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Parameter identification of spring-mass-damper model for bouncing people
Journal of Sound and Vibration 456 (2019) 13e29
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Parameter identification of spring-mass-damper model for bouncing people Haoqi Wang a, b, Jun Chen a, *, Tomonori Nagayama b a b
College of Civil Engineering, Tongji University, Shanghai, 200092, China Department of Civil Engineering, The University of Tokyo, Tokyo, 113-8656, Japan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 October 2018 Received in revised form 18 April 2019 Accepted 14 May 2019 Available online 18 May 2019
Bouncing, which refers to the up-and-down movement of the human body with both feet remaining on the ground, is one of the most common human-induced dynamic loads for civil structures such as sports stadiums and concert halls. Although researchers have focused on the modeling of bouncing-induced loads, studies on the excitation source (the bouncing person) have not gained much attention. The spring-mass-damper model with a pair of internal biomechanical forces is the simplest model to represent a bouncing person. The model parameters, such as stiffness and damping, though very important for application and for HSI (Human Structure Interaction) analysis, have been rarely reported. This study utilized the particle filter (a step-by-step system identification method) to identify human SMD (Spring-Mass-Damper) model parameters from measured bouncing forces obtained by a wireless insole system. A total of 173 continuous bouncing force signals were recorded and divided into 6800 bouncing cycles, from which the SMD model parameters were identified. The results indicated that human natural frequency and high-order biomechanical load factors have a linear trend against the bouncing frequency, while the damping ratio and first-order BLF (Biomechanical Load Factor) share a parabolic relation. For each model parameter, a skew normal distribution was fitted, and the fitting parameters were found to be dependent on the bouncing frequency range. The influence of HSI on the model parameters was also experimentally investigated, indicating that a vibrating surface leads to a lower human natural frequency and a higher damping ratio value. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Bouncing Spring-mass-damper model Biomechanical load factor Parameter identification Particle filter Human-structure interaction
1. Introduction Human-induced vibrations of long-span civil structures including floors, cantilever stands, and pedestrian bridges have attracted the attention of researchers for many years [1,2]. Most human activities are periodic or near-periodic. Therefore, if the dominant frequency of the activity falls into a range close to the natural frequencies of the structure, excessive resonantlike vibrations may occur, leading to serviceability problems or even safety problems in some extreme cases [3e5]. For instance, in 2003, Leeds Town Hall in England was evacuated after only 30 min of a rock concert when 1000 fans induced large vibrations, causing visible cracks on the floor they occupied [6]. In 2011, a group of 17 people caused a 39-story building in
* Corresponding author. E-mail address: [email protected] (J. Chen). https://doi.org/10.1016/j.jsv.2019.05.034 0022-460X/© 2019 Elsevier Ltd. All rights reserved.
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South Korea to vibrate for 10 min by doing aerobics [7]. More recently, in 2013, an “earthquake” was recorded at a seismographic station near a sports stadium where fans were celebrating their team's victory [8]. These cases, among others, indicate that human-induced excitation can give rise to severe problems in civil structures, making it important to investigate the nature of human-induced excitation. Among many types of human activities, the bouncing motion, which refers to the up-and-down movement of a human body with both feet remaining on the ground [5], is more realistic and more likely to occur during sports or musical events than other types of motion. This is because of its lower consumption of energy and the easiness of coordination among multiple individuals [9,10]. Although jumping motions usually generate higher amplitudes of excitation on human-occupied structures [11,12], bouncing effects can last much longer, giving enough time for a structure to reach a high level of vibration. Moreover, it was reported that the bouncing frequencies can be as high as 5 Hz [9], making the occurrence of resonant vibration more probable for many large-span structures. For the reasons stated above, the bouncing load has been studied by many researchers [9,13e20]. A correct and reliable model reflecting load properties is a prerequisite for the study of bouncing-induced vibration. The Fourier-harmonic-based model, despite its drawbacks [21e24], is the most studied and widely accepted mathematical model because of the nearperiodic feature of the bouncing load [10,16,21]. Other efforts have been made regarding the variability of bouncing loads from the perspective of stochastic excitation [11,25,26]. However, most existing bouncing load research aimed to propose a modeling method that describes the ground reaction force (GRF) curves most similar to the recorded ones. However, these studies failed to reflect the properties of the excitation source, i.e., the bouncing person. In fact, it was established that human beings can change the structural dynamic properties of the buildings they occupy through various activities owing to their individual mass, damping, and stiffness effects [27e31]. Therefore, structural vibrations will change owing to HSI. This justifies the importance of studying human vibration-causing properties and their effects on structural vibration. Although some researchers addressed this issue in cases of human walking [32e39], studies about the dynamic properties of bouncing people (as in dancing or response to music) are meager. To the best of the authors’ knowledge, the only attempt to study human bouncing parameters was made by Dougill et al. [40], where the bouncing person was modeled as a spring-mass-damper (SMD) system with a pair of biomechanical forces exciting the human system, while the GRF was related to the human system output. Through a numerical analysis, the authors proposed a mean natural frequency of 2.3 Hz and a damping ratio of 0.25 for a bouncing person. This paper introduces a method to obtain the SMD parameters of a bouncing person from recorded GRF signals. A particle filter method that estimates the system state in a step-by-step manner [41] is adopted. The particle filter method is applied to each cycle of bouncing-induced GRFs recorded in an experiment to identify human SMD parameters and the unknown biomechanical force input simultaneously, assuming that the parameters remain constant for each bouncing cycle. Therefore, the human body responses can be determined by biomechanical force properties and human SMD parameters. The influence from HSI is also investigated. The structure of this paper is as follows. In Section 2, an experiment to collect individual human bouncing load signals using sensor-equipped insoles is described, and a comparison between signals measured from insoles and from force plates is provided. In Section 3, the procedure of the proposed method for SMD parameter identification is explained. Section 4 provides the identification results for one individual, and Section 5 provides a statistical analysis on all of the experimental data. In Section 6, the effect of HSI is investigated experimentally. Finally, some conclusions are drawn in Section 7. 2. Experiments on individual bouncing loads 2.1. Experimental setup To identify human parameters correctly, real bouncing force signals are indispensable. Experiments aimed at obtaining real GRFs generated by bouncing motions were conducted at Tongji University in China. The test participants were equipped with shoe insoles that were tracked by the Novel Pedar system (Novel Co., Germany) to measure continuous GRF signals. The insoles were connected to a receiver attached to the bouncing person, and the signals were wirelessly transmitted to a nearby data acquisition PC through the Bluetooth technique at a sampling frequency of 100 Hz. Twenty-five individuals participated in the experimental tests. Each participant was asked to bounce on a rigid floor at six different frequencies guided by a metronome. The provided frequencies were from 1.5 to 3.5 Hz with an increment of 0.4 Hz. In addition to these guided bouncing frequencies, one more bouncing frequency was freely chosen by participants according to their own preference. For each bouncing frequency, participants were asked to keep bouncing for approximately 30 s. In total, 173 time histories of continuous bouncing signals were obtained in this experiment. The selection of the test participants and the test protocol satisfied the requirements of the Medical Ethics Committee of Tongji University. 2.2. Comparison of signals from insole system and force plates Although the validity and accuracy of the insole system, which has been widely adopted for biomechanical and biomedical uses, are already well acknowledged [42e44], a validation test specially designed for the bouncing GRF measurements in this study was carried out (see Fig. 1). A test subject wearing the insole system was asked to bounce on a force plate (AMTI OR6-7) whose records were taken as references. The force plate had a sampling rate of 1000 Hz, and the signals recorded by the force
H. Wang et al. / Journal of Sound and Vibration 456 (2019) 13e29
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plate and the insole system were synchronized according to their correlation function. The test subject performed fourbouncing frequencies of 1.5, 1.75, 2.0, and 2.25 Hz. Typical recorded signals with their corresponding Fourier amplitude spectra are shown in Fig. 2 for comparison. Note that the GRFs are normalized to the body weight of the test subject. The signals recorded by the insole system coincide well with those from the force plate. However, a slight difference between the signals of the two systems can still be observed, which is explained by the influence of the footwear [11]. This is because the insole system is directly contacted with the feet, while the force plate measures the forces under the shoes. The influence from this difference is given in later sections. In the frequency domain, the main features in the amplitudes of the bouncing-induced loads are reflected up to at least 10 Hz, which covers the frequency range of interest for most cases. Therefore, the data obtained from the experimental tests is used as GRFs for the human parameter identification. 3. Theory of identification 3.1. Spring-mass-damper model of a bouncing person The SMD model is the simplest single degree-of-freedom model for a bouncing person, and concentrates all of the human mass at the center of mass (COM) of the body. A spring and a damper are attached to represent the mechanism of the human dynamic system, as shown in Fig. 3. Despite its simplicity, the SMD model can reflect the main dynamic properties of the human system, including the natural frequency and the damping ratio. Owing to the movement of human muscles, a pair of equal but opposite biomechanical forces is assumed to appear in the SMD model as the internal excitation of the human system [34]. The movement of the human mass is determined by the SMD model equation of motion as
Mh u€ þ Ch u_ þ Kh u ¼ Fbio ;
(1)
where Mh, Ch, and Kh are the static human whole-body mass, damping coefficient, and stiffness, respectively. Term Fbio is the biomechanical forces exciting the system, and u is the displacement of the human COM from the static equilibrium position. When divided by Mh at both sides of Eq. (1), the equation of motion of the SMD model becomes Eq. (2). The natural frequency and damping ratio of the human body are explicitly expressed in the equation as follows:
u€ þ 2xh uh u_ þ u2h u ¼ Fbio =Mh ;
(2)
in which xh is the damping ratio, and uh is the circular natural frequency that equals 2pfh, where fh is the human natural frequency. As indicated in Fig. 3, the GRF is equal to the summation of the spring force, damping force, biomechanical force, and the body static weight. Therefore, the following relation between the measured GRF and the movement of the body's COM is found:
u€ ¼ ðGRF Mh gÞ=Mh ;
(3)
Fig. 1. Experimental setup.
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Fig. 2. Typical recordings of validation test: (a) time history of 1.5 Hz, (b) spectrum of 1.5 Hz, (c) time history of 1.75 Hz, (d) spectrum of 1.75 Hz, (e) time history of 2.0 Hz, (f) spectrum of 2.0 Hz, (g) time history of 2.25 Hz, and (h) spectrum of 2.25 Hz.
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Fig. 3. SMD model of bouncing person with pair of biomechanical forces.
in which g is the gravitational acceleration. As shown in the typical recordings in Fig. 2, the GRF has a near-periodic feature. Therefore, according to Eq. (1), it is reasonable to assume that the biomechanical force Fbio is also near-periodic. A Fourier series model is then applied to each cycle of the biomechanical force normalized to the human body weight, as shown in Eq. (4).
