**5. Conclusions**

In this paper, a modified Duhem model (MDM) is proposed to describe rate-dependent hysteresis behaviors at high-frequency and high-amplitude excitations. The MDM combines trigonometric functions and derivatives of input signal based on the classical Duhem model (CDM). The MDM parameters can be identified easily by the nonlinear least squares method. Six groups of experiments were conducted and the experimental and simulation results show that the MDM can more precisely describe rate-dependent hysteresis behaviors at high-frequency and high-amplitude excitations than the CDM. It is demonstrated that the MDM is effective and useful.

**Author Contributions:** Conceptualization, J.G., Z.M., X.C., M.-F.G. and Y.Z.; methodology, J.G. and Z.M.; software, J.G and Z.M.; validation, J.G. and Z.M.; formal analysis, J.G. and Z.M.; investigation, J.G.; resources, J.G. and Z.M.; data curation, J.G. and Z.M.; writing—original draft preparation, J.G and Z.M.; writing—review and editing, J.G., Z.M., X.C., M.-F.G. and Y.Z.; visualization, J.G. and Z.M.; supervision, J.G. and M.-F.G.; project administration, J.G. and M.-F.G.; funding acquisition, J.G and M.-F.G.

**Funding:** This work was supported by the National Natural Science Foundation of China (Grant Nos. 51805494 and 61703374), the Fundamental Research Funds for the Central Universities (Grant Nos.CUGL180819), and the Open Foundation of Guangdong Provincial Key Laboratory of Precision Equipment and Manufacturing Technology (PEM201702).

**Conflicts of Interest:** The authors declare no conflict of interest.
