# One-Dimensional Mathematical Model for Kayak Propulsion
## 3 Analytical Solutions for Periodic Propulsion
### Stroke as Quadratic Function
This approximation allows us to solve analytically and estimate its overall contribution to the result. Expanding the square term, we have the following:  
\[ \begin{split} m\sqrt{\frac{f^{\prime}}{d}}\sum_{n}(a_{n}w_{n} cos(w_{n}t)-b_{n}w_{n}sin(w_{n}t))&=-f^{\prime}\Big{(}a_{1}^{2} sin^{2}(wt)+b_{1}^{2}cos^{2}(wt)+a_{1}b_{1}sin(2 wt)\Big{)}+\\ &+\sum_{n=1}^{6}(A_{n}sin(w_{n}t)+B_{n}cos(w_{n}t))\end{split}  \tag{43}\]  
where we have simplified the crossed term at double frequency \(2a_{1}b_{1}sin(wt)cos(wt)=a_{1}b_{1}sin(2 wt)\), which is precisely the crossed term mentioned above. We group terms and rewrite them to arrive at the following:  
\[ \begin{split} m\sqrt{\frac{f^{\prime}}{d}}\sum_{n}(a_{n}w_{n} cos(w_{n}t)-b_{n}w_{n}sin(w_{n}t))=\\ =-f^{\prime}\left(\frac{a_{1}^{2}+b_{1}^{2}}{2}+\frac{a_{1}^{2} -b_{1}^{2}}{2}cos(2 wt)+a_{1}b_{1}sin(2 wt)\right)+\sum_{n=1}^{6}(A_{n}sin(w_{n }t)+B_{n}cos(w_{n}t)).\end{split}  \tag{44}\]  
Figure 8: (**a**) Piecewise function for men’s category, defined by expression (38) and representative Fourier series. (**b**) Piecewise function for women’s category, defined by expression (39) and representative Fourier series. the expression from which we can equal terms of the same frequency to arrive at the following:  
\[ \begin{array}{llll}n=0&f^{\prime}(\frac{a_{1}^{2}+b_{1}^{2}}{2})=0&--\\ n=1&muw_{1}=B_{1}&-mwb_{1}=A_{1}\\ n=2&m2 wa_{2}=B_{2}+\frac{f^{\prime}}{2}(a_{1}^{2}-b_{1}^{2})&-m2 wb_{2}+f^{ \prime}a_{1}b_{1}=A_{2}\\ n=3&m3 wa_{3}=B_{3}&-m3 wb_{3}=A_{3}\\ n=4&m4 wa_{4}=B_{4}+\frac{f^{\prime}}{2}(a_{2}^{2}-b_{2}^{2})&-m4 wb_{4}=A_{4}\\ n=5&m5 wa_{5}=B_{5}&-m5 wb_{5}=A_{5}\\ n=6&m6 wa_{6}=B_{6}+\frac{f^{\prime}}{2}(a_{3}^{2}-b_{3}^{2})&-m6 wb_{6}=A_{6} \end{array}  \tag{45}\]  
Thus, we arrive at the following general solution:  
\[v=\sqrt{\frac{f^{\prime}}{d}}tanh\left(\sqrt{\frac{d}{f^{\prime}}}\left( \frac{f^{\prime}}{m}t+K^{\prime}\right)\right)+\sum_{n=1}^{6}(a_{n}\cdot sin( nwt)+b_{n}\cdot cos(nwt)) \tag{46}\]  
where now \(f^{\prime}=A_{0}/2\) and the coefficients \(a_{n}\) and \(b_{n}\) are cleared from the previous expressions. With the intention of evaluating the precision of the approximation assumed in the analytical resolution, in Figure 9, we represent this solution for men and women against the numerical resolution of the differential equation in both cases (see the detail of the figures). It is observed that the result is quite approximate, as only a small displacement appears at the beginning of the transient and especially in men. The weight of the crossed term \(f^{\prime}a_{1}b_{1}\) on \(A_{2}\) (see row \(n=2\) in the expressions (45)) is estimated at 6.6% for men and less than 1% for women; therefore, its influence on the solution is limited. In any case, it is observed that the agreement between the numerical and analytical solutions is quite good in both cases. The characteristics of the represented solutions in Figure 9 are again given by the stroke model used. The stroke frequency is about 109 strokes/minute for women and 117 strokes/minute for men. The limit velocity is given by \(v_{limit}=\sqrt{f^{\prime}/d}\), about 6.98 m/s in men and about 5.41 m/s in women. In addition, the value of \(f^{\prime}=A_{0}/2\) fits quite well the value of average force, which could be estimated for the Gomes table in Table 1. For instance, we find a peak force for women of 153N, which, modeled by a quadratic function and taking the relative weight of aquatic versus aerial phases, gives \(153\cdot 0.666\cdot 0.781=79.58\) N, which is a very close approximation of the value \(f^{\prime}=A_{0}/2=79.746\) N obtained from  
Figure 9: Analytical solution for maximum frequency and quadratic stroke: women and men. In detail, the comparison with the numerical solution (in dashed line) for both cases and both in the initial transient and in the final part of the curves. Table 5 for women. A quite similar result can be obtained for men.