## 3 Relationship between Pressure and Output Force  
Consider an infinitely-long thick-walled cylinder with an inner diameter \(r_{i}\) and the outer diameter \(r_{o}\), as shown in Figure 3. The following conditions are given, from the structure of the soft actuator. Condition 1.: The pressure \(P\) is applied uniformly to the inner wall of the cylinder. Condition 2.: Radial and circumferential deformation are zero because the outer wall of the cylinder is restrained by a fiber or a spring. Condition 3.: Elongation of the cylinder, or axial strain, is also zero for simplicity. The following equations are also valid when the cylinder is elongated, by considering the deviation from a reference point (arbitrary length without external forces). Figure 1: Structure of a coil-reinforced soft actuator. Figure 2: Prototype of a soft actuator. In a small volume of the thick-walled cylinder shown in Figure 4, the force equilibrium in polar coordinates \((r,\theta,l)\) along \(r\)-axis is written as follows:  
\[(\sigma_{r}+d\sigma_{r})(r+dr)d\theta\,dl-\sigma_{r}r\,d\theta\,dl-2\sigma_{ \theta}\,dr\,dl\sin(d\theta/2)=0. \tag{1}\]  
The first term is the force from the outer face, and the second is from the inner face, which represents the product of pressure and area. The last term represents the force from the sides, given that the side faces have angles, and the force perpendicular to these faces results in the radial force: \(\sigma_{\theta}\,dr\,dl\sin(d\theta/2)\) from each side. By ignoring high-order terms (\(dr^{2}\)) and approximating \(\sin(d\theta/2)\) as \(d\theta/2\), this equation can be rearranged as follows:  
\[r\frac{d\sigma_{r}}{dr}+\sigma_{r}-\sigma_{\theta}=0. \tag{2}\]  
Figure 4: Small volume of a cylinder. Figure 3: Infinitely-long cylinder model. The basic equations of stress and strain are:  
\[\varepsilon_{r} =\frac{1}{E}(\sigma_{r}-\
u\sigma_{\theta}-\
u\sigma_{l}) \tag{3}\] \[\varepsilon_{\theta} =\frac{1}{E}(\sigma_{\theta}-\
u\sigma_{r}-\
u\sigma_{l})\] (4) \[\varepsilon_{l} =\frac{1}{E}(\sigma_{l}-\
u\sigma_{r}-\
u\sigma_{\theta}), \tag{5}\]  
where \(\varepsilon_{l}\), \(\sigma_{l}\), \(E\), and \(\
u\) are the strain, the stress, Young's modulus, and [PERSON]'s ratio, respectively. The subscript \(i\) denotes the axis index. Condition 3 is represented as \(\varepsilon_{l}=0\). Substituting it into Equation (5) yields:  
Figure 1:0  
The following differential equation is obtained by substituting it into Equation (2):  
Figure 1:1  
This can be rewritten as:  
Figure 1:2 Figure 1:3  
and then, we get:  
Figure 1:4 Figure 1:5  
where \(C_{1}=\sigma_{l}/2\
u\) and \(C_{2}\) are the constants of integration. Next, we consider the circumferential strain \(\varepsilon_{\theta}\). The circumferential length of the material changes by \(2\pi u_{r}\) when the radial displacement is \(u_{r}\). Thus,  
Figure 1:6  
Substituting it into Equation (4) yields:  
Figure 1:7 Figure 1:8  
From Condition 2, that the outer wall is radially constrained, \(u_{r}=0\) when \(r=r_{o}\). The preceding equation becomes:Figure 1:9 Figure 2:0 (16) Figure 2:1 (17) Figure 2:2  
Thus, the constant is derived as follows:  
Figure 2:3 Figure 2:4  
Then, we use Condition 1 that \(\sigma_{r}=-P\) at \(r=r_{i}\) and obtain the following:  
Figure 2:5  
It should be noted that the signs for stress and air pressure are different, because stress is defined to be positive for tension, but air pressure is positive for compression. In this case, we introduce a new variable \(\eta\) defined as:  
Figure 2:6  
and (21) becomes the following simple expressions. Figure 2:7  
The variable \(\eta\) denotes the pressure transmission ratio. When \(\eta=1\), the rubber material fully transmits the pressure to the axial direction in the same manner as the fluid. A dimensionless expression of the preceding equation is given as follows by letting \(\alpha=r_{i}/r_{o}\):  
Figure 2:8  
Figure 5 shows the relationship between \(\alpha\) and \(\eta\) for different [PERSON]'s ratio, where a larger \(\alpha\) indicates a thinner rubber wall.