1. State Key Laboratory of Reliability and Intelligence of Electrical Equipment Hebei University of Technology Tianjin 300130 China; 2. Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability of Hebei Province Hebei University of Technology Tianjin 300130 China
Abstract：The Villari effect describes the magnetization change of magnetostrictive materials induced by mechanical loading. Magnetostrictive devices based on the Villari effect can realize electrical energyconverted from mechanical energy to harvest vibration energy in the environment. The output characteristics of magnetostrictive materials under stress excitation conditions depend on the bias conditions (prestress, bias magnetic field) and excitation frequency. Previous studies establishthe Villari effect model according to linear piezomagnetic equations, ignoring the material’s nonlinearity, hysteresis properties, and eddy current effects. This paper uses the Armstrong energy model to derive a hysteresis-free magnetization strength equation of magnetostrictive materials under stress. The hysteresis-free equation is expanded into a static hysteresis model combined with the basic ideologies of the J-A hysteresis model. Then, considering the eddy current effect inside the material under high-frequency stress, the static model is expanded into a dynamic nonlinear Villari hysteresis model by introducing a first-order differential equation with a frequency-dependent time coefficient. The particle swarm and genetic optimization algorithms extract the model parameters through 3 progressive steps. The particle swarm genetic algorithm calculates the average static model parameter values for the quasi-static B-σ major loops of the Galfenol bar with 3.88~7.17 kA/m bias magnetic fields under 31.4 MPa stress amplitude, 1 Hz frequency, and -31.9 MPa prestress. The values of the quasi-static B-σ major and minor loops with bias magnetic fields of 0.73~13.76 kA/m are simulated using the average static model parameter values, which agree with experimental data. The bias conditions of the maximum piezo magnetic coefficient of the quasi-static B-σ major loop are -1.65 MPa prestress and 1.42 kA/m bias magnetic field. Based on the average static model parameter values, the dynamic parameter values τp for the dynamic B-σ major loops are extracted with 2.46 kA/m bias magnetic field under 31.4 MPa stress amplitude, 10~200 Hz frequency, and -31.9 MPa prestress. The dynamic parameter values τm for the dynamic B-σ minor loops are extracted with 2.46 kA/m bias magnetic field under 2.88 MPa stress amplitude, 10~200 Hz frequency, and -7.96 MPa prestress. The dynamic parameters τp, τm, and the stress frequency are distributed according to a power function law. The model calculation values of the dynamic major loop and minor loops under 400 Hz and 600 Hz stress frequencies are obtained. The maximum error is 6.28%, which verifies the accuracy of the proposed dynamic model. The results show that the nonlinear dynamic Villari hysteresis model can effectively predict the magnetic property response of magnetostrictive materials when the external compressive stress changes. It can provide a reference for the optimal design and performance regulation of electromagnetic devices, such as magnetostrictive energy harvesters, sensors, and dampers.
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2024, Vol. 39 
