The three-dimensional packing problem is generally on how to pack a set of models into a given bounding box using the smallest packaging volume. It is known as an NP-hard problem. When discussing the packing problem in mechanical field, the space utilization of a mechanism is low due to the constraint of mechanical joints between different mechanical parts. Although such a situation can be improved by breaking the mechanism into components at every joint, it burdens the user when reassembling the mechanism and may also reduce the service life of mechanical parts. In this paper, we propose a novel mechanism packing algorithm that deliberately considers the DOFs (degrees of freedom) of mechanical joints. With this algorithm, we construct the solution space according to each joint. While building the search tree of the splitting scheme, we do not break the joint, but move the joint. Therefore, the algorithm proposed in this paper just requires the minimal number of splits to meet the goal of space utilization. Numerical examples show that the proposed method is convenient and efficient to pack three-dimensional models into a given bounding box with high space utilization.