# Ground Reaction Forces Generated by Runners--Harmonic Analyses and Modelling
## 2 Materials and Methods  
The \(VGRF\)s generated during running were measured during laboratory tests of a group of 13 healthy volunteers with full mobility and with no injuries and disabilities (5 males and 8 females, weight 51.6 \(\div\) 108.4 kg (mean \(\pm\) SD: 70.2 \(\pm\) 16.2 kg), height 158-187 cm (mean \(\pm\) SD: 172.3 \(\pm\) 9.8 m) age 22-45 years (mean \(\pm\) SD: 25 \(\pm\) 6.8 years)). Volunteers were chosen from the 22-45 age group using a convenience sampling method among students and employees of the University of Physical Education in Krakow and Cracow University of Technology. All running tests were carried out in the laboratory of the Biomechanics Section of the University of Physical Education in Krakow. The force measurements were realized using two AMTI 6-degrees of freedom force plates (40 \(\times\) 60 cm) located in line, one after the other, in the centre of a wooden walkway (Figure 2). The data were collected at a sample rate of 1.0 kHz. A high sampling rate was necessary to accurately measure the impact peak of the _VGRF_ generated by heel-strike runners. During the tests, the volunteers ran in sports shoes. All test participants were asked to run at the pace indicated by the electronic metronome. The tests were performed in the frequency range 2.40-3.40 Hz to simulate slow, moderate and fast running pace. The running frequency increment was 0.2 Hz. The running frequencies were recalculated from [Hz] to [BPM] (beats per minute). The tested frequency range corresponds to the frequencies of the steps occurring during recreational and marathon running. Cases of very fast running or sprinting were not studied. All volunteers were asked to land with their feet in the area of the force plates during the tests. Prior to conducting the force measurement, each volunteer practiced running with the frequency indicated by an electronic metronome to better match the running pace during the final measurement. All test participants performed two series of runs for each running frequency. The duration of each series was 60 s. After two series, the examined person rested during the series performed by other participants of the studies. The frequency was subsequently increased by 0.20 Hz, and the measurement was repeated for the new running frequency. The recorded _VGRF_ signals were filtered using a fourth-order low-pass Butterworth digital filter with a cut-off frequency of 60 Hz. The cut-off frequency of 60 Hz was established during signal processing tests as the frequency that allowed removal of unwanted noise and did not change the course of the _VGRF_ waveform. In analyses, the normalised (dimensionless) ground reaction forces \(F_{VGRF}\), defined as \(F_{VGRF}=VGRF/G\), were used (where: _VGRF_--recorded and filtered vertical component of the ground reaction force, \(G\)--weight of running person). To calculate the amplitudes of real _VGRF_, the amplitudes of normalised \(F_{VGRF}\) should be multiplied by the weight of running person \(G\). The normalised \(F_{VGRF}\) forces were parameterised by the following indicators: the normalised amplitude of the impact peak (\(A_{ip}\)), the normalised amplitude of the propulsive peak (\(A_{pp,h}\) or \(A_{ppf}\) for heel-strike and forefoot-strike runner respectively), the time of occurrence of \(A_{ip}\) (\(t_{ip}\)), the time of occurrence of \(A_{pp,h}\) or \(A_{ppf}\) (\(t_{pp,h}\) or \(t_{pp,f}\)) and the approximated contact time of the foot with the ground (\(t_{\alpha,h}\) or \(t_{\alpha f}\) for heel-strike and forefoot-strike runner respectively, see Figure 1). The times \(t_{ip}\), \(t_{pp,h}\) and \(t_{ppf}\) associated with the force curve peaks were extracted by finding the force curve peaks in the selected region of the curve along with the time values corresponding to identified peaks.