Mechanical Engineering 2021 Paper I 50 marks Calculate

Q2

(a) In a slider-crank mechanism, the lengths of the crank and connecting rod are 150 mm and 600 mm respectively. Locate all the I-centres of the mechanism for the position when the crank has turned 30° from IDC. Also, find the velocity of the slider and the angular velocity of the connecting rod, if the crank rotates at 30 rad/s. (15 marks) (b) A mass weighing 100 N is suspended from a spring of constant k = 4000 N/m. At time t = 0, it has a downward velocity of 1 m/s as it passes through the position of static equilibrium. Determine the following : (i) The static spring deflection. (ii) The natural frequency of the system. (iii) The displacement (x) of the mass as a function of time, where x is measured from the position of static equilibrium. (iv) The maximum acceleration attained by the mass. (15 marks) (c) Write the equations for shearing force and bending moment for various sections and draw SFD and BMD for the beam supported at A and B as shown in the figure. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) एक सर्पी-क्रैंक शेप्ट यंत्र युक्ति में क्रैंक तथा संयोजी दंड की लंबाई क्रमशः 150 mm तथा 600 mm है। क्रैंक के IDC से 30° घूमने के बाद की स्थिति के लिए यंत्र युक्ति के सभी I-केंद्रों का स्थान निर्धारित कीजिए। यदि क्रैंक 30 rad/s पर घूर्णन कर रहा हो, तो स्पर्क (स्लाइडर) का वेग तथा संयोजी दंड का कोणीय वेग भी ज्ञात कीजिए। (15 अंक) (b) एक द्रव्यमान जिसका भार 100 N है, एक ऐसी स्प्रिंग से लटका है जिसका स्थिरांक k = 4000 N/m है। समय t = 0 पर स्थैतिक स्थिरता (संतुलन) के स्थान से गुजरते समय इसका नीचे की ओर वेग 1 m/s है। निम्न का मान ज्ञात कीजिए : (i) स्थैतिक स्प्रिंग विस्थापन। (ii) तंत्र की स्वाभाविक आवृत्ति। (iii) समय के फलन के रूप में द्रव्यमान का विस्थापन (x), जहाँ x का मापन स्थैतिक संतुलन के स्थान से किया जाता है। (iv) द्रव्यमान द्वारा लब्ध अधिकतम त्वरण। (15 अंक) (c) चित्र में दर्शाए अनुसार टेक A और B पर आधारित (स्थित) एक धरन के विभिन्न काटों (सेक्शन्स) के लिए अपरूपक बल और बंकन आघूर्ण हेतु समीकरण लिखिए तथा SFD एवं BMD आरेखित कीजिए। (20 अंक)

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Directive word: Calculate

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How this answer will be evaluated

Approach

Calculate systematically across all three parts: for (a) locate I-centres using Kennedy's theorem and apply velocity analysis (spend ~30% time, 15 marks); for (b) solve the free vibration problem using energy methods or direct integration (spend ~30% time, 15 marks); for (c) derive SF and BM equations for all beam sections and construct diagrams (spend ~40% time, 20 marks). Begin each part with clear free-body diagrams, show all derivations, and conclude with properly labelled diagrams and physical interpretations.

Key points expected

  • Part (a): Locate all 6 I-centres (I12, I13, I14, I23, I24, I34) using Kennedy's theorem; I13 found via auxiliary points or direct construction
  • Part (a): Velocity of slider = ω₂ × (I12I24) × (I13I34)/(I12I13) or equivalent; angular velocity of connecting rod ω₃ = ω₂ × (I12I13)/(I13I34)
  • Part (b)(i): Static deflection δ_st = W/k = 100/4000 = 0.025 m = 25 mm
  • Part (b)(ii): Natural frequency ω_n = √(k/m) = √(4000×9.81/100) = 19.81 rad/s or f_n = 3.15 Hz
  • Part (b)(iii): x(t) = (v₀/ω_n)sin(ω_nt) = 0.0505 sin(19.81t) m, since x₀=0, v₀=1 m/s downward
  • Part (b)(iv): Maximum acceleration a_max = ω_n² × X_max = (19.81)² × 0.0505 = 19.81 m/s²
  • Part (c): Identify loading pattern on beam (UDL, point loads, moments as shown in figure); determine reactions at supports A and B
  • Part (c): Derive piecewise V(x) and M(x) equations for each region; locate points of zero shear and maximum moment

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10For (a): correctly applies Kennedy's three-centre theorem and velocity ratio theorem; for (b): recognizes this as free vibration with initial velocity, uses correct energy formulation mẍ + kx = 0; for (c): correctly identifies beam type, loading, and sign conventions for SF/BM.Applies correct formulas but with minor conceptual gaps (e.g., confuses absolute and relative I-centres, or treats forced vibration as free vibration); beam analysis mostly correct but sign convention inconsistent.Fundamental errors: treats I-centres as fixed points, uses wrong vibration equation (e.g., includes damping term), or confuses hogging/sagging moments in beam analysis.
Numerical accuracy20%10All numerical values correct: (a) slider velocity ≈ 3.75 m/s, ω_rod ≈ 5.77 rad/s; (b) δ_st = 25 mm, ω_n = 19.81 rad/s, x(t) coefficient = 0.0505 m, a_max = 19.81 m/s²; (c) reactions and critical values accurate to 2 decimal places; units consistent (mm, m, rad/s, m/s, m/s²).Most final answers correct but with minor arithmetic slips in intermediate steps; or correct method but wrong substitution (e.g., uses g=10 instead of 9.81); units mostly consistent.Multiple calculation errors; wrong order of magnitude (e.g., slider velocity 37.5 m/s instead of 3.75 m/s); units missing or grossly inconsistent (mm mixed with m without conversion).
Diagram quality20%10(a) Space diagram with all 6 I-centres clearly marked, velocity diagram to scale; (b) displacement/velocity/acceleration sketches with proper phase relationships; (c) complete SFD and BMD with all salient values labelled, dimensions shown, linear/parabolic/cubic curves correctly identified.Diagrams present but incomplete: I-centres diagram missing labels, SFD/BMD shapes correct but values omitted, or scale inconsistent.Diagrams missing or seriously flawed: no velocity diagram for (a), no SFD/BMD for (c), or curves drawn arbitrarily without mathematical basis.
Step-by-step derivation20%10(a) Shows complete I-centre location process using Kennedy's theorem, velocity polygon construction; (b) Derives from Newton's 2nd law or energy method, applies initial conditions systematically; (c) Shows FBD, equilibrium equations, derives V(x) and M(x) for each section, identifies critical points by differentiation.Derivations present but with gaps: jumps from I-centre location to velocity formula without showing velocity polygon; states differential equation solution without showing characteristic equation; beam equations correct but integration constants not fully justified.No derivations shown: states final formulas without justification; or fundamental errors in derivation (e.g., wrong differentiation of moment equation).
Practical interpretation20%10Links (a) to IC engine design (quick-return mechanisms, balancing); (b) relates to machine foundation isolation, critical speed avoidance; (c) connects to structural design of beams (section modulus selection, reinforcement placement), mentions Indian Standard codes (IS 456 for RCC, IS 800 for steel).Brief mention of practical relevance for each part but lacks depth; no code references or specific engineering applications.No interpretation; treats as pure mathematical exercise; or irrelevant/wrong applications cited.

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