## 1 Introduction  
In the field of intelligent manufacturing, micro-displacement and microactuation technology with an actuation stroke of less than 1 mm and a resolution of less than 1 um has a wide range of applications [1,2]. At present, there are some functional materials used in the field of microactuation technology such as piezoelectric materials, shape memory alloys, and giant magnetostrictive material [3, 4, 5, 6, 7]. As a new type of functional material, giant magnetostrictive material have some advantages of a strong magnetostrictive effect, a high electromechanical coupling coefficient, high response speed, noncontact drive, and so on, and they are widely used in aerospace, precision control, and intelligent manufacturing fields. Giant magnetostrictive material (GMM) is ferromagnetic material that were developed by [PERSON] and [PERSON] et al. In 1971[8]. Currently, the type of GMM used most commonly is TB\({}_{\mathrm{X}}\)Dy\({}_{1-\mathrm{x}}\)Fe\({}_{\mathrm{y}}\), when \(x~{}=~{}0.23{\sim}0.5\) and \(y~{}=~{}1.9{\sim}2.0\). The trade name of GMM is Terfenol-D [8; 9]. The giant magnetostrictive actuator (GMA) developed by the giant magnetostrictive material (GMM) has the characteristics of a large magnetostriction coefficient, large output power, fast response speed, and high magnetic(electric)-mechanical conversion efficiency, and it is widely used in the fields of ultraprecision machining, micromotors, vibration control, fluid machinery, sonar, and so on. It is one of the hotspots in the field of microactuation [10]. [PERSON], who worked for Toshiba, had developed a new type of tool microfeeding device with a positioning accuracy on the nanometer scale. Tsinghua university [11] designed a linear peristaltic mechanism used the giant magnetostrictive material with a displacement accuracy of \(\pm 15\) nm. Successful development of this actuator has promoted the development of research fields such as Micro Electro Mechanical System (MEMS). Under the action of the external driving magnetic field, the giant magnetostrictive material rod will output displacement and force, and the input electrical signal and output displacement signal of the actuator have corresponding eigen relations. Due to the nonlinear strong coupling properties between the machine electromagnetic heat of the material, it is very difficult to construct a magnetic-mechanical coupling model. [PERSON] [12] proposed a piezomagnetic constitutive equation of a giant magnetostrictive actuator based on a large volume of research, which became the basic equation for the study of magnetic-mechanical coupling characteristics of the giant magnetostrictive actuator. [PERSON] [13] used finite element simulations to calculate nonlinear dynamic characteristics of the giant magnetostrictive rod, which can guide the design and optimization of the magnetic circuit of the giant magnetostrictive actuator. However, the model is a two-dimensional planar model, and its dynamic characteristics are obtained from static characteristics of the giant magnetostrictive material, in addition, there are too many parameters, due to static modeling, which lead to complicated calculations and susceptibility to other field-related parameters, therefore, application of the model has certain limitations. In the research of multiphysics dynamic magnetic-mechanical coupling models, [PERSON] [14] established a three-dimensional generalized finite element model based on the magnetic machine coupled constitutive equation, and simulations of the solenoid coil driving the GMM rod were made.