The elastic bar of Figure P10.13 is undergoing longitudinal vibrations. Let \(u(x, t)\) be the time-dependent displacement of a particle along the centroidal axis of the bar, initially a distance \(x\) from the left support.

(a) Draw free-body diagrams showing the external and effective forces acting on a differential element of thickness \(d x\), a distance \(x\) from the left end of the bar at an arbitrary instant of time.

(b) Show that the governing partial differential equation is

\[ E \frac{\partial^{2} u}{\partial x^{2}}=ho \frac{\partial^{2} u}{\partial t^{2}} \]

(c) Introduce non dimensional variables to derive a nondimensional partial differential equation.

Transcribed Image Text:

u(x, t) dx -p, A, E X FIGURE P10.13 L