Resonant-Frequency-Distribution of Internal Mass Inferred From Mechanical Impedance Matrices, With Application to Fuzzy Structure Theory

The partitioning of internal mass among bands of resonant frequencies is addressed for a prototype internal vibrating structure with small damping, attached via an arbitrary number NA of attachment points to an external structure. Insofar as the dynamics of the latter are concerned, the internal structure is adequately described by a frequency-dependent impedance matrix, any given column of which lists the ratios of the 3NA force components induced by one of the attachment points’ velocity components when all of the other velocity components are held to zero. The properties of matrix elements and their frequency dependence are discussed in relation to principles of mechanics, especially the requirements of translational and rotational invariance of the potential energy functions. Among the deductions are that modal masses can be defined with values calculable solely from the impedance matrix measurements, and that the modal masses sum to the total mass of the internal vibrating system.