JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year DYNAMICS OF MACHINERY Previous Question Paper (July/August - 2021 - R16)
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Main
Derive the condition for the limiting value of the Angle of heel (θ) to avoid skidding of the vehicle.
Answer
When a vehicle takes a turn, it is subjected to a centrifugal force acting outwards through the center of gravity (CG). To avoid skidding, the frictional force between the tires and the road must be sufficient to balance this centrifugal force. Let m be the mass of the vehicle, v the velocity, r the radius of the turn, h the height of the CG from the ground, and b the track width.
The centrifugal force Fc is given by:
For the vehicle to avoid skidding, the limiting condition is determined by the coefficient of friction μ. The total normal reaction is mg. Skidding occurs when the centrifugal force exceeds the maximum frictional resistance:
Substituting the expression for Fc:
When a vehicle is banked at an angle θ, the normal reaction R and the frictional force F act to provide the necessary centripetal force. Resolving forces perpendicular and parallel to the road surface:
For the limiting case of skidding, F = μR. Substituting this into the force equations and simplifying for the angle of heel θ, the condition to avoid skidding is defined by the relationship between the banking angle and the coefficient of friction:
If the vehicle is not banked (θ = 0), the condition simplifies to v2 / gr ≤ μ. To avoid overturning, the angle of heel must also satisfy the condition tan θ < 2h / b, ensuring the resultant force passes within the track width.
What are the conditions for a body to be in equilibrium under the action of two forces, three forces, and two forces and a torque? Explain.
Answer
For a body to be in equilibrium under the action of external forces, the resultant force and the resultant moment about any point must be zero. The specific conditions for the stated cases are:
- Two forces: The two forces must be equal in magnitude, opposite in direction, and act along the same straight line (collinear).
- Three forces: The forces must be coplanar and concurrent (their lines of action must intersect at a single point), or they must be parallel. Furthermore, the vector sum of the three forces must be zero, forming a closed triangle in the force polygon.
- Two forces and a torque: The two forces must be equal in magnitude, parallel, and opposite in direction (forming a couple) such that their moment is equal and opposite to the applied torque. The resultant force of the system is zero, and the resultant moment of the forces must balance the applied torque.
Explain how the gyroscopic couple for an aeroplane can be evaluated.
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The turning moment diagram for a 4-stroke engine may be assumed to be represented by four isosceles triangles. The areas of the triangles are: Suction = –0.5 cm2; Compression = –2.1 cm2; Expansion = –8.1 cm2; and Exhaust = –0.8 cm2. 1 cm2 of area represents 1400 N–m of work. Find the mass M.I. of the flywheel to keep the fluctuation of speed within 1% of the mean speed, if mean speed is 400 rpm.
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Derive a relation for the turning moment at the crankshaft in terms of the piston effort and the angle turned by the crank.
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Find an expression for the braking torque for a simple block (or) shoe brake when the brake drum is rotating anti-clock wise, and the friction force is at a distance ‘b’ above the fulcrum.
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In the differential band brake, shown in figure, the diameter of the drum is 900 mm, and the coefficient of friction between the drum and the band is 0.3. The angle of contact is 240°. When a force of 650 N is applied at the free end of the lever, find the maximum and the minimum force in the band, and the torque which can be applied by the brake.
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In a porter governor, the upper and lower arms are each 250 mm long, and pivoted on the axis of rotation. The mass of each rotating ball is 3 kg, and the mass of sleeve is 20 kg. The sleeve is in its lowest position when the arms are inclined at 30° to the governor axis. The lift of sleeve is 36 mm. Find the force of friction at the sleeve, if the speed at the moment it rises from the lowest position is equal to the speed at the moment it falls from the highest position. Also find the range of speed of the governor.
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A vertical steel shaft, 15 mm in diameter, is held in long bearings 1 m apart, and carries at its middle a disc of mass 15 kg. The eccentricity of the C.G. of the disc from the centre of the rotor is 0.3 mm. The modulus of elasticity for the shaft material is 200 GN/m2, and the permissible stress is 70 MN/m2. Find the (a) critical speed of the shaft, and (b) range of speed over which it is unsafe to run the shaft. Neglect the mass of the shaft.
Critical speed of the shaft
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Range of speed over which it is unsafe to run the shaft
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