Fault tree analysis (FTA) — top-down, AND - OR gates

3.6.30 · Physics › Spacecraft Structures & Systems Engineering

Yeh top-down approach bottom-up methods (jaise FMEA) se alag hai jo component failures se shuru karke upar ki taraf jaate hain.

Fault Tree Analysis Kya Hai?

Key components:

• Top Event: Woh system failure jo hum analyze kar rahe hain (e.g., "Loss of spacecraft attitude control")
• Intermediate Events: Subsystem failures jo top event mein contribute karte hain
• Basic Events: Component-level failures (is analysis mein aage decompose nahi ho sakte)
• Logic Gates: AND/OR/NOT gates jo dikhate hain ki events kaise combine hote hain

Teen Gate Types

Independent events ke liye, output probability 1 minus woh probability hai ki SAARE inputs safe hain.

AND Gate: Output tab hi hota hai jab saare inputs hों $P(\text{Output})={\prod }_{i=1}^n{P}_i$

Output probability saare input probabilities ka product hai (independent events ke liye).

NOT Gate (Inhibit): Event ko invert karta hai $P(\text{Output})=1-P(\text{Input})$

YEH FORMULAS KYUN?

• OR gate: System fail hota hai agar kam se kam ek path fail ho → "koi fail nahi hota" ka complement
• AND gate: Saari conditions simultaneously true honi chahiye → joint probability
• Spacecraft ke liye, AND gates redundancy represent karte hain (accha!), OR gates single points of failure represent karte hain (bura!)

Fault Tree Kaise Banate Hain

Step 1: Top Event Define Karo

[Top Event: Total Power Loss]

Step 2: Immediate Causes Identify Karo (First Level) Total power loss tab hota hai jab:

• Solar panels fail AUR batteries fail (dono ka fail hona zaroori)

        [Total Power Loss]
               |
            [AND Gate]
           /         \
   [Solar Fail]   [Battery Fail]

AND KYUN? Hamare paas redundancy hai — koi bhi ek source craft ko power de sakta hai.

Step 3: Solar Fail Ko Decompose Karo Solar panels fail hote hain agar:

• Deployment mechanism jam ho YA
• Solar cells threshold se zyada degrade ho jaayein YA
• Sun tracking system fail ho

    [Solar Fail]
         |
     [OR Gate]
    /    |    \
[Deploy] [Degrade] [Tracking]

OR KYUN? Koi bhi single failure solar power khatam kar deta hai — yahan koi redundancy nahi.

Step 4: Battery Fail Ko Decompose Karo Batteries fail hoti hain agar:

• Cell 1 fail ho AUR Cell 2 fail ho (dono battery banks ka marr jaana zaroori)

    [Battery Fail]
         |
     [AND Gate]
       /     \
  [Cell1]  [Cell2]

Step 5: Basic Events Tak Pahuncho Ruko jab tum un events tak pahuncho jिनके failure rates known hain:

• [Deploy]: P = 0.001 (deployment mechanism jam)
• [Degrade]: P = 0.01 (solar degradation <50% over mission)
• [Tracking]: P = 0.005 (sun sensor failure)
• [Cell1]: P = 0.02 (battery bank 1 failure)
• [Cell2]: P = 0.02 (battery bank 2 failure)

Complete Tree:

Quantitative Analysis: Top Event Probability Calculate Karna

Solar Fail (OR gate): $P_{\text{Solar}}=1-(1-0.001)(1-0.01)(1-0.005)$ $=1-(0.999)(0.99)(0.995)=1-0.98405=0.01595\approx 0.016$

Battery Fail (AND gate): $P_{\text{Battery}}=P_{\text{Cell1}}\times P_{\text{Cell2}}=0.02\times 0.02=0.0004$

Total Power Loss (AND gate): $P_{\text{Top}}=P_{\text{Solar}}\times P_{\text{Battery}}=0.016\times 0.0004=6.4\times 10^{-6}$

Interpretation: 6.4 failures per million missions. Top level par AND gate (solar/battery ke beech redundancy) risk ko drastically reduce kar deta hai!

Architecture:

• Do transponders (A, B) cold redundancy mein
• Ek high-gain antenna (HGA)
• Loss tab hota hai agar: (Transponder A fail ho AUR Transponder B fail ho) YA HGA fail ho

Fault Tree:

        [Loss of Comm]
              |
          [OR Gate]
         /         \
    [AND Gate]   [HGA Fail]
      /    \
  [Trans-A] [Trans-B]

Given:

• P(Trans-A) = 0.01
• P(Trans-B) = 0.01
• P(HGA) = 0.05

Calculate: $P(\text{Both Transponders})=0.01\times 0.01=0.0001$ $P(\text{Top})=1-(1-0.0001)(1-0.05)$ $=1-(0.9999)(0.95)=1-0.94991=0.05009$

YEH STRUCTURE KYUN?

