This paper presents a novel method for the computation of nonlinear normal modes (NNMs) of nonlinearly damped dynamical systems based on the combination of force appropriation technique and efficient path following method (EPFM) which is a variant of pseudo arc length continuation method. Pseudo arc length continuation method is one of the most powerful methods for the computation of NNMs but its application is limited to conservative system. The authors extended the application of the EPFM for the computation of linearly damped nonlinear normal by the aids of force appropriation technique (FAT) previously. The method is called FAT-EPFM. In this manuscript the FAT-EPFM is further extended to compute nonlinearly damped normal modes. It was observed that the previously FAT-EPFM fails to compute the nonlinearly damped normal modes, so a mode indicator function (MIF) was utilized in the method to compute nonlinearly damped normal modes. In order to investigate the capability of the method, the nonlinear normal modes of vanderpole system was computed. The results show that FAT-EPFM could calculate the nonlinear normal modes and limit cycle of vanderploe system very well.

Finite element models of realistic nonlinear structures are characterized by very high dimensionality that renders simulations of the full system infeasible. The recent theory of spectral submanifolds enables a locally exact, nonlinear model reduction of the full system's variables in a mathematically justifiable fashion. In this work, we demonstrate recent advances towards the nonintrusive computation of SSMs that enable the treatment of realistic finite-element models.

Gauss's Principle of Least Constraint allows to interpret the equations of motion of a constrained mechanical system as the ones arising from the minimization of a well defined deviation function which, mathematically, is a least-squares problem. Following previous approaches in which a recursive least-squares (RLS) formulation for this problem was discussed, this work introduces a new strategy in which, through a loop involving three minimization problems, a fixed time-step numerical integration algorithm for multibody systems is obtained. This strategy simultaneously enforces the necessary conditions for the accelerations to be consistent with the system constraints and suppresses constraint drifts in configuration and velocity levels. The proposed algorithm is tested on a benchmark problem involving a rectangular Bricard mechanism. Results demonstrate the prevention of constraint violations and mechanical energy drifts, indicating the potential of this strategy for the computational treatment of multibody system problems.