Harmonics multiple to the driving frequency of damped bilinear oscillators

Field observations show that power spectra of the response to high amplitude harmonic excitation contain peaks at frequencies multiple to the driving frequency, e.g. [1]. This phenomenon is conventionally attributed to the effect of mechanical non-linearity of the Earth’s crust. Given that there exist various types of non-linearity it is important to identify the types of non-linearities that can produce multiple harmonics in response to high power excitation and thus ensure the correct interpretation of the monitoring data.

One type of non-linearity capable of producing multiple resonances is bilinearity of stiffness, the simplest representation of which is a bilinear oscillator – the oscillator with different stiffnesses for compression and tension. In the Earth’s crust the role of bilinear oscillators can be played by pre-existing fractures initially closed by the in-situ compression but capable of being opened by the tensile phase of the applied high amplitude harmonic excitation.

Bilinear oscillators possess multiple resonances, e.g. [2], however these are multiples of the natural frequency. We note that extreme damping effected by the presence of fluids in fractures and porous rocks can quickly eliminate the effect of the natural frequency leaving only the stationary oscillations with the driving frequency in each linear (tensile or compressive) stage of oscillations. The transition from one stage to another is characterised by short transients, which gives rise to multiple spectral peaks. This mechanism is investigated in asymptotics of high damping ratio. It is shown the existence of the following spectral peaks: if f₀ is the driving frequency, the peaks will be observed at 2f₀, 3f₀, 5f₀ and further at all odd multiples of f₀.

The theory developed is essential for identifying the prevailing mechanisms of non-linearity in the Earth’s crust and determining their parameters.

1. Yurikov, A., B. Gurevich, K. Tertyshnikov, M. Lebedev, R. Isaenkov, E. Sidenko, S. Yavuz, S. Glubokovskikh, V. Shulakova, B. Freifeld, J. Correa, T.J. Wood, I.A. Beresnev and R. Pevzner, 2022. Evidence of nonlinear seismic effects in the earth from downhole distributed acoustic sensors. Sensors 2022, 22, 9382.

2. Dyskin, A.V., E. Pasternak and E. Pelinovsky, 2012. Periodic motions and resonances of impact oscillators. Journal of Sound and Vibration 331(12) 2856-2873.

Acknowledgement. EP, AVD and BG acknowledge support from the Australian Research Council through project DP190103260. RP and BG acknowledge financial support from the Australian Department of Industry, Science and Resources for the 2021 Global Innovation Linkage (GILIII000114) grant and the Sponsors of the Curtin Reservoir Geophysics Consortium.