## 2 Traditional Mechanical Model of the GMA System  
Figure 1 shows the structure of the GMA system. The GMA working principle: the permanent magnet generates bias magnetic field, so that the driver can avoid the problem of frequency doubling; the disc spring and output rod sleeve form the preload structure, which generates the axial preload to make the GMM bar work under the compressive stress condition; the magnetic guide ring, magnetic guide sleeve and magnetic guide sheet can form the closed magnetic circuit, which can improve the magnetic field utilization and reduce the leakage, and can also reduce the external environment on the internal magnetic field. The influence of the external environment on the internal magnetic field can also be reduced. The drive coil generates a magnetic field through the current, which causes the GMM bar to change its length and push the output rigid rod to move in the form of output shaft force to realize the displacement and force output. Considering the connection stiffness of the output part and the stiffness characteristics of the pre-pressed disc spring, the output of the GMA system is equated to a single degree of freedom k-m-c system, and its simplified model is shown in Figure 2. The upper half of the simplified model is a cross-shaped output rod, and the lower half is a GMM rod with a lower end face displacement of 0. Where N is the output axis force generated by the  
Figure 1: GMA structure: 1. output rod; 2. disc spring; 3. drive coil; 4. GMM bar; 5. permanent magnet; 6. lower guide ring; 7. base; 8. upper end cap; 9. upper permeable ring; 10. permeable sleeve; 11. outer wall sleeve; 12. permeable sheet; 13. coil. GMM rod under the excitation magnetic field, F and \(\omega\) are the amplitude and frequency of the external excitation signal on the output rod, c is the system damping factor, k is the coefficient of elasticity of the disc spring, \(u\) and \(i\) are the voltage and current of the drive coil, respectively, and x is the acceleration of the output rod. The spring module consists of the pre-pressed disc spring and the GMM bar in the GMA system, and the system damping includes the internal damping of the GMM bar and the damping of the output bar. The forces on the output rod in the GMA system are analyzed as shown in Figure 3. Where K(x) is the prestress applied by the disc spring, and C(\(\dot{\mathrm{x}}\)) is the damping force on the output rod. Under the action of static uniform magnetic field, the upper face of the GMM rod and the output rod have the same displacement x, velocity \(\dot{\mathrm{x}}\), and acceleration x between them. Under the action of a static stable magnetic field, according to the first piezomagnetic equation, the magnetostrictive effect, the linear piezomagnetic equation along the length direction of the GMM bar can be obtained as follows:  
\[\left\{ \begin{array}{l}\mathrm{B}=\mathrm{d}_{33}\mathrm{T}_{3}+\mu_{33}^{ \mathrm{T}}\mathrm{H}\\ \zeta=\mathrm{s}_{33}^{\mathrm{T}}\mathrm{T}_{3}+\mathrm{d}_{33}\mathrm{H} \end{array} \right. \tag{1}\]  
Figure 3: Equivalent mechanical model of the GMA output rod. Figure 2: Simplified model of the GMA system. where B is the magnetic induction intensity, d\({}_{33}\) is the piezomagnetic constant, T\({}_{3}\) is the stress on the GMM bar, \(\mu_{33}^{\rm T}\) is the magnetic permeability, H is the magnetic field strength inside the GMM bar, and s\({}_{33}\) is the flexibility factor. According to [PERSON]'s second law, the force analysis of the output rod, the dynamics equation of the GMA system, is obtained as follows:  
\[\rm{mx}=N+Fcos\omega t\ -\ K\ -\ C \tag{2}\]  
where m is the mass of the output rod. The displacement of the output lever of the GMA system is as follows:  
\[\rm{x}=\zeta \tag{3}\]  
where \(\zeta\) is the strain of the GMM bar, and l is the length of the GMM bar.