Complex Modes in Non-Classically Damped Systems

An interactive companion to:

C. Chern and O. M. O’Reilly, On Complex Modal Inverse Analysis: The Cautionary Tale of the Single Surviving Mode, ASME Journal of Applied Mechanics, 2026. doi:10.1115/1.4072378

For a classically damped system, the modes are real and mutually orthogonal, and each mode traces a straight line in configuration space. Moreover, the number of modes (harmonic, synchronous solutions) equals the number of degrees of freedom in the system.

When damping is non-classical, this picture can break down. The animation below shows the two modes of a simple damped oscillator for three cases.

from mdof.utilities import visualization
from IPython.display import HTML

fig = visualization.plot_simplest_oscillator(nt=175, speed_up=True)
HTML(fig.to_html(include_plotlyjs="cdn", full_html=False, auto_play=False))

For the governing equation of motion,

\[\begin{split}\begin{bmatrix} {x}'' \\ {y}'' \end{bmatrix} + \begin{bmatrix} 2\zeta_{1} & 0 \\ 2\zeta_{3} & 2\zeta_{2} \end{bmatrix} \begin{bmatrix} {x}' \\ {y}' \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 0 & \Delta \end{bmatrix} \begin{bmatrix} {x} \\ {y} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix},\end{split}\]

• Ex 1 — ζ₃ = 0 (classical): modes are orthogonal and trace straight lines.

• Ex 2 — Δ = 1 (monophase): both modes are still monophase (straight-line) trajectories, but they are no longer orthogonal.

• Ex 3 — Δ = 1.5 (non-monophase): the second mode becomes non-monophase. Its trajectory becomes an elliptical spiral.

The lower panels show the phase-normalized displacement eigenvector in the complex plane for each mode.