## 3 Mechanical Model of the GMA System with Fractional Damping Term  
It is well known that the step response depends on the structural parameters of the system itself, which can well express the influence of the structural parameters of the system on its dynamic performance. [PERSON] [28] derived the system transfer function of the GMA system with the current as the input and the displacement as the output, and the specific transfer function expression was obtained by substituting the system model parameters in this literature as follows:  
\[\text{G(s)}=\frac{\text{x(s)}}{\text{I(s)}}=\frac{137,170}{19\text{s}^{2}{+}2.48 \ \times\ 10^{5}\text{s}+2.22\ \times\ 10^{9}} \tag{9}\]  
Figure 4 shows the step response curves of the GMA system measured by the experiment and the theoretical curve of the simulation. From the Figure, it can be seen that the step response curves of the system obtained from the experiment and the theory not only have some errors in the overshoot and peak time, but also have large differences in the curve overlap. Since the fractional order derivatives of the Grunwald-Letnikov form are convenient for discretization and easy to perform numerical operations, it is appropriate to include the fractional damping term defined in its form for a more accurate study of the stability of the GMA system. The defining equation of the Grunwald-Letnikov derivative of the function f(t) is as follows:  
\[\left\{ \begin{array}{c}\text{P}=\xi_{1}\frac{\text{d}^{\alpha}}{\text{d}t^{ \alpha}}\\ \left.\frac{\text{d}^{\alpha}\text{f(t)}}{\text{d}t^{\alpha}}\right|_{\text{t =k}_{\alpha}\text{d}}=\lim_{\Delta t\to 0}\frac{1}{\text{d}t^{\alpha}} \sum_{\text{j}=0}^{\text{k}}(-1)^{\text{j}}\left(\begin{array}{c}\alpha\\ \text{j}\end{array}\right)\text{f(k}_{\text{n}}\Delta t\ -\ \text{j}\Delta t) \end{array} \right. \tag{10}\]  
where \(\alpha\) is the order of derivation, \(\Delta\)t is the calculated time step, \(\delta_{1}\) is the fractional damping coefficient, considering the actual engineering context of the GMA system, and \(\alpha\) is taken as \(0\ <\ \alpha\ <\ 2\). The binomial coefficients are as follows:  
\[\left( \begin{array}{c}\alpha\\ \text{j}\end{array} \right)=\frac{\Gamma(\alpha+1)}{\Gamma(\text{j}+1)\Gamma( \alpha+1\ -\ \text{j})} \tag{11}\]  
where \(\Gamma(\cdot)\) is the gamma function, defined as follows:  
\[\Gamma(\alpha+1)=\int_{0}^{+\infty}\text{e}^{-\text{t}}\text{f}^{-1}\text{d}t,\ \text{Re(z)}\ >\ 0 \tag{12}\]  
and satisfies the equation: \(\Gamma(\text{z}+1)=\text{z}\Gamma(\text{z})\). When fractional damping is added to the GMA system, the linear damping term describes the damping of the output rod and the fractional damping term describes the  
Figure 4: Experimental and theoretical curves of step response of the GMA. internal damping of the GMM bar, and the above two terms, together, constitute the damping term of the system. Equation (9) is transformed to obtain the transfer function of the system containing the fractional order calculus as follows:  
\[\mathrm{G(s)}=\frac{\mathrm{x(s)}}{\mathrm{I(s)}}=\frac{137,170}{19\mathrm{s}^{2 }{+}2.48\times 10^{5}\;(\mathrm{s}\;-\;\mathrm{s}\;^{\alpha})\;+\;2.22\times 10^{9}} \tag{13}\]  
Figure 5 shows the step response curves and experimental curves of the GMA system at different fractional orders \(\alpha\). From the Figure, it can be seen that when the order \(\alpha\) gradually changes from 0.84 to 0.89, the overshoot and peak time of the system gradually approach to the experimental value. Compared with the theoretical curve of the integer-order GMA system, the step response curve of the fractional-order GMA system gradually tends to the experimental curve with a higher degree of overlap. Therefore, adding the fractional order calculus to describe the damping characteristics of the GMA system can more accurately characterize the physical change process of the GMA system, which is more consistent with the actual situation.