In Problem 6, using the governing equations of motion given below, assume a harmonic base motion and find the undamped response:

\[ \begin{aligned} (m+M) \ddot{x}+M l\left(\ddot{\theta} \cos \theta-\dot{\theta}^{2} \sin \theta\right) & \\ -k(y-x)-c(\dot{y}-\dot{x}) & =0 \\ \left(I_{G}+M l^{2}\right) \ddot{\theta}+M l \ddot{x} \cos \theta & \\ +K \theta-M g l \sin \theta & =0 \end{aligned} \]

Problem 6:

A simple lumped-parameter model of a building, shown as the inverted link in Figure 6.68, can be used for preliminary study of earthquake dynamics. Derive the equations of motion for this two degreeof-freedom system using

(a) Newton's second law, and

(b) Lagrange's equation. Let \(y(t)\) be the prescribed ground motion, and \(x(t)\) be the absolute position of the mass \(m\).

Transcribed Image Text:

c/2 00000 m k/2 M. IG K c/2 C 00000 k/2 Figure 6.68: An earthquake loading model showing a simple model of building (the inverted beam) under base excitation.