Sheet metal is produced by repeatedly feeding the sheet between steel rollers to reduce the thickness. In the article "On the characteristics and mechanism of rolling instability and chatter," (Y-J Lin et al., J. of Manufacturing Science and Engineering, 125 (2003) 778-786) the authors nd that the distance between rollers is well approximated by h + y , where h is the nominal output thickness and y is the solution of the dierential equation y + 2 y + 2 y = 0. The elasticity of the sheet and the rollers provides the restoring force, and the plastic deformation of the sheet eectively provides damping.

a. For high-speed operation, the system is under-damped. Solve the initial value problem consisting of the dierential equation and the initial conditions y (0) = 0.001h , y (0) = 0.

b. For an input speed of 25.4 m/s, it is observed that 600 Hz or 1200 radians/s and = 0.103 . Using these values, obtain a graph of the solution of the problem above, over the range 0 < t < 0.02 s. How far does the sheet move horizontally in 0.02 s.?