Calculating latent frequencies of systems with local damping

Modal analysis of damped systems often cannot proceed with common real-eigenvalue techniques. The system of equilibrium equations leads to a matrix with elements being quadratic functions of a parameter λ. The values of that which make the matrix singular are the latent roots, while the solutions of the associated homogenous equation are the latent vectors. They are the (generally complex) characteristic frequencies and the mode shapes of the system, respectively. Although the theory is well developed, the numerical application is open to refinements yet. A reduction to better-known real-domain subtasks deserves attention. With a theorem of Popper and Gáspár, a n × nλ-matrix problem can be cut in two: into n-size asymmetric real matrices having as eigenvalues the n lower and n upper latent roots, ranked by absolute value. This approach may be of use for systems with high number of degrees of freedom while damped by a relatively few concentrated devices. It might fit also an earthquake analysis, where the lower portion of eigenvalues is customarily what counts. The dampers appear in the splitting algorithm as restricted-size modifications, ready for use by the Sherman–Morrison–Woodbury identity. The task is re-traced this way to a more usual real-asymmetric eigenproblem. A requirement of convergence is that the lower and upper n-set of latent values must be distinct. With odd-number degrees of freedom and neither over-damped, i.e. all latent roots being complex, this condition is surely violated. For such cases, a supplemental algorithm is proposed.