Mechanical Engineering 2024 Paper I 50 marks Solve

Q4

(a) A governor of Hartnell type has each ball of weight 20 N and the lengths of vertical and horizontal arms of the bell crank lever are 125 mm and 65 mm, respectively. The fulcrum of the bell crank lever is at a distance of 100 mm from the axis of rotation. The maximum and minimum radii of rotation of the balls are 125 mm and 80 mm, and the corresponding equilibrium speeds are 325 rpm and 300 rpm, respectively. Find the stiffness of the spring and the equilibrium speed when the radius of rotation is 100 mm. (20 marks) (b) Determine the deflection at a point 1 m from the left-hand end of the beam loaded as shown in the figure below. Use double integration method. The beam is having a constant flexural rigidity of 0·6 MN-m² : (20 marks) (c) Enumerate different types of cams and followers commonly used. State their relative merits and demerits. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक हार्टनेल गवर्नर के प्रत्येक बॉल का वजन 20 N तथा बेल क्रैंक लीवर की उद्धवर्ध तथा क्षैतिजीय भुजाओं की लम्बाइयाँ क्रमशः 125 mm तथा 65 mm हैं। बेल क्रैंक लीवर का फलक्रम, घूर्णन अक्ष से 100 mm की दूरी पर स्थित है। बॉलों की अधिकतम तथा न्यूनतम घूर्णन त्रिज्याएँ 125 mm तथा 80 mm हैं, एवं संबंधित संतुलन गतियाँ क्रमशः 325 rpm तथा 300 rpm हैं। स्प्रिंग की दृढ़ता तथा उसकी संतुलन गति ज्ञात कीजिए, जब घूर्णन त्रिज्या 100 mm है। (20 अंक) (b) नीचे चित्र में दर्शाए गए भारित धरन के बाएँ किनारे से 1 m की दूरी पर विचलन ज्ञात कीजिए। दिशा: समाकल विधि का प्रयोग कीजिए। धरन की स्थिर आनमनी दृढ़ता 0·6 MN-m² है : (20 अंक) (c) सामान्यतः इस्तेमाल होने वाले विभिन्न प्रकार के कैम और फॉलोअर (अनुगामियों) को एक-एक कर बताइए। इनके सापेक्ष गुणों और दोषों को बताइए। (10 अंक)

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How this answer will be evaluated

Approach

Solve the three sub-parts sequentially, allocating approximately 40% time to part (a) due to its computational complexity, 40% to part (b) requiring double integration setup, and 20% to part (c) for enumeration. Begin with clear free-body diagrams for (a) and (b), show all equilibrium and integration steps, and conclude with practical applications for each mechanism.

Key points expected

  • Part (a): Correct governor geometry with a/b = 125/65 = 1.923; sleeve lift h = 45 mm for r = 80 to 125 mm; spring stiffness s = 2S/h where S found from moment equilibrium at both speeds
  • Part (a): Centrifugal forces F1 = mω1²r1 and F2 = mω2²r2; solving simultaneous moment equations yields spring stiffness ≈ 8.5-9.5 N/mm and equilibrium speed at r = 100 mm ≈ 308-312 rpm
  • Part (b): Correct boundary conditions for double integration; Macaulay's method or direct integration with appropriate singularity functions for the given loading (UDL/point loads as per typical figure)
  • Part (b): Deflection at x = 1 m from left end using y = ∫∫M/EI dx dx; integration constants from deflection = 0 at supports and continuity conditions; final answer typically 2-5 mm
  • Part (c): Cams: radial/face/cylindrical/conical/spiral; Followers: knife-edge/roller/flat/mushroom; merits/demerits covering wear, cost, pressure angle, and manufacturing complexity
  • Part (c): Specific applications: roller followers in IC engines (Maruti/Diesel locomotives), flat followers in textile machinery, knife-edge for precision instruments but high wear

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies Hartnell governor principle (sleeve movement proportional to speed change), recognizes that spring stiffness relates to speed regulation; for beam, identifies correct bending moment equation and boundary conditions; for cams, distinguishes kinematic pairs and pressure angle effects.Uses correct governor formula but confuses a/b lever ratio application; beam M(x) correct but boundary conditions partially wrong; cam types listed with generic descriptions.Treats Hartnell as Porter governor or ignores bell-crank geometry; beam integration without proper sign convention or missing boundary conditions; confuses cam types with gear types.
Numerical accuracy20%10Part (a): Spring stiffness 8.8-9.2 N/mm, speed at r=100mm ≈ 310 rpm; Part (b): Deflection 2.5-4.5 mm with correct units; all ω in rad/s, consistent mm/m conversion; 3-4 significant figures.Correct methodology but arithmetic slip in final answer (10-15% error); units mixed but convertible; one part fully correct, others partially.Order of magnitude errors (stiffness in N/m instead of N/mm, speed in rpm not rad/s); deflection unrealistic (>50 mm or <0.1 mm); no unit consistency.
Diagram quality20%10Clear FBD of Hartnell governor showing all forces (F, mg, spring force) with geometry labeled; beam FBD with reactions, loading, and deflected shape; cam-follower schematic with pressure angle marked; all diagrams with proper lettering and dimensions.Governor diagram present but forces not fully labeled; beam loading shown but deflected shape missing; cam types described without sketches.No diagrams despite explicit need for FBD in (a) and (b); or diagrams with wrong geometry (e.g., straight arms instead of bell-crank).
Step-by-step derivation20%10Part (a): Full moment equilibrium about fulcrum at both speeds, elimination of variables to find S and s, then interpolation for intermediate speed; Part (b): M(x) written, integrated twice with constants C1, C2 determined from boundary conditions, final substitution; Part (c): Structured comparison table.Key steps present but some algebraic manipulation skipped; integration constants stated without derivation; cam merits/demerits in prose without systematic comparison.Final answers only with no derivation; or major steps missing (e.g., no integration for beam, direct formula quoting without context).
Practical interpretation20%10Governor: interprets stiffness in terms of isochronism and sensitivity, relates to steam turbine speed regulation (NTPC/BHEL context); Beam: discusses serviceability limits (L/250 to L/360 per IS 800); Cams: links follower choice to wear, speed, and cost for Indian manufacturing (SME sector, automotive).Brief mention of governor sensitivity or beam deflection limits without code reference; cam applications mentioned generically.No interpretation; pure mathematical exercise with no engineering context or relevance to Indian industry standards.

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