Jumping Loads Research Articles

A stochastic approach for generating individual jumping loads considering different jumping force patterns

This study establishes a stochastic model for generating individual jumping force signals based on a database of individual jumping force records. A total of 1296 vertical jumping loads at a jumping frequency of 1.5–3.5 Hz were experimentally measured with high quality. To generate a jumping ground reaction forces (GRFs) curve, the proportions of different force patterns were first obtained based on the database and subsequently used to assign the numbers of individual jumping pulses for different force patterns. Then, a half-sine square function, three Gauss superposition functions, and two Gauss superposition functions were respectively used to generate ‘single-peaked’, ‘double-peaked’, and ‘merging’ shape jumping forces. To reflect the near-periodic nature of the jumping forces, the function parameters and coefficients were randomly generated from their normal distributions, and these parameters and coefficients were assigned to each jumping force. Finally, the generated individual jumping pulses were randomly arranged and connected to form a jumping GRF signal. A comparison of the jumping load–time history curves and Fourier amplitude spectra between the measured and generated results verified the reliability of the model. Compared with the conventional half-sine model, the stochastic approach for generating jumping loads in this study can better reflect the temporal characteristics of the real jumping GRF signals, and the model is convenient to be applied in the vibration serviceability assessment of civil engineering structures, such as stadium stands. • A stochastic model to generate individual jumping loads was established based on a database of 1296 vertical jumping force records. • Different jumping force shapes can be described by the model. • Comparison of jumping load time-history curves and the Fourier amplitude spectra between measured and generated results verified the model's reliability. • The coefficient of correction for the jumping impact factor parabolically varies with the increase of the jumping frequency from 1.5 to 3.5 Hz.

Multi-mode response spectrum for multi-harmonic crowd jumping loads

A major challenge in the vibration performance assessment of structures under crowd jumping loads is the lack of efficient prediction models. With the challenge in mind, this study develops a multi-mode response spectrum for multi-harmonic crowd jumping loads to predict the root mean square value of structural accelerations according to the stochastic vibration theory. The response spectrum is applicable to structures with arbitrary vibration modes, because crowd jumping loads are regarded as multi-point excitations. A random field model proposed by the author is used to derive the closed-form expression of the single-mode response induced by individual jumping loads of a single-harmonic. Furthermore, the closed-form expressions of the correlation coefficients for different load harmonics, different vibration modes, and different jumping persons are established based on the random field model. The performance of the response spectrum is evaluated for two floors, and the results indicate the feasibility of the proposed approach to predicting structural accelerations subjected to crowd jumping loads.

Equivalent Dynamic Load Factor of Different Non-Exceedance Probability for Crowd Jumping Loads

Existing dynamic load factors (DLF) for crowd jumping loads are modeled by identifying the peaks and energies from the corresponding Fourier amplitude spectrum, which may underestimate and overestimate structural responses, respectively. Based on the principle of equal structural responses, this study herein develops an equivalent DLF, using the frequency response function to weight and integrate the power spectral density (PSD) matrix of crowd jumping loads. Firstly, massive PSD matrices of different crowd sizes and different metronome frequencies are simulated using a random field model of crowd jumping loads. Thereafter, the equivalent DLF of different structural damping ratios, crowd sizes, metronome frequencies, and non-exceedance probabilities are investigated, and a model of the equivalent DLF is established accordingly. It is believed that this model could provide an efficient and accurate way to conduct reliability design for structures subjected to crowd jumping loads.

Crowd jumping load simulation with generative adversarial networks

To mathematically represent crowd jumping loads, the features of the jumping load of each person, including pulse curve patterns, pulse interval sequences, and pulse energy sequences are considered. These features are essentially highdimensional random variables. However, they have to be represented in a practically simplified model due to the lack of mathematical tools. The recently emerged generative adversarial networks (GANs) can model high-dimensional random variables well, as demonstrated in image synthesis and text generation. Therefore, this study adopts GANs as a new method for modelling crowd jumping loads. Conditional GANs (CGANs) combined with Wasserstein GANs with gradient penalty (WGANs*—GP) are used in pulse curve pattern modelling, where a multi-layer perceptron and convolutional neural network are selected as the discriminator and generator, respectively. For the pulse energy sequence and pulse interval sequence modelling, similar GANs are used, where recurrent neural networks are selected as both the generator and discriminator. Finally, crowd jumping loads can be simulated by connected the pulse samples based on the pulse energy sequence samples and interval sequence samples, generated by the three proposed GANs. The experimental individual and crowd jumping load records are utilized in training GANs to ensure their output can simulate real load records well. Finally, the feasibility of the proposed GANs was verified by comparing the measured structural responses of an existing floor to the predicted structural responses.

