JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III DYNAMICS OF MACHINERY Previous Question Paper (March/April - 2021 - R16)
Answer any five questions. All questions carry equal marks.
Main
A four wheel trolley car of total mass 2000 kg running on rails of 1 m gauge, rounds a curve of 30 m radius at 45 km/h. The track is banked at 10°. The wheels have an external diameter of 0.6 m and each pair of the axle has a mass of 250 kg. The radius of gyration of each pair is 250 mm. The height of C.G. of the car above the wheel base is 1 m. Allowing for centrifugal force and gyroscopic couple action, determine the pressure on each rail.
Answer
Given data: Mass of car (M) = 2000 kg, Gauge (w) = 1 m, Radius of curve (R) = 30 m, Velocity (v) = 45 km/h = 12.5 m/s, Banking angle (θ) = 10°, Wheel diameter (D) = 0.6 m (radius r = 0.3 m), Mass of each axle pair (m) = 250 kg, Radius of gyration (k) = 0.25 m, Height of C.G. (h) = 1 m.
1. Centrifugal Force (P): P = Mv²/R = 2000 × (12.5)² / 30 = 10416.67 N.
2. Gyroscopic Couple (C): Angular velocity of wheels (ω) = v/r = 12.5 / 0.3 = 41.67 rad/s. Angular velocity of precession (ωp) = v/R = 12.5 / 30 = 0.4167 rad/s. Moment of inertia of one axle (I) = mk² = 250 × (0.25)² = 15.625 kg·m². Total gyroscopic couple (C) = 2 × I × ω × ωp = 2 × 15.625 × 41.67 × 0.4167 = 542.5 N·m.
3. Vertical Forces: Total weight (W) = Mg = 2000 × 9.81 = 19620 N. Normal reaction (R1 + R2) = W cos θ + P sin θ = 19620 cos 10° + 10416.67 sin 10° = 19322.5 + 1809.1 = 21131.6 N.
4. Moments about the outer rail: Taking moments of all forces about the outer rail contact point: (R1 × w) + (C/cos θ) = (W cos θ × w/2) - (W sin θ × h) + (P cos θ × h) + (P sin θ × w/2). Substituting values: (R1 × 1) + (542.5 / cos 10°) = (19620 cos 10° × 0.5) - (19620 sin 10° × 1) + (10416.67 cos 10° × 1) + (10416.67 sin 10° × 0.5). R1 + 550.9 = 9661.2 - 3406.4 + 10258.5 + 904.5 = 17417.8 N. Thus, R1 = 16866.9 N (Inner rail pressure).
5. Outer rail pressure (R2): R2 = 21131.6 - 16866.9 = 4264.7 N. Pressure per wheel on inner rail = 8433.45 N; Pressure per wheel on outer rail = 2132.35 N.
Formulate expressions for the velocity and acceleration of a piston in a slider crank mechanism.
Answer
In a slider-crank mechanism, let r be the crank radius, l be the connecting rod length, θ be the crank angle from the inner dead center, and φ be the angle of the connecting rod with the line of stroke. The ratio of lengths is defined as n = l/r.
The displacement of the piston x from the inner dead center is given by:
Using the relation sin φ = (r/l) sin θ = (1/n) sin θ, and approximating cos φ ≈ 1 - (sin² φ)/2, the displacement simplifies to:
To find the velocity v, differentiate the displacement with respect to time t, where ω = dθ/dt:
To find the acceleration a, differentiate the velocity with respect to time t:
In these expressions, ω is the angular velocity of the crank in rad/s, and n is the obliquity ratio. The first term ω²r cos θ represents the primary acceleration, while the second term ω²r (cos 2θ / n) represents the secondary acceleration.
An engine runs at a constant load at a speed of 480 rpm. The crank effort diagram to a scale of 1 cm = 2 kN m torque and 1 cm = 360° crank angle. The areas of the diagram above and below the mean torque line are measured in square cm and are in the following order: +1.1, -1.32, +1.53, -1.66, +1.97, -1.62. Design the flywheel if the total fluctuation of speed is not to exceed 10 rpm and the centrifugal stress in the rim is not to exceed 5 N/mm². You may assume that the rim breadth is approximately 3.5 times the rim thickness and 90% of the moment of inertia is due to the rim. The density of the material of the flywheel is 7250 kg/m³.
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In a single-acting four-stroke engine, the work done by the gases during the expansion stroke is three times the work done during the compression stroke. The work done during the suction and exhaust strokes is negligible. The engine develops 14 kW at 280 rpm. The fluctuation of speed is limited to 1.5 % of the mean speed on either side. The turning-moment diagram during the compression and the expansion strokes may be assumed to be triangular shape. Determine the inertia of the flywheel.
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What are uniform pressure and uniform wear theories? Deduce expressions for the friction torque considering both the theories for a flat collar.
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Describe the working of a band and block brake with the help of a neat sketch. Deduce the relation for ratio of tight and slack side tensions.
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Describe the function of a simple watt governor. What are its limitations?
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A 1.8 m long hollow shaft is supported in flexible bearings at the ends. It carries two wheels each of 60 kg mass, one at the centre of the shaft and the other at 450 mm from the centre. The external and internal diameters of the shaft are 80 mm and 50 mm respectively. Determine the lowest whirling speed of the shaft. The density of the shaft material is 7500 kg/m³ and the modulus of elasticity is 210 GN/m².
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