Keywords

42.1 Introduction

Low-frequency structures, such as footbridges, long span floors and grandstands, frequently exhibit strong vibration responses to human-generated dynamic forces, such as walking and jumping. The vibration serviceability state of these structures is often the governing design criterion. To successfully estimate the vibration response the development of reliable, high fidelity models of human-induced forces is required. These models are expected to represent the force waveform adequately, i.e. to replicate the narrow-band nature of the excitation typical of human activities [1] which is a consequence of inherent randomness (also called intra-subject variability) in the parameters describing the dynamic force.

Significant developments in force modelling of both walking and jumping excitations have been made over the last decade. In particular, understanding of the inter-subject variability in key parameters (such as the activity frequency) in the population of structural users has been improved. Some of these developments have already been included in design guidance [2]. Significant advances in modelling intra-subject variations have also been made [3–6], often resulting in complex models that might be difficult to implement in design practice. Quantifying randomness in key force parameters can contribute towards better understanding of human actions and development of the effective modelling strategies. This paper intends to provide some insight into the randomness of human walking and jumping and examine the significance of capturing this type of information in force modelling.

The outline of the paper is as follows. The experiments involving the walking and jumping on a rigid surface are described in Sect. 42.2, followed by a brief overview of acquired data and basic processing in Sect. 42.3. Randomness in the key parameters is presented in Sect. 42.4. A subset of walking experiments on lively structures is analysed in Sect. 42.5 and compared with the data recorded on the rigid surface. This is followed by an insight into the sensitivity of the vibration response to randomness in the force in Sect. 42.6. Finally, conclusions are presented in Sect. 42.7.

42.2 Experiments on Rigid Surface

Experiments were conducted in the Gait Laboratory at the University of Warwick. Ten test subjects volunteered to perform walking trials [7], while eight test subjects took part in tests involving jumping [8]. In both cases, each test subject performed the actions on their own, i.e. without involvement and interaction with other test participants. Both sets of experiments were approved by the Biomedical and Scientific Research Ethics Committee at the University of Warwick.

To study the walking activity, test subjects were instrumented with 18 reflective markers and monitored using a Vicon motion capture system (equipped with 12 infra-red cameras) while walking on a treadmill (Fig. 42.1a). Utilising recorded marker trajectories and well-known model for mass distribution and geometry of body segments [9], the ground reaction force (GRF) was calculated as the sum of inertial forces of individual body parts [7]. Every test subject was asked to walk at a series of treadmill speeds, which followed this pseudo-random order: 1.15, 1.56, 1.36, 1.88, 1.67, 2.08, 1.76, 1.04, 1.24, 0.84, 0.93, 1.97 and 1.45 m/s. Overall, 130 tests were performed, each with about 450 steps.

Fig. 42.1
figure 1

Test setup in the Gait Laboratory for experiments involving human (a) walking and (b) jumping

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In the jumping tests, subjects were instrumented with 17 markers (Fig. 42.1b) to track the trunk displacements, and asked to jump on a force plate (also shown in Fig. 42.1b) at frequencies of 1, 2 and 3 Hz. Each test lasted 20 s [8].

In both sets of experiments the test subjects were offered short breaks during which the test setup and data acquired were checked. The data were sampled at frequency of 200 Hz.

42.3 Data Processing

In the walking tests, marker trajectories were low-pass filtered using a fourth-order zero-phase-shift Butterworth filter with a cut-off frequency of 10 Hz. An example of a derived GRF is shown in Fig. 42.2a. The period of each force cycle was determined as shown in the same figure, and all the periods were extracted for each time history. The reciprocal value of the period was then calculated to find the pacing frequency. The imposed treadmill speed was divided by the pacing rate to get the step length in each walking cycle. All parameters (pacing frequency, step length and step width) were extracted on a cycle-by-cycle basis and the average value and the coefficient of variation (CoV) were obtained. A more detailed description of the extraction of these parameters is provided elsewhere [7, 10].

Fig. 42.2
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Ground reaction force waveforms measured whilst (a) walking and (b) jumping

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The time-domain GRF signals acquired using the force plate whilst jumping were filtered using a low‐pass fifth order Butterworth filter, with the cut‐off frequency having value of either 1 Hz above the frequency of the 3rd harmonic or 7 Hz, whichever was larger [8]. A filtered GRF is shown in Fig. 42.2b. The parameters extracted on a jump-by-jump basis were the period, contact ratio (i.e. contact time CT divided by period T) and impulse (i.e. the area enclosed by the force signal), all of which are shown in the figure.

