21. Extraction of Wave Dispersion Characteristics in a Discrete Chain Using Complex Modal Decomposition

The dispersion relationship of a discrete chain of masses is extracted from numerically simulated data by applying complex modal decomposition. When an impulse excitation is applied to one end of a semi-infinite mass-spring chain, a wave is generated and propagates down the chain. This wave consists of various modes. The time record for the generated data is limited such that the wave reflection does not return to the “sensed” masses. For example, a 250-mass chain is simulated, and we consider (or sensed) the time record of the first 100 masses. The data collected from the numerical simulation consists of the displacements of each mass at each time step. This data is then used to extract complex modes using the complex orthogonal decomposition (COD) and smooth complex orthogonal decomposition. The extracted complex modes accommodate modal traveling waves. We then compute the frequencies and wave numbers from modal coordinates and mode shapes, respectively. The amplitudes and frequencies of the modes are also estimated using Rayleigh’s quotients. The COD extracted dispersion relationship matched the analytical prediction of the dispersion curve for the linear mass chain.