• Transponders AND mein: redundancy acchi hai (dono ka fail hona zaroori)
• OR with HGA: antenna single point of failure hai (risk par dominate karta hai!)
• Insight: HGA 99.8% risk drive karta hai (0.05 vs 0.001). Reliability improvements wahan focus karo!

FTA Mein Common Mistakes

Galti Ko Steel-man Karo: Jab hum kehte hain "power fail hoti hai agar solar AND battery dono fail hों," ek student sochta hai "dono ka fail hona unlikely hai." SAHI HAI! Isliye hum AND gates CHAHTE hain. Confusion tab aata hai jab "solar fail hoti hai agar tracking OR deployment fail ho" analyze karte hain — yahan OR ka matlab hai "akela ek hi use khatam kar deta hai," jo WORSE hai.

The Fix:

• AND gate → Redundancy → Zyada kam probability (chhote numbers multiply karo → aur chhota)
• OR gate → Single points of failure → Zyada probability (probabilities add karo → bada)

Kyun Sahi Lagta Hai: Formula simpler hai, aur hum shared dependencies bhool jaate hain.

The Fix:

• Common cause failures (CCF) identify karo
• Shared components ke liye alag basic events add karo
• Example: Agar dono transponders ek power regulator share karte hain, to "Regulator Fail" ko ek OR input ke roop mein add karo

Kyun Sahi Lagta Hai: Hum analysis khatam karna chahte hain.

The Fix: Tab tak decompose karte raho jab tak tum un events tak nahi pahunch jaate jinke measurable failure rates databases mein hoon (MIL-HDBK-217, NASA parts library). Resolution ka level available reliability data se match hona chahiye.

FTA Kab Use Karein Baaki Methods Ki Jagah

FTA Ki Strengths:

• Effort ko specific critical failures par focus karta hai
• AND gates (acchi redundancy) vs OR gates (weak points) reveal karta hai
• Quantitative probability calculation
• Visual logic structure

FTA Ki Limitations:

• Yeh jaanna zaroori hai ki kaunsa failure analyze karna hai (assume karta hai ki top event identify ho chuka hai)
• Chosen top event se connected nahi failures miss ho sakti hain
• Time-dependent failures (sequence matters) model karna mushkil hai

Spacecraft Design Se Connections

James Webb Space Telescope:

• Sunshield deployment ke liye FTA (107 release mechanisms)
• Massive OR gate structure reveal hua (koi bhi mechanism fail → mission fail)
• Per-mechanism reliability of 0.999+ ki requirement padi

ISS Life Support:

• "Loss of cabin atmosphere" ke liye FTA
• Redundant pumps/valves AND gates create karte hain
• Quantitative analysis ne critical paths mein triple redundancy justify ki

Active Recall Practice

Fault Tree Analysis ek tasveer banana hai un saare tareekon ki jिनसे kuch bura (jaise ek rocket ka power khona) ho sakta hai. Tum upar se buri cheez se shuru karte ho aur baar baar poochhhte rehte ho "Woh kya cause karega?" jab tak simple parts tak nahi pahunch jaate jo bas toot jaate hain. Phir tum AND aur OR use karte ho yeh dikhane ke liye ki kya tumhare paas acchhe backups hain (AND) ya dangerous single problems hain (OR). Yeh tasveer engineers ko yeh dekhne mein help karti hai ki kahan zyada backups add karein ya parts ko zyada reliable banayein.

Redundancy AND gates create karta hai = accha armor.
Single points of failure OR gates create karte hain = ek rupture mission khatam.

Connections

• Failure Modes and Effects Analysis (FMEA) - FTA ka Bottom-up complement
• Reliability Block Diagrams - Fault trees ka success-oriented dual
• Common Cause Failure Analysis - FTA mein shared dependencies handle karta hai
• Boolean Algebra and Logic Design - Gates ka mathematical foundation
• Monte Carlo Reliability Simulation - Alternative probabilistic approach
• Spacecraft Redundancy Architectures - Designs jo AND gates create karte hain
• Minimum Cut Sets - Sabse chhote combinations jo top event cause karte hain
• Single Point Failure Review - Basic events se connected OR gates dhundhna

#flashcards/physics

Fault Tree Analysis ki primary direction kya hai? :: Top-down (deductive): ek undesired system-level event se shuru hota hai aur component-level causes tak neeche jaata hai

Independent events ke liye, P₁, P₂, .., Pₙ inputs wale OR gate ke liye probability formula kya hai? :: P(Output) = 1 - ∏(1 - Pᵢ), saare inputs ke safe hone ki probability ka complement

FTA mein common cause failure (CCF) kya hai? :: Ek single failure jo multiple components ko simultaneously affect karta hai (e.g., shared power supply failure), independence assumptions violate karta hai

Concept Map