The tangent stiffness matrix for an absolute interface coordinates floating frame of reference formulation

In this work, a full and complete development of the tangent stiffness matrix is presented, suitable for the use in an absolute interface coordinates floating frame of reference formulation. For simulation of flexible multibody systems, the floating frame formulation is used for its advantage to describe local elastic deformation by means of a body's linear finite element model. Consequently, the powerful Craig-Bampton method can be applied for model order reduction. By establishing a coordinate transformation from the absolute floating frame coordinates and local interface coordinates corresponding to the Craig-Bampton modes to absolute interface coordinates, it is possible to satisfy kinematic constraints without the use of Lagrange multipliers. In this way, the floating frame does not need to be located at an interface point and can be positioned close to the body's center of mass, without requiring an interface point at the center of mass. This improves simulation accuracy. In this work, the expression for the new method's tangent stiffness matrix is obtained by taking the variation of the equation of equilibrium. The global tangent stiffness matrix is expressed as a local tangent stiffness matrix, consisting of both material stiffness and geometric stiffness terms, transformed to the global frame by the rotation matrix of the floating frame. Simulations of static and dynamic validation problems are performed. These simulations show the importance of including the tangent stiffness matrix for both convergence and simulation efficiency.