stochastic approach to modelling of near-periodic jumping loads

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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 24 (2010) 3037–3059

0888-32

doi:10.1

� Cor

E-m1 Te

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Stochastic approach to modelling of near-periodic jumping loads

V. Racic �, A. Pavic 1

Department of Civil and Structural Engineering, University of Sheffield, Sir Frederick Mappin Building, Sheffield S1 3JD, United Kingdom

a r t i c l e i n f o

Article history:

Received 10 June 2009

Received in revised form

9 February 2010

Accepted 27 May 2010Available online 2 June 2010

Keywords:

Vibration serviceability

Human–structure dynamic interaction

Jumping forces

Stadia

Grandstands

70/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ymssp.2010.05.019

responding author. Tel: +44 114 222 5727; fa

ail addresses: [email protected] (V. Racic

l.: +44 114 222 5721; fax: +44 114 222 5700

a b s t r a c t

A mathematical model has been developed to generate stochastic synthetic vertical

force signals induced by a single person jumping. The model is based on a unique

database of experimentally measured individual jumping loads which has the most

extensive range of possible jumping frequencies. The ability to replicate many of the

temporal and spectral features of real jumping loads gives this model a definite

advantage over the conventional half-sine models coupled with Fourier series analysis.

This includes modelling of the omnipresent lack of symmetry of individual jumping

pulses and jump-by-jump variations in amplitudes and timing. The model therefore

belongs to a new generation of synthetic narrow band jumping loads which simulate

reality better. The proposed mathematical concept for characterisation of near-periodic

jumping pulses may be utilised in vibration serviceability assessment of civil

engineering assembly structures, such as grandstands, spectator galleries, footbridges

and concert or gym floors, to estimate more realistically dynamic structural response

due to people jumping.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Predicting dynamic performance of civil engineering structures due to crowd-induced loading is an increasingly criticalaspect of vibration serviceability design process for assembly structures, such as grandstands, spectator galleries andconcert halls, which are becoming more slender and lighter than ever before [1]. Broadly speaking, the procedure forpredicting dynamic response of a structure under consideration involves specifying design load and determining dynamicproperties of the structure in terms of modal mass, stiffness and damping obtained from a structural model. There aredegrees of uncertainty and latitude in each element and different degrees of guidance on procedures. However, of all theseelements, determining the design load has the greatest uncertainty.

A vast majority of relevant design guidelines around the world has recognised jumping as the most important type ofcrowd-induced load on an assembly structure [2,3]. This is because it is the most severe and frequent to happen in practice.There have been cases of impaired vibration serviceability under crowds jumping on footbridges (due to vandal loading asdescribed by Zivanovic et al. [4]), grandstands [5] and concert halls [6]. Therefore, there is a need for a reliable model ofjumping forces to facilitate vibration serviceability checks of these structures.

It is now well established that jumping by individuals and crowds is not a deterministic and ‘perfectly’ periodic,but rather a stochastic and narrow band process [7,8]. This is because of the so called intra- and inter-subjectvariability between naturally imperfect humans taking part in the jumping. However, the vast majority of jumping

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), [email protected] (A. Pavic).

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dx.doi.org/10.1016/j.ymssp.2010.05.019

mailto:[email protected]

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Nomenclature

fs sampling rateDt time stepFi measured forceN number of jumping pulsesTi real periods of jumping cyclesT0

i synthetic periods of jumping cyclesTi variations of real jumping periodsT0

i variations of synthetic jumping periodsAT Fourier amplitudes of Ti seriesA0T Fourier amplitudes of T

0

i seriess2

Tvariance of Ti

ST(fm) the ASD of Ti

ST(f) fit of the ASD ST(fm)

Iw,i real weight normalised impulsesI0w,i synthetic weight normalised impulsesIs,i real unity normalised impulsesai real scaling factorsa0i synthetic scaling factorsr0 and r1 autoregression coefficientsDIw,i autoregression erroroi angular frequency of rotationy angular coordinateZi(t) and Zi(y) shapes of unity normalised jumping pulsesWj and Air Gaussian weightscj, tir and yr Gaussian centresdj, bir and br Gaussian widthsFFT fast Fourier transformZ correlation coefficient

V. Racic, A. Pavic / Mechanical Systems and Signal Processing 24 (2010) 3037–30593038

models in the published literature and design guidelines assume that jumping forces are deterministic andperiodic, presenting their modelling as sinusoids capable of exciting pure resonance of structures [9]. Thisassumption is often over-conservative. The resulting excessive levels of vibration predicted can therefore precludeefficient design. The same problem was observed in the past for walking as another type of human-induced dynamicforce [10].

Recently, there have been serious attempts to resolve this issue by developing a new generation of near-periodicmathematical models for jumping forces induced by a single person, groups or crowds [7,8]. Sim’s modelling is the mostrecent and relevant step in the right direction, but has shortcomings which need addressing. To fit individual jumpingpulses, it utilises a cosine squared function symmetric about a vertical axis through its peak. Therefore, it does not take intoaccount the lack of pulse’s symmetry observed in the real force measurements [11]. Also, only a limited number of jumpingfrequencies are modelled, the majority of them being in the range of moderate and fast jumping rates. As will be seen later,the key reason for this is a considerably more complex shape of the pulses generated for jumping at slow motion, featuringtwo peaks, which cannot be modelled using a squared cosine function. Subsequent pulses for a single jumper are shifted intime by different amounts determined by autoregressive modelling based on approximately 1000 force records. The sameprocedure is repeated for additional persons yielding a model for a crowd dynamic loading as a simple sum of individualsynthetic forces.

It is clear that a key ingredient of a reliable crowd jumping model is an accurate model for a single person jumping.Therefore, considering advances made by Sim et al. [8], there is still a need to develop a good quality jumping model for asingle person applicable to the whole range of possible jumping frequencies. This model has to be narrow band andrandom taking into account all aspects of the inter- and intra-subject variability.

This paper offers a solution to this problem by proposing a novel pulse modelling technique which makes use of anextensive database of measured jumping forces gathered at the University of Sheffield in 2008 and 2009. This database hasan impressive coverage of 14 jumping frequencies in the range 1.4–2.8 Hz, which are observed to be comfortable forrhythmic, repeated motion [12]. The technique proposed is motivated by an existing procedure for modelling recordings ofelectric waves being generated during heart activity [13], generally known as electrocardiogram (ECG) signals. Thesesignals have similar near-periodic features as the force signals due to jumping. Therefore, the paper utilises technology formeasuring jumping forces, which has been available for more than 20 years, and addresses the current lack of anappropriate mathematical modelling to simulate accurately what is being measured. Moreover, the paper presentsa logical extension of the force models published recently by the authors [11,14]. Having analysed additionalabove mentioned jumping force records, major differences between the models can be observed in the selection ofmodelling parameters, the way they are mathematically described and mutually related to simulate more reliably themeasured data.

2. Background reviews

Two periods can be clearly identified in the history of developments in jumping force modelling. First, consisting ofmodels developed prior to about 2004, utilised methods for modelling perfectly periodic signals. Subsequent methodsrecognised the need for a narrow band modelling. These are also characterised by rapid advancements in computationalpower (whereby complex models can be better utilised), structural analysis, and advanced measurement technology whichbecame affordable.

The next two sections present a critical overview of the force models which marked these two periods.

V. Racic, A. Pavic / Mechanical Systems and Signal Processing 24 (2010) 3037–3059 3039

2.1. Periodic models

Dynamic forces generated by people jumping on a structure are commonly determined by direct measurement of theinterface forces between the feet and the structure itself, hence they are known as ground reaction forces (GRFs). Themeasured force signal for individuals jumping is typically a series of distinctive pulses (Fig. 1), which are the reaction to theforce the body exerts on the supporting ground during ‘contact phase’ of jumping. The pulses are separated by zero-forceintervals which indicate ‘aerial phases’ of jumping when both feet leave the ground.

A common practice until now is to idealise a continuously measured jumping force signal as periodic with the periodbeing the average time between two consecutive jumps. This means that actual forces due to continuous jumping can berepresented by a sequence of identical pulses on a jump-by-jump basis.

A number of authors [15,16] fitted simple half-sine function to the average measured jumping pulse. Fig. 2 shows anexample of directly measured pulse extracted from Fig. 1 and the associated half-sine model based on modellingparameters suggested elsewhere [16]. As illustrated in Fig. 2, the symmetric half-sine pattern cannot fit a typicallyasymmetric shape of the real jumping pulse with visually apparent good matching with measured data [11].

