Worked examples — Verification methods — analysis, test, inspection, demonstration

3.6.28 · D3 · Physics › Spacecraft Structures & Systems Engineering › Verification methods — analysis, test, inspection, demonstra

Yeh page char verification methods ki practice arena hai. Parent note ne bataya tha ki analysis, test, inspection, aur demonstration kya hote hain. Yahan hum har tarah ka case throw karte hain — har method, woh edge cases jahan ek method fail hoti hai, degenerate "zero" inputs, limiting values, ek real word problem, aur ek nasty exam twist — aur har ek ko ground up se solve karte hain.

Kuch bhi compute karne se pehle, do words jo baar baar aayenge, unhe samajhna zaroori hai:

Figure (s01) ko concretely samjho. Upar wala mint green bar allowable hai — maano ek material $8$ units of stress tak le sakta hai yield karne se pehle. Neeche wala coral bar applied hai — launch sirf $5$ units produce karta hai. Kyunki mint bar lambi hai, uska tip coral tip se aage nikalti hai; woh stick-out piece lavender double arrow ke roop mein draw ki gayi hai jis par "margin" likha hai. Dashed vertical line bilkul coral (applied) tip par baithi hai — yahi "MoS zero line" hai. Picture aise padho: agar mint tip dashed line ke daayein utri, toh PASS; agar bilkul uski par utri, toh MoS $=0$; agar coral bar kabhi mint bar se aage nikal gayi, toh applied overshoot kar gaya aur FAIL. Neeche ke har example mein bas yahi picture hai alag bar lengths ke saath.

1 kyun ghataate hain? Kyunki $\frac{\text{allowable}}{\text{applied}}$ akela ek ratio hai: jab dono barabar hon toh yeh 1 hota hai. 1 ghatana us "just barely equal" point ko zero par shift kar deta hai, toh MoS ka sign instantly pass/fail bata deta hai. Yahi shift hai jis wajah se figure mein dashed zero line applied tip par hi baithi hai.

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Beam-stress tool ko scratch se banana

Neeche ke kai examples ko beam ki bending stress chahiye. Parent note ne collapsed formula $\sigma =\frac{6M}{b{h}^2}$ quote kiya tha; aao har symbol ko earn karein taaki koi example unexplained jump na kare.

Figure s02 walk-through. Slate rectangle beam ka cross-section hai (width $b$ across, height $h$ upar). Uske middle mein dashed horizontal line neutral axis hai — wahan ke fibres na stretched hain na squashed. Lavender arrow us middle line se top edge tak point karta hai; uski length $c=h/2$ hai, sabse door wala fibre. Height ke saath stacked chhote coral tick marks yaad dilate hain ki area ka har sliver $y^{2}dA$ contribute karta hai $I$ mein — jitna upar (bada $y$), utna zyada count, isliye $I$ $h^3$ ki tarah badhta hai.

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Scenario matrix

Is topic ka har problem in cells mein se kisi ek mein aata hai. Neeche har worked example us cell se tagged hai jo woh hit karta hai, aur sab milkar poora grid cover karte hain. Flow-schema neeche (figure s03) dikhata hai ki cells ek single starting question se kaise fan out karti hain, taaki tum dekh sako ki hum poori space cover karte hain, na ki bas ek list.

Figure s03 walk-through. Butter box on the left woh single root question hai jisse har verification shuru hoti hai: "Main kaunse verification scenario mein hun?" Isse daayein nikalti hain eight arrows, table ki har row ke liye ek — arrow ka colour bas related cells group karta hai (mint = passes, coral = fails/wrong-method, lavender = degenerate/limit, butter = statistical). Upar se neeche padhte hue leaf boxes hain cells A se H tak; har box case class ka naam aur uski one-line "trick" batata hai. Picture ka point coverage hai: chahe kisi bhi node par land karo, neeche ek example usse solve karta hai, isliye fan mein koi missing branch nahi hai.

