Nonlinear mapping of non-conservative forces for reduced-order modelling

Abstract

Non-intrusive or indirect reduced-order modelling strategies, such as the implicit condensation and expansion method, are applicable to geometrically nonlinear structures modelled using commercial finite-element (FE) software. Traditionally, the non-conservative forces acting on the structure are reduced via a linear projection onto the space spanned by the reduced modeshapes. As such, only the forces acting directly on these reduced modes can be captured, while any energy gained or dissipated by the statically condensed modes is neglected. This can lead to significant inaccuracies in the reduced-order model (ROM) predictions, which is demonstrated here using a 2-degrees-of-freedom (DOF) oscillator, and an FE model of a cantilever beam. It is shown that the non-conservative forces acting on the statically condensed modes can be captured using a nonlinear mapping of the physical DOFs into the reduced coordinates. This introduces additional terms in the reduced equations of motion, which we describe as force compensation . Excellent agreement is observed between the forced response curves of the full-order models and those of our proposed ROMs, both for the oscillator as well as the cantilever beam under different external excitation conditions (i.e. a constant-direction force and a follower force).

Reduced-order model of geometrically nonlinear flexible structures for fluid–structure interaction applications

“…If the FE code provides the quadratic and cubic components of f nl separately, the nonlinear coefficients can be identified individually by solving two different systems. Otherwise, the coefficients have to be identified simultaneously by solving the system (22) that can be written as follows:…”

“…Increasing the number of linear normal modes in the reduction basis for the ICE method to 8 and even 10 did not improve the solution since the additional modes are still bending modes. The solution may be improved by introducing membrane modes, or, as suggested in the literature, to treat follower forces as an extension of the ICE method [22]. Unlike the ICE solution, the ICDual ROM perfectly captures the nonlinear FOM solution as shown in Figure 10 and its precision is quantified in Table 6.…”

“…An Inertial Compensation [21] has been suggested in the literature to consider the in-plane dynamics into the equations of the bending modes, changing the nature of the reduced equations. A Force Compensation [22] was also recently proposed to take into account follower forces with known shape.…”

Reduced Order Model of Nonlinear Structures for Turbomachinery Aeroelasticity

“…This modification changes the nature of the equation that becomes less convenient to integrate. Such a method was recently enhanced by a Force Compensation [15] in order to tackle following forces. In the present paper we choose to add dual modes to the reduction basis in order to compute the in-plane dynamics directly in the reduced equations of the motion.…”