# Mechanical Vibrations Questions and Answers – Harmonically Excited Vibrations

This set of Mechanical Vibrations Multiple Choice Questions & Answers (MCQs) focuses on “Harmonically Excited Vibrations”.

1. When does resonance occur?
a) The external excitation has the same frequency as that of the natural frequency of the system
b) The external excitation has a higher frequency as that of the natural frequency of the system
c) The external excitation has a lower frequency as that of the natural frequency of the system
d) The external excitation has zero frequency

Explanation: Resonance occurs when the external excitation has the same frequency as that of the natural frequency of the system, leading to a large displacement of the amplitude from the mean position.

2. Which of the following case is not at all practiced?
a) Militants marching while crossing a bridge
b) 2 cars honking in synchronized manner
c) A singer singing on a wooden dais
d) 2 bikes cruising through a bridge running at same speed

Explanation: Militants are not allowed to march while crossing a bridge due to chances of occurrence of resonance. If the marching troops have the same frequency as that of the natural frequency of the bridge then there will be a huge displacement and can cause the system to exceed its elastic limit and will eventually fail.

3. State True or False.
For a single frequency, the harmonic excitation force may take the form of a sine or cosine function.
a) True
b) False

Explanation: Considering a single mass system with negligible damping on a frictionless surface, for the case of harmonic excitation F(t) = F0 cos⁡(ωt) or F0sin⁡(ωt).

4. What is the particular solution of an oscillation equation of a single degree of freedom excited by f0cos⁡(ωt)?
a) xp = X cos⁡(ωt)
b) xp = X2 sin⁡(ωt)
c) xp = 2X cos⁡(ωt)
d) xp = 2X sin⁡(ωt)

Explanation: From the general equation for oscillation of a single degree of freedom we have: m$$\ddot{x}$$(t) + ωn2.x(t)=f0cos⁡(ωt). A solution of these types of equations can be determined by method of undetermined co-efficient by summing the homogeneous solution and particular solution. The particular solution is given by xp = X cos⁡(ωt), where X is the amplitude of force response.

5. Find the magnification factor (amplitude ratio) when the damping factor is 0.3 and the angular velocity ratio is 0.5.
a) 1.2379
b) 2.5674
c) 3.6644
d) 0.9152

Explanation: The value of magnification factor is given by the formula M = $$\frac{1}{\sqrt{(1-r^2)^2+(2ζr)^2}}$$. Putting the values of ζ = 0.3 and r = 0.5 in the equation we get the value of M as 1.2379.

6. What is the phase angle (in degree) when the damping factor is 0.3 and the angular velocity ratio is 0.5?
a) 19.7
b) 20.8
c) 21.8
d) 22.7

Explanation: The phase angle is given by φ = tan-1$$(\frac{2ζr}{1-r^2})$$. Putting the values of ζ = 0.3 and r = 0.5 in the equation we get the value of φ as 21.8°.

7. Find the amplitude of vibration (in m) of an un-damped system when the natural frequency and forced frequency of vibration are 20 rad/s and 15 rad/s respectively, excited by a force of 200 N.
a) 1.142
b) 2.428
c) 3.617
d) 0.012

Explanation: For an un-damped system we have X=$$\frac{f_0}{ω_n^2-ω^2}$$ where X is the amplitude of vibration. Putting the value of f0 = 200 N, ωn = 20 rad/s and ω = 15 rad/s in the above equation, we get X = 1.142m.

8. What is the period of beating?
a) The time period between one minimum and zero amplitude
b) The time period between one maximum and zero amplitude
c) The time period between two points of zero amplitude
d) The time period between one minimum and maximum amplitude

Explanation: The time period between two point of zero amplitude or the time period between two points of maximum amplitude is known as the period of beating.

9. What is the maximum value of amplitude when damping factor is 0.5?
a) 1
b) 2
c) 3
d) 4

Explanation: The equation for the maximum value of amplitude (X) is given by X = $$\frac{1}{2ζ}$$ where ζ is the damping factor. So, putting the value of ζ as 0.5 in the above equation, the value of X becomes 1.

10. State true or false.
For an un-damped system the phase angle is 180 degree for angular velocity ratio greater than 1.
a) True
b) False

Explanation: From the equation φ=tan-1($$\frac{2ζr}{1-r^2})$$, for an un-damped system ζ=0, and by the condition r>1, we have the value of φ = 180°. This implies that excitation is out of phase for r>1.

Sanfoundry Global Education & Learning Series – Mechanical Vibrations.

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