PorPri]
+MPi2 + Pi3)[Por + Po2?27 ] Di = 1 - (Pi2 + Pi3)P2o(Poi + PorPri) + P2r((Pri + ProPoi) - PorPro.
4. Availability and Maintenance Analysis of the System
Using probabilistic argument and through reliability models design as figure 1, the relations related to availability and maintenance analysis of the system are obtained as:
Xo = Zo + qoi© Xi + +qoe©Xe + +qor©Xr. Xi = Zi + qi2© X2 + qi3© X3 + qu© X4 + qi5© X5 + qm© Xf. X2 = Z2 + q2o© Xo + q24© X4 + q2r© Xr. X3 = Z3 + q3o© Xo + q35© X5 + ^r© Xr. X4 = Z4 + q42© X2. X5 = Z5 + q5 3© X3. Xe = Ze + qei©Xi + qe2©X2 + qe3©X:i + qe4© X4 + qe5©X5. Xr = Zr + qro©Xo + qri©Xi + qre© Xe. By solving above equations, we get the value of reliability parameters as: Availability:
N2(1)
Ao = -7^, Zi = 0 fori = 4, 5, 6. Busy schedule of Inspection:
Bo = - , ^ = 0 fori = 0, 2, 3, 4, 5, 7. Busy schedule of Repairman rr.
B'o = -^¡^, Zi = 0 fori = 0,1, 3, 4, 5, 6, 7. Busy schedule of Repairman r2:
(1)
B'0 = -, zt = 0 fori = 0,1, 2.
where
N2(1) = Ml - РгвРвгШо + P27P70] + Ы(1 - P24)[Poi + РовРвг + Po7(P7i + P7ePei)] +P27(PoeP7i - PoiP7e)(1 - Pei)] + (1 - PiePei)[(^2 + + Mil - P07P70 + P27 + P20P07)]
N3(1) = (1 - P24)[M[Poi + PoePei + P07(Pri + P7ePei)] + P27(PoeP7i - PoiP76) (1 - Pei)) + M[Poe + PoiPie + Po7(P7e + PriPie)] + P27(PoiP7e - PoePfi)(1 - Pei))]
N4(1) = (1 - PiePei)M1 - P07P70)
N5(1) = (1 - PiePei)^s(1 - P07P70) + №[(Poi + P07P71 )[Pi4 + P15 + P16(P64 + P65) +P2i(Pi2 + Pis + Piq(PQ2 + Pes))] + P06 + P07P76 [P&4 + P65 + Pel (P14 + P15) +P24(P62 + Pes + Pei(Pi2 + Pis))] + ?27(PoeP7i - P01 P7e)[(Pe4 + Pen)(1 - Pie) (P14 + Pi5)(1 - Pei)]] + Ve([Poe + PoiPie + Po7(?7e + PriPie)] +P27(PoiP7e - PoeP7i)(1 - Pei)) + »7(1 - P07P70 + P27 + P20P07)
D2(1) = -po(1 - PiePei)[P20 + P27P70] - ^i[(1 - P2A)[POI + PoePei + Po7(P7i + P7ePei)] +P27(PoeP7i - PoiPre)(1 - Pei)] - (1 - PiePei)[(^2 + + - P07P70 + P27 + P20P07)] -Vi[(Poi + Po7P7i)[Pi4 + Pi5 + Pie(Pe4 + Pen) + P2i(Pi2 + Pi3 + Pie(Pe2 + Pes))] +Poe + Po7P7e[Pe4 + Pen + Pei(Pu + P15) + P2i(Pe2 + Pes + Pei(Pi2 + Pis))] +P27(PoeP7i - PoiP7e)[(Pe4 + Pen)(1 - Pie)(Pu + Pis)(1 - Pei)]] -Ve([Poe + PoiPie + Po7(P7e + PriPie)] + P27(PoiP7e - PoePn)(1 - Pei))]
5. Conclusion
The total profit of system to its steady state will be calculated by using:
P = C1A0 - C2B0 - Cs[B0 + B'0]
where C^. be the per unit up time revenue by the system. C2,Cs: be the per unit down time expenditure on the system. As per the data analysis, the performance of profit function was analyzed through having some fixed parameters C-^, C2 and Cs &s C0 = 1500, C1 = 300, C2 = 350 and P2 = 0.6.
