# The Impact of a Flexible Stern on Canoe Boat Maneuverability and Speed

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## Abstract

**:**

^{®}. We built down-scaled boat models and tested them in a water channel. The similarities between experimental and original setup were evaluated by means of a dimensional analysis. (Thermoplastic) elastomers with various flexibility were used for the stern construction. In the experiments conducted in the water channel, we determined the forces acting on the boat with different stern models. The results reveal that the flexible stern induced a torque counteracting the boat’s deflection, while the stiff stern caused a torque enhancing it. A paddle boat with a flexible stern could hence be a promising new method to reduce the boat’s yawing movement.

## 1. Introduction

^{®}shark-skin swimsuit is certainly the most successful example; in the 2000 Summer Olympics in Sydney, 83% of medals were won by swimmers using this suit.

^{®}: In 1997, during fishing holidays in Norway, Leif Kniese saw that fish fins bent around his finger when he applied pressure instead of bending away [7]. The Fin Ray Effect

^{®}(Evologics GmbH, Berlin, Germany) was discovered [8]. The principle of this effect is based on a characteristic triangular geometry consisting of a rigid skeleton (cross braces) and flexible side walls (Figure 1). The most famous application is the Fin Ray gripper. It can handle delicate objects, easily conforms to an object, and can be applied for robots with human contact [9,10].

^{®}back into water. To reduce the yawing effect described above, a boat with a flexible stern was designed. The stern is based on the Fin Ray Effect

^{®}, meaning that it moves towards the force induced by the paddle. To study the impact on boat speed, down-scaled boat models were designed and tested in a water channel. For the construction of the stern, elastomers with various flexibility (ranging from stiff to highly flexible) were used. We measured the lateral force that would lead to a yawing motion on different boat models and compared the results. They show that the flexible stern damps the lateral boat movement. We believe that it is a promising method to enhance canoe boat speed.

^{®}damps the boat’s yawing, and (b) to design a boat model based on an original canoe boat and test its applicability towards design, material, and flexibility.

## 2. Materials and Methods

#### 2.1. Dimension Analysis

^{®}material (Y in $\frac{\mathrm{kg}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{2}}$), length of the flexible boat stern ( ${l}_{r}$ in $\mathrm{m}$), and wall thickness of the boat stern model (${d}_{wt}$ in $\mathrm{m}$). Using the above variables, the problem could be characterized by the following dimensionless parameters (Table 1):

#### 2.2. Models

^{®}. Therefore, the models were designed without a bottom membrane. After successful testing of the simple Fin Ray geometries (A and B), we designed the down-scaled boat model C based on an original canoe boat.

#### 2.2.1. Stern Models A and B

^{®}945 with a Young’s modulus of ${Y}_{A}$ $=1.79\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ $\frac{\mathrm{kg}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{2}}$, and for model B Ecoflex

^{TM}00-30 (both Smooth-on, Macungie, PA, USA) with a Young’s Modulus, ${Y}_{B}$ $=6.895\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ $\frac{\mathrm{kg}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{2}}$ was used. Both models were cast with a mold to obtain the characteristic Fin Ray shape. Since the silicone rubber used in model B is highly flexible, the cross braces were shored up with Poly(methyl methacrylate) (PMMA) elements to retain the Fin Ray effect. We conducted alternating experiments, where we measured the forces acting on both the stiff and the flexible model. To guarantee the same measurement conditions between the experiments, the stiff models were designed by adding stiff braces (Figure 3) to the flexible model. These braces were held in place by friction and prevented any flexion of the stern, thereby creating a stiff reference model. The boat hull is a down-scaled canoe boat model (scale factor 1:5).

#### 2.2.2. Stern Model C

^{®}, NinjaTek

^{®}, Manheim, PA, USA) with a Young’s modulus of ${Y}_{C}$ $=1.242\times {10}^{7}$ $\frac{\mathrm{kg}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{2}}$ [14]. The stiff version was printed from polylactic acid (PLA), a stiff bio-polymer. The front of the boat was also printed from PLA (Figure 4). While we were testing models A and B, it was shown that even a thin skin at the bottom eliminates the Fin Ray Effect

^{®}. Therefore, model C was designed bottomless to guarantee a reduction of the yawing.

#### 2.3. Experimental Setup

## 3. Results and Discussion

^{®}reduces the paddle-stroke-induced yawing, we designed and manufactured a simple geometry (models A and B) reflecting the classic Fin Ray gripper [15]. Then, a down-scaled boat model with a flexible, Fin Ray-stern was drafted based on an original canoe boat to verify that only a slight adaptation of a standard boat can have a significant impact on the yawing movement.

