The dynamics of pedestrian-bridge interaction is given (Newland, 2004) by

\[ M \ddot{y}+K y(t)+m \alpha \beta \ddot{y}(t-\Delta)+2 \xi \sqrt{K M} \dot{y}=-m \beta \ddot{x}(t) \]

where \(M\) : Mass of the bridge, \(K\) : Stiffness of the bridge, \(\xi\) : damping ratio, \(Y\) : displacement of the pavement, \(x\) : movement of the center of mass caused by pedestrian walking, \(m\) : mass of the pedestrian, and \(\Delta\) : time lag. \(\alpha\) and \(\beta\) are constants.

a. Derive the transfer function \(G(s)=\frac{y(s)}{x(s)}\)

b. Plot \(|\alpha G(j \omega)|\) versus frequency ratio \(\frac{\omega}{\omega_{n}}\) where \(\omega\) : input frequency, \(\omega_{n}=\sqrt{\frac{K}{M}}\) for \(\omega \Delta=0,-\frac{\pi}{2},-\pi\) and \(\frac{\pi}{2}\). Assume that \(\frac{\alpha \beta m}{M}=0.1\) and \(\xi=0.1\).