In substructuring methods, the substructures are independently analyzed under free-free conditions. For a free-free substructure, its stiffness matrix is singular and rank deficient due to rigid body motion. The variables associated with the inverse of the stiffness matrix are not easy to be accurately determined in the usual manner. This study expands on the previous research on the substructuring methods by taking a deeper look at the analysis of a free-free substructure. A well-conditioned stiffness matrix is constructed for the analysis of a free-free structure. Some difficulties associated with the analysis of the free-free substructures can be solved in a simple and effective way. The substructural eigensolutions and eigensensitivity are solved from the well-conditioned stiffness matrix, other than the singular stiffness matrix. The proposed well-conditioned eigenequation is accurate and efficient to calculate the substructural eigensolutions and eigensensitivity. The properties addressed in this paper are not limited to be used for the analysis of a free-free substructure in many substructuring methods, and they are promising to be generalized to a range of analysis relevant to a free-free structure.