Measuring Ground Reaction Force and Quantifying Variability in Jumping and Bobbing Actions

Vibration serviceability assessment under human-induced dynamic loading is one of the most challenging aspects of structural engineering design due to inherent randomness in frequency, duration, and type of human actions to which the structure could be exposed. Walking, running, rising up from sitting, sitting down from standing, swaying, jumping, and bobbing are frequently observed human activities that generate three-dimensional dynamic forces (Bachmann and Ammann 1987). This paper is concerned with two activities, rhythmic jumping and bobbing, which are of most interest in the design of stadia and concert venues (Jones et al. 2011b). Only the largest (i.e., vertical) component of the force, whose peak value could be up to seven times larger than the body weight (Bachmann and Ammann 1987), will be considered. The vertical force is often responsible for excessive vibrations that could lead to damage of nonstructural components, unwanted noise, discomfort, or panic among occupants, overstressing the structure, and, in rare cases, compromising the structural integrity.

The key difference between jumping and bobbing is that a jumping cycle consists of a contact and a flight phase, while bobbing can be seen as akin to an attempt to jump while maintaining continuous contact with the ground (Jones et al. 2011b). Distinctions are often made between two different styles of bobbing: bouncing and jouncing. Bouncing is a more controlled action during which heel contact is maintained with the ground at all times (Sim et al. 2005). In this case, the majority of the movement is caused by the subject bending their knees. Jouncing is a more energetic alterative in which the subject rises on to their toes, breaking contact between their heels and the floor (Jones et al. 2011b). The action of jouncing is more complex as maintaining balance while rising on to the toes requires the engagement of additional muscles and joints. While jumping is more severe in terms of loading amplitude (Ellis and Ji 1994), bobbing is more common at concerts and other events, as it requires less energy (and less space) to be maintained (Racic et al. 2013a; Dougill et al. 2006). In addition, bobbing is an important loading case as the person is more likely to feel the structural movement and potentially react, consciously or unconsciously, by synchronizing with it, which could lead to a prolonged and excessive vibration response.

Body kinematics while jumping and bobbing, as well as the resulting dynamic force, varies significantly within a human population. Moreover, a single individual produces different kinematic and kinetic outputs when performing nominally the same activity on different occasions, or even from one jumping/bobbing cycle to another (Sim et al. 2008; Racic and Pavic 2010a). Tackling limited understanding of dynamic loads with unnecessarily conservative and uneconomical designs is no longer acceptable in the age of performance-based design and expectation of minimum use of natural resources. Hence, modern design is somewhat contradictory: it favors structures that are lightweight and slender (and inherently vibration sensitive) while expecting vibration serviceable design solutions. Anticipating and preventing vibration serviceability failures requires detailed characterization of human actions and the intrinsic randomness in both the population and an individual’s dynamic loading, neither of which is currently readily available to structural designers.

Typically, jumping can be performed at frequencies between 1 and 4 Hz (Rainer et al. 1988; Pernica 1990). The peak value of the generated ground reaction force (GRF) is usually 2.0–4.5 times larger than the weight of the person jumping (Sim et al. 2008). This GRF results from trunk motion characterized by peak-to-peak displacements around 10–30 cm and acceleration around $15–35\mathrm{m}/{\mathrm{s}}^2$ (McDonald 2015). Bobbing can be performed at frequencies up to 6 Hz, albeit the upper limit of 4.0–5.0 Hz is more frequently encountered (Yao et al. 2004). Typical peak-to-peak trunk displacements while bobbing are about two times lower than while jumping as body movement is limited by the continuous contact with the ground. As a consequence, the GRF generated while bobbing is lower compared to jumping.

In design applications, the force is traditionally decomposed into the main harmonics using Fourier transform. The harmonics are then normalized by the body weight to calculate a dimensionless quantity called dynamic loading factor (DLF). For jumping, the upper limit of the DLF is typically 1.8 for the first, 1.0 for the second, and 0.3 for the third harmonic (Rainer et al. 1988), while the bobbing DLFs are typically about two times lower. The strength of the forcing harmonics normally reduces with an increase in harmonic frequency. As a result, the structures considered to be at risk from excessive vibrations are those with a natural frequency of up to 6 Hz (IStructE 2008). Accurate modeling of this low-frequency content of the force is considered to be most important for vibration serviceability assessment. Use of traditional models for this purpose has been found to be inadequate in vibration-prone structures as they cannot model the narrow band nature (i.e., randomness) of human-induced dynamic loading (Brownjohn et al. 2004).

