This paper presents a conjugate-pair decomposition (CPD) method for offline damage inspection and online health
monitoring of dynamical systems. Responses of damaged dynamical systems are often nonlinear and nonstationary. For
a nonlinear non-stationary signal, empirical mode decomposition (EMD) uses the apparent time scales revealed by the
signal's local maxima and minima to sequentially sift intrinsic mode functions (IMFs) of different time-varying scales,
starting from high- to low-frequency ones. For offline detailed damage inspection, CPD uses one or more pairs of
windowed adaptive harmonics and function orthogonality to track time-varying frequency and amplitude of each IMF.
Because CPD processes only time-domain data, it is free from the edge effect caused by Gibbs' phenomenon and other
mathematical and numerical problems caused by the use of Hilbert transform. Hence, results from CPD are valuable for
accurate identification of dynamical systems. For parametric identification, one can compare the time-varying frequency
and amplitude from CPD with those from perturbation analysis to determine the type and order of nonlinearity and
system parameters. For online health monitoring, CPD tracks the instantaneous frequency of an arbitrary signal without
signal decomposition by processing three or more most recent data to estimate its instantaneous frequency and
amplitude. Numerical results show that CPD is versatile for system identification, damage inspection, and health
monitoring of different linear/nonlinear dynamical systems.