Low-frequency spectral properties under true Coulomb friction

Determine whether the low-frequency properties of the energy matrix \(\tilde H_{sym}\) for frictional granular systems governed by true Coulomb friction are similar to those of frictionless granular systems.

Background

The paper analyzes the spectrum of the symmetric part of the linearized dynamical matrix, H~sym\tilde H_{sym}, as a proxy for the energy landscape and viscoelastic response of frictional granular packings. The intermediate- and high-frequency regimes display qualitative similarities across friction coefficients and packing fractions, including a plateau in the density of states and localized rotational modes.

The low-frequency regime contains phonon bands and localized modes reminiscent of frictionless systems, but the discrete-element model introduces spurious instabilities through its history-dependent tangential spring and nonsymmetric dynamical matrix. Because these artifacts disproportionately affect low-frequency modes, the paper leaves unresolved whether the apparent similarities with frictionless systems persist for the exact Coulomb-friction law.

References

As discussed in the previous section, however, the low-frequency data should be treated with caution due to the spurious instability and we cannot determine whether the low-frequency properties of $\tilde H_{sym}$ are indeed similar to those of frictionless systems in the case of true Coulomb friction.

— Linear stability and viscoelastic response in weakly jammed frictional granular systems  (2608.30151 - Shimada et al., 31 Aug 2026) in Section 4.3, “Energy matrix” (subsection label: sec: density of states)