Vibration and buckling of flexible rotating beams

Perturbation techniques are employed to estimate the free vibration characteristics and buckling limit of a flexible rotating Bernoulli-Euler beam. The normalized beam stiffness e=EI/mΩ 2 R 4 is introduced and treated as a small parameter. Then, singular perturbation solutions to the governing eigenvalue problem are derived that are valid up to and including order e for any given hub radius. The special case of a zero hub radius is then considered and the corresponding solutions are presented. Next, a transformation is introduced which leads to a regular perturbation formulation of the problem the solution of which is presented. The natural frequency/mode shape predictions and buckling limits obtained from both the singular and regular perturbation formulations are compared with «exact» values obtained from a power series solution of the eigenvalue problem. The singular perturbation solution matches well with the «exact» values for small stiffnesses whereas the regular perturbation solution provides an excellent accuracy for all beam stiffnesses and hub radii considered

[1]  M. Rothman.  TABLES OF THE INTEGRALS AND DIFFERENTIAL COEFFICIENTS OF Gi(+χ) AND Hi(−χ) , 1954 .

[2]  R. Mathon,et al.  Buckling and vibration of a rotating spoke , 1978 .

[3]  W. Lakin,et al.  Vibration and buckling of rotating flexible rods at transitional parameter values , 1979 .

[4]  Stephen Wolfram,et al.  Mathematica: a system for doing mathematics by computer (2nd ed.) , 1991 .

[5]  David A. Peters,et al.  In-Plane Vibration and Buckling of a Rotating Beam Clamped Off the Axis of Rotation , 1980 .

[6]  W. Lakin,et al.  A FOURTH-ORDER EIGENVALUE PROBLEM WITH A TURNING POINT AT THE BOUNDARY , 1975 .

[7]  W. Lakin,et al.  Unstable vibrations and buckling of rotating flexible rods , 1978 .

[8]  W. Lakin,et al.  Uniform Real-Variable Asymptotic Approximations for Vibrations of a Rotating Flexible Rod , 1982 .

[9]  W. Lakin.  On the differential equation of a rapidly rotating slender rod , 1974 .

[10]  W. Lakin.  Vibrations of a rotating flexible rod clamped off the axis of rotation , 1976 .

[11]  A. Nayfeh.  Introduction To Perturbation Techniques , 1981 .

[12]  Charles E. Smith,et al.  Vibration Modes of Centrifugally Stiffened Beams , 1982 .

[13]  Raymond G. Kvaternik,et al.  Buckling of rotating beams , 1979 .

[14]  V. Giurgiutiu,et al.  Semi-analytic methods for rotating Timoshenko beams , 1975 .

[15]  Dewey H. Hodges,et al.  An Approximate Formula for the Fundamental Frequency of a Uniform Rotating Beam Clamped Off the Axis of Rotation , 1981 .