A uniform bar is clamped at x 0 and is initially at rest. If a constant force F0 is

Chapter 15, Problem 7

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A uniform bar is clamped at x = 0 and is initially at rest. If a constant force \(F_{0}\) is applied to the free end at x = L, the longitudinal displacement u(x, t) of a cross section of the bar is determined from

\(a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}\),  0<x<L, t>0

u(0, t)=0,  \(\left.\quad E \frac{\partial u}{\partial x}\right|_{x=L}=F_{0}\),  E a constant, t>0

u(x, 0)=0,   \(\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0\),  0<x<L

Solve for u(x, t). [Hint: Expand \(1 /\left(1+e^{-2 s L / a}\right)\) in a geometric series.]