An Optimization Problem for a Vibrating Beam.
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OHIO SCHOOL OF ENGINEERING
Pagination or Media Count:
The mass distribution of a clamped-clamped beam of square cross section is optimized to produce the maximum first natural frequency. An action integral is constrained with a mass integral creating an isoperimetric formulation. Variations are taken with respect to displacement and beam cross sectional area. The resulting differential equations are solved via Galerkins technique applied to both the displacement and area functions. The algorithm described satisfies the constraint and area variation equations for each step of the iterative process. Negative values of the Lagrange multiplier were found to indicate an increase in frequency. The beam was constrained to have a minimum area value of 25 of the uniform beam used for comparison. The assumptions made allowed only continuously differentiable area shapes to be generated. The optimum frequency found represented an increase of 38 over the frequency of the uniform beam. Author
- Structural Engineering and Building Technology