Silvano Erlicher

Towards a footwear design tool: Influence of shoe midsole properties and ground stiffness on the impact force during running

Several spring-damper-mass models of the human body have been developed in order to reproduce the measured ground vertical reaction forces during human running (McMahon and Cheng, 1990; Ferris et al., 1999; Liu and Nigg, 2000). In particular, Liu and Nigg introduced at the lower level of their model, i.e. at the interface between the human body and the ground, a nonlinear element representing simultaneously the shoe midsoles and the ground flexibility. The ground reaction force is modelled as the force supported by this nonlinear element, whose parameters are identified from several sets of experimental data. This approach proved to be robust and quite accurate. However, it does not explicitly take into account the shoe and the ground properties. It turns out to be impossible to study the influence of shoe materials on the impact force, for instance for footwear design purposes. In this paper, a modification of the Liu and Niggs model is suggested, where the original nonlinear element is replaced with a bi-layered spring-damper-mass model: the first layer represents the shoe midsole and the second layer is associated with the ground. Ground is modelled as an infinite elastic half-space. We have assumed a viscoelastic behaviour of the shoe material, so the damping of shoe material is taken into account. A methodology for the shoe-soles characterization is proposed and used together with the proposed model. A parametric study is then conducted and the influence of the shoe properties on the impact force is quantified. Moreover, it is shown that impact forces are strongly affected by the ground stiffness, which should therefore be considered as an essential parameter in the footwear design.

Lateral Vibration of Footbridges under Crowd-Loading: Continuous Crowd Modeling Approach

In this paper, a simple 1D crowd model is proposed, which aim is to properly describe the crowd-flow phenomena occurring when pedestrians walk on a flexible footbridge. The crowd is assumed to behave like a continuous compressible fluid and the pedestrian flow is modeled in a 1-D framework using the (total) mass (of pedestrians) conservation equation. This crowd model is then coupled with a simple model for the dynamical behavior of the footbridge and an optimized modeling of synchronization effects is performed. Numerical simulations are presented to show some preliminary results.

Instantaneous Identification of Bouc-Wen-Type Hysteretic Systems from Seismic Response Data

This paper presents a technique for identification of non-linear hysteretic systems subjected to non-stationary loading. In the numerical simulations, a Bouc-Wen model was chosen for its ability to represent the properties of a wide class of real hysteretic systems. The parameters of the model are computed instantaneously by approximating the internal restoring force surface through an “ad hoc” polynomial basis. Instantaneous estimates result from time-varying spectra of the response signals. A numerical application of interest to earthquake engineering is finally reported.

Instantaneous Identification of Degrading Hysteretic Oscillators Under Earthquake Excitation

This article presents a technique for the structural identification of hysteretic oscillators that are characterized by degradation in stiffness. The main assumption of the proposed procedure is that it is possible to replace the expression of the time derivative of the restoring force with a polynomial approximation, characterized by time-varying coefficients. The work aims at generalizing a method that the authors have proposed for hysteretic nondegrading systems: system parameters are evaluated from instantaneous estimates of the time-varying coefficients. The instantaneous estimation, based on optimization techniques, is made possible through the temporal localization of frequency components, i.e., the representation in the joint time—frequency domain. A numerical application consisting of the instantaneous identification of a Bouc-Wen model with stiffness degradation under earthquake excitation is presented and discussed. Although the sensitivity of the estimation to exogenous noise is greater than in the nondegrading case, the global accuracy of the identification is satisfactory.

Phase Difference in Lateral Synchronization of Pedestrian Floors Using a Modified Hybrid Van der pol/Rayleigh Oscillator

The modified hybrid Van der Pol/Rayleigh (MHVR) oscillator was originally proposed by the authors to model the lateral oscillations of a pedestrian walking on a rigid floor and it was shown that for the autonomous case, the MHVR oscillator can correctly fit the experimental data. The case of a pedestrian walking on a laterally moving floor is modeled by a nonautonomous oscillator. The case of a floor subjected to a harmonic lateral motion has been then studied by the authors, with focus on the amplitude and stability of the entrained response, i.e. the response having the same frequency as that of the given periodic excitation. For the nonautonomous (moving floor) case, the main focus of this paper is on the analysis of the phase difference between the oscillator entrained response and the external excitation. Both analytical and numerical calculations have been performed. The approximate analytical method is the harmonic balance method. Then, the model is used to represent the experimental results for the pedestrian lateral oscillations during walking. Comparison is made for the examples along with discussions.

Discrete approaches for crowd movement modelling

This article is devoted to the modelling of the movements of an assembly of particles. Our aim is to develop a model capable of reproducing the behavior of a crowd of people in walking situations (free motion, emergency evacuation, etc.). The final model must be able to handle local interactions such as pedestrian-pedestrian and pedestrian-obstacle in order to reproduce the global dynamic of pedestrian traffic. Three already existing discrete methods, originally proposed to simulate a granular assembly, are first analyzed and compared. These methods are able to manage collisions between rigid particles. They are then adapted for representing pedestrians together with their willingness to move. Their numerical implementation allows for the performance of simulations in various specific configurations.

A recursive finite element method for computing tyre vibrations

A numerical method is described for computing tyre vibrations over a large frequency range. It is based on a recursive finite element method for building the dynamic stiffness matrix of a complete tyre from the knowledge of a finite element model of a small part of the structure. The present method is compared to full three-dimensional finite element solutions showing a perfect agreement for low frequencies. However, this method allows computations for medium and high frequencies which are needed for the analysis of noise generated by a tyre. The influence of various parameters on the frequency response functions like the positions where the responses are computed, the mechanical parameters of the tyre or the internal air-pressure are described.

On the use of Continuous Wavelet Analysis for Modal Identification

This paper reviews two different uses of the continuous wavelet transform for modal identification purposes. The properties of the wavelet transform, mainly energetic, allow to emphasize or filter the main information within measured signals and thus facilitate the modal parameter identification especially when mechanical systems exhibit modal coupling and/or relatively strong damping.

A Nonlinear Oscillator Model to Generate Lateral Walking Force on a Rigid Flat Surface

This study proposes a single degree of freedom nonlinear oscillator to model the lateral movement of the body center of mass of a pedestrian walking on a flat rigid surface. Experimentally recorded ground reaction force of a dozen of pedestrians in the lateral direction is used to develop the model. In detail, the hardening and softening effects are observed in the stiffness curve as well as higher odd harmonics are present in the frequency spectrum of the lateral force signals. The proposed oscillator is a modification of the Rayleigh and the Van der Pol oscillators with additional nonlinear softening and hardening terms. To obtain an approximation of the limit cycle of the oscillator and its stability, two methods are studied: the energy balance method and the Lindstedt–Poincare perturbation technique. The experimental force signals of pedestrians at four different walking speeds are used for the identification of the values of the model parameters. The results obtained from the proposed model show a good agreement with the experimental ones.

Crowd-Structure Interaction in Laterally Vibrating Footbridges: Comparison of Two Fully Coupled Approaches

Two models that deal with the crowd-structure interaction have been developed. The first is a 1D continuous model and the other is a 2D discrete one. In this paper, a summary of the formulation of these two models is presented. Both approaches used to represent the pedestrian-structure coupling phenomenon are detailed and compared. We start by introducing the partial and ordinary differential equations that govern the dynamics of both the continuous and the discrete models. First, the equation of dynamics of the footbridge for the case of lateral vibrations is recalled. Then, the Kuramoto phase equation is implemented for describing the coupling between the pedestrians and the laterally moving deck of a footbridge. Results obtained from numerical simulations are presented and compared with available experimental data.