Damping of Structures - Part 1: Theory of Complex Damping
University at Buffalo
This report, consisting of two parts, presents a complex energy-based damping theory and its applications. The theory is formulated by considering the dissipative and the conservative energy components of damped vibrating systems simultaneously by complex-valued quantities. It provides a theoretical foundation for the analyses of generally (non-proportionally) damped systems. At the same time, the complex damping theory offers new approaches to model the dynamic responses of multiple-degree-of-freedom systems (a relatively underdeveloped area in Newtonian mechanics) and to deal with vibration control problems in structural engineering. Chapter 1 selectively reviews the basic concepts of structural dynamics and damping, as they will be needed in subsequent presentations in the report. Chapter 2 presents the theory of complex damping. This theory gives a unified representation of energy dissipation and energy transfer by means of one complex quantity. Chapter 3 is concerned with lightly damped systems. In such systems, the real part of the eigenvalue of the state matrix and the damping ratio possess a special linearity. This property is directly deduced from the complex damping theory. Chapter 4 deals with evaluation methods for the damping matrices. Chapter 5 first presents the reasons that the quantitative value of the damping matrix depends not only on damping configurations, but also on the mass and stiffness matrices of the structure. This explains why conventional finite element and/or other methods cannot model a damping matrix directly. Then, based on the complex damping theory and by using the lightly damped approach, a method of directly assembling general damping matrices is presented.
Complex Damping, Nonproportionally Damped Systems, Damping Matrices, Lightly Damped Systems, Damping Coefficient Matrices, and Earthquake Engineering.