Some recent results on MDGKN-systems

The linearized equations of motion of finite dimensional autonomous mechanical systems are normally written as a second order system and are of the MDGKN type, where the different n × n matrices have certain characteristic properties. These matrix properties have consequences for the underlying eigenvalue problem. Engineers have developed a good intuitive understanding of such systems, particularly for systems without gyroscopic terms (G‐matrix) and circulatory terms (N‐matrix, which may lead to self‐excited vibrations). A number of important engineering problems in the linearized form are described by this type of equations. It has been known for a long time, that damping (D‐matrix) in such systems may either stabilize or destabilize the system depending on the structure of the matrices. Here we present some new results (using a variety of methods of proof) on the influence of the damping terms, which are quite general. Starting from a number of conjectures, they were jointly developed by the authors during recent months. The linearized equations of motion of finite dimensional autonomous mechanical systems are normally written as a second order system and are of the MDGKN type, where the different n × n matrices have certain characteristic properties. These matrix properties have consequences for the underlying eigenvalue problem. Engineers have developed a good intuitive understanding of such systems, particularly for systems without gyroscopic terms (G‐matrix) and circulatory terms (N‐matrix, which may lead to self‐excited vibrations). A number of important engineering problems in the linearized form are described by this type of equations. It has been known for a long time, that damping (D‐matrix) in such systems may either stabilize or destabilize the system depending on the structure of the matrices.