| In this thesis a flexible and powerful model is presented for the in-plane free vibration of a beam whose curvature and thickness are arbitrary, such a model being the one-dimensional analog of a two-dimensional fan blade model. A beam's shape may have almost any conceivable representation; curvature and thickness are represented by quartic polynomials in the centerline arc length. The vibration model includes the effects of shear deformation, rotary inertia, and centerline extension; the equations of motion are solved by an alternate form of the Rayleigh-Ritz method. Integral formulas for stiffness and mass matrices are evaluated on the computer by a set of very simple routines that do symbolic algebra and calculus. These routines are central to this approach, allowing beam support schemes to be easily changed and obviating the need for any simplifying assumptions. The software developed is practicable for unfamiliar users, being reliable and accepting simple input. Results for Bernoulli-Euler and Timoshenko models are compared to the results of theoretical studies representing a wide variety of authors, models, solution methods, beam geometries, and support schemes, with excellent agreement. Predictions of the Timoshenko model are compared to experimental results, with good agreement. A detailed study of the cantilever beam of variable curvature and thickness is done, showing that frequency is relatively insensitive to curvature, either constant or variable. Vibration frequencies and mode shapes are fare more sensitive to variable thickness than to curvature. Finally, the solution procedure for the fan blade vibration problem is outlined. |