For dynamic analysis, a set of identical half-sine pulses can be represented more efficiently if expressed in terms ofFourier series with the fundamental harmonic having frequency identical to the jumping rate [9,15]. Such models can befound in the current British Standard BS 6399-1 [2] and Commentary D of the National Building Code of Canada [3]. Inthese, jumping loads are defined for use in the design of structures likely to be subjected to significant vertical occupantmotion, such as footbridges, grandstands, and concert or gym floors, where human-induced serviceability issues maygovern design.

The main disadvantage of this method is that it requires many terms to describe satisfactorily the original half-sineapproximation. For ease of use, only the first three harmonics (including six coefficients, i.e. three amplitudes and threephases) are typically considered [3,9,16]. However, even the sum of the first six Fourier harmonics (12 coefficients), which

measured data

half-sine model

Time [s]

3500

0 0.1 0.2 0.3 0.4 0.5

3000

2500

2000

1500

1000

500

0

Forc

e [N

]

Fig. 2. Measured jumping pulse extracted from Fig. 1 vs. corresponding half-sine model based on modelling parameters suggested by Ellis and Ji [16]. The

shaded areas represent the difference.

3000

2000

1000

0

Forc

e [N

]

0 2015105

Time [s]

Fig. 1. Typical measured GRF signal generated by a single person jumping at 2 Hz.


is the maximum number reported in the literature [16,17], cannot match adequately enough the original half-sine forcingfunction for all contact times (Fig. 3), let alone real jumping force measurements (Fig. 1).

Brownjohn et al. [10] showed that there were significant differences between the resonant responses due to realwalking forces (which are narrow band and therefore ‘imperfectly’ periodic) and the equivalent periodic simulation. Theeffect is more pronounced for higher harmonics of walking loading where the simulated vibration response is regularlyoverestimated. This issue has not been researched in great detail for jumping excitation, so the available literature islimited. Nevertheless, a similar analysis can demonstrate that ‘synthetic’ periodic jumping forces generate a considerablyhigher resonant response compared with its more realistic narrow band counterpart. The reason for this becomes evidentif, instead of an average jumping pulse, a window comprising a number of consecutive jumping pulses is used to derivetheir Fourier amplitude spectra. Fig. 4 shows the dominant harmonics around 2, 4 and 6 Hz of real jumping at an averagerate of 2 Hz. Other spectral lines, having lower amplitudes around the dominant harmonics, are a consequence of thenarrow band nature of the actual force signal. Also, the effect is frequency dependant: the higher harmonic centrefrequency the greater the ‘spread’ of excitation energy. This phenomenon has been observed for walking forces elsewhere[10,18]. Hence, if resonance of a single degree of freedom system representing a mode of interest is assumed, the spread ofenergy into nearby frequencies results in reduced structural response for higher harmonics compared with the predictionusing a ‘perfectly’ periodic model. Parametric studies can also demonstrate that the differences reduce with increase indamping of the structure, but are still significant even for damping as high as 3% found, say, in modern grandstands [19]. Amathematical characterisation of naturally irregular jumping pulses from individuals is hence one of the key problemswhich needs addressing by modern design guidelines if they are to represent realistically dynamic structural response dueto individual people, groups and crowds jumping.

If reliable models existed representing the complete Fourier spectrum of continuously measured jumping forces (seeFigs. 5a and b), the reconstruction of the force in the time domain would be possible. Although a quite good model of theFourier amplitudes (Fig. 5a) could be obtained (e.g. as suggested for the walking forces elsewhere [10,18]), variations in the

0.0

Time [s]

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0

Forc

e am

plitu

des/

body

wei

ght

-1

1

2

3

4

5 Fourier series model

Half-sine model

Fig. 3. Half-sine function and sum of the first six Fourier terms to represent the force history given in Fig. 1 (after Ellis and Ji [17]).

Frequency [Hz] Frequency [Hz] Frequency [Hz]

1 1.5 2

1.6

1.4

1.2

1

0.8

0.6

Four

ier

ampl

itude

/bod

y w

eigh

t

Four

ier

ampl

itude

/bod

y w

eigh

t

Four

ier

ampl

itude

/bod

y w

eigh

t

0.4

0.2

02.5 3 3 3.5 4

0.6

0.5

0.4

0.3

0.2

0.1

04.5 5 5 5.5 6

0.02

6.5 7

0.04

0

0.06

0.08

0.10

Fig. 4. Fourier amplitudes of measured jumping force (black) and corresponding periodic model (grey) due to jumping at 2 Hz in the vicinity of the

dominant harmonics at a) 2 Hz, b) 4 Hz and c) 6 Hz.

0 2015105Time [s]

4000

1000

0

Forc

e [N

]

3000

2000

-1000

1.5

1

0.5

00 102

Frequency [Hz]

Frequency [Hz]

4 6 8

Four

ier

ampl

itude

s/bo

dyw

eigh

t

0 102 4 6 8

4

-2

Phas

e [r

ad] 2

0

-4

Fig. 5. (a) Amplitude and (b) phase FFT of real-life measured jumping data, demonstrating apparently ‘random’ measured phases; (c) regenerated time

history based upon Fourier amplitudes in (a) and phase lags based upon the uniform random domain [�p, p].


jumping force between subsequent jumps (Fig. 1) yielding widely varying harmonic phase lags (Fig. 5b) would be verydifficult to characterise analytically. If they are, however, assumed to be uniformly distributed in the range [�p, p] (whichis currently the only known modelling strategy for this phenomenon [18]), the sum of dominant Fourier series sinusoidsnormally does not match the real jumping force time history. This can be clearly illustrated by comparison of Figs. 1–5c.Therefore, randomising phases is not the way forward. Their variation seems to be more subtle than a simple set ofuniformly distributed random numbers.

Bearing all this in mind, a more advanced modelling strategy than Fourier series approach is needed to approximatereliably the narrow band nature of the actual jumping loading.

2.2. Narrow band models

First attempts to account for the near-periodic nature of jumping forces can be attributed to studies by Ellis and Ji [20]and Kasperski and Agu [21]. They modelled the jumping frequency and Fourier coefficients of each jumping pulse usingprobability density functions. However, although the parameters generated by such models are random in nature, they areindependent from values of the parameters calculated for the preceding jumps. This seems not to be the case in reality.


With regard to this, Sim et al. [8] showed that there is some structure in the slight variation of timing between peaks ofsubsequent jumps, which can be predicted by the first order autoregression model. This implies that a jumper adjusts thetiming of the current jump according to the timing of the previous jump. The variations in the timing were further relatedto variations in amplitudes of jumping pulses yielding near-periodic GRF time series. However, Sim and co-investigatorsdid not manage to model reliably the full frequency range contained in measured jumping forces. A good compatibilitybetween frequency spectra of measured and modelled force signals was found only for the first two dominant harmonics.The reason for this was the assumption that a very smooth, symmetric, cosine-squared fitting function could represent theirregular shape of measured pulses (Fig. 6). As such, it requires smaller number of Fourier components, thus cannot modelaccurately high frequency content. For vibration serviceability assessments, the number of Fourier harmonics to be takeninto account depends on their contribution to the vibration response and type of structure under consideration. The latestBRE digest 426 [16] suggests that even small energy of the force signal around the sixth dominant harmonic can causevibration serviceability issues. This might not be relevant for design of grandstands and spectator galleries, but it isrelevant for vibration serviceability assessment of, say, multi-story apartment buildings including a fitness centre.

