Eulerian and lagrangian formulations of steady rotating annuli

It is interesting to ask if there are any fundamental differences between Eulerian and Lagrangian formulations in structural dynamics problems. Steady rotating annuli formulated in both Eulerian and Lagrangian coordinates are exemplified for the present study. Nonlinear strain-displacement relationships for radial vibration only without circumferential motion in polar coordinates are derived. Linear and nonlinear strain-displacement relationships are implemented to form the governing equations and are compared to check for the validity of small strain analysis. By including the Poisson's effect, the resulting governing equations are nonlinear equations with material density redistribution and plate thickness variation. Analytical solutions of the linearized versions of nonlinear Eulerian and Lagrangian governing equations by keeping material density and plate thickness constant are offered. Fundamental issues in existence and uniqueness of analytical solutions are discussed between linear Eulerian and linear Lagrangian formulations. Furthermore, the fundamental difference of imposing free edge boundary conditions in Eulerian and Lagrangian coordinates is emphasized. Results show that there are in general insignificant differences between Eulerian and Lagrangian descriptions for circular plate problems, although the approaches to solve the problems in the two formulations are quite different. © 2008 WILEY-VCH Verlag GmbH & Co. KGaA.