An approximate analytical solution of beam vibrations during axial motion

An approximate analytical solution for the transverse vibrations of a beam during axial deployment is derived. The approach relies on removing the time dependency from the boundary conditions, and transferring it to the differential equation. The resulting partial differential equation with time-dependent coefficients is solved by employing the classical method of separation of variables and the method of multiple scales. The approximate analytical solution is obtained for axially moving beams with different end conditions, at constant deployment rates. Except for those assumptions associated with slow axial movement and Euler's beam theory, no further assumptions were necessary. To verify the validity of the approximate analytical solution, comparisons with available numerical solutions are presented. Some numerical results were presented for axially moving beams with moving end supports.