The cosine-squared function could not also represent a wide variety of the pulse shapes which can be generated atdifferent jumping rates. As noted by Sim [7], there are three characteristic pulse shapes: double peaked, merging and singlepeaked, as illustrated in Fig. 6. The double peaked shape is often generated by jumping at rates below 2 Hz, when thelanding and launching actions are separated visibly by a long bounce. The majority of people generate merging shapeswhen jumping around 2 Hz. They are either due to a brief bounce or because the feet hit the ground at slightly differenttimes. Also, a sharp peak may appear at the beginning of the landing phase if the heels are the first to strike the ground,depending on the style of jumping. At rates higher than 2 Hz, the pulse shapes are mostly smooth and single peaked. Thishappens when people hit the ground with the left and right toes ‘simultaneously’ and propel straight away into the air

0.5

0

1

1.5

2

2.5

3

0.1 0.2 0.3 0.4 0.5 0.6 0.70

Time [s] Time [s] Time [s]



Forc

e/B

ody

Wei

ght [

-]

Forc

e/B

ody

Wei

ght [

-]

Forc

e/B

ody

Wei

ght [

-]Fo

rce/

Bod

y W

eigh

t [-]

Forc

e/B

ody

Wei

ght [

-]Fo

rce/

Bod

y W

eigh

t [-]

Forc

e/B

ody

Wei

ght [

-]Fo

rce/

Bod

y W

eigh

t [-]

Forc

e/B

ody

Wei

ght [

-]

0.5

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1

1.5

2

2.5

0.1 0.2 0.3 0.4 0.5 0.6 0.70

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1

1.5

2

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3

3.5

0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1 0.2 0.3 0.4 0.50

0.50

11.5

22.5

3

5

3.54

4.5

0.1 0.2 0.3 0.4 0.50

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0

1

1.5

2

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3

0.1 0.2 0.3 0.4 0.50

0.5

0

1

1.5

2

2.5

3

0.05 0.1 0.15 0.2 0.25 0.3 0.350 0.4

0.5

0

1

1.5

2

2.5

3

0.5

0

1

1.5

2

2.5

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3.5

4

0.05 0.1 0.15 0.2 0.25 0.3 0.350 0.4

0.50

11.5

22.5

33.5

44.5

0.05 0.1 0.15 0.2 0.25 0.3 0.350 0.4

Shape 1 Shape 2 Shape 3

1.5 Hz

2.0 Hz

2.5 Hz

Fig. 6. Examples of different shapes of measured force traces due to jumping at 1.5, 2 and 2.5 Hz.

2000

1500

1000

500

0

Time [s]

Pi Pi+1Pi-1 Pi+2

Ii Ii+1Ii-1 Ii+2

TP,i-1

Ti Ti+1Ti-1

Tc,i-1 Tc,i Tc,i+1 Tc,i+2

TP,i TP,i+1 TP,i+2

4.4 4.8 5.2 5.6 6.0 6.44.6 5.0 5.4 5.8 6.2

Forc

e [N

]

Fig. 7. A portion lasting 2 s of measured force signal due to a single person jumping at 2 Hz. The complete signal lasts 25 s.


having no time for the heel contact. Bearing all this in mind, the quality of Sim’s model depended on jumping rate. Athigher rates, the cosine-squared function could approximate the shape of actual pulses better [7,8]. Hence, the best fit wasfor jumping when people generate landing and launching impulses which are almost the same, yielding fairly symmetricalsingle peaked shapes. However, because of symmetry and smoothness of the fitting function, double peaked pulses werenot considered in the modelling at all.

More recently, Racic and Pavic [11,14] made a step forward and used a sum of two Gaussian functions to account for theomnipresent lack of symmetry of the single peaked shapes. Although by varying the overlapping between the Gaussianstheir sum could fit reasonably well main features of all three pulse shapes (such as the number of peaks and the generalform), modelling local irregularities yielding higher Fourier harmonics (e.g. the sharp peak in Shape 3 at 2 Hz in Fig. 6) stillremains a point of concern. A way to overcome this problem is to increase the number of Gaussians in the sum, as it will beshown in Section 3.5.

Not only can the pulses change shape when a person is jumping at different rates, but successive pulses can also takedifferent shapes for jumping at a single frequency. This typically happens for moderate rates around 2 Hz when thesuccessive pulses switch the shape randomly between merging and single peaked profiles (see Fig. 7). To the bestknowledge of the authors, there is no model available in the literature which takes into account changing the shapebetween successive jumping pulses.

Sim developed her stochastic model using the most comprehensive database of human jumping loads availableworldwide, which comprises approximately 1000 force histories from about 100 individuals jumping alone on a rigidforce plate [22]. Such a large number of the GRF time series provided a statistically reliable platform for the study of theinter-subject variability. However, the poor resolution of the measured jumping rates, which included only four temposat 1.5, 2.0, 2.67 and 3.5 Hz, resulted in a lack of statistical rigour when studying certain aspects of the intra-subjectvariations. For example, a study of changes of the force patterns for different jumping rates would be very rudimentaryand fragmented if based on such a coarse dataset. This means that there remains a requirement to establish a sufficientlylarge database of GRF time series which will also have a fine resolution of measured jumping rates to provide statisticalreliability in the study of both inter- and intra-subject variability. Establishment of such a database is the key aspectoutlined in the rest of this paper, together with the utilisation of the database as a solid foundation for developing a morerealistic mathematical model of jumping forces which can be used reliably in vibration serviceability assessment ofassembly structures.

3. Concept: from measured to synthesised force

The purpose of this section is to describe the concept of the new model development, so that the reader can follow therationale for the remaining parts of the paper. This will be done by demonstrating, step-by-step, how a single measuredforce trace can be utilised to generate its synthetic counterpart. Two components are needed: a good quality measuredforce trace from a single test subject and an appropriate mathematical model of this trace. Therefore, the methodpresented in this section accounts only for the intra-subject variability. It will be extended later, in Section 5, to account forinter-subject variability as well. This will be done based on utilisation of a database of measured jumping force historiesgenerated by a diverse human population, as presented in Section 4.


3.1. Step 1: Measured force records

Fig. 7 illustrates a portion of a continuously measured force time history generated by a single person jumping for 25 sin response to a regular metronome beat at 2 Hz. The force signal was recorded at a sampling frequency of 200 Hz.

By looking at this signal, it is apparent that there is some irregularity on a jump-by-jump basis in terms of variation ofshape of jumping pulses, duration and force amplitudes. Therefore, a mathematical model for characterisation of irregularjumping pulses must include modelling of the signal parameters which can represent this natural variability best. Selectionof these parameters is the key aspect outlined in the next section.

3.2. Step 2: Basic processing of measured force time history parameters

A jumping cycle is the period of time between any two nominally identical events in the jumping process. In the contextof this paper, the instant at which the feet hit the ground (also known as ‘initial contact’) yielding a new pulse was selectedas starting (and completing) event (Fig. 7).

From the 25 s long force signal illustrated in Fig. 7 yielding about 50 jumping pulses, a window comprising 42successive jumping cycles was selected for further analysis. A total of eight cycles were discarded from the start and end ofthis time history. The force threshold marking the start of the pulse was set to 15 N.

From each of the 42 cycles, contact time Tc,i, period Ti (i.e. duration of a jumping cycle), peak timing TP,i, peak amplitudePi and impulse Ii are extracted on a cycle-by-cycle basis (Fig. 7). In the past, statistical models of these parameters and theirmutual relationships were used to describe intra-subject variability of jumping forces. For example, Sim et al. [8] derivedtheir model based on the theoretically derived linear relationship between the impulse size Ii and timingTP,i ¼ ðTP,iþ1�TP,i�1Þ=2. However, using the measured values extracted from the 42 cycles, Fig. 8a illustrates a lack oflinear correlation between these two parameters. This is probably because the model proposed by Sim et al. [8] was basedon an assumption that the impulse is of very short duration i.e. instantaneous, whereas in reality it lasts as long as the

0.48 0.49 0.50 0.51 0.52

Period of jumping cycle T [s]i Period of jumping cycle T [s]i

1950

1900

1850

1800

1750

1700

1650

Peak

ampl

itude

P[N

]i

0.48 0.49 0.50 0.51 0.52

0.34

0.33

0.32

0.31

0.30

0.29

0.28

Con

tact

time

Tc,

i[s]

326

324

322

320

318

316

312

310

314

308

3060.47 0.48 0.49 0.50 0.51 0.52 0.53

Impu

lse

I i [

Ns]

Timing T [s]P,i

0.54

Fig. 8. Examples of poor correlation between (a) timing TP,i and impulse size Ii (the correlation coefficient Z=0.104), (b) periods Ti and peak amplitudes Pi

(Z=0.235)and (c) periods Ti and contact times Tc,i (Z=0.418).

