  In this article, we present an extension of the formulation recently
developed by the authors (A Framework for Data-Driven Computational Mechanics
Based on Nonlinear Optimization, arXiv:1910.12736 [math.NA]) to the structural
dynamics setting. Inspired by a structure-preserving family of variational
integrators, our new formulation relies on a discrete balance equation that
establishes the dynamic equilibrium. From this point of departure, we first
derive an "exact" discrete-continuous nonlinear optimization problem that works
directly with data sets. We then develop this formulation further into an
"approximate" nonlinear optimization problem that relies on a general
constitutive model. This underlying model can be identified from a data set in
an offline phase. To showcase the advantages of our framework, we specialize
our methodology to the case of a geometrically exact beam formulation that
makes use of all elements of our approach. We investigate three numerical
examples of increasing difficulty that demonstrate the excellent computational
behavior of the proposed framework and motivate future research in this
direction.
