Mechanical Engineering 2025 Paper I 50 marks Solve

Q3

(a) (i) The recoil mechanism of a gun consists of a critically damped spring-damper system. The maximum permissible recoil distance of the gun is specified as 0·5 m. If the initial recoil velocity of the gun is 10 m/s and the mass of the gun is 500 kg, determine the spring stiffness of the recoil mechanism and the critical damping coefficient of the damper. (10 marks) (ii) A punching press executes 10 holes per minute in a 25 mm thick plate. The diameter of each hole is 20 mm. The punch has a stroke of 60 mm and the punch moves with uniform velocity throughout. A flywheel is attached to the press and the mean speed of the flywheel is 25 m/s. If punching requires 10 N-m of energy per mm² of the sheared area, find the power needed to operate the punching press. Further determine the mass of the flywheel required if the total fluctuation of speed is restricted to 5% of the mean speed. (10 marks) (b) A beam AB under loading is shown in the figure. The rigid bar DEF is welded at a point D. The Young's modulus of the beam material is 200 GPa and its moment of inertia is 20 × 10⁶ mm⁴. (i) Draw shear force diagram. (ii) Draw bending moment diagram. (iii) Find the maximum bending moment and its location. (iv) Find deflection at C. (20 marks) (c) Identify the different types of defects and their causes in a steel article after it is hardened. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) एक बंदूक का प्रतिक्षेप तंत्र एक महत्वपूर्ण रूप से डैम्प्ड स्प्रिंग-अवमंदक (स्प्रिंग-डैम्पर) प्रणाली से बना है। बंदूक की अधिकतम अनुमत प्रतिक्षेप (रिकॉइल) दूरी 0·5 m के रूप में निर्दिष्ट की गई है। यदि बंदूक की प्रारंभिक प्रतिक्षेप वेग 10 m/s तथा बंदूक का द्रव्यमान 500 kg है, तो प्रतिक्षेप तंत्र की स्प्रिंग दृढ़ता तथा अवमंदक (डैम्पर) का कांतिक अवमंदन गुणांक निर्धारित कीजिए। (10 अंक) (ii) एक छिद्रक प्रेस (पंचिंग प्रेस) 25 mm मोटी प्लेट में 10 छिद्र प्रति मिनट निष्पादित करती है। प्रत्येक छिद्र का व्यास 20 mm है। छिद्रक का आघात (स्ट्रोक) 60 mm है तथा छिद्रक सदैव एकसमान बेग के साथ चलता है। प्रेस के साथ एक गतिपालक चक्र (फ्लाइव्हील) जुड़ा हुआ है तथा गतिपालक चक्र की औसत गति 25 m/s है। यदि पंचिंग को अपरूपण क्षेत्रफल के 10 N-m प्रति mm² की ऊर्जा की आवश्यकता होती है, तो पंचिंग प्रेस के संचालन में लगने वाली शक्ति (पावर) ज्ञात कीजिए। यदि गति का कुल उतार-चढ़ाव औसत गति के 5% तक सीमित है, तो गतिपालक चक्र के द्रव्यमान की आवश्यकता निर्धारित कीजिए। (10 अंक) (b) भार सहित एक धरन (बीम) AB चित्र में दर्शाई गई है। एक दृढ़ दंड DEF बिंदु D पर वेल्डित है। धरन सामग्री का यंग मापांक 200 GPa तथा इसका जड़त्व आघूर्ण 20 × 10⁶ mm⁴ है। (i) अपरूपण बल आरेख बनाइए। (ii) बंकन आघूर्ण आरेख बनाइए। (iii) अधिकतम बंकन आघूर्ण तथा इसकी स्थिति ज्ञात कीजिए। (iv) C पर विस्थेप ज्ञात कीजिए। (20 अंक) (c) किसी कठोरीकृत इस्पात वस्तु में विभिन्न प्रकार के दोषों तथा उनके कारणों की पहचान कीजिए। (10 अंक)

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Approach

Solve each sub-part systematically with clear section headers. For (a)(i)-(ii), apply vibration theory and flywheel energy equations respectively, showing all substitutions. For (b), draw SFD/BMD first, then use Macaulay's method or moment-area for deflection at C—allocate ~40% time here as it carries 20 marks. For (c), enumerate heat treatment defects with metallurgical causes. Conclude each numerical part with units and physical reasonableness checks.

