Pentru a realiza studiul fiabilitatii s-a realizat interogarea a zece variante ale aceluiasi model de autovehicul (Skoda Octavia 1.9 TDI), s-au notat in tabel, defectele aparute si numarul de kilometri la care a aparut fiecare defect.

Nr crt. Km Defectiune VIN

1 20000Filtru motorina

WAUZZZ8EX2A088254

2 50000 BujiileWAUZZZ8EX2A088458

3 90000 Pompa de uleiWAUZZZ8EX2A088795

4 110000 PlanetareWAUZZZ8EX2A084578

5 150000 Pompa de apaWAUZZZ8EX2A088437

6 175000 AmbreajWAUZZZ8EX2A088972

7 230000 TurbinaWAUZZZ8EX2A089820

8 270000Pompa injectie

WAUZZZ8EX2A088862

9 350000 InjectoareWAUZZZ8EX2A088021

10 370000Filtru de particule

WAUZZZ8EX2A080215

Determinarea intervalului de esantionare Δ, se calculeaza cu ajutorul relatiei:

80831.40878

tmin si tmax= limitele intervalului de timp masurate in KmN= numarul de interogari

Se adopta ∆t=81000 [km]

∆𝑡=(𝑡_𝑚𝑎𝑥−𝑡_𝑚𝑖𝑛)/(1+3,33∙lg𝑁)=(370000−20000)/(1+3,33∙𝑙𝑔10)

Determinarea intervalelor de timp si nr de defectiuni din tabelul:

Interval 0-8100081000-162000 162000-243000 243000-324000

324000-370000

ti 20000 90000 150000 270000 370000

Ni 2 3 2 1 2ΣN 2 5 7 8 10

F(ti) 0.2 0.5 0.7 0.8 1

F(ti)[%] 20 50 70 80 100

Se calculeaza frecventa cumulate a defectiunilor F(t) cu relatia: Rezultatele sunt notate in tabel

Se estimeaza cu ajutorul diagramei, parametrii constanti ai legii Weibull astfel: -se marcheaza mijlocul fiecarui interval si frecventa cumulata a defectarilor corespunzatoare, in procente;

-daca punctele astfel obtinute se afla aproximativ pe o dreapta, atunci parametrul de pozitie γ este zero; -se intersecteaza dreapta formata de puncte cu dreapta η din diagrama si se citeste valoarea de pe abscisa corespunzatoare punctului de intersectie. Astfel se obtine valoarea parametrului de scara η, care va reprezenta viata caracteristica a produsului pentru un procent cumulat al defectarilor de 63%; -parametrul de forma β se estimeaza prin trasarea unei paralele prin trasarea unei paralele prin punctul de coordonate (1,63%) la dreapta valorilor experimentale. Valoarea lui β se citeste la intersectia acestei paralele cu dreapta β.

Valorile obtinute:η 233840 parametrul vietii caracteristice β 2.898476 parametru de forma (daca e<1, pericolul scade, curba descrescatoare)γ 0 parametru de pozitie

𝐹(𝑡𝑖)=(Σ𝑁_𝑖)/𝑁

Cu valorile parametrilor legii Weibull, se determina fiabilitatea R(t), functia de repartitie a timpului de functionare F(t), densitatea de probabilitate a timpului de functionare f(t) si intensitatea de defectare z(t).Pentru determinarea valorilor se folosesec relatiile:

F(t)= 1-R(t)

Valorile obtinute se afla in tabelul:

t(km)ExponentR(t) R(t) F(t) f(t) Z(t)

0 0 1 0 0 02500 -0.00000194 0.999998063 0.00000194 0.00000000 0.000000005000 -0.00001444 0.999985556 0.00001444 0.00000001 0.000000017500 -0.00004678 0.999953218 0.00004678 0.00000002 0.00000002

10000 -0.00010770 0.999892304 0.00010770 0.00000003 0.0000000312500 -0.00020564 0.999794379 0.00020562 0.00000005 0.0000000515000 -0.00034883 0.999651227 0.00034877 0.00000007 0.0000000717500 -0.00054533 0.999454816 0.00054518 0.00000009 0.0000000920000 -0.00080306 0.999197259 0.00080274 0.00000012 0.0000001222500 -0.00112983 0.998870805 0.00112920 0.00000015 0.0000001525000 -0.00153335 0.998467825 0.00153217 0.00000018 0.0000001827500 -0.00202124 0.997980806 0.00201919 0.00000021 0.0000002130000 -0.00260104 0.997402345 0.00259766 0.00000025 0.0000002532500 -0.00328022 0.996725151 0.00327485 0.00000029 0.0000002935000 -0.00406621 0.995942045 0.00405795 0.00000034 0.0000003437500 -0.00496635 0.995045959 0.00495404 0.00000038 0.0000003840000 -0.00598795 0.994029942 0.00597006 0.00000043 0.0000004342500 -0.00713825 0.992887164 0.00711284 0.00000048 0.0000004945000 -0.00842447 0.991610918 0.00838908 0.00000054 0.0000005447500 -0.00985376 0.990194631 0.00980537 0.00000060 0.0000006050000 -0.01143324 0.988631867 0.01136813 0.00000066 0.0000006652500 -0.01317001 0.986916333 0.01308367 0.00000072 0.0000007355000 -0.01507111 0.985041892 0.01495811 0.00000078 0.00000079

