On linear vibrational systems with one dimensional damping

Abstract

We consider the quadratic eigenvalue problem of linear damped oscillations with one-dimensional damping. We solve the following inverse spectral problem: For given eigenvalues in the left half plane find the undamped frequencies and the components of the damping matrix in the undamped eigenbasis. The solution turns out to be explicit and essentially unique. Moreover, multiple eigenvalues are defective of the highest possible order. As a continuous limit we obtain a homogeneous vibrating string, linearly damped at one end and having no spectrum at all. The obtained matrices could serve as highly non-trivial, exactly solvable examples for testing numerical algorithms.

AMS(MOS):

• 70J20
• 70J25
• 15A18

Keywords:

• Linear vibration
• matrix theory
• 70J20
• 70J25
• 15A18