# Ground Reaction Forces Generated by Runners--Harmonic Analyses and Modelling
## 1 Introduction
These proposals can be considered correct for modelling the \(VGRF\) generated during forefoot running for the needs of dynamic analysis performed in civil engineering tasks. To determine the \(VGRF\) generated by heel-strike runners, consideration of approximately 10-15 harmonics is required [16; 17]. For a frequency of running of \(f_{r}\) = 2.80 Hz the frequency range corresponding to 15 harmonics is 2.8-45.0 Hz (see Section 4 for details). Figure 1: Examples of time courses of normalised ground reaction forces \(F_{VGRF}\) for different foot-strike patterns for running frequency \(f_{r}\) = 2.80 Hz: (**a**) heel-strike pattern, (**b**) midfoot-strike pattern, (**c**) forefoot-strike pattern (based on the author’s research results). Another type of the \(VGRF\) model, called the half-sine model, has been proposed in [7; 13]. The half-sine model allows determination of the \(VGRF\) generated during the forefoot-strike running technique. It should be also mentioned that the half-sine model is very sensitive to the value of contact time of the foot with the ground \(t_{c}\). The amplitude of the \(VGRF\) in the half-sine model is inversely proportional to \(t_{c}\). The assumption of the underestimated \(t_{c}\), e.g., \(t_{c}=0.5\ T_{r}\) recommended in [7] for ordinary running activity (where: \(T_{r}\)--running period, \(T_{r}=1/\!f_{r}\)), results in a large overestimation of the \(VGRF\) amplitude [1; 15] (see Section 4 for details). The load model proposed in [14] is another example of a model of the \(VGRF\) generated by a runner, enabling determination of the \(VGRF\) in the form of a half-sine curve. The model allows the reliable value of the dynamic response of the structure subjected to the dynamic action of running people to be estimated, but it cannot be directly compared with the measured \(VGRF\)s. The model is characterized by a constant amplitude of 1250 N, as well as the duration of the foot contact time with the ground being significantly shorter than contact time values measured during running tests. The model proposed in [14] describes the force which generates an impulse causing a dynamic response of the structure comparable to the response induced by real \(VGRF\)s, but the parameters of the modelled \(VGRF\)s are not directly comparable to the parameters of the measured \(VGRF\)s.  
A different methodology for the \(VGRF\) modelling, presented in [18], uses the sum of the Gaussian functions to reconstruct the \(VGRF\) time course. The proposed procedure does not distinguish between running techniques but can easily be adapted to model different running styles. The correct selection of the number and parameters of the Gaussian functions enables determination of the template curve of the \(VGRF\) separately for forefoot-strike and heel-strike running. Another method of the \(VGRF\) modelling has been presented in [19; 20]. In the model, the runner's body was divided into two masses: \(m_{1}\)--the mass of the lower limb which contacts with the ground during running (\(m_{1}=0.08\ G\), where: G--total body mass of the runner)--and \(m_{2}\)--the mass of the rest of the runner's body (\(m_{2}=0.92\ G\)). In order to create the \(VGRF\) curve, the sum of the cosine functions describing the forces created during the motion of the two parts of the runner's body was used. With properly selected parameters, the proposed model allows the \(VGRF\) curves generated during heel-strike and forefoot-strike running to be determined. Most recreational runners, around 90-95%, are heel-strike runners [21; 22; 23]. In [24], it was reported that in a group of professional marathon runners, nearly 75% of runners, were also heel-strike runners. The heel-strike running style can be considered to be the most common running style.