The constant drive to improve the performance of aeronautic structures is leading to new designs where nonlinearity is ubiquitous. Accurately predicting the dynamic behavior of nonlinear systems is very challenging because they can exhibit a wide range of behaviors that have no linear equivalent and are very sensitive to parameter changes. In this work, we consider a physics-based model to capture the underlying linear behavior of the system. This linear model is then augmented with a data-driven, machine-learnt model that captures the nonlinearities present in the system. Standard ML models have, however, several important shortcomings from an engineering point of view. They often require large training datasets, do not generalize well to unseen conditions, and can even be physically inconsistent. To overcome these limitations, we investigate the use of Lagrangian Neural Networks (LNNs) where a neural network is used to directly model the Lagrangian function of the system. To enforce physical consistency, the Euler-Lagrange equations of motion of the system are obtained by differentiating this neural network using automatic differentiation techniques. The potential of this modeling approach is numerically and experimentally shown on a range of systems with stiffness and damping nonlinearities.