Computing Natural Frequencies and Mode Shapes of an Axially Moving Nonuniform Beam

The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass, and material properties has been derived. Using the theory of linear time-varying systems and numerical optimization, a general algorithm has been developed to compute complex eigenvalues/natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by nonuniformity in a beam's cross section.