0.48Period of jumping cycle Ti [s]

0.49 0.50 0.51 0.520.48

0.49

0.50

0.51

0.52

]s[i,wIseslupmi

desilamronthgie

W

measured datalinear fit

Fig. 9. Correlation between periods Ti and weight normalised impulses Iw,i (Z=0.801).


contact period. Also, Sim et al. [8] assumed that a launching impulse which projects the jumper into the air and thesubsequent landing impulse are the same. However, this is only true for the frictionless impact between two infinitely stiffi.e. rigid bodies. In reality, it is possible for the stiffness of the legs to be adjusted between successive jumps. For example,Farley et al. [23] showed that when humans hop in place, the neuromuscular system can significantly alter the stiffness ofthe leg to accommodate variations in hopping height at a given frequency. Differences in launching and landing pulsesmight also be the most likely explanation for the asymmetry of jumping pulses. More examples of the lack of correlationsbetween other jumping force parameters are given in Figs. 8b and c.

However, certain correlations between the jumping parameters seemingly do exist. In the study presented here, it wasfound that there is some relationship between period Ti and weight normalised impulses Iw,i calculated over integrationtime Tc,i as

Iw,i ¼1

mg

Xn

i ¼ 1

Fi Dt, Dt¼1

fsð1Þ

where m is the body mass, g is gravity (g=9.81 m/s2), Fi is the force, fs is the sampling rate and n is the total number ofsamples of Tc,i. This relationship shows a linear trend (Fig. 9), hence provides an opportunity to describe weight normalisedimpulses as a function of the period. This is an important feature for the proposed methodology, as will be shown later inSection 3.4.

3.3. Step 3: Analysis of periods of jumping pulses

The problem now is to model slight variations of period Ti (i=1, y, 42) between each jump. This effect can berepresented by a sequence of numbers Ti calculated as

Ti ¼Ti�mT

mT

mT ¼meanðTiÞ ð2Þ

Two methods were tried for modelling Ti data. It was first assumed that the current Ti value is a linear combination ofprevious k values Ti�1,. . .,Ti�k, which could be described by an autoregressive model [24]. This method proved unreliablesince weak correlation was found between the previous and subsequent Ti values for increasing order of regression fromone to four (k=1, y, 4). Therefore, the rest of this section will be focused on the alternative and more successful approachbased on utilisation of the auto spectral density (ASD) of Ti.

Given the single-sided spectral density Sx(f) of the real random process x(t), the variance of x(t), sx2, can be computed by

using the relation [25]:

s2x ¼

Z 10

Sxðf Þ df ð3Þ

The point here is to use the ASD (as illustrated in Fig. 10) to artificially generate synthetic T0

i series having the samestandard deviation as the actual Ti series. When doing this, the newly generated set of numbers is assumed to have the

c5 c10 c15 c20

0.10 0.2 0.3 0.4 0.5

Quasi-frequency

2.5

3

2

1.5

1

0.5

0

x10-3

ASD

[1

/Hz]

measured

Gaussian fit

Fig. 10. Single-sided ASD of discrete data Ti .


same ASD as the measured set of numbers. This applies regardless of the number of data points in the old and new sets,which will be demonstrated later in this section. This assumption also means that the standard variation of Ti does notchange for the given jumper, jumping rate and jumping interval. Moreover, the ASD preserves a ‘frequency structure’between Ti values (if there is any), which the autoregressive models could not represent.

The ASD of Ti can be calculated as

ST ðfmÞ ¼A2

TðfmÞ

2Df, fm ¼

m

42, m¼ 0,. . .,20 ð4Þ

where AT ðfmÞ is a single-sided discrete Fourier amplitude spectrum having spectral line spacing Df=1/42.The ASD ordinates do not depend on the number of discrete data points Ti used for the calculation of A

TðfmÞ but it is

coarse due to the limited number of points (21 single-sided FFT). More points might reveal a much richer structure but thisrequires a longer jumping force record. However, continuous jumping demands significant effort of a test subject, so theduration is limited to avoid causing fatigue which can influence the force records. This will be discussed further in Section4.2.The ASD ST(fm) can be analytically described by a series of Gaussian functions (Fig. 10):

S0Tðf Þ ¼

X21

j ¼ 1

Wje�ððf�cjÞ

2=2d2j Þ ð5Þ

Here, parameter Wj is the height of the jth Gaussian peak, cj is the position of the centre of the peak, and dj controls thewidth (i.e. time duration) of the corresponding bell-shaped curve. The Gaussian centres cj, j=1, y, 21, are placed in eachsample on the quasi-frequency axis in order to fit exactly the measured ASD (Fig. 10). For such fixed positions of Gaussiancentres cj and predefined widths dj=Df, Gaussian heights Wj (also called weights) can be computed using the non-linearleast-square method [26]. The results of the fitting are given in Table 1.

Representation of the discrete ASD ST(fm) in the form of the continuous function S0T(f) enables calculation of the ASD

ordinates for an arbitrary spectral line spacing Df. As the ASD ordinates do not depend on the number of discrete data

points used for its calculation via FFT, the continuous function S0T(f) can be used to generate a set of data points T0

k

(k=1, y, N) which would have the same ASD as the measured data. This is done by calculating the new discrete spectral

Table 1Parameters of function S0T.

j cj (dimensionless) Wj (dimensionless) j cj (dimensionless) Wj (dimensionless)

1 0.0000 �0.0040 12 0.2619 0.0003

2 0.0238 0.0027 13 0.2857 �0.0001

3 0.0476 �0.0001 14 0.3095 0.0002

4 0.0714 0.0008 15 0.3333 0.0001

5 0.0952 0.0020 16 0.3571 0.0000

6 0.1190 0.0008 17 0.3810 0.0001

7 0.1429 �0.0001 18 0.4048 0.0001

8 0.1667 0.0009 19 0.4286 0.0001

9 0.1905 0.0002 20 0.4524 �0.0001

10 0.2143 0.0004 21 0.4762 0.0000

11 0.2381 �0.0001

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

1 5 10 15 20 25 30 35 40 42

Jumping cycle [-]

Var

iatio

nof

jum

ping

peri

ods

T[-

]i

measured

simulated

Fig. 11. Variations of peak-to-peak intervals on a cycle-by-cycle basis.


line spacing Df 0 ¼ 1=N. For a sequence of discretely spaced frequency points fn=n Df0 (where n¼ 0,. . .,N=2�1)

corresponding to the new set of data points to be generated, a new set of ASD amplitudes is calculated using S0TðfnÞ.

This is then used in Eq. (4) to generate a new set of Fourier amplitudes A0TðfnÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Df 0S0

TðfnÞ

q. Finally, assuming a set of

randomly distributed phases in the range [�p, p], a new set of N variations T0

k can be generated by inverse Fourier transform

making use of A0TðfnÞ and the phases. Different realisations of the random phases may be specified by varying the seed of the

random number generator, hence many different series T0

k may be generated with the same spectral properties (Fig. 11).

Because of the relation defined by Eq. (3), all series T0

k generated also have the same variance sT2 regardless of their

length N. Bearing all this in mind, it is possible to generate T0

k series of arbitrary length (e.g. N542), which will havestatistically the same properties of variations on the sample-by-sample basis as the measured set of 42 actual Ti datapoints. According to Eq. (2), scaling T

0

k by mT and adding the offset value mT calculated from the measured data, results in aseries of synthesised jumping periods T

0

k, as would be generated by the test subject during nominally identical jumpingexercises. Empirical evidence for this is presented in Section 3.6.

3.4. Step 4: Analysis of impulses of jumping pulses

As previously demonstrated (Fig. 9), the relationship between the weight normalised impulses Iw,i and the durations ofthe jumping cycles Ti is approximately a linear function. Therefore, this relationship can be used to generate a new set ofsynthetic impulses I0w,k based on the generated set of synthetic periods T0k from the previous section.

The measured weight normalised impulse values Iw,i can be expressed as a function of period Ti (Fig. 9) using thefollowing linear regression model [24]:

Iw,i ¼ r1Tiþr0þDIw,i ð6ÞHere, r1=0.832 and r0=0.082 are regression coefficients and DIw,i is the subsequent error (also known as a disturbanceterm), which is a random variable. Given the regression coefficients r1 and r0, DIw,i can be calculated from the real data as

DIw,i ¼ Iw,i�r1Ti�r0 ð7Þ

It is common to model DIw,i as Gaussian noise [24] having a probability density function jDI(x) given by [27]

jDIðxÞ ¼1

sI

ffiffiffiffiffiffi2pp e�ððx�mI Þ

2=2s2IÞ, x 2 R ð8Þ

where x is a synthetic random variable which corresponds to measured DIw,i, mI=0 is the mean value and sI is the standarddeviation of real DIw,i.