#	Cell (case class)	Tricky kyun hai	Example
A	Analysis, positive margin	Seedha number-crunch, PASS	Ex 1
B	Analysis, negative margin (FAIL)	Sign negative ho jaata hai → redesign	Ex 2
C	Zero / degenerate input	Divide-by-zero, undefined MoS	Ex 3
D	Limiting value (MoS = 0)	Bilkul boundary par	Ex 4
E	Test, qualification vs acceptance	Do alag levels, same unit	Ex 5
F	Wrong method chosen	Requirement test nahi ho sakti → inspect karni padegi	Ex 6
G	Real-world word problem	Prose se numbers extract karo	Ex 7
H	Exam twist: combined + statistical	Do uncertainties combine hoti hain, phir verify	Ex 8

Physics tools ko jaise jaise aate hain link karte hain: Margin Philosophy, Finite Element Analysis, Thermal Math Modeling, Vibration Testing, Acceptance Testing, Traceability Matrix.

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Cell A — Analysis, positive margin

Forecast: compute karne se pehle guess karo — kya 8g par ek 1 cm-thick aluminium bar par 2 kg mass pass karega, ya yeh bahut thin hai? PASS ya FAIL abhi likho.

1. Inertial force. $F=m(ng)=2\times (8\times 9.81)=156.96\text{ N}$. Yeh step kyun? Launch ke dauran acceleration extra gravity ki tarah feel hoti hai (ek quasi-static load); Newton's second law load factor "$n=8$" ko ek real force mein convert karta hai jo bracket carry karni padti hai.
2. Bending moment. $M=FL=156.96\times 0.10=15.696\text{ N⋅m}$. Yeh step kyun? Fixed root se door ek force bar ko bend karti hai. Moment $M$ (upar define kiya) woh bending leverage measure karta hai — mass jitna door, root par utna zyada twist.
3. Bending stress transparent form $\sigma =\frac{6M}{b{h}^2}$ use karke (upar derive kiya, jahan $M$ step 2 ka bending moment hai): $\sigma =\frac{6\times 15.696}{0.02\times (0.01{)}^2}=\frac{94.176}{2\times 10^{-6}}=4.7088\times 10^7\text{ Pa}=47.088\text{ MPa}.$ Yeh step kyun? Stress bending effort hai jo sabse door fibre par concentrated hoti hai (neutral axis se doori $c=h/2$); top fibre sabse zyada stretches hai, isliye hum wahan check karte hain — agar worst fibre survive kar le, sab kar lete hain.
4. Margin. $\text{MoS}=\frac{270}{47.088}-1=5.734-1=4.734$. Yeh step kyun? Yeh pass/fail verdict hai. Positive aur bada → bracket bahut zyada over-strong hai.

Verify: Units — Pa = N/m². $47.088\text{ MPa}$ $270\text{ MPa}$ se bahut neeche hai (allowable ka sirf ek-chhata), isliye MoS $\approx 4.73>0$ → PASS. s01 picture mein, coral "applied" bar tiny hai aur mint "allowable" bar uske upar tower karti hai. Sanity: ek chunky 1 cm bar jo sirf 2 kg carry kare easily strong honi chahiye, jo bade positive margin se match karta hai.

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Cell B — Analysis, negative margin (FAIL)

Forecast: thinner aur higher g — kya tumhara khayal hai yeh phir bhi pass karega? Guess karo.

1. New force. $F=m(ng)=2\times (12\times 9.81)=235.44\text{ N}$. Yeh step kyun? Higher load factor ka matlab zyada inertial force — same Newton's-law conversion, ab $n=12$ ke saath.
2. New bending moment phir stress. $M=FL=235.44\times 0.10=23.544\text{ N⋅m}$, toh $\sigma =\frac{6M}{b{h}^2}=\frac{6\times 23.544}{0.02\times (0.004{)}^2}=\frac{141.264}{3.2\times 10^{-7}}=4.4145\times 10^8\text{ Pa}=441.45\text{ MPa}.$ Yeh step kyun? Notice karo $h$ denominator mein squared appear karta hai — $h$ ko 10 mm se 4 mm tak shrink karna ($2.5\times$ thinner) stress ko $2.5^2=6.25\times$ blast kar deta hai. Isliye thickness itni matter karti hai.
3. Margin. $\text{MoS}=\frac{270}{441.45}-1=0.6116-1=-0.3884$. Yeh step kyun? Negative! s01 picture mein coral "applied" bar ab mint "allowable" bar se aage nikal gayi hai — tip mint tip ke baayein land karti hai — toh part yield kar jaata hai.

