Eigenvalue-counting methods for non-proportionally damped systems. (Q1430801)

As a rule, the dynamical response of a civil structure can be simulated by using only the few lowest modes. So, the numerical methods which can treat only the lowest eigenvalues, and modes at minimal computational cost are very important from the practical point of view. In this paper the authors consider a numerical method which uses the argument principle to bound the region in the complex plane where the lowest eigenvalues of the system are placed. The method uses a relationship between the characteristic polynomial and factorized matrices. The authors consider the systems with non-proportional damping typical of soils. As a test problem, the authors investigate the complex eigenvalues when the dissipation matrix has the form \(C=\alpha M+\beta K\), where \(M\) is mass matrix, \(K\) is stiffness matrix, \(\alpha\) and \(\beta\) are constant coefficients. In this case the complex eigenvalues can be expressed as \[ \lambda_{2i-1,2i}= -\xi_i\omega_i \pm j\omega_i\sqrt{1-\zeta^2_i},\quad\text{for }i=1,\dots,n,\;\zeta_i =\frac 12\left(\frac {\alpha} {\omega_i}+\beta\omega_i\right), \] \(\omega_i\) being the natural frequencies of the undamped system. So the numerical results can be obtained by considering only the undamped system.