Corrected Formulas for Natural Frequencies of Cantilever Beams Under Uniform Axial Tension

Abstract

N ATURAL frequencies of beams under axial tensile force are of interest in many applications such as rotating fan blades, turbine blades, etc. There have been extensive studies on beam dynamics considering the effects of rotation. Rosales and Filipich [1], for example, studied the dynamic stability of a spinning beam with an axial dead load.Naguleswaran [2] presented a study of lateral vibration of a centrifugally tensioned uniform Euler–Bernoulli beam. He developed the database of nondimensional parameters of speed and natural frequencies for various offset ratios of a beam. Lee [3] studied the dynamic response of a rotating Timoshenko beam subjected to axial forces and a moving load. Closed-form solutions for the natural frequencies of rotating pinned–pinned beams are available (for example, Timoshenko et al. [4]).However, other boundary conditions are not readily amenable to such a treatment. Gorman [5] presented the variation of the frequency parameter with axial tension parameter for the six modes of clamped–pinned and clamped–clamped beams. Bokaian [6] presented a comprehensive study of the effect of a constant axial tensile force on the natural frequencies andmode shapes of a uniform single-span beam, with different combinations of the end conditions. Although simple relations could capture the behavior of beams with various boundary conditions, clamped–free beams showed deviant behavior. Thus, the present study focuses on the natural frequencies of a clamped–free beam under uniform axial tension. The relations available for finding the fundamental frequency of a cantilever beam under uniform axial tension are summarized next: The Bokaian [6] characteristic equation is