*Article* **A Modified Duhem Model for Rate-Dependent Hysteresis Behaviors**

#### **Jinqiang Gan, Zhen Mei, Xiaoli Chen, Ye Zhou and Ming-Feng Ge \***

School of Mechanical Engineering and Electronic Information, China University of Geosciences, Wuhan 430074, China; ganjq@cug.edu.cn (J.G.); mz@cug.edu.cn (Z.M.); gysjdd@163.com (X.C.); yezhou72@163.com (Y.Z.) **\***

 Correspondence: fmgabc@163.com; Tel.: +86-027-6788-3273

Received: 18 September 2019; Accepted: 7 October 2019; Published: 9 October 2019

**Abstract:** Hysteresis behaviors are inherent characteristics of piezoelectric ceramic actuators. The classical Duhem model (CDM) as a popular hysteresis model has been widely used, but cannot precisely describe rate-dependent hysteresis behaviors at high-frequency and high-amplitude excitations. To describe such behaviors more precisely, this paper presents a modified Duhem model (MDM) by introducing trigonometric functions based on the analysis of the existing experimental data. The MDM parameters are also identified by using the nonlinear least squares method. Six groups of experiments with different frequencies or amplitudes are conducted to evaluate the MDM performance. The research results demonstrate that the MDM can more precisely characterize the rate-dependent hysteresis behaviors comparing with the CDM at high-frequency and high-amplitude excitations.

**Keywords:** piezoelectric ceramic materials; Duhem model; hysteresis model
