Analysis of equations of motion with complex stiffness mode superposition method applied to systems with many degrees of freedom

Abstract

According to past experimental studies, the damping effects of a structure may be considered to be structural damping mechanics in the elastic range, the damping factors or logarithmic damping ratios of which are independent of the frequencies of the disturbing force or the eigenvalues of the structure. In this paper, the authors propose a simple method of expressing the above-mentioned damping effects using complex numbers, although viscous damping mechanics is generally used as the conventional mathematical method. Thus, the spring constant can be expressed as k = k 0 exp(i sgn ωø), where sgn ω = 1 for ω > 0; sgn ω = 0 for ω = 0; sgn ω = −1 for ω < 0. The concept of complex damping is described comparing it with the most common ‘Voigt model’ for a system with a single degree of freedom and it is concluded that both solutions are exactly identical under the conditions of free and forced vibration when both systems have equivalent natural periods and damping ratios. Furthermore, the authors attempt to apply the above complex stiffness to multi systems with many degrees of freedom and investigate their mathematical and dynamical characteristics. The fundamental mathematical characteristics can be described as follows: 1. (1) Any n degrees of freedom system that has 2n distinct eigenvalues occurring as n complex conjugate pairs and n complex conjugate pairs of corresponding eigenvectors according to the definition of ‘sgn ω’. 2. (2) The eigenvectors establish the orthogonality of the frequency domain when either ω > 0 or ω < 0, but they do not establish this property over the domains for both ω > 0 and ω <. 3. (3) By using the above properties, the equations of motion can be reduced to n conjugate pairs of first order differential equations and the solution is obtained by the superposition of n complex conjugate pairs. The fundamental dynamic characteristics can also be described as follows: 1. (1) When the same damping values are assigned to all structural elements making up a vibrational system, the reduced equations form an estimate of the constant damping ratio over all of the modes from the lowest to the highest. 2. (2) Furthermore, when the different damping values are assigned to each individual structural element, the damping values of the reduced equations denote the value which is equivalent to that of any structure of which the mode shapes are predominant. Finally, the authors present the computed results of a system with 18 degrees of freedom consisting of four individual structural elements with different damping values. Through the above studies, it is concluded that the authors' method is reasonable for estimating the damping effects of the structure.