Verify: $441.45\text{ MPa}$ $270\text{ MPa}$ se bada hai → MoS $<0$ → FAIL. Finite Element Analysis is region ko red flag karega. Fix: $h$ mota karo, ribs add karo, ya stronger alloy lo. Sanity check: ratio $\frac{441.45}{47.088}=9.375=1.5\times 6.25$ — 1.5 load-factor increase hai ($12/8$), 6.25 thickness effect hai. Physics self-consistent hai.

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Cell C — Zero / degenerate input

Forecast: MoS $=\frac{{\sigma }_y}{\sigma }-1$. $\sigma =0$ daalo… kya hoga? Trap guess karo.

1. Formula directly try karo. $\text{MoS}=\frac{270}{0}-1$. Yeh step kyun? Zero se divide karna mathematically undefined hai — yeh koi real number nahi hai, aur isse "infinity" kehna sirf informal shorthand hai "grows without bound as denominator shrinks" ke liye. Yahan koi finite margin nahi likh sakte.
2. Physically interpret karo. Stress ka exactly zero hona matlab hai ki member is load case mein bilkul loaded nahi hai. Yeh step kyun? Ek picture: figure s01 mein coral "applied" bar ki zero length hai, toh mint "allowable" bar bina kisi upper limit ke usse aage nikal jaati hai. Ratio $\frac{270}{\sigma }$ ki koi finite value nahi hai — jaise $\sigma$ zero ke paas jaata hai, woh bas badhta rehta hai.
3. Sahi engineering statement. Report karo: "MoS not applicable (denominator zero — undefined); member unloaded in this load case; verification satisfied by inspection of the load path, not by margin computation." Yeh step kyun? Traceability Matrix par kabhi literal "$\infty$" ya koi banaya-hua bada number mat likho. Zero applied load ek alag verification story hai — tum inspect karte ho ki load path genuinely is member ko bypass karta hai.

Verify: Limit check — jaise $\sigma \to 0^{+}$, ratio $\frac{{\sigma }_y}{\sigma }$ without bound badhta hai (koi finite limit exist nahi karta). Formula isliye trap confirm karta hai: MoS $\sigma =0$ par undefined hai, isliye special reporting rule. Kabhi koi finite lie nahi, aur paperwork mein literal $\infty$ kabhi nahi.

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Cell D — Limiting value (MoS exactly zero)

Forecast: applied exactly allowable ke barabar hai. Pass ya fail? Requirement mein "$>0$" ke baare mein socho.

1. MoS compute karo. $\text{MoS}=\frac{200}{200}-1=1-1=0$. Yeh step kyun? Figure s01 mein coral aur mint bars ab identical length ke hain; lavender extra-length arrow ki zero length hai; mint tip bilkul dashed zero line par land karti hai.
2. Requirement dhyan se padho. Usne kaha MoS $>0$ (strictly greater), na ki $\ge 0$. Yeh step kyun? Boundary case ek inequality symbol par turn karta hai. $\text{MoS}=0$ ka matlab hai part precisely design load par yield karta hai — manufacturing scatter, temperature effect, ya model error ke against zero protection.
3. Verdict: MoS $=0$ "MoS $>0$" satisfy nahi karta. FAIL (marginal). Yeh step kyun? Isliye hi Margin Philosophy positive margin demand karti hai — reality mein hamesha scatter hoti hai, aur zero-margin part ek coin flip hai.

Verify: $\frac{200}{200}-1=0$ exactly; $0>0$ false hai → correctly strict requirement fail karti hai. Limiting behaviour confirm: yeh Ex 1 (positive) aur Ex 2 (negative) ke beech ka razor's edge hai.

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Cell E — Test: qualification vs acceptance levels

Forecast: kaun sa level higher hai, aur tum flight unit ko design proof se lower level par kyun test karoge? Padhne se pehle guess karo.