#### 3.1. Dimension Analysis

^{®}-based, flexible stern could reduce the boat’s yawing motion and hence enhance maneuverability and speed.

#### 3.2. Experimental Results

## 4. Conclusions

^{®}would experience less yawing due to its deflection movement.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ICF | International Canoe Federation |

PMMA | Poly(methyl methacrylate) |

Str | Strouhal number |

Re | Reynolds number |

Fr | Froude number |

Ca | Cauchy number |

TPU | Thermoplastic urethane |

PLA | Polylactic acid |

## References

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**Figure 2.**Stern models (

**A**–

**C**). Models (

**A**) and (

**B**) were cast from silicone rubber with different Young’s modulus to proof our hypothesis. Model C was 3D-printed from thermoplastic polyurethane (TPU, NinjaFlex

^{®}, NinjaTek

^{®}, Manheim, PA, USA) and represents the stern of a down-scaled canoe boat model.

**Figure 3.**The draft for stern models A and B (in violet) with braces (in beige) to create the stiff configuration of the stern models.

**Figure 4.**The draft for model C with hull in orange and stern in blue and the interlocking system for a convenient connection.

**Figure 5.**Measurement setup. The boat was deflected by ${5}^{\circ}$ to measure the forces ${F}_{plus}$ and ${F}_{minus}$ by means of two strain gauge measurement bridges.

**Figure 6.**Histograms of the total torque of the three stern models and their stiff counterparts. While in model A the mean value is not reduced by much, the torque’s overall amplitude is reduced significantly. In model B, the mean shifts to lower values, but the total amplitudes are increased. This is caused by the very flexible model, resulting in high vibrations. In model C, the shift to lower torque values, and therefore less yaw, is significant. The amplitude of the oscillations is also slightly reduced.

**Table 1.**Four known and three geometric dimensionless quantities. Str denotes the Strouhal number, Re the Reynolds number, Fr the Froude number, and Ca is the Cauchy number.

${\pi}_{1}=Str=\frac{l}{T\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}v}$ | ${\pi}_{2}=Re=\frac{l\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}v}{\nu}$ | ${\pi}_{3}=\frac{A}{{l}^{2}}$ | ${\pi}_{4}=Fr=\frac{{v}^{2}}{l\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}g}$ |

${\pi}_{5}=Ca=\frac{{v}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\rho}{Y}$ | ${\pi}_{6}=\frac{{l}_{r}}{l}$ | ${\pi}_{7}=\frac{{d}_{wt}}{l}$ |

Model | $\mathit{Re}$ | $\mathit{Str}$ | $\mathit{Fr}$ | ${\mathsf{\Pi}}_{3}$ | $\mathit{Ca}$ | ${\mathsf{\Pi}}_{6}$ | ${\mathsf{\Pi}}_{7}$ |
---|---|---|---|---|---|---|---|

A | $2.8\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$ | 1 | $7.99\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $5.50\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $4.36\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | $0.27$ | $1\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

B | $2.8\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$ | 1 | $7.99\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $5.50\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $1.13\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $0.27$ | $1\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

C | $1.4\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$ | 1 | $1.60\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $2.00\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $6.29\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | $0.27$ | $2\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

**Table 3.**Experimental results, showing the mean values ± standard deviation of the total torque M for the stiff and the flexible stern models.

Model A | Model B | Model C | |
---|---|---|---|

${M}_{stiff}$ ($\mathrm{mNm}$) | $22.76\pm 19.04$ | $16.54\pm 3.96$ | $13.27\pm 11.70$ |

${M}_{flex}$ ($\mathrm{mNm}$) | $18.62\pm 14.70$ | $9.44\pm 6.27$ | $-0.72\pm 2.54$ |

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**MDPI and ACS Style**

Stadler, A.T.; Schönauer, M.; Aslani, R.; Baumgartner, W.; Philippi, T.
The Impact of a Flexible Stern on Canoe Boat Maneuverability and Speed. *Biomimetics* **2020**, *5*, 7.
https://doi.org/10.3390/biomimetics5010007

**AMA Style**

Stadler AT, Schönauer M, Aslani R, Baumgartner W, Philippi T.
The Impact of a Flexible Stern on Canoe Boat Maneuverability and Speed. *Biomimetics*. 2020; 5(1):7.
https://doi.org/10.3390/biomimetics5010007

**Chicago/Turabian Style**

Stadler, Anna Theresia, Martin Schönauer, Roozbeh Aslani, Werner Baumgartner, and Tillmann Philippi.
2020. "The Impact of a Flexible Stern on Canoe Boat Maneuverability and Speed" *Biomimetics* 5, no. 1: 7.
https://doi.org/10.3390/biomimetics5010007