Mechanical Engineering 2023 Paper I 50 marks Calculate

Q4

(a) Two involute gears in mesh have pressure angle of 20° and module of 6 mm. The number of teeth on pinion is 28 and the larger gear has 52 teeth. Calculate the following : (20 marks) (i) Contact ratio (ii) Angle of action of gear wheel and pinion (iii) Ratio of sliding velocity to rolling velocity at (I) Starting of contact, (II) Pitch point, (III) End of contact. Consider the addenda on pinion and gear wheel as equal to one module. (b) A solid circular shaft made of steel is subjected to a bending moment of 12 kNm and a twisting moment of 16 kNm. In a simple uniaxial tensile test of the same material it gave the stress at yield point = 300 N/mm². Assuming factor of safety = 2, estimate the minimum diameter required for the circular shaft using (i) Maximum Principal Stress theory, and (ii) Maximum Shear Stress theory. (20 marks) (c) Write short notes on the following : (10 marks) (i) Ceramics (ii) Nano-materials

हिंदी में प्रश्न पढ़ें

(a) दो अंतर्वलित गियर (मेश में) का दाब कोण 20° तथा प्रमाणक (मॉड्यूल) 6 mm है । पिनियन पर दांतों की संख्या 28 तथा बड़े गियर पर 52 दांते हैं । निम्नलिखित की गणना कीजिए : (20 अंक) (i) संपर्क अनुपात (ii) गियर पहिये और पिनियन का क्रिया कोण (iii) सर्पण वेग और रोलिंग वेग का अनुपात (I) संपर्क के प्रारंभ में, (II) पिच बिंदु पर, (III) संपर्क की समाप्ति पर । पिनियन और गियर व्हील पर ऐडेंडा एक प्रमाणक (मॉड्यूल) के बराबर मान लीजिए । (b) एक इस्पात के ठोस वृत्तीय शाफ्ट पर 12 kNm का बंकन आघूर्ण और 16 kNm का ऐंठन आघूर्ण लगाया जाता है । एक सामान्य एक-अक्षीय तनन परीक्षण में इस पदार्थ का प्रारंभ बिंदु प्रतिबल 300 N/mm² पाया गया । सुरक्षा गुणक को 2 मानते हुए इस वृत्तीय शाफ्ट के न्यूनतम आवश्यक व्यास का आकलन (i) अधिकतम मुख्य प्रतिबल सिद्धांत, तथा (ii) अधिकतम अपरूपण प्रतिबल सिद्धांत का प्रयोग करते हुए कीजिए । (20 अंक) (c) निम्नलिखित पर संक्षिप्त टिप्पणियाँ लिखिए : (10 अंक) (i) मृत्तिका-शिल्प (ii) नैनो-पदार्थ

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Directive word: Calculate

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How this answer will be evaluated

Approach

Calculate requires systematic numerical solutions with clear derivations. Allocate ~40% time to part (a) gear calculations (20 marks) covering contact ratio, angle of action, and sliding/rolling velocity ratios; ~35% to part (b) shaft design using failure theories (20 marks); and ~25% to part (c) short notes on ceramics and nano-materials (10 marks). Begin each numerical part with given data listing, show all formulae with substitutions, and conclude part (c) with applications relevant to Indian manufacturing (ISRO, DRDO, Make in India).

Key points expected

  • Part (a): Base circle radii rb1 = 78.93 mm, rb2 = 146.59 mm; addendum circle radii ra1 = 90 mm, ra2 = 162 mm; path of contact KP = 15.79 mm, PL = 16.21 mm; contact ratio = 1.34
  • Part (a): Angle of approach = 11.47°, angle of recess = 11.79° for pinion; angle of action = 23.26° for pinion, 12.48° for gear wheel
  • Part (a): Sliding/rolling velocity ratios — start: 0.518, pitch point: 0, end of contact: 0.518 (magnitudes equal, directions reverse)
  • Part (b): Equivalent bending moment Me = 17.09 kNm, equivalent twisting moment Te = 20 kNm; diameter by MPS theory = 87.6 mm, by MSST = 95.8 mm
  • Part (b): Correct application of σ_max = (16/πd³)[Me + √(Me² + Te²)] for MPS and τ_max = (16/πd³)√(Me² + Te²) for MSST with σ_y/2N
  • Part (c): Ceramics: ionic/covalent bonding, high hardness, low toughness, thermal shock resistance, applications in ISRO thermal protection tiles, cutting tools
  • Part (c): Nano-materials: size effects (1-100 nm), quantum confinement, high surface area-to-volume ratio, applications in DRDO nano-composites, drug delivery, energy storage
  • All parts: Clear statement of assumptions, standard formulae cited (AGMA/IS standards for gears, IS 800 for shafts), units in mm, N, MPa consistently

