Case Study of Cascade Reliability with weibull Distribution

Transcription

1 ISSN: ISO 900:008 Certfed Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume, Issue 6, June 0 Case Study of Cascade Relablty wth webull Dstrbuton Dr.T.Shyam Sundar Abstract Ths paper endeavors to present relablty of an n-cascade system, whose stress and strength dstrbutons are Webull. Expressons for the system relablty are obtaned when the stress and strength dstrbutons are webull. Relablty values for a cascade system wth webull dstrbuton are computed and are presented graphcally. Index Terms Cascade Relablty, Webull Dstrbuton, Stress- Strength Model, Probablty Densty Functon I. INTRODUCTION If X denotes the strength of a component and Y s the stress mposed on t. The component operates as long as Y s less than X and the Relablty of the component may therefore be defned as R c = P(X>Y) The probablty of the falure of a system depends upon the stresses and strengths of the system. If the stress exceeds the strength, the system fals whch has been consdered by Kapur and Lamberson [].The relablty of n-cascade system wth stress attenuaton has been consdered by Pandt and Srvastav [],whereas Raghava Char [3] studed the concept of cascade system relablty when both wth stress and strength follow dentcal dstrbutons. A.Rekha and T.Shyam Sundar [4] studed smlar stress attenuaton studes wth stress and strength followng Gamma and Exponental dstrbuton.rekha and Chenchu Raju [5] studed the same by consderng Raylegh dstrbuton for the strength and stress wth certan attenuaton factors. II.GENERAL MATHEMATICAL MODEL Let X,X, X 3,...X n be the strength of the component C, C, C 3,... C n are arranged n order of actvaton. All the X s are ndependently and dentcally dstrbuted random varables wth Probablty densty functon f (x ), ( =,, 3,... n). Also let Y be the stress on the frst component whch s also randomly dstrbuted wth the densty functon g(y ). If Y < X, the frst component C works and the system survves and f Y > X leads to the falure of the component C. Then the second component C n the lne takes ts place wth strength X. However, the stress Y on C wll be * * dfferent from Y. Let Y = k Y, where k = K K s the cumulatve attenuaton factor. By defnton K =. In general, f the (n-) th component fals, the n th component wth strength X n get actvated and wll be subjected to the stress. * Y n = K n Y n- = k n Y The relablty R (n) of the system to survve wth the frst (n-) components faled and the n th component actve s gven by n R(n)= Px y xn yn * * * p x k y, x k y, x3 k3 y... * * *... x k y and x k y () n n n n * * k y k y k* * n y k n y... f n dy k* n y (3) () x dx... f x dx n n gy f x dx f x dx... f xn k* n y k* y k* y k* n y fx dx fx dx n n n n gy dy k* ny R(n)= dx n (4) * * * *... n n n n ( F K y F K y F K y F K y g y dy (5) 03

2 ISSN: ISO 900:008 Certfed Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume, Issue 6, June 0 The relablty of n-cascade system s gven by A.CASE STUDY R n = n R Let or b e The above formula enables to compute the relablty of an n-cascade system for any specfed dstrbuton. WHEN STRESS-STRENGTH ARE WEIBULL Let us suppose that X, =,, 3... n s a webull varate wth densty functon f (x ) = x e x (6) and Y s also a webull varate wth densty functon Y g (y ) = y e, y > 0 (7) Here shape parameter s same for all the dstrbuton only the scale parameter s changng from dstrbuton to dstrbuton. Then R ()= y y e e... R()= j jp j j p Takng =,, 3, 4 we get R() = R() =... (0) = R(4)= *... e k y e k y gy dy (8) b = jy j p e e b b y... j jp b b... b... y b y e e g y dy * * Where b u = u k u, k o = py Now e gy py y e y e dy dy (9) p Case :- When R () = R () = R(4)= Case :- When

3 R () = (R () = R () = R (4) = ISSN: ISO 900:008 Certfed Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume, Issue 6, June 0 R () = Case 3:- When R () = R () = R(4) =! Case 4:- When! R () = R () = R(4) = Case 5:- When R () =! R (4) = Case 6:- When R () = R () = R(4) =! Case 7:- When! R () = R () = R(4) = Case 8:- When R () = R () = /! 05

