A hierarchical non-iterative extension of the Guyan condensation method for damped structures

Reduction methods are commonly employed for solving the eigenvalue problems of systems with a large number of degrees of freedom. These methods are based upon dividing the system׳s degrees of freedom into masters and slaves, and obtaining a reduced system which is in terms of the masters only. Since 1965 when the Guyan condensation method for undamped structures was presented, which neglects the dynamic effects of the slaves entirely, there have been many efforts to overcome this by proposing various forms of dynamic condensation methods. These methods take into account the dynamics of the slaves through an iterative procedure. In this paper, a hierarchical, non-iterative reduction method has been proposed for damped dynamic systems. The method results in explicit forms of the effective stiffness, viscosity and mass, and also introduces higher order properties when third and higher order approximations are used. Furthermore, a procedure for the automatic selection of master degrees of freedom has been proposed which assures the convergence and increases the efficiency of the method. Application of the method for obtaining low-frequency eigenvalues of two example structures, with and without damping, reveals that results with good accuracy are obtained by using higher order approximations, as they consider the dynamics of the slaves properly.