Table 2 and Figure 2 reflects that the profit function will decrease as the failure rate p1 increases from 0.1 to 0.9 for certain r^ r2, ps values. Whereas Table 3 and Figure 3 reflects its opposite behaviour for certain p^ ps with increasing r^ r2. Hence, with the help of numerical and graphical analysis it has been proved that the profit function increases with increasing repair and decreasing failure rate. In other words, the research paper will verify its objectives of benefiting the industries by developing new techniques using prescribed repairng techniques for different failure.
6. Acknowledgement
The authors are grateful to the Editor, Co-Editor and Reviewers for their constructive suggestions.
Таблица 2: Reliability parameters w.r.t repair r\,r2, Failure Rate p3
Repair, Failure Rate MTSF Ac Bc B'c + B'0 Profit
13.34816 0.439386 0.071212 0.88796 326.9289262
8.26087 0.382619 0.076136 0.915303 230.7321359
6.718898 0.363914 0.076619 0.926769 198.5157803
Г1 = 0.06, 6.024452 0.355656 0.076143 0.93332 183.9797496
Г2 = 0.04, 5.650493 0.351492 0.075428 0.937648 176.4327009
P3 = 0.2 5.426997 0.349245 0.074682 0.940759 172.1972711
5.283912 0.347999 0.073974 0.943121 169.7147604
5.187718 0.347315 0.073322 0.944984 168.231833
11.95662 0.730824 0.130052 0.710412 808.5761745
6.44206 0.628585 0.179353 0.769325 619.8084424
4.611465 0.57487 0.205185 0.800348 520.627402
n = 0.3 3.701681 0.541837 0.221014 0.819481 459.6323401
Г2 = 0.1, 3.160083 0.519522 0.231661 0.832451 418.4274372
P3 = 0.5 2.802469 0.503476 0.239279 0.841815 388.7952611
2.549898 0.491411 0.244974 0.84889 366.5126824
2.362903 0.48203 0.249373 0.854418 349.1869919
11.17526 0.867294 0.131712 0.594928 1053.202165
5.860592 0.795704 0.206111 0.647095 905.2398548
4.068428 0.752177 0.254561 0.679733 813.9911542
r 1 = 0.65 3.154762 0.724026 0.289165 0.701777 753.6677721
Г2 = 0.15, 2.590463 0.70537 0.315597 0.717387 712.2912115
P3 = 0.8 2.198814 0.693146 0.336889 0.728752 683.589792
1.903705 0.685638 0.35483 0.737123 664.0154099
1.666667 0.681853 0.370562 0.743258 651.4703642
Таблица 3: Reliability parameters w.r.t Failure rate p\, p3
Failure Rate pi,рз MTSF Ac Bc B'c + B'0 Profit
5.187718 0.347315 0.073322 0.944984 168.231833
5.346386 0.525387 0.147298 0.854984 444.6463936
5.527778 0.631806 0.207931 0.768851 616.2317649
Pi = 0.8 5.728331 0.703059 0.256255 0.693314 735.0530471
Рз = 0.2 5.946468 0.754788 0.295017 0.628232 823.7964199
6.181694 0.794554 0.326554 0.57213 893.6185798
6.434177 0.826391 0.352603 0.523485 950.586493
6.704545 0.852641 0.374421 0.481001 998.2854495
2.509642 0.170905 0.09413 0.984894 -116.5943514
2.604938 0.303493 0.156777 0.947438 76.602372
2.702541 0.412269 0.203008 0.897451 243.3935662
Pi = 0.6 2.802469 0.503476 0.239279 0.841815 388.7952611
Рз = 0.5 2.904762 0.580645 0.268817 0.784946 515.5913978
3.009473 0.646232 0.293449 0.729478 625.9958088
3.116667 0.702148 0.314321 0.67686 722.0244016
3.226415 0.749953 0.332213 0.627792 805.5379368
2.691851 0.120397 0.11433 0.994673 -201.8382064
2.765625 0.234297 0.177468 0.975075 -43.07078764
2.835925 0.341 0.215672 0.941976 117.1070403
Pi = 0.4 2.903458 0.438468 0.240567 0.899243 270.7977208
Рз = 0.8 2.96875 0.525427 0.257924 0.850976 412.9208633
3.032202 0.601609 0.270802 0.800524 540.990066
3.094124 0.667504 0.280893 0.750277 654.3913683
3.154762 0.724026 0.289165 0.701777 753.6677721
Рис. 2: Profit vs Failure rate
Рис. 3: Profit vs Repair rate
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Получено 26.04.2020 г.
Принято в печать 21.02.2021 г.
