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

A novel application of the FPM to the buckling differential equation of non-uniform beams

“…As introduced in the introduction this method is capable of solving bending, strength and Eigenvalue problems as introduced here. It is also capable of solving vibration problems [17] and was used to solve boundary value problems of linear materials with non-homogeneous properties [5]. Of course using the FPM for solving much more complicated problems like plates, shells and structures are not trivial missions and are good for future research.…”

A new approximation method to solve boundary value problems by using functional perturbation concepts

“…In 2006, a one dimensional stochastically heterogeneous rod embedded in a uniform shear resistant elastic medium is solved in [3]. The solution of natural frequencies and mode shapes of non-homogeneous rods and beams was studied based on the FPM in 2007 [14]. Also in the same year the buckling load of heterogeneous columns has been found by applying the FPM directly to the buckling differential equation in [18].…”

Free Vibration of Axially Inhomogeneous Beams that are Made of Functionally Graded Materials

“…2 Nachum and Altus used the functional perturbation method to determine the natural frequencies and mode shapes of non-homogeneous rods and beams. 3 By applying the Adomian modified decomposition method, Hsu et al converted the governing differential equation to a recursive algebraic equation and kept the boundary conditions within simple algebraic frequency equations which were suitable for symbolic computation. 4 Based on the fact that a nonuniform beam can be partitioned into multi homogeneous uniform sub-beams, Singh et al 5 developed a numerical method for determining the natural frequencies of a nonuniform beam.…”