# Ground Reaction Forces Generated by Runners--Harmonic Analyses and Modelling
## 4 Discussion
The value of the phase angle for the 3 rd harmonic (\(\phi_{3}\)) can be assumed to be in the range of 5.5-6.3 rad. The values of the phase angle for the higher harmonics (harmonics 4-15) should be linearly reduced every 0.5 rad (compare Figure 7c). The constant term of the Fourier series (\(\alpha_{0}\)) is another important parameter for modelling any curves by means of the Fourier series. This constant term \(\alpha_{0}\) represents the average value of the function over its domain. In the case of the \(F_{VCRF}\) generated by forefoot-strike runners the value of the normalised constant term can be assumed \(\alpha_{0,f}\) = 1.0 \(\pm\) 0.034. For the \(F_{VCRF}\) generated by heel-strike runners, the value of the normalised constant term is \(\alpha_{0,h}\) = 0.978 \(\pm\) 0.035. Figure 9 presents the impact of the number of harmonics included in the \(F_{VCRF}\) modeling on the accuracy of the \(F_{VCRF}\) curve fitting. The residual plots, presented below the \(F_{VCRF}\) charts, show the accuracy of fitting of the \(F_{VCRF}\) to the example laboratory test result. To achieve high accuracy of the mapping of the \(F_{VCRF}\) curve generated by forefoot-strike runners, it is sufficient to use 3-5 harmonics (Figure 9e,f). In the case of the \(F_{VCRF}\) generated by heel-strike runners, sufficient accuracy can only be achieved if 10-15 harmonics are used (Figure 9c,d). Figure 9: Impact of the number of harmonics on the accuracy of the \(F_{VCRF}\) curve modeling: (**a–d**) heel-strike running; (**e–h**) forefoot-strike running: (**a,e**) model containing 3 harmonics; (**b,f**) model containing 5 harmonics; (**c,g**) model containing 10 harmonics; (**d,h**) model containing 15 harmonics (upper chart—the \(F_{VCRF}\) curves, lower chart—residual plots; dot line—result of laboratory test, solid line—result of the \(F_{VCRF}\) fitting). Due to the large variability of the Fourier series parameters, especially the phase angles values, modelling of the \(F_{VCRF}\) generated by heel-strike runners using the Fourier series can be burdensome and can lead to large inaccuracies. An alternative way of modelling of the \(F_{VCRF}\) generated by heel-strike runners should be proposed. This alternative method may be the method presented in [18; 19; 20]. This could also be the model proposed by the author and based on the half-sine model. The \(F_{VCRF}\) generated by heel-strike runners can be simulated using the half-sine model (Equation (3) [7; 13]) assuming the correct contact time value \(t_{\alpha,be}\) according to Equation (2) and adding an impact peak function \(\phi_{ip}(t)\) (Equation (6), author's proposal) responsible for producing the impact peak occurring in the \(F_{VCRF}\) generated by heel-strike runners. \[F_{VCRF}(t)=\left\{ \begin{array}{cl}A_{r}\sin\biggl{(}\frac{\pi \cdot f_{r}}{k}\cdot t\biggr{)}&\mbox{for}\ \ i\cdot T_{r}<t\leq(i+k)\cdot T_{r}\ \ i=0,1,2\ldots\\ 0&\mbox{for}\ \ (i+k)\cdot T_{r}<t\leq(i+1)\cdot T_{r}\end{array} \right. \tag{3}\]  
\[k=\frac{t_{\alpha,be}}{T_{r}} \tag{4}\]  
\[A_{r}=\frac{\pi}{2k} \tag{5}\]  
\[\frac{\phi_{ip}(t)}{G}=A_{\phi}\sin\biggl{(}\alpha_{\phi}\cdot\pi \cdot f_{r}\cdot t\biggr{)}^{\beta_{\phi}}\ \ \ \ \ \ \ \ \ \mbox{for}\ \ \ 0\leq t\leq\lambda\cdot T_{r} \tag{6}\]  
where \(F_{VCRF}(t)\)--normalised vertical component of the ground reaction force \(F_{VCRF}(t)=VGRF/G\), \(G\)--weight of running person (note: \(VGRF(t)=F_{VGRF}(t)\cdot G\)), \(A_{r}\)--the \(F_{VGRF}\) amplitude acc. to Equation (5), \(k\)--coefficient of contact time of the foot with the ground acc. to Equation (4),\(f_{r}\)--step frequency during running, \(T_{r}\)--step period during running \(T_{r}=1f_{r}\), \(\phi_{ip}\)--the impact peak function, \(A_{\phi}\)--the impact peak amplitude, \(\alpha_{\phi}\)--coefficient of the impact peak location, \(\beta_{\phi}\)--coefficient of the impact peak slenderness, \(\lambda\)--coefficient of the impact peak duration \(\lambda\approx(0.17\)-\(0.20)T_{r}\) (only the first cycle of the impact peak function should be used).