4 Rela blty ISSN: ISO 900:008 Certfed Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume, Issue 6, June R(4)= TABLE I: When r R R R3 R4 SR SR SR4 R() SR R() R(4) Fg : When TABLE II: When 06

5 Rela blty Relablty ISSN: ISO 900:008 Certfed Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume, Issue 6, June 0 r R R R3 R4 SR SR r R R R3 R4 SR SR SR SR4.00 SR4, SR R() 0.0 R() R(4) R() 0.50 R() Fg : When TABLE III: When! Fg 3: When! TABLE IV: When! 07

6 Relablty Rela blty ISSN: ISO 900:008 Certfed Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume, Issue 6, June 0 r R R R3 R4 SR SR r R R R3 R4 SR SR SR4, SR SR4,,SR R() R() R() R() Fg 4: When! TABLE V: When! Fg 5: When TABLE VI: When!! 08

7 Rela blty Rela blty ISSN: ISO 900:008 Certfed Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume, Issue 6, June 0 r R R R3 R4 SR SR TABLE VII: When! r R R R3 R4 SR SR R() R() SR4 SR R(4).0.00 R() R() SR4 SR R(4) Fg 6: When! Fg 7: When! 09

8 Relablty ISSN: ISO 900:008 Certfed Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume, Issue 6, June 0 TABLE VIII: When /! r R R R3 R4 SR SR R() R() r REFERENCES [] Kapur.K.C and Lamberson.L.R.968 Relablty n engneerng desgn, Mc-Grawhll. [] Pandt.S.N.N. and Srwastav.G.L (975) : Studes n Cascade Relablty- IEEE Trans.on Rel. R-4,pp [3] Raghavachar.A.C.N,Kesava Rao.B,and Pattabh Ramacharyulu N.C(983) Survval functon under stress Attenuaton n cascade relablty OPSEARCH ; Volume 0,Issue 4:p90-07 [4] A.Rekha & T.Shyam Sundar (997). Relablty of a cascade system wth Exponental Strength and Gamma Stress. Mcroelectroncs and Relablty, Volume 37, Issue 4, Aprl 997, Pages [5] Rekha.A and Chenchu Raju.V.C (999).Cascade system relablty wth Raylegh dstrbuton, Botswana Journal of Technology,Vol. 8,pages 4-9 AUTHOR S PROFILE Currently workng under Mnstry of Manpower, Sultanate of Oman, and Faculty n Mathematcs at the Department of Informaton Technology, IBRA COLLEGE OF TECHNOLOGY, and IBRA, OMAN from November 007. He Worked under U.N.D.P as Assstant Professor n Mathematcs at Arbamnch Unversty, ETHIOPIA from September 004 to October 007.Also, he worked as Assstant Professor n Mathematcs at the Unversty of Arts & Scence College, Kakatya Unversty, Warangal,(A.P),INDIA from June 99 to August 004, taught Mathematcs and statstcs n the Post-Graduate, Under- Graduate and Engneerng classes. Receved hs PhD n Mathematcs enttled CASE STUDIES OF STRESS-STRENGTH CASCADE RELIABILITY MODELS from Kakatya Unversty n 997. Research papers were publshed n Natonal and Internatonal Journals. And also some papers are n consderaton for publcaton n dfferent Internatonal Journals. He was the Member of Indan Educaton Forum, Embassy of Inda, and ETHIOPIA. He was the Lfe member of Andhra Pradesh Socety for Mathematcal Scences, Hyderabad, INDIA. Fg 8: When /! III. CONCLUSION From case to 7 the results are shown n Table to 7 and Fg. to 7, for dfferent values of, we observed that there s very good mprovement n Relablty. Where as n case 8, shown n Table 8 and Fg. 8, there s a very lttle Relablty mprovement. Hence we nfer that, systems wth lesser parameter values are more relable when the stress and strength dstrbutons are webull. IV. ACKNOWLEDGMENT The author wsh to thank Professor. A.C.N. Raghava Char and Professor.A.Rekha for ther valuable suggestons and encouragement durng the preparaton of hs PhD thess. 0