Lecture 2: Vibration of Continuous System

Overview

• Introduced the course and technical terms in vibration study in the last class.
• Discussed discrete systems vs. continuous systems.
• Explored single degree freedom model: undamped vs. damped vibrations.
• Today’s focus: Time domain analysis of linear systems under harmonic input.

Today's Outline

1. General approach for time domain analysis of linear systems.
2. Decaying nature of response in undamped models.
3. Damped oscillators under harmonic motion (trimester analysis).
4. Magnification factor and resonance phenomena.
5. Half power bandwidth for determining damping ratio.

Time Domain Analysis of Linear Systems

• System Parameters:
  • m (mass), c (damping), k (stiffness) - time invariant.
  • x, y, z (spatial domains), t (time).
• Differential Operators:
  • Linear systems are represented by second order ordinary differential equations for discrete systems.
  • Continuous systems may involve partial differential equations of order 2 or 4 depending on the system considered.

Input and Output in Linear Systems

• Inputs:
  • Exciting forces and initial conditions defined in spatial coordinates and time.
• Outputs:
  • Response measured at specific locations.
• Convolution:
  • Output R is obtained by convoluting input with impulse response function which includes system parameters.

General Approach to Time Domain Analysis

• Involves boundary value and initial value problems.
• Continuous systems are treated as discrete systems with infinite generalized coordinates.
• Similar methods of solution apply to initial value problems in both discrete and continuous systems.

Damping in Systems

• Three types of damping:
  1. Underdamped
  2. Overdamped
  3. Critically damped
• Damping ratio defined as ξ.
• Key Factor:
  • Higher damping results in faster decay of motion after force withdrawal.

Free Vibration Analysis

• For free vibration (F(t) = 0):
  • Equation: x'' + 2ξω_n x' + ω_n² x = 0
  • Solution: Amplitude decays exponentially due to damping.

Damped Oscillator under Harmonic Motion

• For harmonic excitation, substitute F(t) = P cos(ωt).
• Method of Undetermined Coefficients:
  • Particular integral Xp chosen as C1 cos(ωt) + C2 sin(ωt).
• Key Results:
  • Impacts of natural frequency and damping on response are analyzed.

Magnification Factor and Resonance

• Magnification Factor (M_f) defined as:
  • M_f = 1 / √((1 - r²)² + 2ξ²)
• Resonance:
  • Occurs when driving frequency approaches natural frequency (r = 1).
  • Damping reduces the peak amplitude and sharpness of the resonance curve.

Half Power Bandwidth Method

• Relates to the bandwidth defined as the difference between lower and upper cutoff frequencies.
• Damping ratio can be determined using half power points on frequency response curves.

Conclusion

• The lecture has covered:
  • Procedures for solving damped oscillators under harmonic excitation.
  • Relationships between magnification factor, phase angle, and frequency ratio.
  • Measurement techniques for damping ratio (logarithmic decrement and bandwidth).

Acknowledgements

• Thank you for your attendance and participation.