1. Qualification level $=\text{Flight Limit}\times 1.25=8\times 1.25=10g$ RMS. Yeh step kyun? Qualification prove karta hai ki design flight se worse survive karta hai, us design se bane saare future units ko cover karta hai. $1.25$ manufacturing + environment scatter ke against margin khareedta hai.
2. Acceptance level $=\text{Flight Limit}\times 1.0=8\times 1.0=8g$ RMS. Yeh step kyun? Acceptance Testing confirm karta hai ki yeh specific unit mein koi build defects nahi hain (cold solder, loose fastener) flight hardware ko over-stress kiye bina — tumne design pehle hi 10 g par qual unit par prove kar diya, isliye flight unit ko kabhi 10 g tak punish nahi karte.
3. Har ek kya verify karta hai. Qual → design margin. Acceptance → ek article ki workmanship. Yeh step kyun? Same requirement, do sawaal: "kya design achha hai?" vs "kya yeh copy sahi build ki gayi hai?" Alag levels alag sawaalon ke jawab dete hain.

Verify: $8\times 1.25=10$; $8\times 1.0=8$. Qual ($10$) $>$ Acceptance ($8$) aur Acceptance Flight ($8$) ke barabar — sahi ordering. Ratio qual/acceptance $=10/8=1.25$, design margin factor se match karta hai.

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Cell F — Wrong method chosen (method-selection twist)

Forecast: kya tum mass test kar sakte ho? Testing ka yahan matlab kya hoga? Sahi method guess karo.

1. Galat method reject karo. Mass ek static property hai — yeh operation ya environment se emerge nahi hoti. Mission simulation ek demonstration hai aur mass ke baare mein kuch reveal nahi karta. Yeh step kyun? Parent note ka intuition: tum mass "test" nahi kar sakte — tum bas ise weigh karte ho, aur weighing inspection hai, test nahi.
2. Sahi method apply karo — inspection. Assembled, dry spacecraft ko calibrated scale par rakho: yeh $174.3\text{ kg}$ read karta hai. Yeh step kyun? Inspection directly physical property ko limit ke against observe karta hai — koi model nahi, koi environment nahi. Weighing inspection ka textbook example hai.
3. Limit ke against verify karo. Kya $174.3\le 180$? Haan. Mass ke liye hum stress-style MoS ke bajaye remaining allocation report karte hain: spare $=180-174.3=5.7\text{ kg}$, jo $\frac{5.7}{180}=3.17\%$ under budget hai. Yeh step kyun? Mass margin spare allocation ke kilograms mein track ki jaati hai, jo seedha mass budget tracking mein jaati hai — na ki stress ke liye use kiye jaane wale ratio-minus-one MoS ke roop mein.
4. Verdict state karo. $174.3\text{ kg}<180\text{ kg}$ → PASS by inspection, $5.7\text{ kg}$ ($3.17\%$) remaining margin ke saath. Yeh step kyun? Requirement meet hui aur method ab sahi hai; valid verification entry ke liye dono number aur method sahi hone chahiye.

Verify: $174.3<180$ → PASS. Method-fit check: static scalar property → inspection ✓ (test nahi, analysis nahi, demonstration nahi). Spare $180-174.3=5.7$ kg; $5.7/180=0.03167\Rightarrow 3.17\%$. Numeric answer aur method selection dono confirm hain.

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Cell G — Real-world word problem

Forecast: battery kaun se limit violate karne ke zyada karib hai — hot ceiling ya cold floor? Guess karo.

1. Prose se numbers extract karo. Allowable band $[0,40{]}^{\circ }\text{C}$; predicted band $[5,38{]}^{\circ }\text{C}$. Yeh step kyun? Word problem numbers sentences mein chhupa deta hai; pehla kaam hamesha applied values (predicted extremes) aur allowable values (do limits) ko name karna hai.
2. Hot-side margin. Ceiling tak doori $=40-38=2^{\circ }\text{C}$ of headroom. Yeh step kyun? Two-sided limit ke liye tum dono ends check karni hoti hain; temperature high ya low fail ho sakti hai. Hot case ceiling test karta hai.
3. Cold-side margin. Floor tak doori $=5-0=5^{\circ }\text{C}$ of headroom. Yeh step kyun? Cold case floor test karta hai. Requirement poore orbit mein hold ho isliye dono margins positive hone chahiye.
4. Overall verdict. Dono $2^{\circ }\text{C}$ aur $5^{\circ }\text{C}$ $>0$ hain → PASS by analysis. Tighter margin hot side hai ($2^{\circ }\text{C}$). Yeh step kyun? Tightest margin report karna model-validation team ko batata hai ki ek chhoti si model error kahin matter karegi — hot ceiling risk driver hai.