Fbio =Mh ¼ a0 þ
n X ðai cosð2pifb tÞ þ bi sinð2pifb tÞÞ;
(4)
i¼1
where a0, ai, and bi are Fourier series coefficients, fb is the bouncing frequency, and n is the number of harmonics considered in the Fourier series model. In this paper, the value of n is chosen to be 4 because the amplitudes of harmonics higher than the fourth order are considered to be very low and are thus neglected. Similar to the definition of the dynamic load factor (DLF) [19], the biomechanical load factor (BLF) is defined as
BLFi ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2i þ b2i g:
(5)
3.2. Particle filter theory and its application to human parameter identification In this study, the human parameters and the BLFs are identified through a method known as a particle filter, which estimates the state of the system by using the measured data in a step-by-step manner. Because the particle filter method has been used in many other problems in civil engineering [45e49], and the details of a similar procedure were discussed in the human parameter identification for walking pedestrians [36], this method is only briefly reviewed herein. In the particle filter method, the system dynamic equation is written in state-space form as
xkþ1 ¼ fk ðxk Þ þ wðkÞ:
(6)
In general, xk is the vector representing the system state at time step k. The state transition function, which shows the relation between the state vector for two continuous steps of k and k þ 1, is represented by the function fk. Further, w(k) is the system error term following a known distribution. For the SMD model identification in this study, the unknown parameters (natural frequency, damping ratio, and biomechanical force) need to be included in the state vector with the system response terms.
_ fh ; xh ; a0 ; a1 ; b1 ; /; an ; bn T : X ¼ ½u; u; The state equation of the human SMD system is represented by
(7)
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X_ ¼ AX þ w;
(8)
T _ u€; 0; /; 0 X_ ¼ u;
(9)
where
and A is the state matrix, which can be derived from the equation of motion as follows:
2
0 6 u2 h 6 A¼6 6 0 4 « 0
1
2xh uh
0 0 0 0 0 1
3 0 0 / 0 0 cosð2pfb tÞ sinð2pfb tÞ / cosð2pnfb tÞ sinð2pnfb tÞ 7 7 7: / 0 7 5 « / 0
(10)
Eq. (8) is discretized using the Euler discretization method to obtain the relation between the system states of one time step and the next, giving
Xk ¼ Ad Xk1 þ wðkÞ;
(11)
in which
Ad ¼ I þ ADt;
(12)
where I is the identity matrix, and Dt is the reciprocal of the sampling frequency. At each time step, the system responses are included in the observation vector yk, which is related to the system state vector xk through the observation equation. The general form of the observation equation in the particle filter method is shown as
yk ¼ hk ðxk Þ þ vðkÞ;
(13)
in which v(k) is the observation error with a known distribution. In this study, the COM displacement, velocity, and acceleration responses are included in the observation vector represented by Yk as follows:
Y k ¼ ½uk ; u_ k ; u€k T :
(14)
The transition function hk can be derived from the SMD equation of motion as
Y k ¼ Cd Xk þ vðkÞ;
(15)
where Cd is the observation matrix, shown as
2
1 Cd ¼ 4 0 u2h
0 1
2xh uh
0 0 0
0 0 0 0 0 1 cosð2pfb tÞ
/ / sinð2pfb tÞ / cosð2pnfb tÞ
3 0 5: 0 sinð2pnfb tÞ
(16)
In the particle filter method, each particle represents a system state vector. When the number of particles becomes sufficiently large, the probability density function (PDF) of the system state of the current step is equivalently represented by these particles using a Monte Carlo expression [50,51]. At time step k - 1, each particle passes through the system equation shown in Eq. (11), giving a predicted value of the state vector for time step k. Observation vectors corresponding to the predicted state vectors are calculated following Eq. (15) and are then compared with the measurement made at time step k. A process known as resampling is conducted based on the fact that the particles giving observations closer to the measurement have a higher chance of being selected. After resampling, the new particles represent the updated PDF of the system state at time step k, from which an identified value can be extracted. This procedure is repeated in the next step until the end of the signal. The measured GRF is converted to COM acceleration responses through Eq. (3) with a known human body mass measured in advance. The COM velocity and displacement responses are obtained through integration and double integration from the acceleration, which is known to be sensitive to measurement noise. To reduce the integration error, the acceleration signal is applied to a high-pass filter with a cutoff frequency of 0.5 Hz, which does not affect the analysis results because the frequency range of interest is usually higher than 1 Hz for normal bouncing frequencies. In addition, as shown in later sections, the duration of the integrated signal is limited within the period of one bouncing cycle, which is considered to be short before large integration errors accumulate.
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Fig. 4. Converging process of human SMD model parameters within one cycle: (a) natural frequency, (b) damping ratio, (c) first-order BLF, (d) second-order BLF, and (e) third-order BLF.
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A total of 5000 particles are generated. The initial range of the uniform distributions for human natural frequency fh and damping ratio xh are 1e4 Hz and 0e1, respectively, while the range for the input Fourier parameters ai and bi in Eq. (7) are set as 0e0.4. These initial ranges are determined based on the reasonable range of human parameters [34]. The system error term w(k) is assumed to follow a Gaussian distribution with a standard deviation of around 1% of the nominal values of each quantity in order to prevent parameters from converging to a wrong value before reaching the true value [46]. The standard deviation of the observation error v(k) is taken as 10% of the RMS values of the corresponding responses. Note that the values of w(k) and v(k) are related to the system modeling error and measurement noises, and are empirically chosen in this study. Robustness was found against small changes of these values.
4. Identification results for one individual Before implementing a particle filter, the measured time history is split into cycles. Each cycle starts from one peak value recorded by the insoles to the next peak value, as shown in Fig. 2, whose duration is around 0.3e0.8 s depending on the bouncing frequency. The particle filtering process is then applied, giving estimates of the human natural frequency, damping ratio, and the biomechanical force parameters for each of these cycles. Owing to the time limitation of each cycle, it is possible that the time duration for one cycle is not sufficient to make all the parameters converge. Therefore, a repeating process is adopted by implementing the particle filter more times in this cycle, with the initial particle distribution replaced by that of the last step of the previous repetition [47,52]. This method is based on the idea that after one filtering process, the particle distribution becomes closer to the real distibution and represents the parameter's real value more accurately. The identification results for the SMD model parameters of an individual with a body mass of 57.4 kg bouncing at 2 Hz are shown in Fig. 4. Note that these results are only for one cycle of bouncing load extracted from the continuous signals recorded by the insoles. As shown in Fig. 4, after 4e5 repetitions of the particle filtering process, the human natural frequency converges at 1.61 Hz, while the damping ratio converges at 0.26. These figures show that if only one particle filtering process is applied, the parameter values still fluctuate. By implementing more repetitions, the estimation values will converge to a fixed value. Because multiple filtering is required, a parameter is considered to have converged if the estimated value given by one filtering process does not exceed ±5% of the value given by the previous filtering. As stated in Section 2, the forces measured by the insoles are used as a surrogate for the GRF. To quantify the influence from this assumption, the estimation procedure was also applied on the forces simultaneously measured by the force plates. The converging process is shown in Fig. 4, and the estimation values are listed in Table 1, showing that using signals measured by the insoles as the GRF does not result in large estimation errors. From the body mass information of this test subject, the stiffness and the damping coefficient are calculated to be 7212.1 Nm1 and 730.9 Nsm1, respectively, through the following Eqs. (17) and (18). Note that this paper assumes that Mh represents the entire body mass, which is same as in Refs. [33,34,37,39].
Table 1 Converging values of one typical cycle.
Force plates Insoles
Frequency
Damping Ratio
BLF1
BLF2
BLF3
1.776 Hz 1.784 Hz
0.586 0.568
0.125 0.121
0.215 0.212
0.064 0.060
Fig. 5. Typical time history of biomechanical force of one bouncing cycle.
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Fig. 6. Human SMD model parameter estimation of one test participant: (a) natural frequency, (b) damping ratio, (c) stiffness, (d) damping coefficient, (e) firstorder BLF, (f) second-order BLF, and (g) third-order BLF.
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Fig. 7. Results for all recorded bouncing cycles from all participants: (a) natural frequency, (b) damping ratio, (c) stiffness, (d) damping coefficient, (e) first-order BLF, (f) second-order BLF, and (g) third-order BLF.
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Fig. 8. Distribution of parameters and fitting: (a) natural frequency, (b) damping ratio, (c) stiffness, (d) damping coefficient, (e) first-order BLF, (f) second-order BLF, and (g) third-order BLF.