𝑍(𝑡)=𝛽/𝜂((𝑡−𝛾)/𝜂)^(𝛽−1)

𝑓(𝑡)=𝛽/𝜂((𝑡−𝛾)/𝜂)^(𝛽−1)∙𝑅(𝑡)𝑅(𝑡)=𝑒^(〖−((𝑡−𝛾)/𝜂)〗^𝛽)

57500 -0.01714355 0.983002566 0.01699743 0.00000085 0.0000008660000 -0.01939431 0.980792545 0.01920745 0.00000092 0.0000009462500 -0.02183036 0.978406201 0.02159380 0.00000099 0.0000010165000 -0.02445860 0.97583809 0.02416191 0.00000106 0.0000010967500 -0.02728593 0.973082966 0.02691703 0.00000114 0.0000011770000 -0.03031922 0.970135793 0.02986421 0.00000122 0.0000012672500 -0.03356532 0.966991747 0.03300825 0.00000130 0.0000013475000 -0.03703103 0.963646234 0.03635377 0.00000138 0.0000014377500 -0.04072315 0.960094897 0.03990510 0.00000146 0.0000015280000 -0.04464845 0.956333625 0.04366638 0.00000155 0.0000016282500 -0.04881367 0.952358564 0.04764144 0.00000163 0.0000017185000 -0.05322555 0.94816613 0.05183387 0.00000172 0.0000018187500 -0.05789079 0.943753014 0.05624699 0.00000181 0.0000019290000 -0.06281606 0.939116196 0.06088380 0.00000190 0.0000020292500 -0.06800805 0.934252952 0.06574705 0.00000199 0.0000021395000 -0.07347340 0.929160865 0.07083913 0.00000208 0.0000022497500 -0.07921872 0.923837836 0.07616216 0.00000218 0.00000236

100000 -0.08525065 0.918282087 0.08171791 0.00000227 0.00000247102500 -0.09157577 0.912492175 0.08750782 0.00000236 0.00000259105000 -0.09820065 0.906466999 0.09353300 0.00000246 0.00000271107500 -0.10513187 0.900205806 0.09979419 0.00000255 0.00000283110000 -0.11237596 0.893708197 0.10629180 0.00000265 0.00000296112500 -0.11993945 0.886974139 0.11302586 0.00000274 0.00000309115000 -0.12782887 0.880003964 0.11999604 0.00000284 0.00000322117500 -0.13605070 0.87279838 0.12720162 0.00000293 0.00000336120000 -0.14461144 0.865358473 0.13464153 0.00000302 0.00000349122500 -0.15351755 0.857685708 0.14231429 0.00000312 0.00000363125000 -0.16277551 0.849781939 0.15021806 0.00000321 0.00000377127500 -0.17239174 0.841649404 0.15835060 0.00000330 0.00000392130000 -0.18237268 0.833290733 0.16670927 0.00000339 0.00000407132500 -0.19272475 0.824708942 0.17529106 0.00000348 0.00000422135000 -0.20345436 0.81590744 0.18409256 0.00000356 0.00000437137500 -0.21456790 0.80689002 0.19310998 0.00000365 0.00000452140000 -0.22607176 0.797660862 0.20233914 0.00000373 0.00000468142500 -0.23797229 0.788224529 0.21177547 0.00000382 0.00000484145000 -0.25027587 0.778585962 0.22141404 0.00000390 0.00000500147500 -0.26298884 0.768750474 0.23124953 0.00000397 0.00000517150000 -0.27611754 0.758723746 0.24127625 0.00000405 0.00000534152500 -0.28966829 0.748511819 0.25148818 0.00000412 0.00000551155000 -0.30364740 0.738121083 0.26187892 0.00000419 0.00000568157500 -0.31806118 0.727558274 0.27244173 0.00000426 0.00000585160000 -0.33291593 0.716830456 0.28316954 0.00000432 0.00000603162500 -0.34821793 0.705945016 0.29405498 0.00000438 0.00000621

