In recent years, many approaches to visual-inertial odometry (VIO) have become available. However, they neither exploit the robot's dynamics and known actuation inputs, nor differentiate between the desired motion due to actuation and the unwanted perturbation due to external force. For many robotic applications, it is often essential to sense the external force acting on the system due to, for example, interactions, contacts, and disturbances. Adding a motion constraint to an estimator leads to discrepancy between the model-predicted motion and the actual motion. Our approach exploits this discrepancy and resolves it by simultaneously estimating the motion and external force. We propose a relative motion constraint combining the robot's dynamics and the external force in a preintegrated residual, resulting in a tightly coupled, sliding-window estimator exploiting all correlations among all variables. We implement our visual inertial model-based odometry system into a state-of-the-art VIO approach and evaluate it against the original pipeline without motion constraints both on simulated and real-world data. The results show that our approach increases the accuracy of the estimator up to 29% as compared to the original VIO, and it provides external force estimates at no extra computational cost. To the best of our knowledge, this is the first approach that exploits model dynamics by jointly estimating motion and external force. Our implementation will be made available open-source.