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Kh ¼ Mh ð2pfn Þ2 :
(17)
C h ¼ 2xMh ð2pfn Þ:
(18)
Once Mh, Kh, and Ch are known or identified, and the acceleration, velocity, and displacement of the COM are measured or integrated, the biomechanical force can be directly calculated from Eq. (1). For the case shown above, the time history of the biomechanical force is plotted in Fig. 5. For other cycles of this signal, as well as other bouncing frequency cases of this individual, the identified values of the human SMD parameters and their relation against the bouncing frequency are given in Fig. 6 (a)e(g). In Fig. 6 (a), an increasing trend for the natural frequency is clearly observed. In addition, when the bouncing frequency is low, the natural frequency values are limited to a small range, indicating that the bouncing process is quite stable and the human subject does not need to change their bouncing pattern frequently. However, to match the required high bouncing frequency, the human body needs to adjust their bouncing patterns through the expansion of muscles during the bouncing process. As a result, the estimated values of the natural frequency become more scattered in the high-bouncing-frequency range, where the large values indicate an increase in human stiffness caused by stressed muscles. In Fig. 6 (b), it is observed that most of the damping ratio values are in the range of 0.2e0.6. The relation against the bouncing frequency is not a simple linear trend. When the bouncing frequency is lower than around 3 Hz, a decreasing trend is observed for damping ratio values. For the cases higher than 3 Hz, the damping ratios increase with the bouncing frequency. Similar trends are also found for the stiffness and damping coefficient of the SMD model, as shown in Fig. 6 (c) and (d), because they have a close relation with the natural frequency and damping ratio through Eq. (17) and Eq. (18). The first-order BLF increases with the bouncing frequency, while a decreasing trend is found for the second- and third-order BLFs, as shown in Fig. 6 (e)e(g), respectively. 5. Statistical analysis The process described above was applied to all 173 records of bouncing GRFs, from which 6800 cycles were extracted, in order to identify the SMD parameters. The results of the human natural frequency, damping ratio, stiffness, damping coefficient, and first three orders of BLF are shown in Fig. 7 (a)e(g), respectively. To show the trend of each parameter against the bouncing frequency more clearly, the mean values for each bouncing frequency range are calculated and plotted in Fig. 7. These bouncing frequency ranges are 1.35e1.75 Hz, 1.75e2.15 Hz, 2.15e2.55 Hz, 2.55e2.95 Hz, 2.95e3.35 Hz, and 3.35e3.75 Hz, respectively. The natural frequency, stiffness, and first-order BLF values have an increasing trend as the bouncing frequency increases, while the second- and third-order BLF values show a decreasing trend. For the damping ratio and damping coefficients, the values first decrease and then increase after approximately 2.5 Hz. Notably, according to Fig. 7 (b) and (e), at approximately 2.5e3 Hz, the damping ratio reaches its lowest value while the highest value of the first-order BLF appears. The BLF represents the level of biomechanical force excitation; hence, the highest
Table 2 Fitted coefficients of probability density function. SMD parameter
PDF parameter
1.35e1.75 Hz
1.75e2.15 Hz
2.15e2.55 Hz
2.55e2.95 Hz
2.95e3.35 Hz
3.35e3.75 Hz
Natural frequency
m (Hz) s (Hz) a m s a m (Nm1) s (Nm1) a m (Nsm1) s (Nsm1) a m s a m s a m s a
1.7344 0.3671 3.5150 0.1229 0.1609 3.9977 6717.5 2714.6 1.3713 116.2537 192.8070 3.9194 0.1968 0.1005 2.6126 0.1549 0.1317 0.7517 0.0116 0.0609 8.0118
2.0981 0.3283 3.8383 0.0963 0.1250 5.0906 10011.5 3386.8 1.5475 122.7600 175.0183 5.1436 0.1808 0.0937 2.3134 0.1046 0.1306 2.8725 0.0140 0.0503 2.8241
2.4314 0.4314 4.1657 0.0781 0.1477 4.9991 13458.9 5014.8 2.430 111.9995 207.2037 4.5361 0.1626 0.1502 3.6431 0.1189 0.1021 1.6303 0.0125 0.0530 8.9577
2.8634 0.7717 8.4628 0.0811 0.1378 7.1938 18118.9 8959.3 3.5828 135.3333 207.7530 4.3247 0.1500 0.1984 4.6241 0.1060 0.0921 2.2067 0.0119 0.0504 4.7477
3.2391 1.2011 14.5146 0.0875 0.1524 8.2652 17581.1 9620.7 0.8246 155.5664 234.4910 4.1732 0.1404 0.2385 2.9479 0.1092 0.0767 1.0348 0.0045 0.0370 7.5927
3.5414 1.3501 16.2412 0.0905 0.1626 5.9160 19718.3 10266.7 0.6680 184.5520 251.6296 3.5600 0.1585 0.2444 3.0493 0.1184 0.0750 1.0399 0.0052 0.0265 6.4426
Damping ratio
Stiffness
Damping coefficient
First-order BLF
Second-order BLF
Third-order BLF
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Fig. 9. Experimental plate: (a) plate model and (b) experimental setup.
Table 3 Modal parameters of concrete plate. Mode
1
2
3
4
5
6
Natural Frequency (Hz) Damping Ratio Modal Mass (kg)
3.48 0.37 8583
6.14 0.51 2587
6.74 0.61 9625
14.12 0.91 2423
15.19 0.67 2898
18.09 1.49 4900
first-order BLF value indicates that the human body has a high “willingness” to bounce. At the same frequency, the human body tends to change its bouncing patterns to achieve a low damping ratio, aiming at a more efficient bouncing motion to avoid energy dissipation. This phenomenon is in accordance with a series of research findings in the field of biomechanics, claiming that bouncing people have a “preferred frequency” at which the metabolic energy cost is at a minimum [53e55]. On the other hand, the increasing trend of the natural frequency is explained by the fact that as the bouncing frequency becomes higher, the tension in the human muscles increases, leading to an increased human stiffness value. Although the trend against the bouncing frequency of each parameter's mean value is clearly observed, the results are found to be very scattered owing to the randomness of the bouncing-induced load. In addition, the level of scatter for each parameter is shown to be dependent on the bouncing frequency. For each frequency range, the probability density of each parameter is calculated from the identified values and is shown in Fig. 8 (a)e(g), respectively. According to the shape of the distribution, a skew normal distribution is adopted. The probability density function f(x) of the skew normal distribution is shown in Eq. (19).
Fig. 10. Typical midpoint acceleration.
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ðxmÞ2 2 f ðxÞ ¼ pffiffiffiffiffiffie 2s2 s 2p
aZðxs mÞ ∞
1 t2 pffiffiffiffiffiffie 2 dt; 2p
(19)
where m, s, and a are the location, scale, and shape coefficients of the distribution, respectively, whose values are determined through a genetic algorithm. The values of the fitted coefficients are given in Table 2, and the fitting curves are shown together in Fig. 8.
Fig. 11. Effect of HSI on SMD model parameters: (a) natural frequency, (b) damping ratio, (c) first-order BLF, (d) second-order BLF, and (e) third-order BLF.
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Table 4 Comparison of estimated human SMD parameters on vibrating plate and on rigid floor.
On rigid floor On vibrating plate Difference
Frequency
Damping ratio
BLF1
BLF2
BLF3
1.81 Hz 1.49 Hz 17.7%
0.33 0.41 þ24.2%
0.241 0.348 þ44.4%
0.335 0.267 20.3%
0.082 0.046 43.9%
6. Investigation of influence of human-structure interaction In the previous sections, the results were based on data measured from experiments conducted on a rigid floor. However, there are cases in which people bounce on a vibrating surface, where HSI may have an effect on the human SMD parameters. This section investigates the influence of HSI on the human SMD model parameters. A pre-stressed concrete plate (10 m long and 6 m wide) with a weight of 16.5 t, as shown in Fig. 9, was constructed for a multipurpose experiment in the structural laboratory of Tongji University. The modal parameters of this plate were obtained through a modal test using an instrumented hammer. The plate's natural frequencies and damping ratios of the first six modes are listed in Table 3. A test participant wearing the insole system was asked to bounce at the midpoint of the plate at a frequency of 1.75 Hz. This bouncing frequency was chosen according to the fundamental frequency of the plate, which is around twice the bouncing frequency, in order to excite near-resonance vibration through the second harmonic of the bouncing load. The typical acceleration response at the midpoint of the plate under bouncing excitation is shown in Fig. 10. The proposed estimation method described in Section 4 was conducted for the bouncing force signals recorded when the test participant was bouncing on the vibrating plate. The estimations of the human SMD parameters are shown in Fig. 11 (a)e(e) for the natural frequency, damping ratio, and BLFs, together with the results from the same test participant bouncing on the rigid floor. Because these results are extracted from the experiment of the same test participant within a short experimental period, these figures clearly show the influence of a vibrating surface on human SMD parameters. A decrease in the human natural frequency and an increase in the human damping ratio is found. Possible reasons for these phenomena stem from people's reactions when they feel the surface vibrating, including relaxing their muscles or changing posture to reduce the uncomfortableness from the vibrating surface. The estimation values of each parameter on the vibrating plate and on the rigid floor are listed in Table 4 with the corresponding differences. 7. Summary and conclusions The human bouncing parameters of an SMD model were studied. Experiments using wireless force insoles were conducted to obtain bouncing-induced GRFs, which were then converted to the acceleration response of COM. A time-domain method known as the particle filter method was applied to the measured data to identify the human natural frequency, damping ratio, and BLFs simultaneously. From the estimation results, a linear trend was found for human natural frequency against the bouncing frequency, while a parabolic curve was fitted with human damping and first-order BLF. For each SMD parameter, a skew normal distribution was fitted. The effect from HSI was also investigated. From this study, the following conclusions were drawn: (1) The accuracy of the wireless insole system was found to be acceptable from the perspective of human parameter identification based on the GRF measurements of human bouncing. (2) The particle filter method with a repeating process within each bouncing cycle was capable of extracting human model parameters. (3) The existence of a “preferred frequency” at approximately 2.5e3 Hz from a biomechanical perspective was found. (4) The vibrating surface tended to lower the human natural frequency and increase the human damping ratio through HSI.
Acknowledgement This work was supported by National Natural Science Foundation of China (51778465, U1711264). Moreover, the authors would like to thank all test subjects for participating in the project making the data collection possible. References [1] S. Zivanovi c, A. Pavic, P. Reynolds, Vibration serviceability of footbridges under human-induced excitation: a literature review, J. Sound Vib. 279 (2005) 1e74, https://doi.org/10.1016/j.jsv.2004.01.019.