Given a set of artificially generated periods of jumping pulses T0k, as explained in Section 3.3, a series of correspondingsynthetic weight normalised impulses I0w,k can be therefore calculated using Eqs. (6) and (8). This will be demonstrated inSection 3.6.

3.5. Step 5: Analysis of shape of jumping pulses

By looking at the force signal given in Fig. 7, each jumping pulse can be observed as a function of time having distinctivesize and shape. Therefore, the aim is to extract as many as possible different jumping pulses from a continuously measuredjumping force time history and mathematically describe them. The models will be then used to artificially generate a forcesignal which includes an arbitrary number of jumping pulses. This assumes that the extracted pulses can represent theindividual’s long term performance for the nominally identical jumping exercises.

A question arises here about the minimum number of successive jumping pulses needed to ensure reliable representation.At given jumping rate, the total number of measured pulses depends on the duration of the force record. Jumping is ademanding activity requiring a lot of effort, so the duration should be just long enough to avoid causing fatigue, physicaldistress or ethical issues. This will be discussed further in Section 4.2. Published research on the subject is very rare and limited.Parkhouse and Ewins [22] claimed that a minimum of 30 consecutive jumping cycles should constitute the shortest duration ofthe force signal to calculate reliably the corresponding Fourier coefficients, provided a person does not tire visibly or obviouslychange their jumping pattern. However, this suggestion was made without giving much justification, hence it requiresverification. Using a statistical technique called sequential estimation analysis, Rodano and Squadrone [28] found that at least12 single and nonconsecutive jumping pulses (i.e. with pauses between jumps) were needed to obtain a stable mean of jointkinetics, such as hip, knee and ankle internal forces and moments, derived from vertical jumping forces. Assuming that at least12 jumping cycles are also necessary to reach stability of the mean of continuously measured jumping pulses, the set of 42successive, weight normalised pulses are used further in the analysis of the corresponding pulse shapes.

Each weight normalised pulse was individually extracted into a half-second segment and scaled by the correspondingpeak amplitude ai yielding unity normalised pulses (Fig. 12), so that differences in pulse shape could be investigatedindependently of amplitude. These scaling factors can be expressed as the ratio:

ai ¼Iw,i

Is,i, i¼ 1,. . .,42 ð9Þ

0.8

1

0.6

0.4

0.2

0

-0.5 0-1

0.5 1

10.5

0-0.5

-1

θ(t)

z

Gaussian functionsGaussian fitGaussian centres

r

Fig. 13. Trajectory Zi(t) in a three-dimensional (3D) space. Figs. 13 and 12 represent the same data.

0.10 0.2 0.3 0.4 0.5Time [s]

1.2

1

0.8

0.6

0.4

0.2

0

Forc

e/bo

dy w

eigh

t [-]

Gaussian functionsGaussian fit

measured signalGaussian centres

Fig. 12. Example of unity normalised jumping pulses extracted from Fig. 7.


where Is,i are impulses of unity normalised pulses. The ratio ai will be used later in Section 3.6 to describe the smoothmodulation of measured amplitudes of subsequent pulses in a synthetic jumping force signal.

For fs=200 Hz sampling rate, each unity normalised pulse consists of 100 samples (Fig. 12). Therefore, the underlyingpattern of the ith pulse can be modelled mathematically as a sum of 100 Gaussian functions Zi(t):

ZiðtÞ ¼X100

r ¼ 1

Aire�ððt�tir Þ2=2b2

irÞ, t 2 ½0,0:5�, i¼ 1,. . .,42 ð10Þ

where the parameter Air is the height of the rth Gaussian peak, tir is the position of the centre of the peak, and bir controlsthe width of the corresponding Gaussians (Fig. 12).

The Gaussian centres tir ¼ tr ¼ n Dt, n=0, y, 99, Dt=0.005 s are placed in each sample on the time axis to fit exactly themeasured pulse amplitudes, thus to reflect completely the corresponding Fourier amplitude spectrum. A less densedistribution of Gaussian exponentials (e.g. centres are placed in every second or third sample) will make the fit smoothercausing the high frequency components to vanish. For such fixed positions of Gaussian centres tir and predefined widthsbir ¼ br ¼Dt, Gaussian heights Air can be optimised using non-linear least-square curve fit [26].

Analytical functions Zi(t) can be used to replicate unity scaled jumping pulses. Furthermore, they can be individuallyscaled in terms of amplitude (vertically) and time (horizontally) to model realistic impulses and periods of jumping cycles,respectively. This will be demonstrated in the next section.

3.6. Step 6: Dynamic model

Periodic behaviour of jumping could be better visualised when individual pulses are ‘wrapped’ around the surface of acylinder (Fig. 13). Now the first and the last sample of each pulse overlap, thus they become closed orbits that encircle thecylinder.


The 42 consecutive jumping pulses can be generated by retracing the Gaussian fits Zi(t), given by Eq. (10), around acircle of unit radius in the (r, y) plane (Fig. 13). The time required to complete one revolution on this circle is equal to theperiod of corresponding jumping cycle and lasts Ti seconds.

The angular frequency of rotation oi can be calculated as

oi ¼2pTi

rad=s� �

ð11Þ

The equations of motion of a point moving around the circle are therefore given by a set of two equations [14]:

rðtÞ ¼ 1

yðtÞ ¼oit, y 2 ½0,2p� ð12Þ

Further, Eq. (10) can be rewritten as a function of the angular coordinate y instead of time t:

ZiðyÞ ¼X100

r ¼ 1

Aire�ðy�yr Þ

2

2b2r , y 2 0,2p½ �, i¼ 1,. . .,42 ð13Þ

where yr ¼oitr and br ¼oibr are in radians.The time positions of the Gaussian peaks tr now correspond to fixed angles yr along the unit circle, as illustrated in

Fig. 13.For pairs ðoi,ZiðyÞÞ sorted in numerical order i=1, y, 42, coupled system of Eqs. (11)–(13) through 42 iterations

generates a synthetic signal which is identical to the real signal comprising the 42 consecutive unity normalised jumpingpulses. However, the aim is to artificially generate a force record of arbitrary duration, i.e. which includes an arbitrarynumber N of jumping pulses (e.g. N542), the test subject could generate during nominally identical jumping exercises.

Let T0k (k=1, y, N) be a series of periods of jumping cycles computed as explained in Section 3.3. The correspondingangular frequency o0k of circular motion is then given by Eq. (11). It can be further assumed that the duration of thejumping cycle does not influence the general shape of unity normalised pulses at a given jumping rate. Under thisassumption, any of Zi(y) can be assigned randomly and equally likely to each o0k yielding pairs ðo0k,ZkðyÞÞ. Even when the

0.80.60.40.2

0

11.2

Uni

ty-n

orm

alis

edpu

lses

[-]

0.80.60.40.2

0

11.2

Uni

ty-n

orm

alis

edpu

lses

[-]

0 5 10 15 20 25 30 35 40 45 50

0.80.60.40.2

0

11.2

0 5 10 15 20 25Time [s]

0 5 10 15 20 25Time [s]

Time [s]

Uni

ty-n

orm

alis

edpu

lses

[-]

Fig. 14. (a) Measured, and examples of synthetic unity-normalised signals when (b) N=50 and (c) N=100. Jumping rate is 2 Hz.


same Zi(y) is assigned to different o0k, the resulting jumping pulses generated by the coupled system of Eqs. (11)–(13) willbe slightly different. This is because higher angular frequencies o0k generate the pulses faster, hence they compress themresulting in shorter intervals of jumping cycles and vice versa. Fig. 14 illustrates examples of 25 and 50 s long syntheticunity normalised signals generated by Eqs. (11)–(13) when the total number of pairs ðo0k,ZkðyÞÞ is N=50 and 100,respectively.

To reflect the changes in both the amplitude and timing of the real jumping force from one pulse to another, amplitudesof each synthetic unity normalised pulse of the kind shown in Fig. 14 need to be multiplied by the corresponding elementin series a0k defined by Eq. (9).

Unity normalised impulses Is,k in the denominator of ratio a0k are fixed values for each unity normalised pulse Zk(y) andcan be calculated as a definite integral of the corresponding function Zk(t) between 0 and 0.5 s:

Is,k ¼

Z 0:5

0ZkðtÞ dt�

X99

n ¼ 0

Zkðn DtÞ Dt ð14Þ

As demonstrated in Section 3.4, for a generated set of synthetic T0k, the corresponding set of the body weight normalisedimpulses I0w,k in the nominator of ratio a0k can be found from Eq. (6).