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies involute geometry fundamentals for part (a): path of contact as sum of approach and recess, contact ratio definition as arc of action/base pitch; for part (b) distinguishes between MPS and MSST failure criteria with proper principal stress derivation; for part (c) accurately describes ceramic bonding types and nano-scale phenomena (quantum confinement, surface effects).Uses correct gear formulae but confuses approach/recess paths or contact ratio vs. contact length; applies failure theories with minor principal stress errors; describes ceramics and nano-materials generically without specific property mechanisms.Confuses involute with cycloid tooth profile, treats contact ratio as linear distance; applies von Mises instead of MSST or uses wrong safety factor application; describes ceramics as 'brittle materials' only or confuses nano with micro-scale.
Numerical accuracy20%10All six numerical sub-parts correct: contact ratio 1.34 (or 1.33-1.35), angles of action within ±0.5°, sliding/rolling ratios ±0.02, shaft diameters MPS ≈ 88 mm and MSST ≈ 96 mm with consistent rounding and unit handling throughout.Contact ratio correct but angle of action errors due to radian/degree confusion; shaft diameter calculations show correct formulae but arithmetic slips in final cube root; one part (a) or (b) sub-calculation significantly off.Multiple numerical errors: contact ratio >2 or <1 (physically impossible), sliding velocity ratio not zero at pitch point, shaft diameters differ by factor of 2 or more from correct values; inconsistent units (kN vs N, mm vs m).
Diagram quality20%10Clear involute gear mesh diagram for part (a) showing: base circles, pitch circles, addendum circles, pressure line, path of contact (KP and PL labelled), pitch point P; for part (b) free body diagram of shaft with bending and torsion moments indicated; part (c) optional nano-structure schematic.Gear diagram drawn but missing labels for key points (K, P, L) or pressure angle not indicated; shaft shown as simple line without moment indications; diagrams present but not aiding solution clarity.No diagrams despite gear geometry being inherently visual; or incorrect diagrams showing non-involute profiles, wrong mesh configuration; shaft without any loading indication.
Step-by-step derivation20%10Every formula stated before substitution: rb = r cos φ, ra = r + a, path of contact = √(ra² - rb²) - r sin φ for each gear, contact ratio = path of contact/(πm cos φ); for shaft, explicit derivation of principal stresses from combined loading, substitution into failure criteria with algebraic simplification shown.Key formulae stated but some intermediate steps skipped (e.g., jumps from √(ra² - rb²) to numerical value); shaft derivation shows equivalent moments but skips stress transformation details; working legible but condensed.Final answers stated without formulae; or incorrect formulae used throughout (e.g., contact ratio = T1/T2, shaft diameter from σ = M/Z ignoring torsion); no derivation for part (c) short notes.
Practical interpretation20%10For gears: interprets contact ratio 1.34 as indicating continuous mesh with 34% overlap, smooth operation; sliding velocity analysis explains wear patterns at approach vs recess; for shaft: comments that MSST gives conservative (larger) diameter, MPS more economical, selection depends on failure mode concern; for materials: cites Indian applications—nano-materials in ISRO/DRDO lightweight structures, ceramics in Bharat Heavy Electricals turbine blades.Brief comment that contact ratio >1 ensures continuous mesh; notes MSST gives larger diameter than MPS without explaining why; mentions general applications without Indian context.No interpretation of numerical results; or incorrect interpretation (e.g., contact ratio <1 acceptable, smaller shaft diameter safer); part (c) lacks any application examples.

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