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A direct time integration method is presented for the solution of the equations of motion describing the dynamic response of structural linear and nonlinear multi-degree-of-freedom systems. It applies also to large systems of second order differential equations with fully populated, non symmetric coefficient matrices as well as to equations with variable coefficients. The proposed method is based on the concept of the analog equation, which converts the coupled N equations into a set of single term uncoupled second order ordinary quasi-static differential equations under appropriate fictitious loads, unknown in the first instance. The fictitious loads are established from the integral representation of the solution of the substitute single term equations. The method is simple to implement. It is self starting, unconditionally stable and accurate and conserves energy. It performs well when large deformations and long time durations are considered and it can be used as a practical method for integration of the equations of motion in cases where widely used time integration procedures, e.g. Newmark's, become unstable. Several examples are presented, which demonstrate the efficiency of the method. The method can be straightforward extended to evolution equations of order higher than two. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]