Vibration free study note

1. Introduction to Vibration

Vibration is the oscillatory motion of a body about its equilibrium (mean) position.

Examples:

• Engine components vibrating
• Machine tools during cutting
• Buildings during earthquakes

Vibration can be useful (e.g., sieving, compacting) or harmful (causing noise, wear, failure).

2. Basic Terminology

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Important Terms:

• Cycle: One complete oscillation
• Amplitude (A): Maximum displacement from mean position
• Time Period (T): Time for one cycle
• Frequency (f): Number of cycles per second (Hz)
• Angular Frequency (ω): $$\omega = 2\pi f$$
• Phase: Relative position in the cycle

3. Types of Vibration

Based on Excitation

1. Free Vibration
   • No external force after initial disturbance
   • Example: Mass-spring system
2. Forced Vibration
   • External force continuously applied
   • Example: Engine vibration

Based on Energy Dissipation

1. Undamped Vibration
   • No energy loss
   • Theoretical case
2. Damped Vibration
   • Energy lost due to friction, resistance

Based on Motion

• Longitudinal Vibration → Along axis
• Transverse Vibration → Perpendicular to axis
• Torsional Vibration → Twisting motion

4. Simple Harmonic Motion (SHM)

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When restoring force is proportional to displacement:

$F = -kx$F=−kx

$k$k

$F = -kx$F=−kx

• $k$k = stiffness of spring
• Negative sign indicates restoring force

Equation of Motion:

$$m\ddot{x} + kx = 0$$

Natural Frequency

$$\omega_n = \sqrt{\frac{k}{m}}$$

5. Damped Vibration

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Types:

1. Underdamped → oscillations gradually decrease
2. Critically damped → fastest return without oscillation
3. Overdamped → slow return, no oscillation

Damping Force:

$$F = -c \dot{x}$$

Damping Ratio:

$$\zeta = \frac{c}{c_c}$$

6. Forced Vibration and Resonance

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Forced Vibration Equation:

$$m\ddot{x} + c\dot{x} + kx = F_0 \sin(\omega t)$$

🔸 Resonance:

• Occurs when forcing frequency = natural frequency
• Leads to very large amplitude

👉 Dangerous in:

• Bridges
• Rotating machinery

🔹 7. Degrees of Freedom (DOF)

• Number of independent coordinates needed to describe motion

Types:

• Single DOF system (mass-spring)
• Multi DOF system (complex machines)

🔹 8. Vibration Measurement and Instruments

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Common Instruments:

• Accelerometer
• Vibrometer
• Seismometer

Used in condition monitoring and fault detection.

9. Effects of Vibration

Harmful Effects:

• Noise generation
• Fatigue failure
• Wear and tear
• Loosening of parts

Useful Applications:

• Vibrating screens
• Compaction of concrete
• Ultrasonic machining

10. Methods of Vibration Control

• Damping (using dashpots, rubber mounts)
• Isolation (spring supports)
• Balancing of rotating parts
• Avoiding resonance

11. Engineering Applications

• Design of engines and turbines
• Structural analysis (earthquake design)
• Machine health monitoring
• Automotive suspension systems

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