165000 -0.36397344 0.694909648 0.30509035 0.00000444 0.00000639167500 -0.38018875 0.683732344 0.31626766 0.00000450 0.00000658170000 -0.39687009 0.672421372 0.32757863 0.00000455 0.00000677172500 -0.41402372 0.66098527 0.33901473 0.00000460 0.00000696175000 -0.43165588 0.649432824 0.35056718 0.00000464 0.00000715177500 -0.44977277 0.637773053 0.36222695 0.00000468 0.00000734180000 -0.46838064 0.626015192 0.37398481 0.00000472 0.00000754182500 -0.48748568 0.614168673 0.38583133 0.00000476 0.00000774185000 -0.50709409 0.602243103 0.39775690 0.00000478 0.00000794187500 -0.52721207 0.590248252 0.40975175 0.00000481 0.00000815190000 -0.54784579 0.578194022 0.42180598 0.00000483 0.00000836192500 -0.56900143 0.566090435 0.43390956 0.00000485 0.00000857195000 -0.59068517 0.553947608 0.44605239 0.00000486 0.00000878197500 -0.61290315 0.541775729 0.45822427 0.00000487 0.00000899200000 -0.63566152 0.52958504 0.47041496 0.00000488 0.00000921202500 -0.65896643 0.51738581 0.48261419 0.00000488 0.00000943205000 -0.68282402 0.505188314 0.49481169 0.00000488 0.00000965207500 -0.70724040 0.493002811 0.50699719 0.00000487 0.00000988210000 -0.73222170 0.480839519 0.51916048 0.00000486 0.00001011212500 -0.75777403 0.468708596 0.53129140 0.00000484 0.00001034215000 -0.78390350 0.456620112 0.54337989 0.00000483 0.00001057217500 -0.81061619 0.444584032 0.55541597 0.00000480 0.00001080220000 -0.83791821 0.432610191 0.56738981 0.00000478 0.00001104222500 -0.86581562 0.420708273 0.57929173 0.00000475 0.00001128225000 -0.89431452 0.408887787 0.59111221 0.00000471 0.00001152227500 -0.92342096 0.397158052 0.60284195 0.00000467 0.00001176230000 -0.95314101 0.385528171 0.61447183 0.00000463 0.00001201233840 -1.00000000 0.367879441 0.63212056 0.00000456 0.00001240235000 -1.01444615 0.362603202 0.63739680 0.00000454 0.00001251237500 -1.04604333 0.35132508 0.64867492 0.00000449 0.00001277240000 -1.07827830 0.340180712 0.65981929 0.00000443 0.00001302242500 -1.11115708 0.329177854 0.67082215 0.00000437 0.00001328245000 -1.14468571 0.318323948 0.68167605 0.00000431 0.00001354246480 -1.16484333 0.311971535 0.68802847 0.00000427 0.00001370247500 -1.17887019 0.307626101 0.69237390 0.00000425 0.00001381250000 -1.21371654 0.297091074 0.70290893 0.00000418 0.00001407252500 -1.24923076 0.286725274 0.71327473 0.00000411 0.00001434255000 -1.28541884 0.276534735 0.72346526 0.00000404 0.00001461257500 -1.32228679 0.266525119 0.73347488 0.00000397 0.00001488260000 -1.35984057 0.256701699 0.74329830 0.00000389 0.00001516262500 -1.39808619 0.247069356 0.75293064 0.00000381 0.00001544265000 -1.43702960 0.237632576 0.76236742 0.00000374 0.00001572267500 -1.47667678 0.228395438 0.77160456 0.00000365 0.00001600

