**5. Conclusions**

In this paper, a generalized Bouc–Wen model is established to characterize both rate-independent and rate-dependent hysteresis behaviors by introducing relaxation functions in the classical Bouc–Wen model. The corresponding parameter was identified by the nonlinear least squares method through Matlab/Simulink. The validity of the developed model is demonstrated by a number of experiments. Comparing the predicted data of the generalized Bouc–Wen model with the experimental data revealed reasonably good agreements. The results showed that the developed model can describe rate-dependent and rate-independent hysteresis behaviors more precisely than the classical Bouc–Wen model. The modeling errors of the generalized Bouc–Wen model can be reduced greatly.

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

**Funding:** This work was supported by the National Natural Science Foundation of China (Grant Nos. 51805494, U1501247 and U1609206), the Fundamental Research Funds for the Central Universities (Grant Nos.CUGL180819), and Open Foundation of Guangdong Provincial Key Laboratory of Precision Equipment and Manufacturing Technology (PEM201702).

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