This paper presents analytical solutions for enhancing the dynamic performance of
functionally graded material bars in axial motion. Optimized designs with maximized
natural frequencies under mass equality constraint are given and discussed. The
composition of the construction material is optimized by defining the spatial
distribution of volume fractions of the material constituents using either continuous
or discrete variations along the bar length. Three cases of boundary conditions
have been examined: fixed-fixed, fixed-free and free-free bars. The major
aim is to tailor the mass and stiffness distributions in the axial direction so
as to maximize the frequencies and place them at their target values to
avoid the occurrence of large amplitudes of vibration without the penalty of
increasing total structural mass. The resulting optimization problem has been
formulated as a nonlinear mathematical programming problem solved by invoking
the Matlab optimization toolbox routines, which implement the method of
feasible directions interacting with the associated eigenvalue problem routines.
The proposed mathematical models have shown that the use of material
grading can be promising in optimizing natural frequencies without mass
penalty.
To Charles and Marie-Louise Steele,
with gratitude.
