Stability properties of some algorithms for the solution of nonlinear dynamic vibration equations

Abstract

This paper describes numerical experiments made to investigate the stability of some time-stepping algorithms applied to the equation ii + P(u) = 0 representing a nonliner elastic spring. These algorithms would be unconditionally stable when applied to linear problems, but here they may be only conditionally stable. Ways of improving the stability are demonstrated; the effect of linearization is also investigated.

Vibration analysis for elastic multi‐beam structures by the C0‐continuous time‐stepping finite element method

“…Hence, estimate (43) follows readily from (33), (45), (46), and the triangle inequality. Similarly, choosing…”

“…The exact period T * and time history of the vibration can be found analytically [46,48] or obtained numerically by using the following C 0 -continuous time-stepping finite element method (54) with very small time steps.…”

Analysis and performance of a predictor‐multicorrector Time Discontinuous Galerkin method in non‐linear elastodynamics

“…Moreover, the analysis of H i+1 =H i vs large values of the t=T ratio, where T is the natural period, allows the behaviour of the scheme with respect to high frequencies to be assessed. Assume S 1 = 100; q 0 = 1:5; p 0 = 0 and di erent values of S 2 to simulate strong nonlinearities [26]. Firstly, the ratio H i+1 =H i is depicted in Figure 2 for a linear case, i.e.…”

“…They can model, for instance, the motion of a lumped mass attached to a taut string (hardening system) or reproduce the motion of a rigid pendulum (softening system). Moreover, such model problems have been exploited both to highlight the e ect of high non-linearities on the stability properties of the trapezoidal rule [26] and to perform convergence studies [20, p. 306]. The second model problem deals with a two-degrees-of-freedom system, viz.…”

Explicit Predictor–multicorrector Time Discontinuous Galerkin Methods for Non-Linear Dynamics

“…Assume M"1, S "100, p "0, q "1)5, S "10 and q "1)7, S "! to simulate strong hardening and softening systems respectively [25]. Moreover, choose the dissipation characteristics corresponding to the spectral radius @ "0)4 (see Figures 2 and 3 …”