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[2] V. Racic, A. Pavic, J.M.W. Brownjohn, Experimental identification and analytical modelling of human walking forces: literature review, J. Sound Vib. 326 (2009) 1e49, https://doi.org/10.1016/j.jsv.2009.04.020. [3] P. Dallard, A.J. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R.M. Ridsdill Smith, M. Willford, The London millennium footbridge, in: Struct. Eng., 2001, pp. 17e33. [4] J.M.W. Brownjohn, S. Zivanovic, A. Pavic, Crowd Dynamic Loading on Footbridges, Footbridge Vibration Design, CRC Press, 2009, pp. 145e176. [5] C.A. Jones, P. Reynolds, A. Pavic, Vibration serviceability of stadia structures subjected to dynamic crowd loads: a literature review, J. Sound Vib. 330 (2011) 1531e1566, https://doi.org/10.1016/j.jsv.2010.10.032. [6] D. Parker, Rock fans uncover townhall floor faults, N. Civ. Eng. (2003), 20.11.2003. [7] S.H. Lee, K.K. Lee, S.S. Woo, S.H. Cho, Global vertical mode vibrations due to human group rhythmic movement in a 39 story building structure, Eng. Struct. 57 (2013) 296e305, https://doi.org/10.1016/j.engstruct.2013.09.035. [8] J. Lear, E. Payne, Rowdy Seattle Seahawks fans register as earthquake during Monday night football, Cable News Network (CNN) (2013), 3.12.2013. [9] S. Yao, J.R. Wright, A. Pavic, P. Reynolds, Experimental study of human-induced dynamic forces due to jumping on a perceptibly moving structure, J. Sound Vib. 296 (2006) 150e165, https://doi.org/10.1016/j.jsv.2006.02.018. [10] IStructE/DCLG/DCMS Working Group, Dynamic Performance Requirements for Permanent Grandstands Subject to Crowd Action: Recommendations for Management, Design and Assessment, The Institution of Structural Engineers, London, UK, 2008. [11] V. Racic, A. Pavic, Mathematical model to generate asymmetric pulses due to human jumping, ASCE J. Eng. Mech. 135 (2009) 1206e1211, https://doi. org/10.1061/(ASCE)EM.1943-7889.0000044. [12] V. Racic, A. Pavic, Mathematical model to generate near-periodic human jumping force signals, Mech. Syst. Signal Process. 24 (2010) 138e152, https:// doi.org/10.1016/j.ymssp.2009.07.001. [13] A. Ebrahimpour, R.L. Sack, Modeling dynamic occupant loads, J. Struct. Eng. 115 (1989) 1476e1496, https://doi.org/10.1061/(ASCE)0733-9445(1989) 115:6(1476). [14] S. Yao, J.R. Wright, A. Pavic, P. Reynolds, Forces generated when bouncing or jumping on a flexible structure, Proc. Int. Conf. Noise Vib. (2002) 563e572. [15] S. Yao, J.R. Wright, A. Pavic, P. Reynolds, Experimental study of human-induced dynamic forces due to jumping on a perceptibly moving structure, J. Sound Vib. 296 (2006) 150e165, https://doi.org/10.1016/j.jsv.2006.02.018. [16] J.G. Parkhouse, D.J. Ewins, Crowd-induced rhythmic loading, Struct. Build. 159 (2006) 247e259, https://doi.org/10.1680/stbu.2006.159.5.247. [17] E. Agu, M. Kasperski, A statistical approach to loads induced by bobbing, 7th Eur. Conf. Struct. Dyn. EURODYN (2008), 2008. [18] A.J. Comer, A. Blakeborough, M.S. Williams, Rhythmic crowd bobbing on a grandstand simulator, J. Sound Vib. 332 (2013) 442e454, https://doi.org/10. 1016/j.jsv.2012.08.012. [19] V. Racic, J. Chen, Data-driven generator of stochastic dynamic loading due to people bouncing, Comput. Struct. 158 (2015) 240e250, https://doi.org/10. 1016/j.compstruc.2015.04.013. [20] J. Chen, L. Wang, V. Racic, J. Lou, Acceleration response spectrum for prediction of structural vibration due to individual bouncing, Mech. Syst. Signal Process. 76e77 (2016) 394e408, https://doi.org/10.1016/j.ymssp.2016.02.032. [21] J.H. Rainer, G. Pernica, D.E. Allen, Dynamic loading and response of footbridges, Can. J. Civ. Eng. 15 (1988) 335e347, https://doi.org/10.1139/l88-007. [22] J.M. Brownjohn, A. Pavic, P. Omenzetter, A spectral density approach for modelling continuous vertical forces on pedestrian structures due to walking, Can. J. Civ. Eng. 31 (2004) 65e77, https://doi.org/10.1139/l03-072. [23] V. Racic, J.M.W. Brownjohn, Mathematical modelling of random narrow band lateral excitation of footbridges due to pedestrians walking, Comput. Struct. 90e91 (2012) 116e130, https://doi.org/10.1016/j.compstruc.2011.10.002. [24] V. Racic, J.B. Morin, Data-driven modelling of vertical dynamic excitation of bridges induced by people running, Mech. Syst. Signal Process. 43 (2014) 153e170, https://doi.org/10.1016/j.ymssp.2013.10.006. [25] J. Sim, A. Blakeborough, M.S. Williams, G. Parkhouse, Statistical model of crowd jumping loads, ASCE J. Struct. Engieering. 134 (2008) 1852e1861, https://doi.org/10.1061/(ASCE)0733-9445(2008)134:12(1852). [26] V. Racic, A. Pavic, Stochastic approach to modelling of near-periodic jumping loads, Mech. Syst. Signal Process. 24 (2010) 3037e3059, https://doi.org/ 10.1016/j.ymssp.2010.05.019. [27] M.J. Griffin, J. Erdreich, Handbook of human vibration, J. Acoust. Soc. Am. 90 (1991) 2213, https://doi.org/10.1121/1.401606. [28] T. Ji, B.R. Ellis, Human-structure interaction in vertical vibrations, Proc. ICE - Struct. Build. 122 (1997) 1e9, https://doi.org/10.1680/istbu.1997.29162. [29] A. Cappellini, S. Manzoni, M. Vanali, Quantification of damping effect of humans on lightly damped staircases, in: Top. Dyn. Civ. Struct. - Proc. 31st IMAC, A Conf. Struct. Dyn. 2013, 2013, pp. 453e460, https://doi.org/10.1007/978-1-4614-6555-3_49. [30] K.A. Salyards, N.C. Noss, Experimental evaluation of the influence of human-structure interaction for vibration serviceability, J. Perform. Constr. Facil. 28 (2014) 458e465, https://doi.org/10.1061/(ASCE)CF.1943-5509.0000436. rez, A. Lorenzana, Walking model to simulate interaction effects between pedestrians and lively structures, J. Eng. Mech. 143 (2017) 1e9, [31] M. Cacho-Pe https://doi.org/10.1061/(ASCE)EM.1943-7889.0001326. [32] K. Van Nimmen, G. Lombaert, G. De Roeck, P. Van den Broeck, The impact of vertical human-structure interaction on the response of footbridges to pedestrian excitation, J. Sound Vib. 402 (2017) 104e121. [33] E. Shahabpoor, A. Pavic, V. Racic, Identification of mass-spring-damper model of walking humans, Structures 5 (2016) 233e246, https://doi.org/10. 1016/j.istruc.2015.12.001. [34] M. Zhang, C.T. Georgakis, J. Chen, Biomechanically excited SMD model of a walking pedestrian, J. Bridg. Eng. (2016) C4016003, https://doi.org/10.1061/ (ASCE)BE.1943-5592.0000910. [35] E. Shahabpoor, A. Pavic, V. Racic, Structural vibration serviceability: new design framework featuring human-structure interaction, Eng. Struct. 136 (2017) 295e311, https://doi.org/10.1016/j.engstruct.2017.01.030. [36] H. Wang, J. Chen, J.M.W. Brownjohn, Parameter identification of pedestrian's spring-mass-damper model by ground reaction force records through a particle filter approach, J. Sound Vib. 411 (2017) 409e421, https://doi.org/10.1016/j.jsv.2017.09.020. [37] M. Bocian, J.H.G. Macdonald, J.F. Burn, Biomechanically inspired modeling of pedestrian-induced vertical self-excited forces, J. Bridg. Eng. 18 (2013) 1336e1346, https://doi.org/10.1061/(ASCE)BE.1943-5592.0000490. [38] M. Bocian, J.H.G. Macdonald, J.F. Burn, Biomechanically inspired modelling of pedestrian-induced forces on laterally oscillating structures, J. Sound Vib. 331 (2012) 3914e3929, https://doi.org/10.1016/j.jsv.2012.03.023. [39] N.A. Alexander, Theoretical treatment of crowd-structure interaction dynamics, Proc. ICE: Structures & Buildings 159 (2006) 329e338. SB6. [40] J.W. Dougill, J.R. Wright, J.G. Parkhouse, R.E. Harrison, Human structure interaction during rhythmic bobbing, Struct. Eng. 84 (2006) 32e39. [41] N.J. Gordon, D.J. Salmond, A.F.M. Smith, Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proc. F Radar Signal Process. 140 (1993) 107, https://doi.org/10.1049/ip-f-2.1993.0015. [42] S. Barnett, J.L. Cunningham, S. West, A comparison of vertical force and temporal parameters produced by an in-shoe pressure measuring system and a force platform, Clin. Biomech. 16 (2001) 353e357, https://doi.org/10.1016/S0268-0033(01)00026-2. [43] H.L.P. Hurkmans, J.B.J. Bussmann, R.W. Selles, H.L.D. Horemans, E. Benda, H.J. Stam, J.A.N. Verhaar, Validity of the Pedar Mobile system for vertical force measurement during a seven-hour period, J. Biomech. 39 (2006) 110e118, https://doi.org/10.1016/j.jbiomech.2004.10.028. [44] K. Nakazato, P. Scheiber, E. Müller, A comparison of ground reaction forces determined by portable force-plate and pressure-insole systems in alpine skiing, J. Sport. Sci. Med. 10 (2011) 754e762. [45] J. Ching, J.L. Beck, K.A. Porter, R. Shaikhutdinov, Bayesian state estimation method for nonlinear systems and its application to recorded seismic response, J. Eng. Mech. 132 (2006), https://doi.org/10.1061/(ASCE)0733-9399(2006)132:4(396). [46] H.A. Nasrellah, C.S. Manohar, A particle filtering approach for structural system identification in vehicle-structure interaction problems, J. Sound Vib. 329 (2010) 1289e1309, https://doi.org/10.1016/j.jsv.2009.10.041. s.
H. Wang et al. / Journal of Sound and Vibration 456 (2019) 13e29
29
[47] H. Wang, T. Nagayama, B. Zhao, D. Su, Identification of moving vehicle parameters using bridge responses and estimated bridge pavement roughness, Eng. Struct. 153 (2017) 57e70, https://doi.org/10.1016/j.engstruct.2017.10.006. [48] H. Wang, T. Nagayama, J. Nakasuka, B. Zhao, D. Su, Extraction of bridge fundamental frequency from estimated vehicle excitation through a particle filter approach, J. Sound Vib. 428 (2018) 44e58, https://doi.org/10.1016/j.jsv.2018.04.030. [49] H. Wang, T. Nagayama, D. Su, Estimation of dynamic tire force by measurement of vehicle body responses with numerical and experimental validation, Mech. Syst. Signal Process. 123 (2019) 369e385, https://doi.org/10.1016/j.ymssp.2019.01.017. [50] J. Carpenter, P. Clifford, P. Fearnhead, Improved particle filter for nonlinear problems, IEE Proc. - Radar, Sonar Navig. 146 (1999) 2, https://doi.org/10. 1049/ip-rsn:19990255. [51] M.S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for online nonlinear/nongaussian bayesian tracking, in: Bayesian Bounds Param. Estim. Nonlinear Filtering/Tracking, 2007, pp. 723e737, https://doi.org/10.1109/9780470544198.ch73. [52] H. Masaru, O. Maruyama, Identification of running load and beam system, J. Eng. Mech. 113 (6) (1987) 813e824. [53] J. Doke, A.D. Kuo, Energetic cost of producing cyclic muscle force, rather than work, to swing the human leg, J. Exp. Biol. 210 (2007) 2390e2398, https://doi.org/10.1242/jeb.02782. [54] C.E. Raburn, K.J. Merritt, J.C. Dean, Preferred movement patterns during a simple bouncing task, J. Exp. Biol. 214 (2011) 3768e3774, https://doi.org/10. 1242/jeb.058743. [55] K.J. Merritt, C.E. Raburn, J.C. Dean, Adaptation of the preferred human bouncing pattern toward the metabolically optimal frequency, J. Neurophysiol. 107 (2012) 2244e2249, https://doi.org/10.1152/jn.00984.2011.
