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# Machine Dynamics MCQ – Effect of Inertia of the Constraint on Torsional Vibrations

A flywheel is mounted on a vertical shaft. The both ends of a shaft are fixed and its diameter is 50 mm. The flywheel has a mass of 500 kg, the modulus of rigidity for the shaft material is 80 GN/m2 and its radius of gyration is 0.5 m. Find the natural frequency of torsional vibrations.

a) 5.32

b) 5.86

c) 5.65

d) 6.66

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Answer: a

Explanation: We know that J = πd4/32 = 0.6×10-6 m4

q1 = 56×103

q2 = 84×103

q = 140×103

therefore using the formula for fn

we get fn = 5.32 Hz.

If a mass whose moment of inertia is Ic/3 is placed at the free end and the constraint is assumed to be of negligible mass, then the kinetic energy is ______

a) 1/6 Icω2

b) 1/2Icω2

c) 1/3Icω2

d) 1/12Icω2

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Answer: a

Explanation: Kinetic energy is given by the equation

0.5Iω2

inserting Ic/3 in place of I will give kinetic energy as

1/6 Icω2.

If Ic = 125 Kg-m2 and ω= 20 rad/s, calculate the kinetic of the constraint.

a) 8333 J

b) 7333 J

c) 6333 J

d) 9333 J

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Answer: a

Explanation: The kinetic energy of the constraint is given by the equation:

K.E = 1/6 Icω2

substituting the value we get

K.E = 8333J.

If Ic = 500 Kg-m2 and ω = 10 rad/s, calculate the kinetic of the constraint.

a) 8333 J

b) 7363 J

c) 4578 J

d) 9333 J

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Answer: a

Explanation: The kinetic energy of the constraint is given by the equation:

K.E = 1/6 Icω2

substituting the value we get

K.E = 8333J.

If the mass of the constraint is negligible then what is the kinetic energy of the system?

a) 0

b) Half the value

c) Double the value

d) Infinite

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Answer: a

Explanation: If the mass of the constraint is negligible, then the mass moment of inertia will be 0. As a result the expression of the kinetic energy will become 0.

If the mass of the constraint is negligible and a mass is attached at the end of the rod having following values: Ic = 500 Kg-m2 and ω= 10 rad/s, then calculate the kinetic of the constraint system.

a) 8333 J

b) 7363 J

c) 4578 J

d) 9333 J

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Answer: a

Explanation: The kinetic energy of the constraint system in the absence of mass of constrant is given by the equation:

K.E = 1/6 Icω2

substituting the value we get

K.E = 8333J.

Free torsional vibrations will occur in a two rotor system only if both rotors have same frequency.

a) True

b) False

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Answer: a

Explanation: In a two rotor system the free torsional vibration will occur only if both the rotors have the same frequency of vibration, it follows the following relation between length and mass moment of inertia: L(A)I(A) = L(B)I(B).

Increasing which of the following factor would result in increase of free torsional vibration?

a) Radius of gyration

b) Mass moment of inertia

c) Polar moment of inertia

d) Length

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Answer: c

Explanation: The free torsional vibration frequency of a single rotor system depends by the following relation

f = 12(C.Jl.I)−−−−−√

Hence increase in polar moment of inertia results in increase in vibration frequency.