Structural Resonance Threshold (Civil Engineering) Solution

Part 1: Directional Approach & Sign Analysis

As $\omega \to 8^-$, the excitation frequency approaches $8\text{ rad/s}$ from below (e.g., $\omega = 7.99\text{ rad/s}$).

• Numerator: The constant value is $+15 > 0$.
• Denominator: Because $\omega < 8$, we have $\omega^2 < 64$, which means $64 - \omega^2 > 0$. The denominator approaches $0$ through strictly positive values ($0^+$).

Part 2: Analytical Limit Calculation

Using directional zero sign analysis:

Part 3: Physical Engineering Significance

The limit diverging to $+\infty$ indicates that without mechanical damping, structural vibration amplitudes grow without bound near the natural frequency $\omega_0 = 8\text{ rad/s}$. The vertical asymptote at $\omega = 8$ represents a catastrophic mechanical resonance boundary where cyclical wind energy pumps unbounded kinetic energy into the structure, necessitating tuned mass dampers to suppress resonant failure.