An analytic method for strong motion studies in layered media†

Non-Linear Forced Vibrations of an Inhomogeneous Layer

“…Among these, the Ramberg}Osgood model is widely used as it can be transformed to a suitable form for analytical solution [12]. Rearranging this transformed model and considering the relation between the stresses and strains, the non-linear stress}strain relation can be written as…”

“…As a result, the response of the layer is assumed to be harmonic with the period 2 / , as the forcing term. Thus, the periodicity condition is written as [12] ; (y, t)"; (y, t#2 / ).…”

“…Boundary conditions (18) and periodicity conditions (19) are not su$cient to obtain this parameter. Instead of the solution of equation (17), the frequency shift is obtained by using the orthogonality condition introduced in references [1,12]. For this purpose, both sides of equation (17) is multiplied by u ( , s) and integrated over the time and co-ordinate.…”

“…Ablowitz et al [7,8] investigated the non-linear vibrations of a layer which is forced along its bottom by using the Lindstedt}PoincareH technique and the method of multiple scales. Engin et al [12] investigated the same problem for a multilayered system by using the orthogonality of the zeroeth order solution to the right-hand side of the "rst order equation of motion. Sridhar et al [13,14] studied non-linear symmetric and asymmetric responses of circular plates by using the Galerkin procedure.…”

“…Transforming the Ramberg}Osgood model to a suitable form for analytical solution, a non-linear partial di!erential equation is obtained as the governing equation. The method of solution of this problem is based on the work of Engin et al [12]. In this study, the solution is expanded to an inhomogeneous medium, which was given by Engin et al [15] only for the inhomogeneity parameter , "0)5.…”