Dynamic Optimization of Functionally Graded Thin-Walled Box Beams

This paper introduces a mathematical model for optimizing the dynamic performance of thin-walled functionally graded box beams with closed cross-sections. The objective function is to maximize the natural frequencies and place them at their target values to avoid the occurrence of large amplitudes of vibration. The variables considered include fiber volume fraction, fiber orientation angle and ply thickness distributions. Various power-law expressions describing the distribution of the fiber volume fraction have been implemented, where the power exponent was taken as the main optimization variable. The mass of the beam is kept equal to that of a known reference beam. Side constraints are also imposed on the design variables in order to avoid having unacceptable optimal solutions. The mathematical formulation is carried out in dimensionless quantities, enabling the generalization to include models with different cross-sectional types and beam configurations. The optimization problem is solved by invoking the MatLab optimization ToolBox routines, along with structural dynamic analysis and eigenvalue calculation routines. A case study on the optimization of a cantilevered, single-cell spar beam made of carbon/epoxy composite is considered. The results for the basic case of uncoupled bending motion are given. Conspicuous design charts are developed, showing the optimum design trends for the mathematical models implemented in the study. It is concluded that the natural frequencies, even though expressed in implicit functions, are well-behaved, monotonic and can be treated as explicit functions in the design variables. Finally, the developed models can be suitably used in the global optimization of typical composite, functionally graded, thin-walled beam structures.