Sim et al. (2008) were among first researchers to include cycle-by-cycle variations when modeling the force amplitude, contact time, and the frequency of people jumping. They proposed a cosine-squared function to represent each jumping cycle and therefore could not describe asymmetry and local irregularities in the forcing signal regularly encountered in measured data. Racic and Pavic (2010a) overcame these limitations by first modeling asymmetry in the force waveform as a sum of two Gaussian functions. They then extended their approach to a detailed modeling of local irregularities by utilizing a sufficiently large number of Gaussian functions and achieved excellent agreement (in both time domain and frequency domain) between simulated force waveforms and those seen in experiments (Racic and Pavic 2010b). Similar successful detailed modeling has been recently achieved for dynamic loading during bobbing (Racic and Chen 2015). These advanced studies, however, rarely report the degree of variability in parameters describing the jumping and bobbing forces and its influence on the structural response—the information that would be of direct interest to structural designers. Some modeling studies also rely on use of publically unavailable databases of measured force time histories to inform the choice of the modeling parameters. Providing direct insight into the variability of the individual parameters would contribute to bridging the gap between traditional deterministic modeling, featuring parameters with clear physical meaning, and more sophisticated and accurate, but also more complex, stochastic modeling approaches. This study aims to contribute towards characterizing variability in both jumping and bobbing actions and evaluating the importance of parameter variability for design. To achieve these aims, GRFs measured using a force plate will be utilized.

Another aim of this paper is to investigate the possibility of measuring the force in such a way as to overcome some limitations associated with other methods. Force plates are usually small (i.e., $400\times 600$ or $600\times 600\mathrm{mm}$) and normally limited to laboratory environments. The former makes targeting the landing area while jumping challenging, potentially leading to an unnatural jumping action that, in turn, affects the GRF waveform. Racic et al. (2013b) overcame these issues by utilizing a motion capture system. This system consists of a number of cameras that monitor either active or passive markers systematically attached to anatomical landmarks of the test subject’s body, with the aim of recording the body kinematics during the observed activity. The GRF is then indirectly found by summing up the inertia forces of individual body segments (Thorton-Trump and Daher 1975; Racic et al. 2013b). This procedure, however, requires the use of a large number of markers, which may be logistically challenging, especially when monitoring several individuals simultaneously. Instead, it would be more convenient if the kinematics of the body center of mass (BCoM) could be directly measured. In this case, the acceleration of BCoM could be multiplied by the test subject’s mass to derive the GRF. The challenge with this approach is that the BCoM is not directly accessible for monitoring as it is normally located within the test subject’s trunk during jumping and bobbing. A potential solution is to identify a point on the human body whose kinematics closely resembles that of the BCoM. Identifying this point is the aim of the second part of this study.

The paper outline is as follows. The experimental procedure is described first. Then the variability in the jumping and bobbing actions is characterized. This is followed by the identification of a single point on the human body suitable for force measurement. Finally, the discussion and conclusions are presented.

Experiments were conducted in the Gait Laboratory at the University of Warwick, Coventry, U.K. The laboratory is equipped with a Vicon (Oxford Metrics, Oxford, U.K.) motion capture system consisting of 12 infrared cameras (Nexus), two digital video cameras and a force plate OR6-7-2000 (AMTI 2007). Eight test subjects (TSs), four males and four females, volunteered to take part in the experiments. Their basic anthropometric properties are presented in Table 1. The TSs were instrumented with 17 reflective markers, positioned at locations with the potential to represent the movement of the BCoM well (Fig. 1). The markers were attached using double-sided tape. The TSs wore tight clothes as well as a muscle wrap around the lower trunk [highlighted in Fig. 1(d)] to minimize soft tissue artefacts, i.e., relative displacement between the markers and underlying bones (Racic et al. 2013b).

Six markers were placed on a TS’s back: B1, B2, and B3 on the L5th, L3rd, and L1st vertebrae on the lower back, respectively, B4 on the T11th vertebrae, B5 on the T6th vertebrae on the middle back, i.e., between shoulder blades, and B6 on the C7th vertebrae at the base of the neck. Four markers were positioned on the hips: RH1 and LH1 on the anterior superior iliac spine on the right and left hip, respectively, while RH2 and LH2 were attached at the position of the greater trochanter on the right and left hip, respectively (Fig. 1). Seven markers were placed on the front of the TS (F1-F7 in Fig. 1) and spaced evenly up the torso. The majority of the markers were placed on the trunk as the BCoM locations for jumping and bobbing postures are known to be within this part of the human body.

The TSs were first asked to jump on the force plate at metronome-controlled frequencies of 1, 2, and 3 Hz. The frequency range was chosen to expose TSs to a wide range of jumping styles: extremely slow jumping (1 Hz) and relatively fast jumping (3 Hz) as well as comfortable jumping at 2 Hz (Yao et al. 2006). The frequency of 2 Hz was also chosen as it is often embedded in pop music (Ginty et al. 2001) acting as an aural stimulus for crowd actions at concert venues. After completion of these experiments, TSs performed bobbing at 1, 2, 3, and 4 Hz. The frequencies of 1–3 Hz were chosen to enable a comparison between bobbing and jumping activities at nominally the same frequencies, while the frequency of 4 Hz was added to reflect the ability of TSs to bob at frequencies above 3 Hz. Before each trial, the TSs were given enough time to familiarize themselves with the metronome beat and the test procedure, which was followed by 20 s of data collection. Between successive trials the TSs were given a few minutes to rest. During this break, the attachment of the markers was checked as were the force time histories, to ensure the TS did not miss the target force plate area. In rare cases when either of the two issues occurred, the previous trial was repeated. A test session with one TS, including preparation and briefing, lasted about 1 h.

