## 2 The Stiffness Analysis of Post-Buckling SGC
### The Characteristic of the Post-Buckling Spring  
In Figure 1, the dynamic equation of the axial compressed support beam is written as [27]:  
\[EI\frac{d^{2}\theta}{ds^{2}}+P\sin\theta=0 \tag{1}\]  
where \(\theta\) is the corner of the beam bending, \(P\) is the axial pressure of the beam, \(s\) is the beam arc length coordinate, and _d/ds_ is differential operator.  
It is essential to consider the large deflection angle in the post-buckling analysis. This is part of the strong, nonlinear problem, which in turn requires two classes of elliptic integral functions [29].  
With the transformation and integration, Equation (1) can be expressed as  
\[\frac{EI}{4P}\bigg{(}\frac{d\theta}{ds}\bigg{)}^{2}+\bigg{(}\sin\frac{\theta}{ 2}\bigg{)}^{2}=c^{2} \tag{2}\]  
The endpoint deflection angle is assumed to be \(\theta_{0}\), and the curvature is zero due to the endpoint of the hinge, so \(d\theta/ds\) = 0; thus, one can obtain  
\[\bigg{(}\sin\frac{\theta_{0}}{2}\bigg{)}^{2}=c^{2} \tag{3}\]  
due to \(\theta\subset\) (0, \(\theta_{0}\)), and introducing \(\varphi\subset\) (0, \(\pi/2\)), we assume  
\[\sin\bigg{(}\frac{\theta}{2}\bigg{)}=\sin\bigg{(}\frac{\theta_{0}}{2}\bigg{)} \sin\varphi=c\sin\varphi \tag{4}\]  
The following Equation (6) will be obtained by Equation (5).  
\[\frac{d}{ds}\bigg{(}\sin\bigg{(}\frac{\theta}{2}\bigg{)}\bigg{)}=\frac{1}{2} \cos\frac{\theta}{2}\frac{d\theta}{ds}\frac{d}{ds}(c\sin\varphi)=c\cos\varphi \frac{d\varphi}{ds} \tag{5}\]  
\[\frac{d\theta}{ds}=\frac{2c\cos\varphi}{\cos(\theta/2)}\frac{d\varphi}{ds} \tag{6}\]  
By substituting the above Equations (3), (4) and (6) into Equation (2), one can obtain  
\[\sqrt{\frac{1}{1-(c\sin\varphi)^{2}}d\varphi}=\sqrt{\frac{P}{EI}}ds \tag{7}\]  
Considering the first class of elliptic integral,  
\[\int_{0}^{\frac{\pi}{2}}\sqrt{\frac{1}{1-(c\sin\varphi)^{2}}}d\varphi=\frac{L} {2}\sqrt{\frac{P}{EI}} \tag{8}\]  
the value \(c\) is obtained and replaced into Equation (7), and using the first class of non-complete elliptical integral,  
\[\int_{0}^{\varphi}\sqrt{\frac{1}{1-(c\sin\varphi)^{2}}}d\varphi=\int_{0}^{l} \sqrt{\frac{P}{EI}}ds \tag{9}\]  
A set of data (\(l\), \(\varphi\)) is derived and converted into (\(l\), \(\theta\)) via Equation (4), where \(l\subset\) (0, \(L/2\)) is the arc coordinate indicating the arc length from the middle to the end-point, and \(\theta\) indicates the deflection angle.  
A symmetrical arc of length \(L\) can be drawn when the length \(L\), Young's modulus \(E\), and Section diameter \(D\) of the SGC are selected as 0.11 m, 3.5 \(\times\) 10\({}^{4}\) Pa, and 0.02 m,respectively. By increasing the axial pressure, which is applied to the right side of the SGC, the deformation evolution process of the beam is as shown in Figure 2.  
According to the stiffness of the stretching process formula \(k_{1}=EA/l_{0}\), the axial deformation length corresponding to different pressures is extracted, and the recovery curve of the SGC is obtained, in which the compression direction is positive. As shown in Figure 3a, the slope of the solid blue line is much smaller than that of the red dotted line, which demonstrates that there is much less post-buckling stiffness than stretch stiffness. The post-buckling spring recovery vs. ordinary spring curve is shown in Figure 3b. For the same static load, the post-buckling spring saves 0.1057 m more vibration stroke than the ordinary linear spring. While the post-buckling design realizes the low-stiffness design under the same material attributes, the post-buckling spring increases the load capacity compared with the ordinary spring and saves the vibration travel space.