After being scaled by series a0k, signals given in Fig. 14 are still dimensionless and feature variations of the jumping forceparameters on a jump-by-jump basis (shape of the pulses, periods, impulses and peak amplitudes), as the test subjectcould generate in reality during nominally identical jumping exercises. These signals become equivalent jumping forcetime histories when their amplitudes are additionally multiplied by the body weight of the test subject (Fig. 15).

The similarity between the real measured and synthetic near-periodic vertical jumping force signals may be seen bycomparison of Fig. 15a–c. This comparison looks much better than the one given in Section 2.1 between Figs. 1 and 5cfeaturing standard Fourier transform approach with randomly generated phases used to recreate the forcing function. Thestandard Fourier amplitude spectra are compared in Fig. 16. For the first four dominant harmonics the relative errors arewithin the range of 73%. Moreover, relative error in the area under the graph of the spectra (i.e. overall energy of the

0 5 10 15 20 25 30 35 40 45 50

00 5 10 15 20 25

Time [s]

0 5 10 15 20 25

Time [s]

Time [s]

2000

1500

1000

500Forc

e[N

]

0

2000

1500

1000

500Forc

e[N

]

0

2000

1500

1000

500Forc

e[N

]

Fig. 15. (a) Measured and examples of synthetic force signals when (b) N=50 and (c) N=100. Jumping rate is 2 Hz.

1.5

1

0.5

00 2 4 6 0

Frequency [Hz]

0 2 4 6 0

Frequency [Hz]

0 2 4 6 0

Frequency [Hz]

Four

ier

ampl

itude

s/bo

dyw

eigh

t [-]

Four

ier

ampl

itude

s/bo

dyw

eigh

t [-]

Four

ier

ampl

itude

s/bo

dyw

eigh

t [-]

1.5

1

0.5

0

1.5

1

0.5

0

Fig. 16. Fourier amplitude spectra of (a) measured and synthetic force signals when (b) N=50 and (c) N=100. Jumping rate is 2 Hz.


signals) is less than 5%. This indicates a good match in the frequency content between the measured and synthesised GRFsignals.

Perfectly identical signals in both time and frequency domains can be generated only by chance. However, the followingproperties are identical between measured and synthesised forces:

1)
Shapes of the jumping pulses are drawn from the same source (i.e. functions Zi(t)), where each shape has the sameprobability of occurrence.
2)
Because of the common ASD, the quality and quantity of variations of jumping periods T0
k are the same for all syntheticsignals. Quality means that the relationship between successive T

0

k data points follow a nominally identical pattern forall signals, whereas the quantity means that the standard deviation of T

0

k is a fixed value.0

3)
The statistical equivalence between Tk values reflects directly equivalence between the corresponding weight-normalised impulses I0w,k according to Eq. (6). This implies that total energies of generated signals are the same.
The modelling procedure presented in this section has been applied only to a single force record. The step forward is toapply the same procedure to a sufficiently large database of force records, to extract the modelling parameters from eachrecord and to use them to develop a stochastic model of individual jumping loads. Establishment of such a database is thekey aspect outlined in the next section.


4. Experimental data acquisition

A database comprising many high-quality jumping force records is an essential component for the development of astochastic model of jumping loads. Section 4.1 describes only facts related to the experimental setup used, so that it is clearhow the data was collected. Section 4.2 explains test protocol including all details which need considering to gather aquality database.

4.1. Experimental setup

Testing was carried out in the Light Structures Laboratory in the University of Sheffield. The GRFs of one person jumpingat a time were recorded by a single AMTI BP-400600 force plate [29] rigidly fixed to the laboratory floor, as illustrated inFig. 17. All forces were sampled at 200 Hz. As well as safety, the platform built around the force plate (Fig. 17) gave theimpression of a bigger jumping space to avoid test subjects deliberately targeting a relatively small plate surface area(0.6 m�0.4 m), which might influence the natural variability of the GRFs.

4.2. Acquisition of quality experimental data

Fifty-five volunteers were drawn from students, academics and technical staff of the University of Sheffield. Participantswere adult people from diverse ethnic groups, different genders (38 males and 17 females), body size and shape (bodymass 73.2720.6 kg, height 1.7270.12 m) and varying age categories (33.176.6 years). The stochastic approach tomodelling jumping loads, presented in Section 5, is based on an assumption that these fifty-five persons can representgeneral human population.

Each participant was asked to perform 15 jumping tests. Each test followed the same pattern:

1)
the participant stood on the force plate, and then 2) was given a constant metronome beat, 3) was allowed a brief practice for a few seconds prior to the data collection, 4) was asked to jump for 30 s following the metronome beat (middle 25 s were recorded), 5) was asked to leave the force plate and rest.
The constant metronome beat was chosen in a quasi-random order from 15 different rates in the range 1.4–2.8 Hz havingfine resolution of 0.1 Hz. The range included slow and fast jumping frequencies which were suggested in the past as beingcomfortable for individuals [12] and at which synchronised jumping of groups and crowds can occur [30]. Therefore, each

force plate safety platform

Fig. 17. Experimental setup.


participant generated 15 force records yielding a database of 825 force time histories in total for all participants. Typicalmeasured force records are given in Fig. 18.

The closely spaced jumping rates gave the so established database a definite advantage over the other similar datasetscollected worldwide so far, such as those published by Parkhouse and Ewins [22,31]. This brings more statistical reliabilityinto the modelling intra- and inter-subject variations of jumping force patterns due to changes in the jumping rate, whichmodels published so far could not represent to such an extent [7,8,20,21].

The longer a jumping test lasts, the better insight it provides into the natural variability of the measured jumping forcehistory. However, there is a limit on ability of people to keep jumping for a long time without tiring or changing theirmotions. As mentioned in Section 3.5, jumping is a physically intensive activity, so ethical concerns limited duration of thetests to 30 s. In feedback from the participants, this duration was commonly considered optimal. The participants were notgiven any explicit instructions about their jumping technique, but they were encouraged to move as if they were enjoyinga lively concert or an aerobic exercise.

Prior to the experiment, the test protocol (approved by the Research Ethics Committee of the University of Sheffield)required that each participant should complete a Physical Activity Readiness Questionnaire and pass a preliminary fitnesstest (by satisfying predefined criteria for blood pressure and resting heart rate) to check whether they were suited to thekind of physical activity required during the measurements. All participants wore comfortable sportswear and trainers.

1500

1000

500

0

Forc

e[N

]

0 2015105

Time [s]

25

1500

1000

500

0

Forc

e[N

]Fo

rce

[N]

0 2015105

Time [s]

25

4000

1000

00 2015105

Time [s]

25

2000

3000

Fig. 18. Examples of measured force signals generated by three different persons jumping in response to a regular metronome beat at (a) 1.5 Hz, (b) 2 Hz

and (c) 2.5 Hz.


5. Development of stochastic jumping model

After a few preparatory steps outlined in Sections 5.1 and 5.2 combines the modelling strategy developed in Section 3and the database of measured GRF signals from Section 4 to create a stochastic model to generate synthetic humanjumping loading.

5.1. Utilisation of existing database

The 825 force signals measured in Section 4 were classified into 15 categories (clusters) with respect to the actualjumping rate (1.4–2.8 Hz with a step of 0.1 Hz). For example, all force records with the actual rate in the range1.950–2.049 Hz are gathered into a cluster at 2 Hz. In fact, giving the participants constant metronome beats does notexplicitly mean that the beat frequency was followed. This is because not all individuals are able to synchronise theirmovements to the beats [21]. If they were able to do so, the clusters would comprise 55 force records each, which is the

40 50 60 70 80 90 100

Body mass [kg] Body mass [kg] Body mass [kg]

110 110 110

3.63.43.2

32.82.62.42.2

21.81.6

40 50 60 70 80 90 100

4.5

4

3.5

3

2.5

240 50 60 70 80 90 100

4.24

3.83.63.43.2

32.82.62.42.2

Ave

rage

[-]

α i

Ave

rage

[-]

α i

Ave

rage

[-]

α iFig. 19. Average ai vs body mass for the clusters (a) 1.5 Hz, (b) 2 Hz and (c) 2.5 Hz representing slow, moderate and fast jumping rates, respectively.

unity normalised impulses I

from the corresponding cluster get aset of the force parameters by chance

input data: jumping rateand duration of jumping

number of jumps N

pulse shapes

random number seed

assign one pulse shapeto each jump

phases [- ]π,π

random number seed

generation of variationsof jumping periods T

coupled system of equations

unity normalised GRFs

dynamic impact factors αrandom number seed

pdf of body mass

body weight synthetic GRF signal

ASD of variations ofjumping periods S (f)

generation of jumping periods T and thecorresponding angular frequencies ω weight normalised impulses I

Fig. 20. Algorithm describing the procedure for generating synthetic GRF signals.


total number of participants. In the present study, clusters for the rates between 1.7 and 2.5 Hz include more than 55measurements (maximum 67 for 2 Hz), whereas those in the range 1.4–1.6 Hz and 2.6–2.8 Hz comprise less than 55measurements (minimum 42 for 2.8 Hz). This might also be an indicator that the rates in the range 1.7–2.5 Hz were themost comfortable jumping frequencies for the majority of participants in this study as they ‘attracted’ jumping rates fromother clusters.

