## 3 Mechanical Model of the GMA System with Fractional Damping Term
To further investigate the effect of fractional damping on the GMA system, fractional damping is added to the simplified model, as shown in Figure 6.  
Figure 5: Experimental step response curves of GMA and step response curves of different fractional orders. Figure 6: Simplified model of GMA fractional damping system. After substituting Equation (10) into Equation (8) and normalizing it, the differential equation for the dynamics of the GMA system with fractional damping terms is obtained as follows:  
\[\mathrm{x}+\omega_{0}^{2}\mathrm{x}=\mathrm{fcos}\mathrm{u}t\ -\ 2\xi\dot{\mathrm{x}}- \delta\frac{\mathrm{d}^{\alpha}}{\mathrm{dt}^{\alpha}}\mathrm{x}\ -\ \beta\mathrm{x}^{2}-\gamma\mathrm{x}^{3} \tag{14}\]  
where \(\omega_{0}=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}\), \(\mathrm{f}=\frac{\mathrm{F}}{\mathrm{m}}\), \(2\xi=\frac{\mathrm{c}}{\mathrm{m}}\), \(\delta=\frac{\mathrm{g}_{1}}{\mathrm{m}}\), \(\beta=\frac{\mathrm{g}_{1}}{\mathrm{m}}\), \(\gamma=\frac{\mathrm{\gamma}_{1}}{\mathrm{m}}\). When the external excitation frequency of the GMA system is close to the intrinsic frequency of the system, the system will undergo resonant response. To facilitate the micro-amplitude vibration analysis of the nonlinear GMA system, adding the weak nonlinear factor \(\varepsilon\) in the system before the nonlinear and damping terms and introducing the tuning parameter \(\sigma\), and \(\omega^{2}\)= \(\omega_{0}^{2}+\varepsilon\sigma\) is substituted into Equation (14) and organized as follows:  
\[\mathrm{x}+\omega^{2}\mathrm{x}=\varepsilon\big{(}\mathrm{fcos}\mathrm{u}t\ -\ \sigma \mathrm{x}\ -\ 2\xi\dot{\mathrm{x}}\ -\ \delta\frac{\mathrm{d}^{\alpha}}{\mathrm{dt}^{\alpha}}\mathrm{x}\ -\ \beta\mathrm{x}^{2}-\gamma\mathrm{x}^{3}\big{)} \tag{15}\]  
The solution of Equation (14) can be set as:  
\[\mathrm{x}=a\cos\phi,\ \dot{\mathrm{x}}=-a\omega\sin\phi \tag{16}\]  
where \(a\) is the amplitude. \(\phi\) (\(\phi=\omega\mathrm{t}+\theta\)) is the phase, which is a slow-varying function of time. Differentiating \(\mathrm{x}\) and \(\dot{\mathrm{x}}\) in Equation (16) with respect to \(\mathrm{t}\) yields the following:  
\[\left\{\begin{aligned} \dot{\mathrm{x}}&=-\dot{a}\cos \phi\ -\Big{(}\ \dot{\theta}\ +\ \omega\Big{)}a\sin\phi\\ \mathrm{x}&=-\dot{a}\omega\sin\phi\ -\Big{(}\ \dot{\theta}\ +\ \omega \Big{)}a\omega\cos\  
single-value solution, and the right side of the resonance bifurcation appears, the left side of the resonance curve is a stable solution, and the right side of the resonance curve is an unstable solution, and the resonance curve \"jumps\" phenomenon occurs. This also shows the trend of decreasing and then increasing with the order \(\alpha\) when the amplitude \(a\) is at the resonance frequency in Figure 7.  
Figure 9 shows the resonance amplitude and frequency characteristics of the GMA system with different external excitation amplitude f when the basic parameter of the GMA system is \(\omega_{0}^{2}=0.4\), \(\gamma=0.1\), \(\beta=-0.4\), \(2\xi=0.2\), \(\delta=0.5\), \(\alpha=0.88\). As can be seen from the Figure, the resonance amplitude of the GMA system shows a trend of increasing and then decreasing with the increase of external excitation frequency \(\omega\). Along with the increase of external excitation amplitude, the resonance peak of the system also becomes larger accordingly. The resonance curve shows the phenomenon of outward expansion, the skeleton curve is deflected to the right, and it is hard nonlinear, and the resonance region and instability region are gradually widened. Figure 10 shows the main resonance amplitude-frequency characteristic curves of the GMA system with different fractional damping coefficients \(\delta\) when the basic parameters of the GMA system are taken as \(\omega_{0}^{2}=0.4\), \(\gamma=0.1\), \(\beta=-0.4\), \(2\xi=0.2\), f = 0.1, and \(\alpha=0.88\).