Closed-form criterion for convergence and stability of pseudo-force method for nonlinear dynamic analysis

The pseudo-force (PF) method has been successfully applied to numerous problems with positive secant stiffness or positive tangential stiffness; however, it is widely believed to diverge for problems with the ratio of actual stiffness to pseudo-tangential stiffness being negative or larger than two. In the present paper, this point of view is investigated by applying PF method to single- and multi-degree of freedom systems with elasto-plastic nonlinearities which involve repeated cycles of loading and unloading, where the elastic behavior is either very high positive tangential stiffness or very high negative tangential stiffness (e.g., in Coulomb friction or velocity-dependent friction) or geometrically nonlinear problems. The objective of this study is to in essence seek the reasons for convergence of PF method in the presence of stiff nonlinearities. Closed-form expressions for convergence and stability criteria are derived. Approximate solution by modified Lindstedt–Poincaré method (L–P method) is formulated. The PF method is also applied to a base-isolated structure with nonlinear sliding or velocity-dependent friction bearings. The results of the study indicate the wide applicability of PF method for highly nonlinear problems, including those with high positive or negative tangential stiffness.