Chapter 22: Vibrations

Key Topics Covered:

1. Undamped Free Vibration
2. Differential Equations in Vibrations
3. Natural Frequency and Harmonic Motion
4. Trigonometric Representation of Solutions
5. Frequency Analysis and System Resonance
6. Examples of Undamped Free Vibration

Undamped Free Vibration

• Defined by a second order linear homogeneous differential equation without the first order term.
• The form of the equation:
  • ( x'' + kx = 0 )
• Solutions are provided without methods but will be solved using Laplace method in Ogata's book.

Differential Equations Overview

• The differential equation format:
  • Second-order differential
  • Homogeneous, meaning the right-hand side is zero.
• Equilibrium position is the reference point for measuring displacement (y).

Solution Characteristics

• Solutions involve constants ( A ) and ( B ) determined by initial conditions.
• Initial position and velocity are key for determining solution constants.
• Trigonometric identities transform solutions from sine/cosine form into simpler sinusoidal functions.

Natural Frequency and Harmonic Motion

• Natural frequency ( \omega_n ) is the square root of ( k/m )
• Solution format: Combination of sine and cosine functions.
• Simplified to a single sine function with phase shift using trigonometric identities.

Trigonometric Representation of Solutions

• Solutions can be expressed as:
  • ( C \sin(\omega_n t + \phi) )
• Amplitude: Maximum displacement ( C ).
• Phase Angle: ( \phi ) represents the displacement from zero at ( t = 0 ).

Frequency Analysis

• Frequency (f): Number of cycles per unit time, reciprocal of the period (( \tau )).
  • ( f = \frac{1}{\tau} = \frac{\omega_n}{2\pi} )
• Angular Frequency: Different from frequency, measured in radians per second. ( \omega_n ) is natural frequency.
• Resonance: Occurs when the frequency of external force matches the natural frequency.

Example: Simple Pendulum

• Governing equation: ( \theta'' + (g/l)\theta = 0 )
• For small angles, ( \sin\theta \approx \theta )
• Natural frequency ( \omega_n = \sqrt{g/l} )
• Period ( \tau = 2\pi \sqrt{l/g} )

Example: Rigid Body Vibration

• Torsional Spring: Torque-related equivalent of a linear spring.
• Governing equation for vibration: ( I_0 \theta'' + k\theta = 0 )
• Example uses the moment of inertia from appendices for calculations.

Important Notes

• Natural Frequency: Indicates the inherent vibrational frequency in the absence of external forces.
• Damping: Though not covered here, damping affects resonance and real-world system behavior.
• Differential Equations: Core to understanding vibrational behavior, foundational in further studies using methods like Laplace.