Each force history within a cluster was processed using concepts described in Section 3. Multiple sets of information(Gaussian weights Wj and Air, Gaussian centres cj and yr, Gaussian widths dj and br, autoregression coefficients r1 and r0,and standard deviation sI) were developed and stored within the cluster as Matlab structural files [32].

Finally, it needs to be shown that the force is independent from the body weight (mass), so that weight normalisedsynthetic signals generated by the dynamic model (Section 3.6) can be scaled by the weight of any person drawn by chancefrom the world’s population. By doing this, randomisation of the modelling parameters will be extended to the maximum.The evidence to support this hypothesis is given in Fig. 19, where the average ratio ai for each GRF signal in clusters 1.5, 2and 2.5 Hz was plotted against the corresponding body mass. The scattered patterns for all three jumping rates indicatedno correlation between the two parameters, thus they can be treated as independent variables in the modelling process(Section 3.6).

As a further random parameter, body mass can be modelled using a probability density function, as illustrated byHermanussen et al. [33] for German, Austrian and Norwegian citizens.

5.2. Procedure for generating synthetic forces

The flow chart in Fig. 20 illustrates the complete process (algorithm) of creating synthetic GRF signals. For a specifiedjumping rate and jumping period, the algorithm first estimates the total number N of jumping cycles included in thesynthetic force signal. From the corresponding frequency cluster, a set of multiple modelling parameters (such ascontinuous ASD S0T(f), regression coefficients r1 and r0 and shapes of jumping pulses) is selected randomly. At this point,the algorithm splits into two parallel actions: creating the shapes and durations of the N jumping cycles.

Shapes of jumping pulses Zi(t) are randomly assigned to each jumping cycle, so they become Zi(t), k=1, y, N. Thecorresponding unity normalised impulses Is,k will be used later as denominators of ratios a0k (Section 3.5).

Having the spectral characteristics of variations of jumping periods S0T(f)and the total number of cycles N, syntheticperiods T0k (Section 3.3) and the corresponding angular frequencies o0k (Section 3.6) can be calculated on the cycle-by-cyclebasis. The set of synthetic periods T0k are also used to calculate weight normalised impulses I0w,k using autoregressivemodel given by Eq. (9). These will later be numerators of ratios a0k.

The next step integrates everything generated so far to run the dynamic model (Section 3.6) and therefore to generatethe unity normalised jumping pulses of a kind shown in Fig. 14. Finally, these will be transformed into a synthetic force

0

2000

1500

1000

500

Forc

e [N

]Fo

rce

[N]

2500

0 5 10 15 20 25 30

Time [s]

0 5 10 15 20 25 30

Time [s]

0

2000

1500

1000

500

2500

Fig. 21. Examples of two synthetic force time histories generated by the model for jumping at 2 Hz. The signals correspond to two different individual

jumpers.

0 2 4 0

Frequency [Hz]

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

1.4

1.2

1

0.8

0.6

0.4

0

0.2

Four

ier

ampl

itude

s/bo

dyw

eigh

t [-]

Four

ier

ampl

itude

s/bo

dyw

eigh

t [-]

6

0 2 4 0

Frequency [Hz]

6

Fig. 22. Fourier amplitude spectra of the synthetic GRFs given in Fig. 21.

1.4

1.2

1

0.8

0.6

0.4

0.2

00 1 2 3 4 5 6 7 8 9 10

Frequency [Hz]

Four

ier

ampl

itude

spec

tra

/bo

dyw

eigh

t [-]

1.4

1.2

1

0.8

0.6

0.4

0.2

00 1 2 3 4 5 6 7 8 9 10

Frequency [Hz]

Four

ier

ampl

itude

spec

tra/

body

wei

ght [

-]

Fig. 23. Cluster at 1.5 Hz—Fourier amplitude spectra of all (a) measured forces and (b) their synthetic counterparts.



time history after they have been scaled by ratios a0k ¼ I0w,k=Is,k and body weight. As a random parameter, the body weightcan be generated using statistical models available elsewhere [33].

Fig. 21 shows examples of the signals generated when the model is run twice in a row for the jumping rate 2 Hz lasting30 s. Each of the signals corresponds to a unique individual jumper due to the inherent randomness of the modellingparameters. A visual comparison of the two signals provides convincing evidence that the model can account for the inter-subject variability in the pulse shapes and force amplitudes. On the other hand, the ability of the model to generate

0 1 2 3 4 5 6 7 8 9 10

Frequency [Hz]

0 1 2 3 4 5 6 7 8 9 10

Frequency [Hz]

1.6

1.2

0.8

0.4

0

Four

ier

ampl

itude

spec

tra/

body

wei

ght[

-]

1.8

1.4

1

0.2

0.6

1.2

0.8

0.4

0

Four

ier

ampl

itude

spec

tra/

body

wei

ght[

-]

1.8

1.4

1

0.2

1.6

0.6

Fig. 24. Cluster at 2 Hz—Fourier amplitude spectra of all (a) measured forces and (b) their synthetic counterparts.

1.4

1.2

1

0.8

0.6

0.4

0.2

00 1 2 3 4 5 6 7 8 9 10

Frequency [Hz]

0 1 2 3 4 5 6 7 8 9 10

Frequency [Hz]

Four

ier

ampl

itude

spec

tra/

body

wei

ght [

-]

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Four

ier

ampl

itude

spec

tra/

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wei

ght [

-]

Fig. 25. Cluster at 2.5 Hz—Fourier amplitude spectra of all (a) measured forces and (b) their synthetic counterparts.


different degrees of variability in the periods of successive jumping cycles for different persons becomes more obviousfrom comparison between the corresponding frequency spectra (Fig. 22). The broader spread of energy around dominantharmonics (i.e. integer multiples of 2 Hz) in Fig. 22b relative to Fig. 22a indicates that the first ‘virtual’ jumper (Fig. 22a)varied the periods less than the second one (Fig. 22b). This was also observed in the actual test data as demonstrated in thenext section.

6. Model verification

The modelling strategy proposed in this paper was validated for each of the 15 frequency clusters. This was done bycomparing between standard Fourier amplitude spectra of all measured forces in a cluster and their synthetic counterparts.

Figs. 23–25 illustrate results for three clusters at 1.5, 2 and 2.5 Hz, representing slow, moderate and fast tempos,respectively. For the first four dominant harmonics the relative errors between the average measured and averagesynthetic spectra for each cluster are within the range of 73%. Moreover, the relative error in the area under the graph ofthe average spectra is less than 7%. All this indicates good match in the frequency content between the measured andsynthesised GRF signals. Therefore, synthetic forces generated by the model can be utilised in vibration serviceabilityassessment of civil engineering assembly structures, such as grandstands, spectator galleries, footbridges and concert orgym floors, to estimate realistically vibration response due to people jumping.