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Parameter identification of spring-mass-damper model for bouncing people Haoqi Wang a, b, Jun Chen a, *, Tomonori Nagayama b a b
College of Civil Engineering, Tongji University, Shanghai, 200092, China Department of Civil Engineering, The University of Tokyo, Tokyo, 113-8656, Japan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 October 2018 Received in revised form 18 April 2019 Accepted 14 May 2019 Available online 18 May 2019
Bouncing, which refers to the up-and-down movement of the human body with both feet remaining on the ground, is one of the most common human-induced dynamic loads for civil structures such as sports stadiums and concert halls. Although researchers have focused on the modeling of bouncing-induced loads, studies on the excitation source (the bouncing person) have not gained much attention. The spring-mass-damper model with a pair of internal biomechanical forces is the simplest model to represent a bouncing person. The model parameters, such as stiffness and damping, though very important for application and for HSI (Human Structure Interaction) analysis, have been rarely reported. This study utilized the particle filter (a step-by-step system identification method) to identify human SMD (Spring-Mass-Damper) model parameters from measured bouncing forces obtained by a wireless insole system. A total of 173 continuous bouncing force signals were recorded and divided into 6800 bouncing cycles, from which the SMD model parameters were identified. The results indicated that human natural frequency and high-order biomechanical load factors have a linear trend against the bouncing frequency, while the damping ratio and first-order BLF (Biomechanical Load Factor) share a parabolic relation. For each model parameter, a skew normal distribution was fitted, and the fitting parameters were found to be dependent on the bouncing frequency range. The influence of HSI on the model parameters was also experimentally investigated, indicating that a vibrating surface leads to a lower human natural frequency and a higher damping ratio value. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Bouncing Spring-mass-damper model Biomechanical load factor Parameter identification Particle filter Human-structure interaction
1. Introduction Human-induced vibrations of long-span civil structures including floors, cantilever stands, and pedestrian bridges have attracted the attention of researchers for many years [1,2]. Most human activities are periodic or near-periodic. Therefore, if the dominant frequency of the activity falls into a range close to the natural frequencies of the structure, excessive resonantlike vibrations may occur, leading to serviceability problems or even safety problems in some extreme cases [3e5]. For instance, in 2003, Leeds Town Hall in England was evacuated after only 30 min of a rock concert when 1000 fans induced large vibrations, causing visible cracks on the floor they occupied [6]. In 2011, a group of 17 people caused a 39-story building in
* Corresponding author. E-mail address: [email protected] (J. Chen). https://doi.org/10.1016/j.jsv.2019.05.034 0022-460X/© 2019 Elsevier Ltd. All rights reserved.
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South Korea to vibrate for 10 min by doing aerobics [7]. More recently, in 2013, an “earthquake” was recorded at a seismographic station near a sports stadium where fans were celebrating their team's victory [8]. These cases, among others, indicate that human-induced excitation can give rise to severe problems in civil structures, making it important to investigate the nature of human-induced excitation. Among many types of human activities, the bouncing motion, which refers to the up-and-down movement of a human body with both feet remaining on the ground [5], is more realistic and more likely to occur during sports or musical events than other types of motion. This is because of its lower consumption of energy and the easiness of coordination among multiple individuals [9,10]. Although jumping motions usually generate higher amplitudes of excitation on human-occupied structures [11,12], bouncing effects can last much longer, giving enough time for a structure to reach a high level of vibration. Moreover, it was reported that the bouncing frequencies can be as high as 5 Hz [9], making the occurrence of resonant vibration more probable for many large-span structures. For the reasons stated above, the bouncing load has been studied by many researchers [9,13e20]. A correct and reliable model reflecting load properties is a prerequisite for the study of bouncing-induced vibration. The Fourier-harmonic-based model, despite its drawbacks [21e24], is the most studied and widely accepted mathematical model because of the nearperiodic feature of the bouncing load [10,16,21]. Other efforts have been made regarding the variability of bouncing loads from the perspective of stochastic excitation [11,25,26]. However, most existing bouncing load research aimed to propose a modeling method that describes the ground reaction force (GRF) curves most similar to the recorded ones. However, these studies failed to reflect the properties of the excitation source, i.e., the bouncing person. In fact, it was established that human beings can change the structural dynamic properties of the buildings they occupy through various activities owing to their individual mass, damping, and stiffness effects [27e31]. Therefore, structural vibrations will change owing to HSI. This justifies the importance of studying human vibration-causing properties and their effects on structural vibration. Although some researchers addressed this issue in cases of human walking [32e39], studies about the dynamic properties of bouncing people (as in dancing or response to music) are meager. To the best of the authors’ knowledge, the only attempt to study human bouncing parameters was made by Dougill et al. [40], where the bouncing person was modeled as a spring-mass-damper (SMD) system with a pair of biomechanical forces exciting the human system, while the GRF was related to the human system output. Through a numerical analysis, the authors proposed a mean natural frequency of 2.3 Hz and a damping ratio of 0.25 for a bouncing person. This paper introduces a method to obtain the SMD parameters of a bouncing person from recorded GRF signals. A particle filter method that estimates the system state in a step-by-step manner [41] is adopted. The particle filter method is applied to each cycle of bouncing-induced GRFs recorded in an experiment to identify human SMD parameters and the unknown biomechanical force input simultaneously, assuming that the parameters remain constant for each bouncing cycle. Therefore, the human body responses can be determined by biomechanical force properties and human SMD parameters. The influence from HSI is also investigated. The structure of this paper is as follows. In Section 2, an experiment to collect individual human bouncing load signals using sensor-equipped insoles is described, and a comparison between signals measured from insoles and from force plates is provided. In Section 3, the procedure of the proposed method for SMD parameter identification is explained. Section 4 provides the identification results for one individual, and Section 5 provides a statistical analysis on all of the experimental data. In Section 6, the effect of HSI is investigated experimentally. Finally, some conclusions are drawn in Section 7. 2. Experiments on individual bouncing loads 2.1. Experimental setup To identify human parameters correctly, real bouncing force signals are indispensable. Experiments aimed at obtaining real GRFs generated by bouncing motions were conducted at Tongji University in China. The test participants were equipped with shoe insoles that were tracked by the Novel Pedar system (Novel Co., Germany) to measure continuous GRF signals. The insoles were connected to a receiver attached to the bouncing person, and the signals were wirelessly transmitted to a nearby data acquisition PC through the Bluetooth technique at a sampling frequency of 100 Hz. Twenty-five individuals participated in the experimental tests. Each participant was asked to bounce on a rigid floor at six different frequencies guided by a metronome. The provided frequencies were from 1.5 to 3.5 Hz with an increment of 0.4 Hz. In addition to these guided bouncing frequencies, one more bouncing frequency was freely chosen by participants according to their own preference. For each bouncing frequency, participants were asked to keep bouncing for approximately 30 s. In total, 173 time histories of continuous bouncing signals were obtained in this experiment. The selection of the test participants and the test protocol satisfied the requirements of the Medical Ethics Committee of Tongji University. 2.2. Comparison of signals from insole system and force plates Although the validity and accuracy of the insole system, which has been widely adopted for biomechanical and biomedical uses, are already well acknowledged [42e44], a validation test specially designed for the bouncing GRF measurements in this study was carried out (see Fig. 1). A test subject wearing the insole system was asked to bounce on a force plate (AMTI OR6-7) whose records were taken as references. The force plate had a sampling rate of 1000 Hz, and the signals recorded by the force
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plate and the insole system were synchronized according to their correlation function. The test subject performed fourbouncing frequencies of 1.5, 1.75, 2.0, and 2.25 Hz. Typical recorded signals with their corresponding Fourier amplitude spectra are shown in Fig. 2 for comparison. Note that the GRFs are normalized to the body weight of the test subject. The signals recorded by the insole system coincide well with those from the force plate. However, a slight difference between the signals of the two systems can still be observed, which is explained by the influence of the footwear [11]. This is because the insole system is directly contacted with the feet, while the force plate measures the forces under the shoes. The influence from this difference is given in later sections. In the frequency domain, the main features in the amplitudes of the bouncing-induced loads are reflected up to at least 10 Hz, which covers the frequency range of interest for most cases. Therefore, the data obtained from the experimental tests is used as GRFs for the human parameter identification. 3. Theory of identification 3.1. Spring-mass-damper model of a bouncing person The SMD model is the simplest single degree-of-freedom model for a bouncing person, and concentrates all of the human mass at the center of mass (COM) of the body. A spring and a damper are attached to represent the mechanism of the human dynamic system, as shown in Fig. 3. Despite its simplicity, the SMD model can reflect the main dynamic properties of the human system, including the natural frequency and the damping ratio. Owing to the movement of human muscles, a pair of equal but opposite biomechanical forces is assumed to appear in the SMD model as the internal excitation of the human system [34]. The movement of the human mass is determined by the SMD model equation of motion as
Mh u€ þ Ch u_ þ Kh u ¼ Fbio ;
(1)
where Mh, Ch, and Kh are the static human whole-body mass, damping coefficient, and stiffness, respectively. Term Fbio is the biomechanical forces exciting the system, and u is the displacement of the human COM from the static equilibrium position. When divided by Mh at both sides of Eq. (1), the equation of motion of the SMD model becomes Eq. (2). The natural frequency and damping ratio of the human body are explicitly expressed in the equation as follows:
u€ þ 2xh uh u_ þ u2h u ¼ Fbio =Mh ;
(2)
in which xh is the damping ratio, and uh is the circular natural frequency that equals 2pfh, where fh is the human natural frequency. As indicated in Fig. 3, the GRF is equal to the summation of the spring force, damping force, biomechanical force, and the body static weight. Therefore, the following relation between the measured GRF and the movement of the body's COM is found:
u€ ¼ ðGRF Mh gÞ=Mh ;
(3)
Fig. 1. Experimental setup.
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Fig. 2. Typical recordings of validation test: (a) time history of 1.5 Hz, (b) spectrum of 1.5 Hz, (c) time history of 1.75 Hz, (d) spectrum of 1.75 Hz, (e) time history of 2.0 Hz, (f) spectrum of 2.0 Hz, (g) time history of 2.25 Hz, and (h) spectrum of 2.25 Hz.
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Fig. 3. SMD model of bouncing person with pair of biomechanical forces.
in which g is the gravitational acceleration. As shown in the typical recordings in Fig. 2, the GRF has a near-periodic feature. Therefore, according to Eq. (1), it is reasonable to assume that the biomechanical force Fbio is also near-periodic. A Fourier series model is then applied to each cycle of the biomechanical force normalized to the human body weight, as shown in Eq. (4).
