Exact solution of supercritical axially moving beams: symmetric and anti-symmetric configurations

We present an exact solution for supercritical configurations of axially moving beams with arbitrary boundary conditions. We take into account the geometric nonlinearity of the traveling beams in supercritical regime, and the nonlinear buckling problem is analytically solved. A closed-form solution for the supercritical configuration in terms of the axial speed is obtained. Some typical boundary conditions, such as fixed-fixed, fixed-pinned and pinned-pinned, are discussed. More importantly, based on the exact solution, we found a new anti-symmetric configuration for the fixed-fixed axially moving beams. The traveling beam may vibrate around the new anti-symmetric configuration at sufficiently high traveling speeds. A good accuracy of the solution is confirmed by a comparison with the data available in the literature, and with our own numerical results.