7. Conclusions

This paper presents a new mathematical model used to generate near-periodic synthetic jumping force signals withspecified jumping rate and morphology. Similar to modelling the near-periodic human heart beats, the near-periodicnature of the jumping force is modelled using closed-loop trajectories throughout 3D space (r, y, z) around a circle of unitradius in (r, y) plane. Each revolution on this circle corresponds to the period of one jumping cycle. The trajectory replicatesthe size and shape of the measured jumping pulses via a sum of Gaussian exponentials. This modelling strategy canrepresent temporal and spectral features of the real human vertical jumping loading more effectively than theconventional half-sine pulses and Fourier series approach yielding more reliable predictions of dynamic structuralresponse due to people jumping. The proposed Gaussian fit, coupled with equations of circular motion, has the followingconsiderable advantages:

1)
A set of Gaussian bell functions in which centres are placed in each sample of measured pulses can fit exactly any pulseshape. This includes a lack of symmetry, double peak patterns and local irregularities yielding high frequencycomponents, as opposed to the symmetric and smooth half-sine and cosine-squared pulses which can reflect only lowfrequency content in the corresponding Fourier amplitude spectra.
2)
Variations of the jump-by-jump intervals can be included by varying the angular frequency for consecutive revolutionsaround the unit circle. For each revolution, it is also possible to change the pulse shape more effectively than using theconventional Fourier series approach. As a result, the amplitude Fourier spectrum of corresponding synthetic jumpingsignal becomes a narrow band random phenomenon showing the leaking of energy in the vicinity of dominant Fourierharmonics.
3)
Impulses and amplitudes of the synthetic jumping force signal can be changed on a jump-by-jump basis in a mannerwhich allows the model to simulate ‘smooth’ energy transfer between consecutive jumps, as it is measured in reality.
Numerous jumping force records generated by different individuals under a range of jumping frequencies resulted in acomprehensive database of jumping forces for the general human population. The database was used to develop andcalibrate a new generation of stochastic models of jumping loading for individuals.

This framework can be extended further to stochastic jumping loads due to groups and crowds. At the moment,individual forces can be summed with random phase lags as suggested elsewhere [34–36]. However, there are indicationsthat this is not what is happening in reality and more research into synchronisation between people jumping is needed.This presents an opportunity to enhance the vibration serviceability assessment of civil engineering structures occupiedand dynamically excited by humans such as grandstands, footbridges, floors and staircases.

Acknowledgements

The authors would like to acknowledge the financial support provided by the UK Engineering and Physical SciencesResearch Council (EPSRC) for grant reference EP/E018734/1 (‘Human Walking and Running Forces: Novel ExperimentalCharacterisation and Application in Civil Engineering Dynamics’) and GR/T03017/01 (‘Stochastic Approach toHuman–Structure Dynamic Interaction’).


References

[1] SCOSS. (2001) Structural Safety 2000–02: Thirteenth Report of the Standing Committee on Structural Safety. The Standing Committee on StructuralSafety, The Institution of Structural Engineers, London, UK.

[2] BS 6399-1: 1996 Incorporating Amendment No. 1. Code of Practice for Dead and Imposed Loads. British Standards Institution, London, UK, 2002.[3] Canadian Commission on Building and Fire Codes, in: User’s Guide—NBC 2005: Structural Commentaries (Part 4 of Division B), National Research

Council Canada, Institute for Research in Construction, Ottawa, Canada, 2006.[4] S. Zivanovic, A. Pavic, P. Reynolds, Vibration serviceability of footbridges under human-induced excitation: a literature review, Journal of Sound and

Vibration 279 (2005) 1–74.[5] D. Rogers, Two more ‘wobbly’ stands. Construction News, August 17, 2000.[6] D. Parker, Rock fans uncover town hall floor faults. New Civil Engineer 20 November, 2003.[7] J.H.H. Sim, Human–structure interaction in cantilever stands, Ph.D. Thesis, Faculty of Engineering, The University of Oxford, UK, 2006.[8] J.H.H. Sim, A. Blakeborough, M. Williams, Statistical model of crowd jumping loads, ASCE Journal of Structural Engineering 134 (12) (2008) 1852–1861.[9] H. Bachmann, A. Petlove, H. Rainer, Dynamic forces from rhythmical human body motions, Vibration Problems in Structures: Practical Guidelines,

Appendix G, Birkhauser, Basel, Switzerland, 1995.[10] J.M.W. Brownjohn, A. Pavic, P. Omenzetter, A spectral density approach for modelling continuous vertical forces on pedestrian structures due to

walking, Canadian Journal of Civil Engineering 31 (2004) 65–77.[11] V. Racic, A. Pavic, Mathematical model to generate asymmetric pulses due to human jumping, ASCE Journal of Engineering Mechanics 135 (10)

(2009) 1206–1211.[12] D. Ginty, J.M. Dervent, T. Ji, The frequency ranges of dance-type loads, The Structural Engineer 79 (6) (2001) 27–31.[13] P.E. McSharry, G.D. Clifford, L. Tarassenko, L.A. Smith, A dynamical model for generating synthetic electrocardiogram signals, IEEE Transactions of

Biomedical Engineering 50 (2003) 289–294.[14] V. Racic, A. Pavic, Mathematical model to generate near-periodic human jumping force signals, Mechanical Systems and Signal Processing 24 (2010)

138–152, doi:10.1016/j.ymssp.2009.07.001.[15] H. Bachmann, W. Ammann, in: Vibrations in Structures Induced by Man and Machines, IABSE–AIPC–IVBH, Zurich, Switzerland, 1987.[16] B.R. Ellis, T. Ji, in: The Response of Structures to Dynamic Crowd Loads, Digest 426, 2004 ed, British Standards Institution, London, UK, 2004.[17] B.R. Ellis, T. Ji, in: The Response of Structures to Dynamic Crowd Loads, Digest 426, British Standards Institution, London, UK, 1997.[18] S. Zivanovic, A. Pavic, P. Reynolds, Probability-based prediction of multi-mode vibration response to walking excitation, Engineering Structures 29

(6) (2007) 942–954.[19] IStructE/DCLG/DCMS Working Group, in: Dynamic Performance Requirements for Permanent Grandstands Subject to Crowd Action:

Recommendations for Management, Design and Assessment, The Institution of Structural Engineers, The Department for Communities and LocalGovernment and The Department for Culture Media and Sport, London, 2008.

[20] B.R. Ellis, T. Ji, Loads generated by jumping crowds: numerical modelling, Structural Engineer 82 (17) (2004) 35–40.[21] M. Kasperski and E. Agu, Prediction of crowd-induced vibrations via simulation, in: Proceedings of 23rd International Modal Analysis Conference

(IMAC XXIII), 31 January–3 February 2005, Orlando, FL, USA.[22] J.G. Parkhouse, D.J. Ewins, Crowd-induced rhythmic loading, Structures and Buildings 159 (2006) 247–259.[23] C.T. Farley, R. Blickhan, J. Saito, C.R. Taylor, Hopping frequency in humans: a test of how springs set stride frequency in bouncing gaits, Journal of

Applied Physiology 71 (1991) 2127–2132.[24] S.M. Pandit, S.-M. Wu, in: Time Series and System Analysis with Applications, Wiley, New York, USA, 1983.[25] D.E. Newland, in: An Introduction to Random Vibrations, Spectral and Wavelet Analysis, 3rd ed, Pearson Education Limited, Harlow, Essex, UK, 1993.[26] D.M. Bates, D.G. Watts, in: Nonlinear Regression and its Applications, Wiley, New York, 1998.[27] D.C. Montgomery, G.C. Runger, in: Applied Statistics and Probability for Engineers, 2nd ed, John Wiley & Sons, Inc., New York, NY, 1999.[28] R. Rodano, R. Squadrone, Stability of selected lower limb joint kinetic parameters during vertical jump, Journal of Applied Biomechanics 18 (2002)

83–89.[29] AMTI Product Manuals, Advanced Mechanical Technology, Watertown, USA, 2007.[30] J.D. Littler, Frequencies of synchronised human loading from jumping and stamping, The Structural Engineer (2003) 27–35.[31] J.G. Parkhouse and D.J. Ewins, Vertical dynamic loading produced by people moving to a beat, in: Proceedings of ISMA2004, Leuven, May 28–31,

2004.[32] MathWorks (2009) /www.mathworks.comS.[33] M. Hermanussen, H. Danker-Hopfe, G.W. Weber, Body weight and the shape of the natural distribution of weight, in very large samples of German,

Austrian and Norwegian conscripts, International Journal of Obesity 25 (2001) 1550–1553.[34] A. Ebrahimpour, R.L. Sack, Modelling dynamic occupant loads, Journal of Structural Engineering 115 (6) (1989) 1476–1496.[35] T. Ji and B.R. Ellis, Evaluation of dynamic crowd effects for dance loads, International Association for Bridge and Structural Engineering (IABSE)

Symposium: Structural Serviceability of Buildings, Goteburg, Sweden, 1993.[36] M. Willford, An investigation into crowd-induced vertical dynamic loads using available measurements, Structural Engineer 79 (12) (2001) 21–25.

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