Lesson 3.5: Vibrations (Free, Forced, Damped, Resonance)

Vibrations are crucial in mechanical systems to ensure safety, stability, and longevity. GATE and PSU exams frequently test free and forced vibrations, damping, and resonance concepts.

🔹 1. Introduction

• Definition: Vibration is the oscillatory motion of a body about an equilibrium position.

• Applications: Engines, turbines, compressors, machine tools, bridges

• Terminology:

  • Amplitude (A): Maximum displacement from equilibrium

  • Time Period (T): Time for one complete cycle

  • Frequency (f): Number of cycles per second, $f=1/T$

  • Angular Frequency ($\omega$): $\omega =2\pi f$

🔹 2. Free Vibrations

• Definition: Oscillation without external force after initial displacement.

• Equation of Motion:

md2xdt2+kx=0m \frac{d^2 x}{dt^2} + k x = 0mdt2d2x​+kx=0

Where:

• $m$ = mass, $k$ = stiffness, $x$ = displacement

• Natural Frequency:

ωn=km;fn=ωn2π\omega_n = \sqrt{\frac{k}{m}} \quad ; \quad f_n = \frac{\omega_n}{2 \pi}ωn​=mk​​;fn​=2πωn​​

Example: Mass-spring system, m = 2 kg, k = 200 N/m → ωn=10\omega_n = 10ωn​=10 rad/s, fn≈1.59f_n ≈ 1.59fn​≈1.59 Hz

🔹 3. Damped Vibrations

• Definition: Vibration with energy loss (friction, air resistance).

• Equation:

md2xdt2+cdxdt+kx=0m \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + k x = 0mdt2d2x​+cdtdx​+kx=0

Where $c$ = damping coefficient

• Types of Damping:

  1. Underdamped ($\zeta <1$) → oscillates with decreasing amplitude

  2. Critically Damped ($\zeta =1$) → returns to equilibrium fastest

  3. Overdamped ($\zeta >1$) → slowly returns to equilibrium

• Logarithmic Decrement: Measures rate of damping

🔹 4. Forced Vibrations

• Definition: Vibration under continuous external periodic force.

• Equation:

md2xdt2+cdxdt+kx=F0sin⁡ωtm \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + k x = F_0 \sin \omega tmdt2d2x​+cdtdx​+kx=F0​sinωt

• Steady-State Solution: Amplitude depends on forcing frequency $\omega$

🔹 5. Resonance

• Definition: Maximum amplitude occurs when forcing frequency = natural frequency

ω=ωn\omega = \omega_nω=ωn​

• Consequences: Excessive vibration → failure of machine or structure

• Applications/Prevention: Design damping, avoid matching operating frequency with natural frequency

Example:
A bridge oscillates dangerously when wind or traffic matches its natural frequency → resonance

🔹 6. Solved Examples (PYQ Style)

Example 1 (GATE ME 2018):
Mass-spring system, m = 1 kg, k = 100 N/m. Find natural frequency.
ωn=100/1=10 rad/s,fn=1.59 Hz\omega_n = \sqrt{100/1} = 10 \text{ rad/s}, f_n = 1.59 \text{ Hz}ωn​=100/1​=10 rad/s,fn​=1.59 Hz

Example 2 (PSU Exam):
Damped system, m = 2 kg, k = 200 N/m, c = 20 N·s/m. Determine damping type.

• Damping ratio ζ=c/(2mk)=20/(22∗200)≈0.5\zeta = c / (2 \sqrt{mk}) = 20 / (2 \sqrt{2*200}) ≈ 0.5ζ=c/(2mk​)=20/(22∗200​)≈0.5 → underdamped

🔹 7. Practice Exercises

1. Determine natural frequency of mass-spring-damper system with m = 3 kg, k = 300 N/m.

2. Identify damping type for given c, m, k values.

3. Compute amplitude at resonance for forced vibration with F0 = 10 N.

4. Explain difference between free and forced vibration.

5. Discuss methods to prevent resonance in mechanical structures.

🔹 8. Summary

• Free Vibration: Oscillation without external force, natural frequency ωn=k/m\omega_n = \sqrt{k/m}ωn​=k/m​

• Damped Vibration: Energy loss reduces amplitude, types: underdamped, critically damped, overdamped

• Forced Vibration: Oscillation under external periodic force

• Resonance: Maximum amplitude when forcing frequency = natural frequency → dangerous, must be avoided

• Applications: Engines, turbines, bridges, rotating machinery