Natural frequencies and mode shapes of deterministic and stochastic non-homogeneous rods and beams

Natural frequencies and mode shapes of non-homogeneous (deterministic and stochastic) rods and beams are studied. The solution is based on the functional perturbation method (FPM). The frequencies and mode shapes are considered as functionals of the non-homogeneous properties. The natural frequency and mode shape of the kth order is obtained analytically to any desired degree of accuracy. Once the functional derivatives (with respect to the non-uniform property) have been found, the solution for any morphology is obtained by direct integration without resolving the differential equation. Several examples with different non-homogeneous properties are solved and compared with exact solutions as an accuracy check. The FPM accuracy range for the frequency ω and the mode shape is less than 1% even for high heterogeneities. In the stochastic case the accuracy of the natural frequencies depends on the stochastic information used/given, on the correlation distance (roughly the “grain size”), on the function around which the perturbation is executed, and on whether we are interested in the properties of ω or of ω2. Moreover, all frequency modes have the same response to heterogeneity as long as their wave length is of the order of the heterogeneity's characteristic distance. In addition, the heterogeneity effect on the average natural frequencies is minimal for the fundamental mode, and may serve as a design tool.