Gopalkrishna Advanced Rural Research Foundation

Comprehensive Dynamics & Structural Health Monitoring (SHM) Theory

1. Transverse Natural Frequency of Cantilever Beams

The natural frequency of a uniform cantilever beam carrying a concentrated tip mass $m$ is derived using Rayleigh's Energy Method or exact Bernoulli-Euler beam equations. Neglecting the beam's own distributed mass for simplified lumped-mass analysis:

$f_n = \frac{1}{2\pi} \sqrt{\frac{3EI}{L^3 (m + 0.23 m_{beam})}}$

Where $E$ is Young's Modulus, $I$ is the area moment of inertia ($\frac{bh^3}{12}$), $L$ is the span length, and $0.23 m_{beam}$ accounts for the effective mass of the vibrating cantilever itself.

2. Free Vibration Decay & Damping Ratio ($\zeta$)

When an initial displacement (pluck) is applied, the beam oscillates with decaying amplitude due to structural and air damping. The displacement response over time $t$ is expressed as:

• Damped Frequency: $f_d = f_n \sqrt{1 - \zeta^2}$
• Amplitude Envelope: $x(t) = X_0 e^{-\zeta \omega_n t} \cos(\omega_d t - \phi)$
• Logarithmic Decrement ($\delta$): Measured between successive peaks, $\delta = \ln\left(\frac{x_i}{x_{i+1}}\right) \approx 2\pi\zeta$. This is a vital parameter in Structural Health Monitoring (SHM) to detect internal material micro-cracking or fastener loosening.

3. Resonance & Amplitude Magnification

When an external harmonic base excitation or periodic force matches the natural frequency ($\omega \approx \omega_n$), the system enters resonance. The dynamic amplification factor peaks sharply, limited only by the damping ratio $\zeta$: at resonance, dynamic magnification reaches $\frac{1}{2\zeta}$. Unmitigated resonance can cause catastrophic fatigue failure in mechanical and civil infrastructure.