All data analysed in this paper are related to the vertical component of the force. The CoV is used as a key measure of the parameter randomness in each time history.

42.4 Randomness of Key Parameters

42.4.1 Walking

Figure 42.3 shows the average and CoV parameters for step length as functions of treadmill (i.e. walking) speed. The figure demonstrates that the average step length increases with increase in walking speed (Fig. 42.3a). The mean and mean ± one standard deviation (STD) lines are also shown in the figure, indicating that the STD does not change significantly across the studied population of test subjects. Figure 42.3b demonstrates that the CoV ranges between 1 and 5 %, with minimum variation achieved at the boundary between the normal (solid circles) and fast (crosses) walking speeds (1.6–1.8 m/s). While the probability distribution of the average step length (and averages of other parameters of interest) within a population of structural users is often available in literature [11–13], the detailed information regarding the intra-subject variability (i.e. CoV) is not frequently available. This is the reason that the CoV of different parameters is the main focus of this paper.

Fig. 42.3
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(a) Average and (b) CoV values for the step length parameter as functions of the walking speed

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The CoV of pacing frequency is shown in Fig. 42.4a. This is very similar to the CoV for step length, due to mutual dependence of the two parameters (the product between the pacing rate and the step length represents the walking speed). A much larger value of CoV (up to 40 %) is seen for the step width (Fig. 42.4b). Variation in this parameter is relatively independent from the walking speed.

Fig. 42.4
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CoV for (a) pacing rate and (b) step width as functions of the walking speed

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The CoV of three angular parameters (angle of attack, end-of-step angle and trunk angle) were also monitored during the experiments and they exhibited low variation (up to 2.3 %) [10].

42.4.2 Jumping

Figure 42.5a shows that the randomness in the frequency whilst jumping is larger than when walking (Fig. 42.4a), and therefore maintaining the cycle consistency is more difficult for the activity of jumping. The CoV values for the contact ratios (Fig. 42.5b) and the normalised impulse values (Fig. 42.5c) are in a similar range (up to 8 %). More detailed characterisation of the jumping action is available elsewhere [8].

Fig. 42.5
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CoV for (a) frequency, (b) contact ratio and (c) normalised impulse whilst jumping

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All the data presented in this section could be used to develop and calibrate the models for humans walking and jumping, as well as develop an understanding of the level of variability in the parameters of interest frequently used to describe human actions.

42.5 Experiments on Lively Surface

Three test subjects who took part in the walking sessions in the Gait Laboratory, also participated in nominally the same set of experiments on the Warwick Bridge (WB). Two configurations of the bridge were utilised: WB1 having span of 16.2 m and WB2 spanning 17.4 m. The bridge and the experimental setup are explained in this section, followed by a comparison of the data with those collected in the Gait Laboratory.

42.5.1 WB Description and Test Setup

The WB is a steel-concrete composite structure with total deck length of 19.9 m and deck width of 2.0 m [14]. The bridge is situated in the Structures Laboratory at the University of Warwick. The fundamental mode of vibration is 2.44 Hz for WB1 and 2.18 Hz for WB2, with damping in both cases being between 0.30 and 0.55 %, depending on the vibration amplitude. The natural frequency is also mildly dependent on the vibration amplitude [7].

The plan view of the bridge in the WB2 configuration is shown in Fig. 42.6a. The figure also shows the position of the treadmill on the deck, as well as the shaker used in some tests. In addition, close up side views of the treadmill and the shaker are shown in Fig. 42.6b, c. The motion capture cameras were attached to the steel frames built around the bridge (Fig. 42.6a). The test subjects were instrumented with reflective markers in the same way as in the previous tests in the Gait Laboratory.

Fig. 42.6
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(a) Plan view of the Warwick Bridge. Photos of (b) walking on treadmill and (c) electrodynamic shaker

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In addition to conducting tests on the bridge with the shaker switched off, where the bridge vibrations were exclusively caused by the test subject walking, the tests were also repeated whilst the bridge was exposed to a steady-state vibration amplitude of 1.2 m/s2 induced by the shaker before (and during) the walking trial. The force induced by the shaker was a harmonic force matching the fundamental frequency of the bridge.