Fbio =Mh ¼ a0 þ
n X ðai cosð2pifb tÞ þ bi sinð2pifb tÞÞ;
(4)
i¼1
where a0, ai, and bi are Fourier series coefficients, fb is the bouncing frequency, and n is the number of harmonics considered in the Fourier series model. In this paper, the value of n is chosen to be 4 because the amplitudes of harmonics higher than the fourth order are considered to be very low and are thus neglected. Similar to the definition of the dynamic load factor (DLF) [19], the biomechanical load factor (BLF) is defined as
BLFi ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2i þ b2i g:
(5)
3.2. Particle filter theory and its application to human parameter identification In this study, the human parameters and the BLFs are identified through a method known as a particle filter, which estimates the state of the system by using the measured data in a step-by-step manner. Because the particle filter method has been used in many other problems in civil engineering [45e49], and the details of a similar procedure were discussed in the human parameter identification for walking pedestrians [36], this method is only briefly reviewed herein. In the particle filter method, the system dynamic equation is written in state-space form as
xkþ1 ¼ fk ðxk Þ þ wðkÞ:
(6)
In general, xk is the vector representing the system state at time step k. The state transition function, which shows the relation between the state vector for two continuous steps of k and k þ 1, is represented by the function fk. Further, w(k) is the system error term following a known distribution. For the SMD model identification in this study, the unknown parameters (natural frequency, damping ratio, and biomechanical force) need to be included in the state vector with the system response terms.
_ fh ; xh ; a0 ; a1 ; b1 ; /; an ; bn T : X ¼ ½u; u; The state equation of the human SMD system is represented by
(7)
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X_ ¼ AX þ w;
(8)
T _ u€; 0; /; 0 X_ ¼ u;
(9)
where
and A is the state matrix, which can be derived from the equation of motion as follows:
2
0 6 u2 h 6 A¼6 6 0 4 « 0
1
2xh uh
0 0 0 0 0 1
3 0 0 / 0 0 cosð2pfb tÞ sinð2pfb tÞ / cosð2pnfb tÞ sinð2pnfb tÞ 7 7 7: / 0 7 5 « / 0
(10)
Eq. (8) is discretized using the Euler discretization method to obtain the relation between the system states of one time step and the next, giving
Xk ¼ Ad Xk1 þ wðkÞ;
(11)
in which
Ad ¼ I þ ADt;
(12)
where I is the identity matrix, and Dt is the reciprocal of the sampling frequency. At each time step, the system responses are included in the observation vector yk, which is related to the system state vector xk through the observation equation. The general form of the observation equation in the particle filter method is shown as
yk ¼ hk ðxk Þ þ vðkÞ;
(13)
in which v(k) is the observation error with a known distribution. In this study, the COM displacement, velocity, and acceleration responses are included in the observation vector represented by Yk as follows:
Y k ¼ ½uk ; u_ k ; u€k T :
(14)
The transition function hk can be derived from the SMD equation of motion as
Y k ¼ Cd Xk þ vðkÞ;
(15)
where Cd is the observation matrix, shown as
2
1 Cd ¼ 4 0 u2h
0 1
2xh uh
0 0 0
0 0 0 0 0 1 cosð2pfb tÞ
/ / sinð2pfb tÞ / cosð2pnfb tÞ
3 0 5: 0 sinð2pnfb tÞ
(16)
In the particle filter method, each particle represents a system state vector. When the number of particles becomes sufficiently large, the probability density function (PDF) of the system state of the current step is equivalently represented by these particles using a Monte Carlo expression [50,51]. At time step k - 1, each particle passes through the system equation shown in Eq. (11), giving a predicted value of the state vector for time step k. Observation vectors corresponding to the predicted state vectors are calculated following Eq. (15) and are then compared with the measurement made at time step k. A process known as resampling is conducted based on the fact that the particles giving observations closer to the measurement have a higher chance of being selected. After resampling, the new particles represent the updated PDF of the system state at time step k, from which an identified value can be extracted. This procedure is repeated in the next step until the end of the signal. The measured GRF is converted to COM acceleration responses through Eq. (3) with a known human body mass measured in advance. The COM velocity and displacement responses are obtained through integration and double integration from the acceleration, which is known to be sensitive to measurement noise. To reduce the integration error, the acceleration signal is applied to a high-pass filter with a cutoff frequency of 0.5 Hz, which does not affect the analysis results because the frequency range of interest is usually higher than 1 Hz for normal bouncing frequencies. In addition, as shown in later sections, the duration of the integrated signal is limited within the period of one bouncing cycle, which is considered to be short before large integration errors accumulate.
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Fig. 4. Converging process of human SMD model parameters within one cycle: (a) natural frequency, (b) damping ratio, (c) first-order BLF, (d) second-order BLF, and (e) third-order BLF.
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A total of 5000 particles are generated. The initial range of the uniform distributions for human natural frequency fh and damping ratio xh are 1e4 Hz and 0e1, respectively, while the range for the input Fourier parameters ai and bi in Eq. (7) are set as 0e0.4. These initial ranges are determined based on the reasonable range of human parameters [34]. The system error term w(k) is assumed to follow a Gaussian distribution with a standard deviation of around 1% of the nominal values of each quantity in order to prevent parameters from converging to a wrong value before reaching the true value [46]. The standard deviation of the observation error v(k) is taken as 10% of the RMS values of the corresponding responses. Note that the values of w(k) and v(k) are related to the system modeling error and measurement noises, and are empirically chosen in this study. Robustness was found against small changes of these values.
4. Identification results for one individual Before implementing a particle filter, the measured time history is split into cycles. Each cycle starts from one peak value recorded by the insoles to the next peak value, as shown in Fig. 2, whose duration is around 0.3e0.8 s depending on the bouncing frequency. The particle filtering process is then applied, giving estimates of the human natural frequency, damping ratio, and the biomechanical force parameters for each of these cycles. Owing to the time limitation of each cycle, it is possible that the time duration for one cycle is not sufficient to make all the parameters converge. Therefore, a repeating process is adopted by implementing the particle filter more times in this cycle, with the initial particle distribution replaced by that of the last step of the previous repetition [47,52]. This method is based on the idea that after one filtering process, the particle distribution becomes closer to the real distibution and represents the parameter's real value more accurately. The identification results for the SMD model parameters of an individual with a body mass of 57.4 kg bouncing at 2 Hz are shown in Fig. 4. Note that these results are only for one cycle of bouncing load extracted from the continuous signals recorded by the insoles. As shown in Fig. 4, after 4e5 repetitions of the particle filtering process, the human natural frequency converges at 1.61 Hz, while the damping ratio converges at 0.26. These figures show that if only one particle filtering process is applied, the parameter values still fluctuate. By implementing more repetitions, the estimation values will converge to a fixed value. Because multiple filtering is required, a parameter is considered to have converged if the estimated value given by one filtering process does not exceed ±5% of the value given by the previous filtering. As stated in Section 2, the forces measured by the insoles are used as a surrogate for the GRF. To quantify the influence from this assumption, the estimation procedure was also applied on the forces simultaneously measured by the force plates. The converging process is shown in Fig. 4, and the estimation values are listed in Table 1, showing that using signals measured by the insoles as the GRF does not result in large estimation errors. From the body mass information of this test subject, the stiffness and the damping coefficient are calculated to be 7212.1 Nm1 and 730.9 Nsm1, respectively, through the following Eqs. (17) and (18). Note that this paper assumes that Mh represents the entire body mass, which is same as in Refs. [33,34,37,39].
Table 1 Converging values of one typical cycle.
Force plates Insoles
Frequency
Damping Ratio
BLF1
BLF2
BLF3
1.776 Hz 1.784 Hz
0.586 0.568
0.125 0.121
0.215 0.212
0.064 0.060
Fig. 5. Typical time history of biomechanical force of one bouncing cycle.
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Fig. 6. Human SMD model parameter estimation of one test participant: (a) natural frequency, (b) damping ratio, (c) stiffness, (d) damping coefficient, (e) firstorder BLF, (f) second-order BLF, and (g) third-order BLF.
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Fig. 7. Results for all recorded bouncing cycles from all participants: (a) natural frequency, (b) damping ratio, (c) stiffness, (d) damping coefficient, (e) first-order BLF, (f) second-order BLF, and (g) third-order BLF.
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Fig. 8. Distribution of parameters and fitting: (a) natural frequency, (b) damping ratio, (c) stiffness, (d) damping coefficient, (e) first-order BLF, (f) second-order BLF, and (g) third-order BLF.
