A relaxation-process model for non-exponential damping in structural vibrations

Abstract Several damping mechanisms contribute to the damping of mechanical structures, with every damping mechanism being interpretable as a relaxation process. While vibration decay in beams and rods is generally associated with exponential envelopes, a power law often dominates decay curves observed in many applications. Here, we derive a linear stress-strain relation that accounts for several relaxation processes in material damping without assuming an a-priori rheological topology. The solution of the wave equation and measurements of vibrations in steel beams and timber rods show the distinct decay behaviour in these structural systems. Flexural vibrations of an impacted cantilever beam decay by a power law. Using variational mode decomposition, vibration decay can be decomposed into several empirical modes, each exhibiting distinct characteristics. We show that persistent vibration energy within a particular mode is an indicator of energy efficiency. This persistence can then be analysed in greater detail through the Hilbert–Huang spectrum, which highlights the mode's time scales and the physical phenomena involved. The methods can be used to identify damping in the time-domain if non-exponential decay is observed, which is of particular significance to determine damping in timber structures, although we found that the mode identification in timber rods is less statistically robust than in the steel beam.