Key points expected

  • (a)(i) Critical damping: uses x_max = (v_0/ω_n)*e^(-1) for critical damping, or solves characteristic equation with ζ=1; finds k = mω_n² and c_c = 2√(mk)
  • (a)(i) Correct values: k ≈ 20,000 N/m (or 20 kN/m), c_c ≈ 6,324 Ns/m (or 6324 kg/s) from x_max = 0.5 m, v_0 = 10 m/s
  • (a)(ii) Energy per hole: E = 10 N·mm/mm² × π×20×25 mm² = 15,708 J; power = (10×15708)/60 = 2,618 W ≈ 2.62 kW
  • (a)(ii) Flywheel mass: uses ΔE = Iω²C_s = ½m(v₂²-v₁²); with C_s=0.05, finds m ≈ 168 kg (or similar based on exact formulation)
  • (b) SFD: shows reactions, constant shear in segments, jump at D due to rigid bar, correct sign convention
  • (b) BMD: linear/parabolic variations, maximum moment location identified (typically near D or under load), values at key points
  • (b) Deflection at C: applies conjugate beam method or Macaulay's integration with boundary conditions; δ_C ≈ 2-5 mm range depending on loading
  • (c) Hardening defects: quenching cracks (thermal gradients), distortion (uneven cooling), retained austenite (insufficient cooling), soft spots (decarburization/uneven heating), oxidation/scaling

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies critically damped vibration theory for (a)(i) with ζ=1 condition; uses energy method for flywheel with proper C_s definition; identifies beam as statically indeterminate or determinate correctly; names all major hardening defects (quench cracks, distortion, retained austenite, decarburization) with correct metallurgical mechanisms.Uses correct general formulas but minor errors in damping regime identification or flywheel coefficient interpretation; beam analysis mostly correct but misses rigid bar effect; lists 3-4 defects with partial cause explanation.Treats (a)(i) as underdamped or overdamped; confuses flywheel energy with power directly; beam analysis completely wrong (cantilever formulas for SS beam); defects listed without causes or confused with tempering defects.
Numerical accuracy20%10All calculations correct to 3 significant figures: (a)(i) k≈20 kN/m, c_c≈6.3 kNs/m; (a)(ii) Power≈2.62 kW, m≈168 kg; (b) reactions, moments, and δ_C consistent with EI=4×10⁶ N·m²; unit conversions (mm⁴ to m⁴, GPa to Pa) flawless.Final answers correct but one intermediate calculation error (e.g., sheared area as πd²/4 instead of πdt); beam deflection order of magnitude correct but 10-20% error; most units correct.Order of magnitude errors (k in MN/m instead of kN/m); power calculation ignores time conversion; EI value used directly as 20×10⁶ without GPa conversion; no units or inconsistent units throughout.
Diagram quality20%10SFD and BMD for part (b) drawn to scale with all salient values labelled, sign convention indicated, dimensions marked; rigid bar DEF effect shown as moment discontinuity at D; neat free-body diagrams for (a) and (b) supporting calculations.SFD/BMD shapes correct but values not labelled or scale inconsistent; FBDs present but incomplete; diagrams readable but lack professional finish.No diagrams for (b) despite 20 marks; or diagrams completely wrong (triangular BMD for distributed load); illegible sketches without labels.
Step-by-step derivation20%10Complete derivations: (a)(i) shows x(t)=(A+Bt)e^(-ω_n t) with initial conditions; (a)(ii) derives ΔE = 2E_k - E_punching and coefficient of fluctuation; (b) shows equilibrium equations, constructs M(x) piecewise, integrates twice with boundary conditions for deflection; (c) explains metallurgical basis for each defect.Key steps shown but skips some algebra (e.g., jumps from M(x) to deflection without integration constants); (c) lists defects with brief causes but no underlying mechanism.Final answers only with no working; or incorrect formulas memorized and applied blindly; (c) as bullet points without explanation.
Practical interpretation20%10Comments on physical reasonableness: recoil distance achievable, gun mass adequacy; flywheel speed fluctuation acceptable for press operation; beam deflection compared to span (L/250 or similar limit); hardening defects linked to Indian industry practices (IS specifications, oil vs water quenching selection); suggests preventive measures.Brief comment on one or two parts (e.g., 'deflection is small'); generic statement about quality control for heat treatment.No interpretation; or absurd comments (e.g., 'flywheel mass of 168 kg is too heavy for a car'); treats all parts as pure mathematics.

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