Generative Adversarial Networks-Based Stochastic Approach to the Modeling of Individual Jumping Loads

Features of jumping loads are essentially high-dimensional random variables but have been simplistically modeled owing to the lack of proper mathematical tools. Generative adversarial networks (GANs) in conjunction with deep learning technology are adopted herein for modeling the jumping loads. Conditional GANs (CGANs) combined with Wasserstein GANs (WGANs) with gradient penalty (WGANs-GP) are adopted in the impulse modeling, where a multi-layer perceptron and a convolutional neural network are employed for the discriminator and generator, respectively. As for the impulse amplitude sequence and interval sequence modeling, similar CGANs combined with WGANs-GP are adopted, where recurrent neural networks are employed for both the generator and discriminator. A large amount of measured individual jumping loads are utilized in training GANs to ensure the generated samples can simulate the real ones well. After training, the individual jumping loads are simulated by connecting the generated impulse samples, based on the generated impulse amplitude sequence samples and interval sequence samples. The simulated jumping loads can be used to assess the vibration performance of assembly structures, such as grandstands, concert halls, and gym floors. Moreover, the established GANs can be extended to the modeling of other stochastic dynamic excitations.

Spectral analysis of human-structure interaction during crowd jumping

Modern slender structures such as cantilever stands and long-span floors are quite sensitive to vibrations induced by crowd jumping. Existing human-structure interaction models cannot consider human body parameter variabilities and apply to jumping-induced vibration analysis of structures with closely-spaced and complex vibration modes and passive crowd. This paper herein aims to derive the equation of motion and its spectral analysis solution of an advanced model for the crowd-structure coupled system. Each person on a structure is treated as a separate spring-mass-damper, and the structure is treated as a multi-degree-of-freedom system. Each person is classified as active crowd or passive crowd according to whether he/she exerts dynamic forces on the structure. Then, the equation of motion of the crowd-structure coupled system is derived, and its spectral analysis solution is established by the stochastic vibration theory and the state space method. The frequency response function is adopted to analyze dynamic properties of the coupled system. With a previously-proposed random field model for crowd jumping loads, vibration responses of the coupled system are analyzed. Comparisons between the simulations and the results from two experiments demonstrate the feasibility of the proposed spectral analysis method.

Analysis of dynamic properties and vibration comfort of large-scale stands

ABSTRACT Large grandstands provide areas for human activities and leisure pursuits. The degree of vibration comfort for people has an important effect on the functionality of grandstands. In this paper, the changes of the dynamic properties of the stand owing to different human postures are demonstrated. The comfort degree for human vibration in the large stand is also examined by full-scale measurements and questionnaire surveys. In addition, the analysis of the influence of different parameters on acceleration responses of the stand is presented via numerical simulation. Thereafter, it has been observed that ISO10137-2007 and BS6472 are more suitable than GB10070 and AISC-11 for evaluating vibration serviceability owing to crowd jumping loads during football matches in the large grandstand environment. Moreover, it has been observed that females and the staff are more likely to feel uncomfortable during matches, whilst students younger than 30 years old are the least sensitive group to vibrations of large stands. When the frequency of the jumping loads is close to the fundamental frequency of the stand or the synergy rate is high, the resonance is more likely to occur, which should be avoided by designers and users.

Jump load and landing patterns of collegiate female volleyball players during practice and competition.

Overuse lower extremity injuries are common in women's court volleyball players and are likely due to the repetitive jumping and landing the sport requires. The purpose of this study was to quantify jump load during collegiate women's volleyball, describe the quantity of double-leg (DL) to single-leg (SL) landing strategies, and compare loads and landing strategies between games and practices. Fourteen collegiate Division-1 women's court volleyball players participated in the study. Volleyball-specific activity demands were quantified using video analysis from three consecutive practices and one match. Investigators recorded the total frequency of jump landings, and the frequency and percentage of double-leg (DL) landings and single-leg (SL) landings of fourteen collegiate Division-1 women's court volleyball players. Repeated measures ANOVAs identified differences in jumping load and percentages of DL and SL landings among practices and between practices and games (P<0.05). On average, there was a significantly higher overall jumping load (P=0.01) and frequency of DL (P=0.03) and SL (P=0.04) landings during practice than games, yet no differences between practices (P>0.05). Approximately 75% of all landings were DL, and individual patterns of DL to SL landings were consistent across events (P>0.05). Women's collegiate volleyball demands high volumes of repetitive jumping and landing with SL and DL support that may make these athletes susceptible to overuse injuries, especially during practice.

