# Impact Force Mechanism of a Trimaran Model and Its Experimental Validation
## 3 Force Analysis of the Trimaran Model  
The process of the model entering the water can be divided into four stages, and a force analysis for each stage is given in this section. **Stage 1: Free fall in the air**  
At this stage, three forces are exerted on the model, namely air resistance \(F_{\rm a}\), gravity \(F_{\rm g}\), and guide rail friction resistance \(F_{\rm f}\). The schematic of force analysis is shown in Figure 3.  
The resultant force is given by  
\[F_{1}=F_{\rm g}-(F_{\rm a}+F_{\rm f}) \tag{1}\]  
which determines the speed \(v\) of the model when it touches water. The air resistance can be estimated by  
\[F_{\rm a}=\frac{1}{2}C\rho_{\rm a}Sv^{2}, \tag{2}\]  
where \(C\) is air dragging coefficient, considered to be slightly smaller than 1 due to the shape of the model; \(\rho_{\rm a}\) is density of air and normally \(\rho_{\rm a}=1.293\) kg/m\({}^{3}\); \(S\) is the windward area of the object and approximately \(S=0.8\times 0.2=0.16\) m\({}^{2}\); and \(v\) is the motion speed of the model relative to the air, and \(v\approx 2\) m/s according to [8]. The air resistance is considered \(F_{\rm a}<0.2\) N and is negligible compared to gravity (240 N) [8]. Figure 3: The schematic of force analysis for stage 1. Figure 2: Two-dimensional parametric model of the trimaran model (in mm). The simplified model is identified by blue lines. By the law of conservation of energy, we have  
\[F_{1}h=\frac{1}{2}mv^{2}, \tag{3}\]  
where \(h\) is the height of the trimaran model relative to the water surface, and \(m\) is the mass. **Stage 2: Main hull entering the water**  
When the bottom of the model enters the water, it will receive a reaction force from the water. At this stage, seven forces are applied to the model, namely, air resistance \(F_{\rm a}\), gravity \(F_{\rm g}\), guide-rail friction resistance \(F_{\rm f}\), the buoyancy of water \(F_{\rm b}\), the water's impact force on the main hull \(F_{\rm w1}\), the impact force on auxiliary hulls \(F_{\rm w2}\), and the impact force on the connection bridge by splatter water \(F_{\rm w3}\). The schematic of force analysis is shown in Figure 4.  
The resultant force \(F_{2}\) is given by  
\[F_{2}=F_{\rm g}-(F_{\rm a}+F_{\rm f}+F_{\rm b}+F_{\rm w1}+F_{\rm w2}+F_{\rm w3 }). \tag{4}\]  
\(F_{\rm w3}\) is hard to calculate, and it was obtained experimentally. In order to calculate the buoyancy of water \(F_{\rm b}\), the main hull and the auxiliary hulls are approximated by three cubes. The bottom areas of the auxiliary hulls are approximately \(0.073\times 0.20=0.0146\) m\({}^{2}\), and that of the main hull is approximately \(0.327\times 0.20=0.0654\) m\({}^{2}\). 1. [label=()]
2. Slamming the flat plate into the water The impact force is given by \[F=0.5\rho_{\rm w}\mathbb{G}_{\rm s}A_{\rm p}v^{2},\] (5) where \(\rho_{\rm w}\) is the water density, \(\mathbb{G}_{\rm s}\) is the slamming coefficient, \(A_{\rm p}\) is the water-contacting area \(0.327\times 0.20=0.0654\)m\({}^{2}\), and \(v\) is the speed of the flat plate relative to water. Equation (5) is used to calculate the slamming force when the main hull enters the water. 1. [label=()]
2. [PERSON]'s theoretical model [PERSON]'s theoretical model is illustrated in Figure 5. The impact force is given as [2] \[F_{1}=F_{\rm g}-(F_{\rm a}+F_{\rm f}) \tag{1}\]0 (6) where \(\rho_{\rm w}\) is water density, \(v\) is the speed of an object relative to water, \(t\) is the width of one side, and \(\alpha\) is the angle between the tangent of the model and the water. \[F_{1}=F_{\rm g}-(F_{\rm a}+F_{\rm f}) \tag{1}\]1 The schematic of force analysis for stage 2. Equation (6) is used to calculate the slamming force when the auxiliary hulls enter the water. In order to use (6), the measuring points on the side of the trimaran model, as shown in Figures 1 and 2, need to be simplified. **Stage 3: Connection bridge entering the water**  
The connection bridge entering water means that the largest sectional area of the trimaran model submerges into the water and will receive a reaction force \(F_{\rm w4}\).