Consider a nonuniform beam subjected to a base motion such as one experienced during an earthquake as shown in the Figure $1.$ If the elastic deflection of the beam, $w(x,t),$ is measured relative to the support motion, $w_g(t),$ the total displacement of the beam is given by

$w_t(x,t)=w_g(t)+w(x,t)$

Derive the equation of motion of the beam, subjected to base motion, considering the inertia force associated with the total acceleration of the beam. Suggest a method of finding the response of a beam subjected to a specified base motion $w_g(t)$ then write the total displacement of the beam as a function of the beam parameters and the base motion $w_g(t).$

$x$

Figure $1$: Beam subjected to support excitation.