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# Machine Dynamics MCQ – Natural Frequency of Free Longitudinal Vibrations

Find the natural frequency in Hz of the free longitudinal vibrations if the displacement is 2mm.

a) 11.14

b) 12.38

c) 11.43

d) 11.34

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Answer: a

Explanation: We know that the natural Frequency of Free Longitudinal Vibration is given by the equation

f = 0.4985/s√

where s is the displacement of the spring.

If the spring displacement is high then the frequency of the spring increases.

a) True

b) False

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Answer: b

Explanation: The natural frequency of the free longitudinal vibration depends on the displacement and gravitational acceleration (g) by the relation:

f= 0.4985/s√

where s is the displacement of the spring.

since it is not directly proportional, the given statement is false.

Find the displacement in mm of the free longitudinal vibrations if the Natural frequency is 15 Hz.

a) 1.1

b) 1.2

c) 1.5

d) 1.6

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Answer: a

Explanation: We know that the natural Frequency of Free Longitudinal Vibration is given by the equation

f = 0.4985/s√

where s is the displacement of the spring.

substituting the given values we get

s=1.1 mm.

Find the displacement in mm of the free longitudinal vibrations if the Natural frequency is 20 Hz.

a) 0.1

b) 0.2

c) 0.5

d) 0.6

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Answer: d

Explanation: We know that the natural Frequency of Free Longitudinal Vibration is given by the equation

f= 0.4985/s√

where s is the displacement of the spring.

Substituting the given values we get:

s=0.6 mm.

Which of the following methods will give an incorrect relation of the frequency for free vibration?

a) Equilibrium method

b) Energy method

c) Reyleigh’s method

d) Klein’s method

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Answer: d

Explanation: Equilibrium, Energy and Reyleigh’s method give the same relation between the natural frequency and displacement of free vibration whereas klein’s method is used to calculate velocity and acceleration of the parts of the mechanisms.

A cantilever shaft has a diameter of 6 cm and the length is 40cm, it has a disc of mass 125 kg at its free end. The Young’s modulus for the shaft material is 250 GN/m². Calculate the static deflection in nm.

a) 0.001

b) 0.083

c) 1.022

d) 0.065

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Answer: a

Explanation: Area = πd^{2}/4 = 0.00282 m^{2}

s = W.l/A.E

= 0.001 nm.

Static deflection and frequency are independent of each other.

a) True

b) False

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Answer: b

Explanation: The natural frequency of the free longitudinal vibration depends on the static displacement and gravitational acceleration (g) by the relation:

f = 0.4985/s√.

A cantilever shaft having 50 mm diameter and length of 300 mm has a disc of mass 100 kg at its free end. The Young’s modulus for the shaft material is 200 GN/m^{2}. Calculate the natural longitudinal frequency in Hz.

a) 575

b) 625

c) 525

d) 550

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Answer: a

Explanation: Area = πd^{2}/4 = 0.00196 m^{2}

s = W.l/A.E = = 0.751 µm

I = 0.3×10^{-6} m^{4}

f = 0.4985/s√

= 575 Hz.

If the mass is of 10 Kg, find the natural frequency in Hz of the free longitudinal vibrations. The displacement is 0.01mm.

a) 44.14

b) 49.85

c) 43.43

d) 46.34

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Answer: b

Explanation: We know that the natural Frequency of Free Longitudinal Vibration is given by the equation

f = 0.4985/s√

where s is the displacement of the spring.

substituting the given values we get

f=49.85 Hz

It is to be noted that mass has no effect on the natural frequency as it only depends on the displacement.