Each trial consisted of simultaneous recording of the vertical component of the GRF (using the force plate) and the three-dimensional displacements of the markers (using the motion capture system). The force plate signal was sampled at 1,000 Hz while the marker displacements were recorded at 200 Hz. The high sampling rates (which were the maximum sampling rates available in the measurement systems) were chosen to ensure that the time step was sufficiently small, and therefore, the numerical errors negligible, in subsequent numerical time-domain simulations (Chopra 1995). All signals were filtered using a low-pass fifth order Butterworth filter in MATLAB. The cut-off frequency of the filter was 1 Hz above the frequency of the third forcing harmonic or 7 Hz, whichever was larger. This cut-off frequency was chosen to allow analysis of the first three harmonics that contain the most excitation energy, as well as all harmonics up to 6 Hz that have potential to strongly excite grandstand structures (IStructE 2008). The force plate signal was decimated to 200 Hz in those analysis cases requiring comparison with the marker signals. The 20-s-long signal consisted of 20–80 cycles, depending on the frequency of the activity. In total 24 trials of jumping and 32 trials of bobbing were recorded.

The experiments were approved by the Biomedical and Scientific Research Ethics Committee at the University of Warwick. The test protocol and health and safety details were explained to TSs initially in the recruitment phase, and then repeated just before the testing took place. Before commencing the tests, TSs completed a physical readiness questionnaire and signed a consent form. Only TSs with no known relevant health problems at the time of testing were allowed to take part in the experiments.

A typical time history of the low-pass-filtered force induced by jumping is shown in Fig. 2(a). Period $T_i$, contact time $C{T}_i$, and peak force for the $i$th jumping cycle are denoted in the same figure. The force waveforms while bobbing are more diverse than those induced while jumping due to two distinct bobbing styles: bouncing and jouncing. Video footage of the TSs’ heels was used to determine which style was preferred by each TS at each frequency. Table 2 shows that three TSs (TS2, TS3, and TS8) preferred the jouncing style while TS4 preferred to bounce at all four frequencies. The other four TSs made use of both styles, as a means of adapting to the imposed bobbing frequency. Interestingly, there is no correlation between the chosen style and the bobbing frequency across the population of four TSs. Overall, jouncing was encountered more frequently, i.e., in 22 out of 32 trials (69%). Fig. 2(b) shows bobbing force profiles at the nominal frequency of 3 Hz: the solid line represents TS1 bouncing while the dashed line depicts TS3 jouncing. The latter activity is usually more energetic resulting in a larger force compared with bouncing. In addition, the jouncing waveform can be similar to that of jumping, with troughs approaching zero (despite the fact that jouncing does not include a flight phase), as can be seen in the figure. However this observation cannot be generalized as low-energy jouncing and high-energy bouncing can also occur.

In the remainder of this section, the intersubject and/or intrasubject variations in the frequency, BCoM displacement, peak force, and DLFs for jumping and bobbing are shown and compared. Then the contact time, a parameter typical of jumping activity, is presented. At the end of the section, the vibration response is calculated and its sensitivity to randomness in the force investigated.

Parameters were extracted on a cycle-by-cycle basis to calculate the average value and the coefficient of variation (CoV) for each time history. To describe variability in the two statistical parameters (average and CoV) across the population of eight test subjects, their mean and standard deviation (SD) were also determined.

The vertical component of the time-domain acceleration signal for the $i$th marker, $a_i(t)$, was calculated by differentiating the measured displacement twice. The corresponding vertical component of the GRF, $F_i$ ($t$), hereafter referred to as the indirect force, was then calculated by multiplying the test subject’s mass by the net vertical accelerationwhere $g$ = acceleration of gravity. The accuracy in measuring this force was then evaluated by comparing it with the benchmark force recorded using the force plate, hereafter referred to as the direct force.

The coefficient of determination $R^2$ (Draper and Smith 1985) was used to quantify how well the indirect force correlated with the direct force in the time domain. Then the influence of the measurement error on the structural response was evaluated.

Electronic format of the data collected in this research can be downloaded free of charge from the University of Warwick webpage http://wrap.warwick.ac.uk/80470/.

This research work was partly supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC) [grant number EP/I03839X/1: Pedestrian Interaction with Lively Low-Frequency Structures]. The first author was supported by EPSRC DTG Scholarship. The authors would like to thank Birmingham Science City and Advantage West Midlands for the access to the Gait Laboratory.