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4.0 Normalized buckling approximately 0.7 arbitrary constants assumed axial load bending stiffness ratios boundary conditions Buckling and Vibration center section characteristic polynomial Column Vibration column with piecewise compressive load constant cross section cube root dimensionless parameters dotted-line curves effect of axial equa equation 26 equation 37 Figure flexural stiffness fundamental vibration frequency gives the following governing equations high-efficiency method increasing the bending jth section Langley Research Center load and vibration mass and bending mass is proportional mass per unit mass savings mass versus normalized method for increasing NASA natural vibration nondimensionalized set Normalized buckling load normalized vibration frequency Numerical Examples numerical solution optimum ratio Parametric structural efficiency Performing Organization piecewise constant cross presented in reference presents plots Research Center Hampton secant method set of governing Simply Supported Column solved to determine stiffness and mass stiffness are considered structural efficiency analyses thin-walled circular cross uniform column unit length vibration characteristics