Foundations — Reliability — MTTF, MTBF, exponential failure model

3.6.31 · D1 · Physics › Spacecraft Structures & Systems Engineering › Reliability — MTTF, MTBF, exponential failure model

Isse pehle ki tum MTTF, MTBF, ya exponential model ko parent note pe touch karo, tumhe har woh symbol samajhna hoga jo woh silently assume karta hai. Yeh page har ek ko kuch nahi se banata hai — ek plain-words meaning, ek picture jo woh represent karta hai, aur yeh kyun yeh topic uske bina kaam nahi kar sakta. Upar se neeche padho; har block uske upar wale pe lean karta hai.

1. Time, aur survivor ka clock — $t$

Ek stopwatch ki picture karo jo usi instant shuru hoti hai jab satellite switch on hota hai. Is topic mein har sawaal is form ka hai: "stopwatch ki is reading par, kya sach hai?" Stopwatch kabhi ulta nahi chalta, isliye $t$ sirf badhta hai: $t\ge 0$.

Yeh topic isse kyun chahta hai: har doosra quantity — reliability, failure rate, average lifetime — is clock ke against measure ki jaati hai. Bina ek agreed zero-point ke, "5-year survival" ka koi matlab nahi.

2. Probability — 0 aur 1 ke beech ek number

Upar bar dekho: yeh "certainty" ki ek full unit hai jo hum slice karte hain. Agar coin heads par girta hai probability $0.5$ ke saath, toh bar ka aadha hissa shaded hai. Ek satellite $0.11$ probability ke saath survive karta hai toh sirf ek sliver shaded hogi.

Probability and Statistics Fundamentals dekho agar yeh shaky lagta hai.

3. Reliability — shrinking survival number $R(t)$

Upar ke dono ideas combine karo: $R$ ek probability hai (Section 2), aur yeh clock reading $t$ par depend karta hai (Section 1). Isliye $R(t)$ ek number nahi hai — yeh ek poori curve hai, har moment ke liye ek probability.

Figure ko left se right padho:

• $t=0$ par curve height $1$ se start hoti hai — part definitely kaam karta hai jab tum ise switch on karte ho. Isliye $R(0)=1$.
• Jaise $t$ badhta hai, curve sirf neeche jaati hai ya flat rehti hai, kabhi upar nahi. Ek dead part life mein wapas nahi aa sakta (jab tak hum repair allow nahi karte, bahut baad mein). Yeh "kabhi nahi uthti" property monotonically decreasing kehlati hai.
• Curve band $0\le R(t)\le 1$ ke andar rehti hai, kyunki yeh ek probability hai.

4. Rate of change — slope $\frac{dR}{dt}$

Parent note $\frac{dR}{dt}$ likhta hai. Woh symbol logon ko dara deta hai, toh chaliye isse earn karte hain.

Figure mein, dashed straight line sirf ek point par curve ko kiss karti hai — yeh tangent line hai, aur iska steepness hi wahan $\frac{dR}{dt}$ hai. Kyunki curve neeche ki taraf jaati hai, yeh slope negative hai.

Topic ko sirf subtraction ke bajaye derivative kyun chahiye. Hum chahte hain "failures per hour abhi," na ki "poore mission mein." Ek slope ek this-instant sawaal ka jawab deta hai — exactly wahi hai jo ek instantaneous rate hai. Isliye calculus aata hai: yeh akela tool hai jo change ko ek single point par measure karta hai.

5. Failure rate — $\lambda$ (lambda)

Parent ki definition: $\lambda (t)=-\frac{1}{R(t)}\frac{dR}{dt}$

Isse piece by piece decode karo jo humne banaya usse use karke:

• $\frac{dR}{dt}$ (Section 4) batata hai reliability kitni tezi se girti hai — lekin yeh drops ko poori original population ke share ke roop mein count karta hai.
• $R(t)$ se divide karna ise abhi tak ke survivors ke share tak rescale karta hai (sirf woh abhi bhi fail ho sakte hain).
• Minus sign negative slope ko ek positive rate mein flip karta hai.

Constant $\lambda$ — ek flat failure rate jo kabhi nahi badalti — woh special assumption hai jo poora exponential model kaam karati hai. Physically yeh Bathtub Curve ka beech wala hissa hai: infant defects ke baad, wear-out se pehle. Jab $\lambda$ constant nahi hoti, tum Weibull Distribution par graduate karte ho.

6. Exponential $e^{-\lambda t}$

Poora model $R(t)=e^{-\lambda t}$ par land karta hai. Unpack karne ke liye do symbols: $e$, aur ise ek power par uthana.

Figure $e^{-\lambda t}$ dikhata hai: $1$ se shuru hota hai, neeche swoop karta hai, aur $0$ ki taraf flatten hota hai (lekin kabhi nahi pahunchta). Do curves compare karo — ek bada $\lambda$ (steeper, less reliable) vs ek chhota $\lambda$ (gentler, more reliable).

Product $\lambda t$ ek pure number hona chahiye. Units: $\left(\frac{\text{failures}}{\text{hour}}\right)\times (\text{hours})=\text{failures}$, ek bare count. Tum kabhi bhi $e$ ko units wali cheez par raise nahi kar sakte, isliye yeh har calculation par ek sanity check hai.

7. Logarithm $\ln$ — undo button

Topic ko yeh kyun chahiye. Parent mein Example 2 poochta hai "$\lambda$ kya hoga jo 95% reliability dega?" Unknown $\lambda$ $e$ ke exponent ke andar phansa hua hai. Woh lock kholne ki akeli chabi $\ln$ hai: $0.95=e^{-\lambda t}\stackrel{\ln }{\to }\ln (0.95)=-\lambda t\Rightarrow \lambda =-\frac{\ln (0.95)}{t}$

Dono sides ka $\ln$ lene se $\lambda$ khulkar bahar aa jaata hai jahan hum iske liye solve kar sakte hain.

8. Expected value $E[T]$ aur integral $\int$

MTTF ke liye dono kyun chahiye. Failure kisi bhi real instant par ho sakta hai, sirf whole hours mein nahi. Possible failure times ki continuous smear mein average karne ke liye, plain addition kaam nahi karega — tumhe integral chahiye, "sum up" ka continuous version. Isliye MTTF ${\int }_0^{\infty }tf(t)dt$ likha jaata hai: har possible failure time $t$, weighted by how likely it is, sab summed. Parent ise grind karke clean result $\text{MTTF}=1/\lambda$ nikalta hai.

Yahan $f(t)$ probability density hai — failure likelihood time mein kaise spread hai; yeh "already-failed" curve ka slope hai, $f(t)=-\frac{dR}{dt}$.

Yeh sab topic ko kaise feed karta hai

Har foundation box unhe point karta hai jo usp depend karte hain, aur sab kuch reliability topic mein funnel ho jaata hai. Yahan se tum Series vs. Parallel System Reliability, Redundancy Design, aur usi maths ke Poisson Process view ke liye ready ho.

Equipment checklist

Apne aap ko test karo — right side cover karo aur reveal karne se pehle har ek ka jawab do.