The analysis of arbitrarily-damped linear mechanical systems

The analysis of arbitrarily-damped linear mechanical systems is the subject of this paper, the main issue being the analysis of nondecouplable systems . It is well known that decouplable systems occur when the damping matrix h appens to be a linear combination of the mass and stiffness matrices. Syste ms with this type of damping are said to have proportional damping, which n evertheless seldom occurs in practice, but many a damped system is analyzed under the assumption that it is proportionally damped. In fact, this prope rty allows the analyst to study these systems using the same approach as th at applicable to their undamped counterparts. In this paper, we show that p roportional damping need not be assumed in order to analyze the system at h and with the same approach as used to analyze undamped systems. Moreover, w e propose an algorithm to determine the natural frequencies, the damped fre quencies and the damping ratios of an n-degree-of-freedom damped system, th at does not require the casting of the system into first-order form. In thi s way, the characteristic equation is derived naturally as a 2n-degree poly nomial, computing its roots being then straightforward. Furthermore, we pro pose a semigraphical method to ease this calculation, which should be attra ctive to practicing engineers.