An Engineering Interpretation of the Complex Eigensolution ofLinear Dynamic Systems

In traditional finite element based modal analysis of linear non-conservative structures, the modal shapes are determined solely based on stiffness and mass. Damping effects are included by implicitly assuming that the damping matrix can be diagonalized by the undamped modes. The approach gives real valued mode shapes and modal coordinates. While this framework is suitable for analysis of most lightly damped structural systems, it is insufficient for interpretation of the free vibration and resonant response of structures with e.g. significant nonclassical damping, gyroscopic or other effects resulting in a complex eigensolution. In this paper, the more general approach based on complex eigenvalues and eigenvectors is employed. We give an interpretation of the complex eigensolution that describes free and resonant vibrations of a generally damped linear structure. The interpretation show how the different parts of the complex eigensolutions; i.e. the complex left and right eigenvectors together with the complex eigenvalues, combines into vibration frequencies and modal damping ratios, mode shape magnitudes and phase angles, and modal coordinate magnitude and phase angles. The presented interpretation relates all elements of the complex valued solution to physical quantities that are well known in structural dynamics as well as other fields studying linear dynamic systems, and complements the already applied interpretations. INTRODUCTION The dynamics of linear structures are traditionally interpreted in terms of classical normal modes. Classical normal modes are defined as the modes belonging to linear undamped systems. T.K. Caughey [1] has also shown that a special class of damped systems have classical normal modes. A necessary and sufficient condition for a damped system to possess classical normal modes is that the damping matrix can be diagonalized by the transformation that uncouples the associated undamped system. We will refer to such systems as being classically damped. Real structures will generally not have a damping matrix that strictly satisfies the requirements for the system to possess classical normal modes. Nevertheless, for practical engineering purposes and lightly damped systems the normal mode approximation may be sufficiently good. However, the small damping assumption does not hold for e.g. deep-water risers. The dynamic behaviour of such structures exhibits progressive waves that cannot appear in a traditional normal mode approach. In evaluation of the dynamic behaviour of structures, experiments have been widely applied to determine the dynamic properties. Since a modal analysis reveals the basic dynamic behaviour, it is a preferred coordinate basis for interpretation of measured dynamic response. Today almost all measurements of structural response are processed by digital computers, yielding a time, amplitude and space discretized representation of the response. The discretization in time caused by the sampling process and the possibly imperfect time synchronisation between different measuring devices may introduce phase modulation into the vector of measured response time histories. Response measurements should therefore be processed and interpreted as coming from a system that permits spatial phase variations, even for cases that in reality are classically damped. Thus, there is a need for an interpretation of the complex eigensolution of generally damped systems in terms of physical quantities such as mode shapes, vibration amplitudes and phase angles. Several textbook authors have treated elements of this topic over the years; see e.g. Hurty and Rubinstein [2], Newland [3], Meirovitch [4] and Ewins [5]. The topic has also been treated or touched in several papers presented on IMAC conferences over the years, e.g. [6 18] Hurty and Rubinstein show how to apply the complex eigensolution to obtain a real-valued forced response. However, they do not give an explicit interpretation of the complex eigensolution for the free and resonant vibration case. Newland interprets the complex eigenvectors as counter rotating phasors, but does not show how they in fact combine into real valued response. The phase shift between elements of the mode shape belonging to different positions on the structure is briefly indicated in his presentation. Meirovitch as well as Ewins presents the complex eigensolution, but does not give a complete physical interpretation in terms of mode shapes, modal amplitudes and corresponding phase angle of the complex quantities that constitutes the complex eigensolution. In this paper, we will give a physical interpretation of the complex eigensolution that decouples a linear, spatially discretized, dynamic structural system, with general, not necessarily symmetric, mass, damping and stiffness properties. We interpret the modal shape as the envelopes and the spatial phase shifts determined by the magnitude and the phase angle of the corresponding normalized complex right eigenvector. We will show that for a damped free vibration, the initial condition given by a state vector containing generalized displacements and velocities, and the complex left eigenvectors of the system uniquely define the initial complex modal coordinates. The initial complex modal coordinate is interpreted as the amplitudes and the phase angles of the modal vibrations. Realizing that each point in a response time series is the initial condition of an ensuing free vibration, one can also compute time series of modal amplitudes and phase angles from vector time series of response. This requires the left eigenvectors of the system to be known. THE EQUATIONS OF MOTION The damped free vibration of a linear time-invariant multi-degree-of-freedom structural system can be approximated by a spatially discrete second order differential equation as follows ( ) ( ) ( ) ( ) t t t t Q Kq q C q M = + + (1) where is the time varying vector of nodal loads. The time varying generalised displacements vector is ( ) t Q ( ) t q , the generalised velocities vector is and the generalised acceleration vector is . The mass matrix M is assumed positive definite. The damping matrix C may contain both viscous damping terms and gyroscopic terms. Gyroscopic terms may occur for e.g. risers with internal flow, and likewise for towed cables, se e.g. Blevins [19], and of course for rotating shafts etc. Thus, the damping matrix may be non-symmetric. The stiffness matrix may contain general stiffness properties. Normally the stiffness matrix will be symmetric. However, in certain flow-induced vibration problems, e.g. the classical flutter problem of airfoils, the equation of motion may be formulated to yield a non-symmetric stiffness matrix. ( ) t q ( ) t q