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Kh ¼ Mh ð2pfn Þ2 :
(17)
C h ¼ 2xMh ð2pfn Þ:
(18)
Once Mh, Kh, and Ch are known or identified, and the acceleration, velocity, and displacement of the COM are measured or integrated, the biomechanical force can be directly calculated from Eq. (1). For the case shown above, the time history of the biomechanical force is plotted in Fig. 5. For other cycles of this signal, as well as other bouncing frequency cases of this individual, the identified values of the human SMD parameters and their relation against the bouncing frequency are given in Fig. 6 (a)e(g). In Fig. 6 (a), an increasing trend for the natural frequency is clearly observed. In addition, when the bouncing frequency is low, the natural frequency values are limited to a small range, indicating that the bouncing process is quite stable and the human subject does not need to change their bouncing pattern frequently. However, to match the required high bouncing frequency, the human body needs to adjust their bouncing patterns through the expansion of muscles during the bouncing process. As a result, the estimated values of the natural frequency become more scattered in the high-bouncing-frequency range, where the large values indicate an increase in human stiffness caused by stressed muscles. In Fig. 6 (b), it is observed that most of the damping ratio values are in the range of 0.2e0.6. The relation against the bouncing frequency is not a simple linear trend. When the bouncing frequency is lower than around 3 Hz, a decreasing trend is observed for damping ratio values. For the cases higher than 3 Hz, the damping ratios increase with the bouncing frequency. Similar trends are also found for the stiffness and damping coefficient of the SMD model, as shown in Fig. 6 (c) and (d), because they have a close relation with the natural frequency and damping ratio through Eq. (17) and Eq. (18). The first-order BLF increases with the bouncing frequency, while a decreasing trend is found for the second- and third-order BLFs, as shown in Fig. 6 (e)e(g), respectively. 5. Statistical analysis The process described above was applied to all 173 records of bouncing GRFs, from which 6800 cycles were extracted, in order to identify the SMD parameters. The results of the human natural frequency, damping ratio, stiffness, damping coefficient, and first three orders of BLF are shown in Fig. 7 (a)e(g), respectively. To show the trend of each parameter against the bouncing frequency more clearly, the mean values for each bouncing frequency range are calculated and plotted in Fig. 7. These bouncing frequency ranges are 1.35e1.75 Hz, 1.75e2.15 Hz, 2.15e2.55 Hz, 2.55e2.95 Hz, 2.95e3.35 Hz, and 3.35e3.75 Hz, respectively. The natural frequency, stiffness, and first-order BLF values have an increasing trend as the bouncing frequency increases, while the second- and third-order BLF values show a decreasing trend. For the damping ratio and damping coefficients, the values first decrease and then increase after approximately 2.5 Hz. Notably, according to Fig. 7 (b) and (e), at approximately 2.5e3 Hz, the damping ratio reaches its lowest value while the highest value of the first-order BLF appears. The BLF represents the level of biomechanical force excitation; hence, the highest
Table 2 Fitted coefficients of probability density function. SMD parameter
PDF parameter
1.35e1.75 Hz
1.75e2.15 Hz
2.15e2.55 Hz
2.55e2.95 Hz
2.95e3.35 Hz
3.35e3.75 Hz
Natural frequency
m (Hz) s (Hz) a m s a m (Nm1) s (Nm1) a m (Nsm1) s (Nsm1) a m s a m s a m s a
1.7344 0.3671 3.5150 0.1229 0.1609 3.9977 6717.5 2714.6 1.3713 116.2537 192.8070 3.9194 0.1968 0.1005 2.6126 0.1549 0.1317 0.7517 0.0116 0.0609 8.0118
2.0981 0.3283 3.8383 0.0963 0.1250 5.0906 10011.5 3386.8 1.5475 122.7600 175.0183 5.1436 0.1808 0.0937 2.3134 0.1046 0.1306 2.8725 0.0140 0.0503 2.8241
2.4314 0.4314 4.1657 0.0781 0.1477 4.9991 13458.9 5014.8 2.430 111.9995 207.2037 4.5361 0.1626 0.1502 3.6431 0.1189 0.1021 1.6303 0.0125 0.0530 8.9577
2.8634 0.7717 8.4628 0.0811 0.1378 7.1938 18118.9 8959.3 3.5828 135.3333 207.7530 4.3247 0.1500 0.1984 4.6241 0.1060 0.0921 2.2067 0.0119 0.0504 4.7477
3.2391 1.2011 14.5146 0.0875 0.1524 8.2652 17581.1 9620.7 0.8246 155.5664 234.4910 4.1732 0.1404 0.2385 2.9479 0.1092 0.0767 1.0348 0.0045 0.0370 7.5927
3.5414 1.3501 16.2412 0.0905 0.1626 5.9160 19718.3 10266.7 0.6680 184.5520 251.6296 3.5600 0.1585 0.2444 3.0493 0.1184 0.0750 1.0399 0.0052 0.0265 6.4426
Damping ratio
Stiffness
Damping coefficient
First-order BLF
Second-order BLF
Third-order BLF
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Fig. 9. Experimental plate: (a) plate model and (b) experimental setup.
Table 3 Modal parameters of concrete plate. Mode
1
2
3
4
5
6
Natural Frequency (Hz) Damping Ratio Modal Mass (kg)
3.48 0.37 8583
6.14 0.51 2587
6.74 0.61 9625
14.12 0.91 2423
15.19 0.67 2898
18.09 1.49 4900
first-order BLF value indicates that the human body has a high “willingness” to bounce. At the same frequency, the human body tends to change its bouncing patterns to achieve a low damping ratio, aiming at a more efficient bouncing motion to avoid energy dissipation. This phenomenon is in accordance with a series of research findings in the field of biomechanics, claiming that bouncing people have a “preferred frequency” at which the metabolic energy cost is at a minimum [53e55]. On the other hand, the increasing trend of the natural frequency is explained by the fact that as the bouncing frequency becomes higher, the tension in the human muscles increases, leading to an increased human stiffness value. Although the trend against the bouncing frequency of each parameter's mean value is clearly observed, the results are found to be very scattered owing to the randomness of the bouncing-induced load. In addition, the level of scatter for each parameter is shown to be dependent on the bouncing frequency. For each frequency range, the probability density of each parameter is calculated from the identified values and is shown in Fig. 8 (a)e(g), respectively. According to the shape of the distribution, a skew normal distribution is adopted. The probability density function f(x) of the skew normal distribution is shown in Eq. (19).
Fig. 10. Typical midpoint acceleration.
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ðxmÞ2 2 f ðxÞ ¼ pffiffiffiffiffiffie 2s2 s 2p
aZðxs mÞ ∞
1 t2 pffiffiffiffiffiffie 2 dt; 2p
(19)
where m, s, and a are the location, scale, and shape coefficients of the distribution, respectively, whose values are determined through a genetic algorithm. The values of the fitted coefficients are given in Table 2, and the fitting curves are shown together in Fig. 8.
Fig. 11. Effect of HSI on SMD model parameters: (a) natural frequency, (b) damping ratio, (c) first-order BLF, (d) second-order BLF, and (e) third-order BLF.
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Table 4 Comparison of estimated human SMD parameters on vibrating plate and on rigid floor.
On rigid floor On vibrating plate Difference
Frequency
Damping ratio
BLF1
BLF2
BLF3
1.81 Hz 1.49 Hz 17.7%
0.33 0.41 þ24.2%
0.241 0.348 þ44.4%
0.335 0.267 20.3%
0.082 0.046 43.9%
6. Investigation of influence of human-structure interaction In the previous sections, the results were based on data measured from experiments conducted on a rigid floor. However, there are cases in which people bounce on a vibrating surface, where HSI may have an effect on the human SMD parameters. This section investigates the influence of HSI on the human SMD model parameters. A pre-stressed concrete plate (10 m long and 6 m wide) with a weight of 16.5 t, as shown in Fig. 9, was constructed for a multipurpose experiment in the structural laboratory of Tongji University. The modal parameters of this plate were obtained through a modal test using an instrumented hammer. The plate's natural frequencies and damping ratios of the first six modes are listed in Table 3. A test participant wearing the insole system was asked to bounce at the midpoint of the plate at a frequency of 1.75 Hz. This bouncing frequency was chosen according to the fundamental frequency of the plate, which is around twice the bouncing frequency, in order to excite near-resonance vibration through the second harmonic of the bouncing load. The typical acceleration response at the midpoint of the plate under bouncing excitation is shown in Fig. 10. The proposed estimation method described in Section 4 was conducted for the bouncing force signals recorded when the test participant was bouncing on the vibrating plate. The estimations of the human SMD parameters are shown in Fig. 11 (a)e(e) for the natural frequency, damping ratio, and BLFs, together with the results from the same test participant bouncing on the rigid floor. Because these results are extracted from the experiment of the same test participant within a short experimental period, these figures clearly show the influence of a vibrating surface on human SMD parameters. A decrease in the human natural frequency and an increase in the human damping ratio is found. Possible reasons for these phenomena stem from people's reactions when they feel the surface vibrating, including relaxing their muscles or changing posture to reduce the uncomfortableness from the vibrating surface. The estimation values of each parameter on the vibrating plate and on the rigid floor are listed in Table 4 with the corresponding differences. 7. Summary and conclusions The human bouncing parameters of an SMD model were studied. Experiments using wireless force insoles were conducted to obtain bouncing-induced GRFs, which were then converted to the acceleration response of COM. A time-domain method known as the particle filter method was applied to the measured data to identify the human natural frequency, damping ratio, and BLFs simultaneously. From the estimation results, a linear trend was found for human natural frequency against the bouncing frequency, while a parabolic curve was fitted with human damping and first-order BLF. For each SMD parameter, a skew normal distribution was fitted. The effect from HSI was also investigated. From this study, the following conclusions were drawn: (1) The accuracy of the wireless insole system was found to be acceptable from the perspective of human parameter identification based on the GRF measurements of human bouncing. (2) The particle filter method with a repeating process within each bouncing cycle was capable of extracting human model parameters. (3) The existence of a “preferred frequency” at approximately 2.5e3 Hz from a biomechanical perspective was found. (4) The vibrating surface tended to lower the human natural frequency and increase the human damping ratio through HSI.
Acknowledgement This work was supported by National Natural Science Foundation of China (51778465, U1711264). Moreover, the authors would like to thank all test subjects for participating in the project making the data collection possible. References [1] S. Zivanovi c, A. Pavic, P. Reynolds, Vibration serviceability of footbridges under human-induced excitation: a literature review, J. Sound Vib. 279 (2005) 1e74, https://doi.org/10.1016/j.jsv.2004.01.019.
