Showing papers on "Multiple-scale analysis published in 1978"

The relation between temporal and spatial stability in three-dimensional flows

Abstract: An analysis is presented of the nonparallel spatial or temporal stability of three-dimensional incompressible, isothermal boundary-layer flows taking into account the transverse velocity component as well as the axial and crossflow variations of the mean flow. The method of multiple scales is used to derive partial differential equations that describe the axial and crossflow variations of the disturbance amplitude, phase and wavenumbers. This equation is used to derive the expressions that relate the temporal and spatial instabilities. These relations are functions of the complex group velocities. Moreover, this equation is used to derive the expression that relates the spatial amplification in any direction to a calculated amplification in any other direction. These relations are verified by numerical results obtained for two- and three-dimensional disturbances in two- and three-dimensional flows.

Second order hyperbolic equations with small nonlinearities

Abstract: A second order partial differential equation which describes the propagation of one-dimensional nonlinear waves in a bounded, inhomogeneous, dissipative medium is analyzed using the method of multiple scales. The conditions under which the oppositely traveling components of the nonlinear motion uncouple to first order are given. The nonlinear interaction occurs at second order. Two examples are given: the first is a nonlinear free vibration problem; the second is a problem of resonant forced oscillations induced by a time-periodic body force. The nonlinear resonant motion is always bounded, but may contain shocks.

A singular perturbation approach to power system dynamics

Abstract: : This report applies singular perturbation techniques to linear and non-linear models of a single machine-infinite bus power system and a three machine power sytem. In the linear realm, we give a method for obtaining state and eigenvalue approximations by computing approximations to the block diagonal system which isolates the fast and slow dynamics. We also give a 'growth of model' method for determining the slow and fast variables in the power system models. For the non-linear systems, we find that the zero-order approximation of the slow variables is inadequate because it does not account for a non-negligible fast component which is present in these variables. Thus we develop a new and simple method to correct he non-linear zero-order approximations. (Author)

A note on the averaging method

Abstract: A simple problem of a two-body system perturbed by the disturbing function epsilon/r-squared is considered to show that the time-averaged equations of the true and mean anomaly are not necessarily equal. Secular effects due to perturbations in the mean and true anomalies are different not only in value but also in sign. A more general case where the disturbing function is periodic with respect to the mean anomaly is also considered. Here a discrepancy in the time-averaged solutions is due to the fact that the mean anomaly is an action-angle variable conjugate to the action variable L, while the true anomaly is not. When the action-angle variables are chosen as dependent variables, the Hamiltonian is a function only of action variables. One must first derive first-order solutions, substitute them into the right-hand side of the equations of motion, and then take time averages.