Industrial Engineering - BATH TUB CURVE

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BATH TUB CURVE 

MOST PRODUCTS GO THROUGH THREE DISTINCT PHASES FROM PRODUCT INCEPTION TO WEAR OUT. FIGURE BELOW SHOWS A TYPICAL LIFE CYCLE CURVE FOR WHICH THE FAIURE RATE IS PLOTTED AS FUNCTION OF TIME. 

INFANCY / GREEN / DEBUGGING / BURN-IN-PERIOD: MANY COMPONENTS FAIL VERY SOON AFTER THEY ARE PUT INTO SERVICE. FAILURES WITHIN THIS PERIOD ARE CAUSED BY DEFECTS AND POOR DESIGN THAT CAUSE AN ITEM TO BE LEGITIMATELY BAD. THESE ARE CALLED INFANT MORTALITY FAILURES AND THE FAILURE RATE IN THIS PERIOD IS RELATIVELY HIGH. GOOD SYSTEM VENDORS WILL PERFORM AN OPERATION CALLED "BURN IN" WHERE THEY PUT TOGETHER AND TEST A SYSTEM FOR SEVERAL DAYS TO TRY TO WEED OUT THESE TYPES OF PROBLEMS SO THE CUSTOMER DOESN'T SEE THEM.

CHANCE FAILURE / NORMAL OPERATING LIFE: IF A COMPONENT DOES NOT FAIL WITHIN ITS INFANCY, IT WILL GENERALLY TEND TO REMAIN TROUBLE-FREE OVER ITS OPERATING LIFETIME. THE FAILURE RATE DURING THIS PERIOD IS TYPICALLY QUITE LOW. THIS PHASE, IN WHICH THE FAILURE RATE IS CONSTANT, TYPICALLY REPRESENTS THE USEFUL LIFE OF THE PRODUCT.

WEAR OUT / AGEING: AFTER A COMPONENT REACHES A CERTAIN AGE, IT ENTERS THE PERIOD WHERE IT BEGINS TO WEAR OUT, AND FAILURES START TO INCREASE. THE PERIOD WHERE FAILURES START TO INCREASE IS CALLED THE WEAR OUT PHASE OF COMPONENT LIFE.

RELIABILITY DETERMINATION
IN THE ADULT OR MATURE PHASE THE FAILURE RATE OF A COMPONENT OR SYSTEM IS CONSTANT. UNDER SUCH CONDITIONS THE TIME TO FAILURE FOLLOWS AN EXPONENTIAL LAW, AND THE PROBABILITY DENSITY FUNCTION OF EXPONENTIAL DISTRIBUTION IS AS GIVEN BELOW:

WHERE? DENOTES THE FAILURE RATE. THE MEAN TIME TO FAILURE MTTF FOR THE EXPONENTIAL DISTRIBUTION IS
MTTF = 1 / ?
IF THE FAILURE RATE IS CONSTANT, THE MTTF IS THE RECIPROCAL OF THE FAILURE RATE. FOR REPAIRABLE SYSTEM IT IS ALSO EQUAL TO MTBF.
THE RELIABILITY AT TIME T, R(T), IS THE PROBABILITY OF THE PRODUCT LASTING UP TO AT LEAST TIME T. IT IS GIVEN BY

EXAMPLE:
ASSUME WE HAVE AN AUTOMOBILE THAT IS OPERATING IN ITS MATURE PHASE AND HAS THE FOLLOWING FAILURE HISTORY:
TIME TO FAILURE (HOURS): 100 800 1280 2600
THE MTBF IS GIVEN BY: [100+800+1280+2600] / 4 = 1195 HOURS/FAILURE
THIS GIVES A CONSTANT FAILURE RATE OF: 1 / 1195 = 0.000836 FAILURES/HOUR.
WHAT RELIABILITY CAN BE EXPECTED FROM THE AUTOMOBILE AFTER 40, 200, 1000, AND 5000 HOURS?

T-HOURS
40
200
1000
5000
RELIABILITY
97%
84.6%
43.4%
1.5%

BEING ABLE TO PREDICT RELIABILITIES IS PARTICULARLY USEFUL IN TERMS OF MAINTENANCE SCHEDULING OF MACHINERY. ASSUME A MINIMUM RELIABILITY OF 0.9 CAN BE ACCEPTED FROM THE AUTOMOBILE, AT WHAT POINT WOULD A SERVICE BE REQUIRED? 

THUS, THE AUTOMOBILE NEEDS SERVICING EVERY 126 HOURS TO KEEP A MINIMUM RELIABILITY OF 0.9

SYSTEM RELIABILITY
SERIES SYSTEM: WHEN COMPONENTS ARE IN SERIES AND EACH COMPONENT HAS A RELIABILITY R I . IF ONE COMPONENT FAILS, THE SYSTEM FAILS.

THE OVERALL RELIABILITY OF A SERIES SYSTEM SHOWN ABOVE IS:
R AB = R1 X R2 X R3
IF R1 = R2 = R3 = 0.95
RAB = R1 X R2 X R3 = 0.95 X 0.95 X 0.95 = 0.86
R TOTAL IS ALWAYS < THAN R1 OR R2 OR R3

PARALLEL SYSTEM : WHEN COMPONENTS ARE IN PARALLEL AND EACH COMPONENT HAS A RELIABILITY RI. IF ONE COMPONENT FAILS, THE SYSTEM DOES NOT FAIL.