42.5.2 Randomness on Lively Bridge

Figure 42.7a shows the CoV of the pacing rate as a function of the walking speed. It can be seen that the CoV for walking over bridge exposed to low-level vibrations (i.e. when excited by the pedestrian only, see thin solid line in the figure) is relatively similar to that recorded on the rigid surface in the Gait Laboratory (thick solid line). When walking on the bridge strongly pre-excited by the shaker, the CoV increased (dotted line). The same conclusions can be drawn for the step length parameter (Fig. 42.7b). While the CoV increase is not too large, it still could be important for modelling human-induced excitation, especially in relation to the pacing frequency parameter, to which the vibration response is most sensitive.

Fig. 42.7
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CoV for (a) pacing rate and (b) step length. Thick solid line: walking on rigid surface; thin solid line: walking on bridge; dotted line: walking on bridge pre-excited by the shaker

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Interestingly, in these tests involving the vertical vibration, the dependence of the CoV of the step width, a parameter important for controlling walking locomotion in the frontal plane [15], on the vibration level could not be established [7]. This supports the hypothesis that the walking locomotion control strategies are direction-dependent [16].

An example of a response time history measured at test point 11 (Fig. 42.6a) on WB2 is shown in Fig. 42.8. Using the measured force, the response was also simulated and its envelope presented in Fig. 42.8. It can be seen that the agreement between the two responses is very good. The bridge response, therefore, can be predicted sufficiently accurately provided the key features of the force are known and the amplitude dependent modal properties of the empty structure are utilised in the simulations.

Fig. 42.8
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A measured vibration response time history and the envelope of the simulated response on WB2

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42.6 Sensitivity of Vibration Response

The CoV of the force parameters shown previously varies between different parameters; of note are the relatively low CoV values for frequency. However, given the known sensitivity of the slender structures to excitation frequency, the sensitivity of the simulated vibration response to the randomness in the frequency will be investigated in this section. For this purpose, a structure is modelled as a single-degree-of-freedom system, with a damping ratio of 1 % and a modal mass of 10,000 kg. The response to the force measured while test subject 2 was jumping at nominal frequency of 1 Hz was calculated. This was done using the measured force and then the simulation was repeated using a synthetic force signal generated in such a way to enforce the same frequency in each jumping cycle (at the same time preserving the measured shape and the amplitude of the force signal). The enforced frequency was set to 0.94 Hz, which is the average pacing frequency achieved in the actual test.

The vibration response was calculated for two structures having natural frequency of 1.88 and 2.12 Hz. For the 1.88 Hz case, the resonance response due to the second harmonic of the force was expected. Figure 42.9a shows that the response to the synthetic force does resemble the resonance shape, while the response to the measured force differs significantly. It is the randomness in the frequency parameter of the measured force which causes such a deviation from the expected response. An out-of-resonance response was also simulated on the 2.12 Hz structure. This time neglecting the randomness in the actual force caused an under-estimation of the response to the actual force (Fig. 42.9b).

Fig. 42.9
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Vibration response time histories under measured and synthetic forces representing test subject 2 jumping at 0.94 Hz for structure with a natural frequency of (a) 1.88 Hz and (b) 2.12 Hz

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These simple examples show that neglecting the randomness in the jumping frequency (CoV = 8 % in the test analysed), can result in both over-estimations and under-estimations of the vibration response. The former is typical of vibration responses in or near the (expected) “resonance” while the latter is usually relevant for responses away from resonance. In the examples analysed, the over-estimation of the peak response is by factor 2.3 while the under-estimation is by factor of 3.0. These findings suggest that the randomness in the frequency of the activity should be accounted for in the force modelling. There is also a need to investigate the sensitivity of the response to randomness in other parameters of interest.

42.7 Conclusions

In this study, randomness in parameters typical of walking and jumping has been investigated. The main conclusions are:

  • Data on randomness of key parameters characterising walking and jumping were collected successfully. The test population consisted of ten test subjects in walking tests and eight test subjects in jumping tests.

  • It is more difficult to preserve consistency whilst jumping than while walking.

  • The coefficient of variation of the cycle-by-cycle frequency could be up to 8 % and 5 % for jumping and walking, respectively.

  • Exposure to vertical vibration increases randomness in the pacing frequency and step length whilst walking.

  • Vibration response is sensitive to the randomness in the frequency of the activity, and it can lead to both over- and under-estimation of the actual vibration response, depending on the relative ratio between the frequency of the relevant forcing harmonic and the natural frequency of the structure.