Determination of stiffness modifications to keep certain natural frequencies of a system unchanged after mass modifications

Structural modifications in the form of mass, stiffness and damping to a structure change the dynamic properties of that structure. However, in some cases, after modifications are made to the structures, certain specified natural frequencies of the structure are desired to remain unchanged. This study is interested in the determination of necessary stiffness modifications in order to keep a certain number of natural frequencies of the system unchanged despite mass attachments. In particular, two methods based on the Sherman–Morrison formula are developed in order to determine the spring coefficients needed to keep one and more than one natural frequency of the structures unchanged. The developed methods directly use the Frequency Response Functions of the unmodified system relating the modification coordinates only and they need neither a physical model nor a modal model. The numerical simulations show that they are very effective. However, due to the nature of the inverse problem, any solution or practical realistic solution may be not found. The existence of the solution depends on also the modification coordinates chosen. A simple sensitivity approach demonstrated by a 3D graph is proposed to be able to choose a suitable modification.