Chapter 5, Problem 5.98P

Joints of high quality can be formed by friction welding. Consider the friction welding of two 40-mm-diameter Inconel rods. The bottom rod is stationary, while the top rod is forced into a back-and-forth linear motion characterized by an instantaneous horizontal displacement, $d\left(t\right)=a\cos \left(\omega t\right)$ where $a=2\text{mm}$ and $\omega =1000\text{rad/s}\text{.}$ The coefficient of sliding friction between the two pieces is $\mu =\mathrm{0.3.}$ Determine the compressive force that must be applied to heat the joint to the Inconel melting point within $t=3\text{s,}$ starting from an initial temperature of $20°\text{C}\text{.}$ Hint: The frequency of the motion and resulting heat rate are very high. The temperature response can be approximated as if the heating rate were constant in time, equal to its average value.