# Improvement of a Tunable Stiffness Organ-Grasping Device by Design of a Wavy-Shaped Beam Structure
## 2 Methods  
### Mechanism of the Device  
Figure 2 shows the structure and basic of organ-grasping device. The device consists of the two soft beams covered by thin membrane. A suction tube and a handle are attached to the top of the device. The bottom of the device is opened to the air, and beams have holes for the airway. The tube is connected to a medical vacuum pump. When the bottom of the device is in the air, air is vacuumed from the bottom, and the two beams make a gap (Figure 2a). In this state, the beams behave as two thin beams and retain flexibility. After adhering to the liver, the device bottom opening is covered by the liver surface, and the two beams are constrained by suction (Figure 2b). In this state, the two beams act as a single thicker beam and become stiffer than the separated beams.  
Figure 1: Tunable stiffness organ-grasping device in a previous work [13]. (**a**) Concept. (**b**) Prototype. (**c**) Experimental scene with porcine liver.  
The stiffness of the device in the two states has been analyzed following beam theory [13]. Beams with homogeneous materials were supposed. In a low-stiffness state, the bending stiffness of the two beams is:  
\[EI_{\text{unconstrained beams}}=2 EI_{\text{single beam}}=2\bigg{(}E\frac{bh^{3}}{12} \bigg{)}=E\frac{bh^{3}}{6}, \tag{1}\]  
where \(E\) is elastic modulus, \(I_{\text{single beam}}\) are the second moment of area for a single beam, \(b\) is the width of beam, and \(h\) is the height of single beam. In a high-stiffness state, the bending stiffness is the stiffness of a beam with double thickness as follows:  
\[EI_{\text{constrained beams}}=E\frac{b(2h)^{3}}{12}=E\frac{2 bh^{3}}{3}. \tag{2}\]  
From the ratio between \(EI_{\text{constrained beams}}\) and \(EI_{\text{unconstrained beams}}\), the stiffness gain is four, theoretically.  
To obtain more stiffness gain, non-homogeneous beam was designed. Figure 3a shows horizontal normal and shear stress in the device working in a high-stiffness state. While the device is pulled by forceps, adhesion force under the device applies bending moment to the device. This causes horizontal normal force and shear force in the device. As shown in Figure 3a, the normal force is maximized on the surface of the device, and the shear force is maximized on the center line of the device. We have designed the beam to resist the stress [13]. Figure 3b shows the designed beam and stress while the device is grasped and pulled. The beam is composed of a wavy-shaped high-stiffness material (rubber) with flat surface on the outer surface of the beams, and the gaps are filled with low-stiffness material (silicone). The high-stiffness material around the surface resist to the normal stress in a high-stiffness state, and the wavy shape retains flexibility in a low-stiffness state. On the inner surface of the beams, friction sheets (silicone) were attached to prevent slipping between the two beams caused by shear stress in a high-stiffness state. Our experiments showed that the device with the structure in a high-stiffness state were 5.48 times harder than the device in low-stiffness state, which was harder than the expected stiffness from the beam theory [13].  
Figure 3: Stress in the device. (**a**) Normal and horizontal shear stress in the grasped device in a high-stiffness state. (**b**) The wavy-shaped beam and stress while the device is grasped and pulled.  
Figure 2: Structure and basic of the organ-grasping device. (**a**) The device in a low-stiffness state. (**b**) The device in a high-stiffness state.