Verify: $40-38=2>0$ ✓ aur $5-0=5>0$ ✓. Predicted band $[5,38]\subset [0,40]$ → requirement met. Tighter margin correctly hot side ke roop mein identified hai ($2<5$).

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Cell H — Exam twist: uncertainties combine karo, phir verify karo

Forecast: do 10% scatters — kya yeh 20% add hote hain? Combine karne se pehle guess karo.

1. Independent uncertainties quadrature mein combine karo. ${\sigma }_{\text{total}}=\sqrt{{\sigma }_{\text{mfg}}^2+{\sigma }_{\text{env}}^2}=\sqrt{0.10^2+0.10^2}=\sqrt{0.02}\approx \mathrm{0.1414.}$ Yeh step kyun? Independent random errors seedha add nahi hote (woh $0.20$ hoga); woh root ke neeche squares ke roop mein add hote hain — uncertainties ke liye Pythagorean rule. Do perpendicular arrows of length 0.10 imagine karo; unka resultant $0.10\sqrt{2}$ hai, 0.20 nahi.
2. High ($99.7\%$) coverage ke liye $3\sigma$ tak jaao. $\text{Test Level}=\text{Mean}+3{\sigma }_{\text{total}}=1.0+3(0.1414)=1.0+0.4243=\mathrm{1.4243.}$ Yeh step kyun? Normal distribution par $3\sigma$ band $99.7\%$ saare units capture karta hai — is level tak testing ka matlab hai ki almost har real flight unit tumhare test se bounded hai.
3. Practical $1.25\times$ factor se compare karo. Yeh step kyun? Pure statistics $\approx 1.42\times$ demand karta hai, lekin industry standard $1.25\times$ hai (Ex 5). Toh $1.25$ full $3\sigma$ se less conservative hai — ek deliberate cost/risk compromise, hardware ko over-test aur over-build karne se bachne ke liye thoda $3\sigma$ se kam coverage accept karna. Yahi Margin Philosophy hai ek number mein.

Verify: $\sqrt{0.10^2+0.10^2}=\sqrt{0.02}=0.14142$; $1+3(0.14142)=1.42426$. Aur $1.25<1.4243$, confirming ki practical factor less-conservative compromise hai. $0.20$ nahi: quadrature ≠ linear sum ✓.

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Recall Quick self-test

Ex 1 bracket MoS (2 kg, 10 mm, 8g) ::: $+4.73$ → PASS Ex 2 bracket MoS (4 mm, 12g) ::: $-0.39$ → FAIL (stress 441 MPa > 270) MoS jab applied stress exactly zero ho ::: undefined (division by zero) — report karo "member unloaded", inspection se verify karo Kya MoS $=0$ "MoS $>0$" satisfy karta hai? ::: Nahi — strict inequality, marginal FAIL 8g flight limit ke liye Qual vs acceptance level ::: 10g (design) vs 8g (workmanship) Mass requirement verify karne ka sahi method ::: Inspection (weigh karo), test nahi Do independent 10% scatters combine karo ::: $\sqrt{0.1^2+0.1^2}=0.1414$, 0.20 nahi $3\sigma$ statistical test level vs industry $1.25\times$ ::: $1.42\times$ (statistics) vs $1.25\times$ (practical, less conservative) Load factor $n$ ka kya matlab hai? ::: ek plain dimensionless number of "g's" — acceleration $n$ times ordinary gravity ki tarah feel hoti hai $\sigma =Mc/I$ kaise $6M/(b{h}^2)$ ban jaata hai? ::: $c=h/2$, $I=b{h}^3/12$ daalo; algebra collapse ho jaata hai $6M/(b{h}^2)$ mein