Matrix analysis of coupled damped vibrations -- two degree-of-freedom analytic solution

Authors: Essén, H.
Document Type: Article
Pubstate: Published
Journal: International Journal of Pure and Applied Mathematics
Volume: 36   115-146
Year: 2007


The linear coupled damped free oscillator problem is solved by means of matrix analysis. The solution can be written by means of exponentials of time evolution matrices multiplied by time. These time evolution matrices are determined as the solutions of quadratic matrix equations. The intricacies of non-proportional damping become particularly clear in this treatment. For the two degree-of-freedom problem an explicit analytic solution is given in terms of a root of a cubic equation. The existence of an algebraically simpler special case (other than proportional damping) is pointed out.