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[2] V. Racic, A. Pavic, J.M.W. Brownjohn, Experimental identification and analytical modelling of human walking forces: literature review, J. Sound Vib. 326 (2009) 1e49, https://doi.org/10.1016/j.jsv.2009.04.020. [3] P. Dallard, A.J. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R.M. Ridsdill Smith, M. Willford, The London millennium footbridge, in: Struct. Eng., 2001, pp. 17e33. [4] J.M.W. Brownjohn, S. Zivanovic, A. Pavic, Crowd Dynamic Loading on Footbridges, Footbridge Vibration Design, CRC Press, 2009, pp. 145e176. [5] C.A. Jones, P. Reynolds, A. Pavic, Vibration serviceability of stadia structures subjected to dynamic crowd loads: a literature review, J. Sound Vib. 330 (2011) 1531e1566, https://doi.org/10.1016/j.jsv.2010.10.032. [6] D. Parker, Rock fans uncover townhall floor faults, N. Civ. Eng. (2003), 20.11.2003. [7] S.H. Lee, K.K. Lee, S.S. Woo, S.H. Cho, Global vertical mode vibrations due to human group rhythmic movement in a 39 story building structure, Eng. Struct. 57 (2013) 296e305, https://doi.org/10.1016/j.engstruct.2013.09.035. [8] J. Lear, E. Payne, Rowdy Seattle Seahawks fans register as earthquake during Monday night football, Cable News Network (CNN) (2013), 3.12.2013. [9] S. Yao, J.R. Wright, A. Pavic, P. Reynolds, Experimental study of human-induced dynamic forces due to jumping on a perceptibly moving structure, J. Sound Vib. 296 (2006) 150e165, https://doi.org/10.1016/j.jsv.2006.02.018. [10] IStructE/DCLG/DCMS Working Group, Dynamic Performance Requirements for Permanent Grandstands Subject to Crowd Action: Recommendations for Management, Design and Assessment, The Institution of Structural Engineers, London, UK, 2008. [11] V. Racic, A. Pavic, Mathematical model to generate asymmetric pulses due to human jumping, ASCE J. Eng. Mech. 135 (2009) 1206e1211, https://doi. org/10.1061/(ASCE)EM.1943-7889.0000044. [12] V. Racic, A. Pavic, Mathematical model to generate near-periodic human jumping force signals, Mech. Syst. Signal Process. 24 (2010) 138e152, https:// doi.org/10.1016/j.ymssp.2009.07.001. [13] A. Ebrahimpour, R.L. Sack, Modeling dynamic occupant loads, J. Struct. Eng. 115 (1989) 1476e1496, https://doi.org/10.1061/(ASCE)0733-9445(1989) 115:6(1476). [14] S. Yao, J.R. Wright, A. Pavic, P. Reynolds, Forces generated when bouncing or jumping on a flexible structure, Proc. Int. Conf. Noise Vib. (2002) 563e572. [15] S. Yao, J.R. Wright, A. Pavic, P. Reynolds, Experimental study of human-induced dynamic forces due to jumping on a perceptibly moving structure, J. Sound Vib. 296 (2006) 150e165, https://doi.org/10.1016/j.jsv.2006.02.018. [16] J.G. Parkhouse, D.J. Ewins, Crowd-induced rhythmic loading, Struct. Build. 159 (2006) 247e259, https://doi.org/10.1680/stbu.2006.159.5.247. [17] E. Agu, M. Kasperski, A statistical approach to loads induced by bobbing, 7th Eur. Conf. Struct. Dyn. EURODYN (2008), 2008. [18] A.J. Comer, A. Blakeborough, M.S. Williams, Rhythmic crowd bobbing on a grandstand simulator, J. Sound Vib. 332 (2013) 442e454, https://doi.org/10. 1016/j.jsv.2012.08.012. [19] V. Racic, J. Chen, Data-driven generator of stochastic dynamic loading due to people bouncing, Comput. Struct. 158 (2015) 240e250, https://doi.org/10. 1016/j.compstruc.2015.04.013. [20] J. Chen, L. Wang, V. Racic, J. Lou, Acceleration response spectrum for prediction of structural vibration due to individual bouncing, Mech. Syst. Signal Process. 76e77 (2016) 394e408, https://doi.org/10.1016/j.ymssp.2016.02.032. [21] J.H. Rainer, G. Pernica, D.E. Allen, Dynamic loading and response of footbridges, Can. J. Civ. Eng. 15 (1988) 335e347, https://doi.org/10.1139/l88-007. [22] J.M. Brownjohn, A. Pavic, P. Omenzetter, A spectral density approach for modelling continuous vertical forces on pedestrian structures due to walking, Can. J. Civ. Eng. 31 (2004) 65e77, https://doi.org/10.1139/l03-072. [23] V. Racic, J.M.W. Brownjohn, Mathematical modelling of random narrow band lateral excitation of footbridges due to pedestrians walking, Comput. Struct. 90e91 (2012) 116e130, https://doi.org/10.1016/j.compstruc.2011.10.002. [24] V. Racic, J.B. Morin, Data-driven modelling of vertical dynamic excitation of bridges induced by people running, Mech. Syst. Signal Process. 43 (2014) 153e170, https://doi.org/10.1016/j.ymssp.2013.10.006. [25] J. Sim, A. Blakeborough, M.S. Williams, G. Parkhouse, Statistical model of crowd jumping loads, ASCE J. Struct. Engieering. 134 (2008) 1852e1861, https://doi.org/10.1061/(ASCE)0733-9445(2008)134:12(1852). [26] V. Racic, A. Pavic, Stochastic approach to modelling of near-periodic jumping loads, Mech. Syst. Signal Process. 24 (2010) 3037e3059, https://doi.org/ 10.1016/j.ymssp.2010.05.019. [27] M.J. Griffin, J. Erdreich, Handbook of human vibration, J. Acoust. Soc. Am. 90 (1991) 2213, https://doi.org/10.1121/1.401606. [28] T. Ji, B.R. Ellis, Human-structure interaction in vertical vibrations, Proc. ICE - Struct. Build. 122 (1997) 1e9, https://doi.org/10.1680/istbu.1997.29162. [29] A. Cappellini, S. Manzoni, M. Vanali, Quantification of damping effect of humans on lightly damped staircases, in: Top. Dyn. Civ. Struct. - Proc. 31st IMAC, A Conf. Struct. Dyn. 2013, 2013, pp. 453e460, https://doi.org/10.1007/978-1-4614-6555-3_49. [30] K.A. Salyards, N.C. Noss, Experimental evaluation of the influence of human-structure interaction for vibration serviceability, J. Perform. Constr. Facil. 28 (2014) 458e465, https://doi.org/10.1061/(ASCE)CF.1943-5509.0000436. rez, A. Lorenzana, Walking model to simulate interaction effects between pedestrians and lively structures, J. Eng. Mech. 143 (2017) 1e9, [31] M. Cacho-Pe https://doi.org/10.1061/(ASCE)EM.1943-7889.0001326. [32] K. Van Nimmen, G. Lombaert, G. De Roeck, P. Van den Broeck, The impact of vertical human-structure interaction on the response of footbridges to pedestrian excitation, J. Sound Vib. 402 (2017) 104e121. [33] E. Shahabpoor, A. Pavic, V. Racic, Identification of mass-spring-damper model of walking humans, Structures 5 (2016) 233e246, https://doi.org/10. 1016/j.istruc.2015.12.001. [34] M. Zhang, C.T. Georgakis, J. Chen, Biomechanically excited SMD model of a walking pedestrian, J. Bridg. Eng. (2016) C4016003, https://doi.org/10.1061/ (ASCE)BE.1943-5592.0000910. [35] E. Shahabpoor, A. Pavic, V. Racic, Structural vibration serviceability: new design framework featuring human-structure interaction, Eng. Struct. 136 (2017) 295e311, https://doi.org/10.1016/j.engstruct.2017.01.030. [36] H. Wang, J. Chen, J.M.W. Brownjohn, Parameter identification of pedestrian's spring-mass-damper model by ground reaction force records through a particle filter approach, J. Sound Vib. 411 (2017) 409e421, https://doi.org/10.1016/j.jsv.2017.09.020. [37] M. Bocian, J.H.G. Macdonald, J.F. Burn, Biomechanically inspired modeling of pedestrian-induced vertical self-excited forces, J. Bridg. Eng. 18 (2013) 1336e1346, https://doi.org/10.1061/(ASCE)BE.1943-5592.0000490. [38] M. Bocian, J.H.G. Macdonald, J.F. Burn, Biomechanically inspired modelling of pedestrian-induced forces on laterally oscillating structures, J. Sound Vib. 331 (2012) 3914e3929, https://doi.org/10.1016/j.jsv.2012.03.023. [39] N.A. Alexander, Theoretical treatment of crowd-structure interaction dynamics, Proc. ICE: Structures & Buildings 159 (2006) 329e338. SB6. [40] J.W. Dougill, J.R. Wright, J.G. Parkhouse, R.E. Harrison, Human structure interaction during rhythmic bobbing, Struct. Eng. 84 (2006) 32e39. [41] N.J. Gordon, D.J. Salmond, A.F.M. Smith, Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proc. F Radar Signal Process. 140 (1993) 107, https://doi.org/10.1049/ip-f-2.1993.0015. [42] S. Barnett, J.L. Cunningham, S. West, A comparison of vertical force and temporal parameters produced by an in-shoe pressure measuring system and a force platform, Clin. Biomech. 16 (2001) 353e357, https://doi.org/10.1016/S0268-0033(01)00026-2. [43] H.L.P. Hurkmans, J.B.J. Bussmann, R.W. Selles, H.L.D. Horemans, E. Benda, H.J. Stam, J.A.N. Verhaar, Validity of the Pedar Mobile system for vertical force measurement during a seven-hour period, J. Biomech. 39 (2006) 110e118, https://doi.org/10.1016/j.jbiomech.2004.10.028. [44] K. Nakazato, P. Scheiber, E. Müller, A comparison of ground reaction forces determined by portable force-plate and pressure-insole systems in alpine skiing, J. Sport. Sci. Med. 10 (2011) 754e762. [45] J. Ching, J.L. Beck, K.A. Porter, R. Shaikhutdinov, Bayesian state estimation method for nonlinear systems and its application to recorded seismic response, J. Eng. Mech. 132 (2006), https://doi.org/10.1061/(ASCE)0733-9399(2006)132:4(396). [46] H.A. Nasrellah, C.S. Manohar, A particle filtering approach for structural system identification in vehicle-structure interaction problems, J. Sound Vib. 329 (2010) 1289e1309, https://doi.org/10.1016/j.jsv.2009.10.041. s.
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[47] H. Wang, T. Nagayama, B. Zhao, D. Su, Identification of moving vehicle parameters using bridge responses and estimated bridge pavement roughness, Eng. Struct. 153 (2017) 57e70, https://doi.org/10.1016/j.engstruct.2017.10.006. [48] H. Wang, T. Nagayama, J. Nakasuka, B. Zhao, D. Su, Extraction of bridge fundamental frequency from estimated vehicle excitation through a particle filter approach, J. Sound Vib. 428 (2018) 44e58, https://doi.org/10.1016/j.jsv.2018.04.030. [49] H. Wang, T. Nagayama, D. Su, Estimation of dynamic tire force by measurement of vehicle body responses with numerical and experimental validation, Mech. Syst. Signal Process. 123 (2019) 369e385, https://doi.org/10.1016/j.ymssp.2019.01.017. [50] J. Carpenter, P. Clifford, P. Fearnhead, Improved particle filter for nonlinear problems, IEE Proc. - Radar, Sonar Navig. 146 (1999) 2, https://doi.org/10. 1049/ip-rsn:19990255. [51] M.S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for online nonlinear/nongaussian bayesian tracking, in: Bayesian Bounds Param. Estim. Nonlinear Filtering/Tracking, 2007, pp. 723e737, https://doi.org/10.1109/9780470544198.ch73. [52] H. Masaru, O. Maruyama, Identification of running load and beam system, J. Eng. Mech. 113 (6) (1987) 813e824. [53] J. Doke, A.D. Kuo, Energetic cost of producing cyclic muscle force, rather than work, to swing the human leg, J. Exp. Biol. 210 (2007) 2390e2398, https://doi.org/10.1242/jeb.02782. [54] C.E. Raburn, K.J. Merritt, J.C. Dean, Preferred movement patterns during a simple bouncing task, J. Exp. Biol. 214 (2011) 3768e3774, https://doi.org/10. 1242/jeb.058743. [55] K.J. Merritt, C.E. Raburn, J.C. Dean, Adaptation of the preferred human bouncing pattern toward the metabolically optimal frequency, J. Neurophysiol. 107 (2012) 2244e2249, https://doi.org/10.1152/jn.00984.2011.