Random field model for crowd jumping loads

A reliable model for crowd jumping loads is a prerequisite to an accurate prediction of a long-span structure’s response when subjected to crowd motion triggered by music and other mass appeal events. Because of the lack of crowd jumping experimental data, crowd jumping load models are scarce with few data available on this type of synchronisation among people. Inspired by the random field model of earthquake excitations, this study attempts to develop a random field model for crowd jumping loads based on extensive records from unique crowd jumping load experiments. A framework for the random field model of crowd jumping is first proposed, in which auto-power spectral density (PSD) and cross-PSD functions of individual and group jumping loads are defined. The auto-PSD was obtained by updating a model in a previous study. Experiments were conducted on a jumping group of 48 persons using a 3D motion capture technology. A cross-PSD model for any two persons in a jumping group was then developed based on the experimental data. Having obtained a PSD matrix for crowd jumping, the structural response could then be predicted by the stochastic vibration theory. Finally, the feasibility of the model was verified by comparing the measured responses of an existing floor with the prediction responses using the proposed model.

Determination of the dynamic load factors for crowd jumping using motion capture technique

The effect of imperfect synchronization is a key factor for modeling crowd jumping loads. Knowledge on this effect is limited due to the lack of crowd jumping experiments. This paper presents an experimental study of crowd jumping of 48 subjects at a frequency range between 1.5 Hz and 3.5 Hz with an interval of 0.1 Hz. The jumping movements of all participants were synchronously and simultaneously monitored using a three-dimensional (3D) motion capture technique. Based on the measured data, dynamic load factors (DLFs) against crowd sizes were determined using the Monte Carlo method. Then, they were converted to the DLF ratios to reduce numerical errors. Results show that the DLF ratios tend to be a constant for a crowd over 48. A simple expression of the ratios is provided using curve fitting. Then the crowd jumping load model was summarized and compared with the measured and published results. Finally, limitations of this study were discussed.

A simplified method for determining acceleration amplitude of prestressed concrete floor under individual jumping load

SummaryModern long‐span prestressed concrete (PC) floors typically possess low damping and low natural frequency, presenting a potential vibration sensitivity problem induced by human activities. Field test and numerical analysis methods are available to study this kind of problems but would be inconvenient for design engineers. This paper proposes a simplified method to determine the acceleration amplitude of a PC floor subjected to individual jumping load, carrying out through a mix of analytical and experimental techniques. To theoretically analyze the acceleration response, the PC floor was simplified as an anisotropic rectangular plate and the mode decomposition method was used. To facilitate the calculation of the acceleration amplitude aP, a frequency‐dependent coefficient βJ is introduced, which depends on the geometry and support condition of the PC floor and dynamic amplification factor KJ. Field tests on 7 PC floors with various boundary conditions and fundamental natural frequencies were conducted to determine βJ coefficient. Based on the test results, practical formulas for βJ coefficient is first proposed, followed by that for acceleration amplitude aP. The proposed simplified method is easy for practical use and will yield safe and conservative structural designs.

Power Spectral Density Function for Individual Jumping Load

Modern slender structures such as long-span floors and cantilever stands are sensitive to jumping-induced vibrations. A conventional deterministic Fourier series model for the human jumping load may overestimate a structure's responses in resonance condition. This paper suggests a power spectral density (PSD) function for the individual jumping load, which was treated as a narrowband stationary stochastic process. Experiments were conducted on individual jumping loads resulting in 334 records from 73 subjects. Statistical analysis of the records led to experimental PSD curves on which a symmetrical bilinear function was suggested. The proposed PSD function is centered on the given jumping frequency and its integer multiples. The function's parameters were determined by equating the total energy of the proposed PSD with that of the experimental PSD. Application of the proposed PSD for predicting a floor's peak response via the stochastic vibration theory was then presented. The predictions for an experimental floor model subjected to individual jumping were compared with the measured peak responses. The comparisons demonstrated that the proposed PSD function was applicable for predicting the floor's response to individual jumping. Finally, the framework for calculating crowd-induced structure vibrations using the suggested PSD was also discussed.

A data-driven wavelet-based approach for generating jumping loads

Abstract This paper suggests an approach to generate human jumping loads using wavelet transform and a database of individual jumping force records. A total of 970 individual jumping force records of various frequencies were first collected by three experiments from 147 test subjects. For each record, every jumping pulse was extracted and decomposed into seven levels by wavelet transform. All the decomposition coefficients were stored in an information database. Probability distributions of jumping cycle period, contact ratio and energy of the jumping pulse were statistically analyzed. Inspired by the theory of DNA recombination, an approach was developed by interchanging the wavelet coefficients between different jumping pulses. To generate a jumping force time history with N pulses, wavelet coefficients were first selected randomly from the database at each level. They were then used to reconstruct N pulses by the inverse wavelet transform. Jumping cycle periods and contract ratios were then generated randomly based on their probabilistic functions. These parameters were assigned to each of the N pulses which were in turn scaled by the amplitude factors β i to account for energy relationship between successive pulses. The final jumping force time history was obtained by linking all the N cycles end to end. This simulation approach can preserve the non-stationary features of the jumping load force in time-frequency domain. Application indicates that this approach can be used to generate jumping force time history due to single people jumping and also can be extended further to stochastic jumping loads due to groups and crowds.