RAB= 1 - PROBABILITY (1 & 2 BOTH FAIL)
THE PROBABILITY OF 1 FAILING IS = (1 - R1 )
THE PROBABILITY OF 2 FAILING IS = (1 - R 2 )
OVERALL RELIABILITY IS R AB =1 - (1 - R 1 ) (1 - R 2)
IF R1= 0.9 AND R2 =0.8
R AB =1 - (1 - 0.9) (1 - 0.8) = 0.98
R TOTALIS ALWAYS > THAN R1 OR R2

MAINTAINABILITY
MAINTAINABILITY IS A DESIGN CHARACTERISTIC DEALING WITH THE EASE, ACCURACY, SAFETY, AND ECONOMY IN THE PERFORMANCE OF MAINTENANCE FUNCTIONS. IT MAY BE DEFINED IN SEVERAL WAYS:
  • THE PROBABILITY THAT AN ITEM WILL BE RESTORED WITHIN A GIVEN PERIOD OF TIME
  • THE PROBABILITY THAT MAINTENANCE WILL NOT BE REQUIRED MORE THAN X TIMES A GIVEN PERIOD OF TIME
  • THE PROBABILITY THAT THE MAINTENANCE COST WILL NOT EXCEED Y RUPEES IN A GIVEN PERIOD OF TIME
MAINTAINABILITY IS THE EASE AND SPEED WITH WHICH ANY MAINTENANCE ACTIVITY CAN BE CARRIED OUT ON AN ITEM OF EQUIPMENT. IT MAY BE MEASURED BY MEAN TIME TO REPAIR.
ONCE A PIECE OF EQUIPMENT HAS FAILED IT MUST BE POSSIBLE TO GET IT BACK INTO AN OPERATING CONDITION AS SOON AS POSSIBLE, THIS IS KNOWN AS MAINTAINABILITY .
TO CALCULATE THE MAINTAINABILITY OR MEAN TIME TO REPAIR (MTTR) OF AN ITEM, THE TIME REQUIRED TO PERFORM EACH ANTICIPATED REPAIR TASK MUST BE WEIGHTED (MULTIPLIED) BY THE RELATIVE FREQUENCY WITH WHICH THAT TASK MUST BE PERFORMED (E.G. NO. OF TIMES PER YEAR).

MAINTENANCE CATEGORIES
  • CORRECTIVE MAINTENANCE: UNSCHEDULED MAINTENANCE TO RESTORE A SYSTEM TO A SPECIFIED LEVEL OF PERFORMANCE
  • PREVENTIVE MAINTENANCE: SCHEDULED MAINTENANCE TO RETAIN A SYSTEM AT A SPECIFIED LEVEL OF PERFORMANCE
MEASURES OF MAINTAINABILITY
MAINTENANCE ELAPSED-TIME FACTORS
  • MEAN CORRECTIVE MAINTENANCE TIME (MC)
  • MEAN PREVENTIVE MAINTENANCE TIME (MP)
  • MEAN ACTIVE MAINTENANCE TIME (MT = MC + MP)
  • LOGISTIC DELAY TIME (LDT)
  • ADMINISTRATIVE DELAY TIME (ADT)
  • MAINTENANCE DOWN TIME (MDT=MT+LDT+ADT)
MAINTENANCE LABOUR-HOUR FACTORS
  • MAINTENANCE LABOR-HOURS PER CYCLE, ETC.
MAINTENANCE FREQUENCY FACTORS
  • MEAN TIME BETWEEN MAINTENANCE
MAINTENANCE COST FACTORS
  • MAINTENANCE COST PER SYSTEM OPERATING PERIOD, ETC.
AVAILABILITY

THE ABILITY OF AN ITEM TO BE IN A STATE TO PERFORM A REQUIRED FUNCTION UNDER GIVEN CONDITIONS AT A GIVEN INSTANT OF TIME OR DURING A GIVEN TIME INTERVAL, ASSUMING THAT THE REQUIRED EXTERNAL RESOURCES ARE PROVIDED. 

AVAILABILITY AT ITS SIMPLEST LEVEL
AVAILABILITY = UPTIME / (DOWNTIME + UPTIME)
THE TIME UNITS ARE GENERALLY HOURS AND THE TIME BASE IS 1 YEAR .  

FROM THE DESIGN AREA OF CONCERN THIS EQUATION TRANSLATES TO
AVAILABILITY(INTRINSIC) A I = MTBF / (MTBF + MTTR)

MTBF = MEAN TIME BETWEEN FAILURES
MTTR = MEAN TIME TO REPAIR / MEAN TIME TO REPLACE.
OPERATIONAL AVAILABILITY IS DEFINED DIFFERENTLY
AVAILABILITY (OPERATIONAL) A O = MTBM/(MTBM+MDT).
MTBM = MEAN TIME BETWEEN MAINTENANCE.
MDT = MEAN DOWN TIME.

SYSTEM AVAILABILITY
SYSTEM AVAILABILITY IS CALCULATED BY MODELING THE SYSTEM AS AN INTERCONNECTION OF PARTS IN SERIES AND PARALLEL. THE FOLLOWING RULES ARE USED TO DECIDE IF COMPONENTS SHOULD BE PLACED IN SERIES OR PARALLEL:
  • IF FAILURE OF A PART LEADS TO THE COMBINATION BECOMING INOPERABLE, THE TWO PARTS ARE CONSIDERED TO BE OPERATING IN SERIES
  • IF FAILURE OF A PART LEADS TO THE OTHER PART TAKING OVER THE OPERATIONS OF THE FAILED PART, THE TWO PARTS ARE CONSIDERED TO BE OPERATING IN PARALLEL. 
THE CALCULATIONS OF SYSTEM AVAILABILITY ARE SIMILAR TO SYSTEM RELIABILITY.



REFERENCES:-  www.nptel.iitm.ac.in/

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