Structural Vibration Resonance & Dynamic Load Thresholds (Civil & Mechanical Engineering)

A structural dynamics engineer at UNLV is evaluating the aeroelastic response of an overhead pedestrian skybridge connecting the Thomas & Mack Center to campus parking structures during high-wind gusts. The steady-state dynamic amplification factor $M(\beta)$ of the bridge deck displacement as a function of the frequency ratio $\beta = \frac{\omega}{\omega_n}$ (where $\omega$ is the wind vortex shedding frequency and $\omega_n$ is the bridge's fundamental natural frequency) is governed by the undamped dynamic magnification model:

When an active mass damper system is engaged, the transient response amplitude near resonance ($\beta \to 1$) is characterized by the modified limit ratio: where $A(\beta)$ represents the normalized modal acceleration ratio.

1. Evaluate the direct substitution behavior of $A(\beta)$ as the frequency ratio $\beta$ approaches resonance ($\beta \to 1$).
2. Apply algebraic conjugate multiplication to evaluate the exact limit $\lim_{\beta \to 1} A(\beta)$ analytically.
3. Investigate the undamped behavior $\lim_{\beta \to 1^-} M(\beta)$ and $\lim_{\beta \to 1^+} M(\beta)$, and explain the structural engineering significance of infinite limits versus finite conjugate limits when sizing tuned mass dampers.