某人行天桥跳跃荷载下振动分析与测试

某人行天桥跳跃荷载下振动分析与测试

Acceleration response spectrum for predicting floor vibration due to occupants jumping

This paper proposes an acceleration response spectrum to predict floors’ responses due to occupants jumping. Experiments were conducted on individual jumping loads resulting in 506 records. Each record was applied to a single-degree-of-freedom system with various frequencies and damping ratios to obtain a corresponding acceleration response spectrum. Statistical analysis of the results led to a representative spectrum, which is further used to derive an analytical design spectrum curve. The suggested design spectrum covers a structural frequency range of 0.5–15Hz and consists of three main parts: the first plateau, the second plateau and the descent. Design values for spectrum parameters were determined by fitting each part’s mathematical function to actual data. The proposed spectrum was verified by comparing its predictions with measured responses from an experimental floor model and floors of existing structures induced by both single individuals and crowds jumping.

Experimental investigation on single person’s jumping load model

This paper presents a modified half-sine-squared load model of the jumping impulses for a single person. The model is based on a database of 22,921 experimentally measured single jumping load cycles from 100 test subjects. Threedimensional motion capture technology in conjunction with force plates was employed in the experiment to record jumping loads. The variation range and probability distribution of the controlling parameters for the load model such as the impact factor, jumping frequency and contact ratio, are discussed using the experimental data. Correlation relationships between the three parameters are investigated. The contact ratio and jumping frequency are identified as independent model parameters, and an empirical frequency-dependent function is derived for the impact factor. The feasibility of the proposed load model is established by comparing the simulated load curves with measured ones, and by comparing the acceleration responses of a single-degree-of-freedom system to the simulated and measured jumping loads. The results show that a realistic individual jumping load can be generated by the proposed method. This can then be used to assess the dynamic response of assembly structures.

Application of a tri-axial accelerometer to estimate jump frequency in volleyball

Patellar tendinopathy is prevalent among athletes, and most likely associated with a high jumping load. If methods for estimating jump frequency were available, this could potentially assist in understanding and preventing this condition. The objective of this study was to explore the possibility of using peak vertical acceleration (PVA) or peak resultant acceleration (PRA) measured by an accelerometer to estimate jump frequency. Twelve male elite volleyball players (22.5 ± 1.6 yrs) performed a training protocol consisting of seven typical motion patterns, including jumping and non-jumping movements. Accelerometer data from the trial were obtained using a tri-axial accelerometer. In addition, we collected video data from the trial. Jump-float serving and spike jumping could not be distinguished from non-jumping movements using differences in PVA or PRA. Furthermore, there were substantial inter-participant differences in both the PVA and the PRA within and across movement types (p < 0.05). These findings suggest that neither PVA nor PRA measured by a tri-axial accelerometer is an applicable method for estimating jump frequency in volleyball. A method for acquiring real-time estimates of jump frequency remains to be verified. However, there are several alternative approaches, and further investigations are needed.

Stochastic approach to modelling of near-periodic jumping loads

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.

Mathematical model to generate near-periodic human jumping force signals

A mathematical modelling procedure has been developed to generate synthetic vertical force signals induced by a single person jumping. The ability to replicate much of the temporal and spectral features of real jumping loads give 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 narrowband jumping loads that simulate reality better. The proposed mathematical concept for characterisation of irregular jumping pulses may be utilised in vibration serviceability assessment of civil engineering assembly structures, such as grandstands, footbridges and concert or gym floors, to estimate realistic dynamic structural response due to people jumping.

Mathematical Model to Generate Asymmetric Pulses due to Human Jumping

A novel mathematical modeling has been proposed to generate synthetic vertical force signal induced by a single person jumping. This model can replicate much of the temporal and spectral features of the real jumping loading more reliably than the existing half-sine models coupled with Fourier series analysis. This includes lack of symmetry of individual jumping pulses and near-periodic nature of consecutive pulses. The model therefore offers way forward as to the development of a new generation of synthetic narrow-band jumping loads. In these, the shape and frequency content of the jumping force can be changed easily on a jump-by-jump basis, which simulates better on what is happening in reality during human jumping. The synthetic jumping loading can be used in assessing vibration serviceability of civil engineering structures for which such dynamic excitation is relevant, such as assembly structures and concert venues.