(SEM V) THEORY EXAMINATION 2022-23 MECHANICAL VIBRATIONS

SECTION A – Short Answer Type Questions (2 Marks each)

(a) Distinguish Between Periodic Motion and Harmonic Motion.

Conclusion:
All harmonic motions are periodic, but not all periodic motions are harmonic.

(b) What Do You Mean by Degree of Freedom (DOF)?

The Degree of Freedom (DOF) is the number of independent coordinates required to describe the motion of a system completely.

Examples:

Single DOF system: Simple pendulum or spring-mass system.

Two DOF system: Double pendulum or two-mass spring system.

Mathematically:
If a system can move in n independent directions, it has n DOF.

 SECTION B – Long Answer Type Questions (10 Marks each)

(a) What Do You Mean by Vibrations? Classify Vibration Systems with Suitable Diagram.

Definition:
Vibration is the oscillatory motion of a body or particle about its equilibrium position due to the restoring force acting on it.

Classification of Vibrations:

Free and Forced Vibrations:

Free vibration: Occurs without external force after initial disturbance.

Forced vibration: Occurs due to an external periodic force.

Damped and Undamped Vibrations:

Undamped: No energy loss (ideal case).

Damped: Energy dissipated by friction or air resistance.

Linear and Non-linear Vibrations:

Linear: Restoring force proportional to displacement.

Non-linear: Restoring force not proportional to displacement.

Periodic and Random Vibrations:

Periodic: Repeats after a fixed time (e.g., rotating machinery).

Random: No fixed repetition (e.g., earthquake vibrations).

Diagram:
A labeled sketch of a spring–mass–damper system showing displacement, damping, and applied forces.

(b) Explain Rayleigh’s Method for Finding the Natural Frequency of a Multi-Degree System.

Rayleigh’s Method is an approximate technique used to determine the natural frequency of complex systems by applying the principle of conservation of energy.

Assumptions:

Maximum kinetic energy (Tmax) = Maximum potential energy (Vmax).

Steps:

Assume a mode shape (displacement pattern) of the system.

Compute maximum kinetic energy:

1. T=12∑mi(xi˙)2T = \frac{1}{2} \sum m_i (\dot{x_i})^2T=21​∑mi​(xi​˙​)2

Compute maximum potential energy:

1. V=12∑ki(xi)2V = \frac{1}{2} \sum k_i (x_i)^2V=21​∑ki​(xi​)2

Apply Rayleigh’s principle:

1. ωn2=VT\omega_n^2 = \frac{V}{T}ωn2​=TV​ ⇒ωn=∑kixi2∑mixi2\Rightarrow \omega_n = \sqrt{\frac{\sum k_i x_i^2}{\sum m_i x_i^2}}⇒ωn​=∑mi​xi2​∑ki​xi2​​​

Convert to natural frequency:

1. fn=ωn2πf_n = \frac{\omega_n}{2\pi}fn​=2πωn​​

Advantage:
Simple and accurate for first natural frequency estimation.

SECTION C – Very Long Answer Type Questions (10 Marks each)

(a) Derive the Equation of Motion Using D’Alembert’s Principle.

D’Alembert’s Principle converts a dynamic problem into a static equilibrium condition by introducing an inertial force.

Consider a mass-spring-damper system:
Let mass = mmm, damping coefficient = ccc, stiffness = kkk, displacement = xxx.

External Force Balance:

Inertial Force+Damping Force+Spring Force=Applied Force\text{Inertial Force} + \text{Damping Force} + \text{Spring Force} = \text{Applied Force}Inertial Force+Damping Force+Spring Force=Applied Force mx¨+cx˙+kx=F(t)m\ddot{x} + c\dot{x} + kx = F(t)mx¨+cx˙+kx=F(t)

Interpretation:

mx¨m\ddot{x}mx¨: Inertia effect

cx˙c\dot{x}cx˙: Damping effect

kxkxkx: Restoring force

F(t)F(t)F(t): External force

This is the general differential equation of motion for a single-degree-of-freedom system.

(b) Derive an Expression for the Magnification Factor and Discuss Its Variation with Frequency Ratio.

Forced Vibration Equation (Undamped):

mx¨+kx=F0sin⁡ωtm\ddot{x} + kx = F_0 \sin \omega tmx¨+kx=F0​sinωt

Steady-State Solution:

x=Xsin⁡(ωt−ϕ)x = X \sin(\omega t - \phi)x=Xsin(ωt−ϕ)

Amplitude (X):

X=F0/k(1−r2)X = \frac{F_0/k}{(1 - r^2)}X=(1−r2)F0​/k​

where r=ωωnr = \frac{\omega}{\omega_n}r=ωn​ω​ = frequency ratio.

Magnification Factor (MF):

MF=XXst=1(1−r2)\text{MF} = \frac{X}{X_{st}} = \frac{1}{(1 - r^2)}MF=Xst​X​=(1−r2)1​

For Damped System:

MF=1(1−r2)2+(2ζr)2\text{MF} = \frac{1}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}MF=(1−r2)2+(2ζr)2​1​

where ζ\zetaζ = damping ratio.

Variation:

At low frequency (r << 1): MF ≈ 1.

At resonance (r ≈ 1): MF = 1/(2ζ) → very high for small damping.

At high frequency (r >> 1): MF → 0.

Graph:
A curve showing MF vs. r, with a sharp peak at r = 1 for lightly damped systems.