270000 -1.51703368 0.219361619 0.78063838 0.00000357 0.00001629272500 -1.55810627 0.210534389 0.78946561 0.00000349 0.00001657275000 -1.59990050 0.201916607 0.79808339 0.00000340 0.00001686277500 -1.64242231 0.193510731 0.80648927 0.00000332 0.00001716280000 -1.68567765 0.185318809 0.81468119 0.00000323 0.00001745282500 -1.72967244 0.177342491 0.82265751 0.00000315 0.00001775285000 -1.77441261 0.169583031 0.83041697 0.00000306 0.00001805287500 -1.81990410 0.16204129 0.83795871 0.00000297 0.00001835290000 -1.86615282 0.154717746 0.84528225 0.00000289 0.00001865292500 -1.91316468 0.147612501 0.85238750 0.00000280 0.00001896295000 -1.96094559 0.14072529 0.85927471 0.00000271 0.00001927297500 -2.00950145 0.134055491 0.86594451 0.00000262 0.00001958300000 -2.05883817 0.127602136 0.87239786 0.00000254 0.00001989302500 -2.10896163 0.121363921 0.87863608 0.00000245 0.00002021305000 -2.15987773 0.115339223 0.88466078 0.00000237 0.00002053307500 -2.21159234 0.109526106 0.89047389 0.00000228 0.00002085310000 -2.26411135 0.103922344 0.89607766 0.00000220 0.00002117312500 -2.31744063 0.098525426 0.90147457 0.00000212 0.00002149315000 -2.37158604 0.093332579 0.90666742 0.00000204 0.00002182317500 -2.42655346 0.088340778 0.91165922 0.00000196 0.00002215320000 -2.48234874 0.083546766 0.91645323 0.00000188 0.00002248322500 -2.53897773 0.078947064 0.92105294 0.00000180 0.00002282325000 -2.59644628 0.074537995 0.92546200 0.00000173 0.00002316327500 -2.65476025 0.070315695 0.92968431 0.00000165 0.00002350330000 -2.71392546 0.066276131 0.93372387 0.00000158 0.00002384332500 -2.77394777 0.062415117 0.93758488 0.00000151 0.00002418335000 -2.83483299 0.058728333 0.94127167 0.00000144 0.00002453337500 -2.89658695 0.055211338 0.94478866 0.00000137 0.00002488340000 -2.95921549 0.051859586 0.94814041 0.00000131 0.00002523342500 -3.02272441 0.048668445 0.95133156 0.00000124 0.00002558345000 -3.08711953 0.04563321 0.95436679 0.00000118 0.00002594347500 -3.15240666 0.04274912 0.95725088 0.00000112 0.00002629350000 -3.21859161 0.04001137 0.95998863 0.00000107 0.00002665352500 -3.28568017 0.037415128 0.96258487 0.00000101 0.00002702355000 -3.35367815 0.034955546 0.96504445 0.00000096 0.00002738357500 -3.42259134 0.032627776 0.96737222 0.00000091 0.00002775360000 -3.49242552 0.030426981 0.96957302 0.00000086 0.00002812362500 -3.56318648 0.028348349 0.97165165 0.00000081 0.00002849365000 -3.63488000 0.026387101 0.97361290 0.00000076 0.00002886367500 -3.70751186 0.024538503 0.97546150 0.00000072 0.00002924370000 -3.78108783 0.022797878 0.97720212 0.00000068 0.00002962

Pentru verificare, în software-ul Weibull se introduc kilometri la care au apărut defectele și se obțin

graficele distribuției Weibull, al fiabilității, al nefiabilității, al intensității de defectare și al densității de probabilitate a timpului de funcționare.

De asemenea, programul va calcula și parametrii constanți ai legii Weibull.Valorile și graficele generate sunt exprimate mai jos

0 50000 100000 150000 200000 250000 300000 350000 4000000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Functia de fiabilitate R(t)

0 50000 100000 150000 200000 250000 300000 350000 4000000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Functia de nefiabilitate F(t)

0 50000 100000 150000 200000 250000 300000 350000 4000000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Functia de fiabilitate R(t)

0 50000 100000 150000 200000 250000 300000 350000 4000000

0.000001

0.000002

0.000003

0.000004

0.000005

0.000006

Densitatea de probabilitate f(t)

0 50000 100000 150000 200000 250000 300000 350000 4000000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Functia de nefiabilitate F(t)

0 50000 100000 150000 200000 250000 300000 350000 4000000

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0.000035

Intensitatea de defectare Z(t)

0 50000 100000 150000 200000 250000 300000 350000 4000000

0.000001

0.000002

0.000003

0.000004

0.000005

0.000006

Densitatea de probabilitate f(t)

0 50000 100000 150000 200000 250000 300000 350000 4000000

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0.000035

Intensitatea de defectare Z(t)

În concluzie, am verificat, numărul de kilometri, parcurși de autovehiculul, marca Skoda Octavia,și am specificat uzura, unor piese, aparută pe parcursul funcționarii motorului.

20000 90000 150000 270000 3700000

0.5

1

1.5

2

2.5

3

3.5

t(km)

Ni

Aceste piese, uzate, au fost înlocuite după un număr de kilometri, de la defectarea lor. Cu diagrama Weibull, am obținut, urmatoarele valori:

parametrii vieții caracteristice parametrii de forma parametrii de poziție.

Cu toate acestea, pentru a verifica cu exactitate, calculele, parcurse de mine, am utilizat programul Weibull, care asigură comparativ, diagramele, deduse din calcule.