MODELLING OF HUMAN INDUCED FORCE ON STRUCTURES

Abstract

This research focuses to the development of self-sustained oscillator which can generate reliably the contact force between feet of a healthy pedestrian and the supporting flat rigid surface during walking. The thesis dealt with some issues of modelling the self-sustained oscillator from measured experimental force data on instrumented treadmill. In detail, the following issues were focused: (i) To derive an appropriate basic self-sustained oscillator(s) and additional nonlinear terms from the analysis of experimentally measured walking force data of a dozen of pedestrians at different walking speeds; (ii) To develop a self-sustained model for pedestrian to generate lateral walking contact force on a rigid flat surface; (iii) To develop a self-sustained model for pedestrian to produce dynamic vertical contact force during walking; (iv) To develop a self-sustained model for pedestrian to generate longitudinal contact force on a rigid flat surface; (v) To determine the stability of the proposed oscillators; and (v) To identify the optimal values of model parameters from the measured data. The modelling approach is motivated by the self-sustained nature of the walking process, i.e. a pedestrian generates the required inner energy to sustain its repetitive body motion. In detail, such an oscillator should be able to produce three phenomena, i.e. (i) self-sustained walking motion; (ii) periodic force signal; and (iii) stable limit cycle. The self-sustained character entails that the autonomous oscillator has a natural amplitude and frequency, representing the natural frequency and amplitude of the motion. The well-known oscillators such as Rayleigh and Van der Pol are self-sustained oscillators. Therefore, human walking motion can be modelled with the help of basic self-sustained oscillators and some additional terms. Human is not a robot so measured walking force signals, due to inter and intra cycle variability, are not exactly periodic but close to periodic. Therefore the walking signal can be represented as truncated Fourier series. The proposed oscillators describe human body as a fictitious mechanical system that is able to replicate dominant Fourier harmonics observed in the experimentally measured walking force records. For single degree of freedom oscillator whole mass is assumed to be concentrated on center of mass (CoM). Fourier force signal acceleration, velocity and displacement of the oscillator motion can be extracted. The motion of the CoM of pedestrian follows x-y-z coordinate, in longitudinal, lateral and vertical directions, respectively. vi From the analysis of the experimental force records as well as the reconstructed force pattern of the extracted dominant Fourier harmonics, it is found that there are odd/even harmonics, symmetry/asymmetry about a point as well as softening and hardening behavior in the restoring force curves. Therefore, a particular model was deduced to satisfy these criteria. For modelling the oscillator in lateral motion different characteristics of lateral walking force are analyzed from time history plot of force, force-velocity and force-displacement plot. These plots reveal that lateral walking force is periodic, symmetric about a point (origin), has nonlinear damping and shows hardening and softening effects. The Fourier spectra of lateral walking force shows the presence of only odd harmonics in signal. Therefore all these properties must be produced by the proposed oscillator. Satisfying these conditions, two self-sustained oscillators are proposed. First one contains third order nonlinear term whereas second one contains fifth order nonlinear terms. Hybrid of Van der Pol and Rayleigh oscillators is the basis for the lateral walking self-sustained oscillators. Hybrid of these two oscillators can cover some required feature in the proposed oscillator such as symmetry about point and periodicity. However, hardening and softening effects and number of harmonics present in experimental data cannot be covered by these hybrid oscillators. To introduce these required properties in the oscillator different nonlinear terms are added. Since lateral walking force is symmetric and have odd harmonic; it reveals that the added terms must be of odd order as even order terms break symmetry and produce even harmonics. To produce the required hardening and softening effects different nonlinear terms are analyzed. In the first model two nonlinear terms 3 y (Duffing oscillator term) and 2 yy with negative coefficients are used to produce softening and hardening effects, respectively. The resulted modified hybrid Van der Pol-Duffing- Rayleigh oscillator produces force pattern close to the measured lateral walking force. However, to produce more odd harmonics accurately, higher order terms are added. Therefore, in the second model two nonlinear terms 4 yy and 5 y are used; negative coefficient of 5 y produces softening effect. Dynamic analysis of both the oscillators is performed using energy balance approach and Lindstedt Poincare perturbation method. Model parameters are expressed in terms of measured quantity and also evaluated using least square optimization process. Both models give satisfactory results on rigid floor. vii Analyzing the different aspects of measured vertical walking force such as Fourier spectra, force time history, stiffness and damping plots, it is inferred that the vertical walking force has both the odd and even harmonics, absence of point symmetry and presence of nonlinear damping. Similar to the lateral walking model, hybrid of Van der Pol and Rayleigh oscillators is the basis for the vertical walking self-sustained oscillator. To produce even harmonics two even order terms 2 z and 2 2 z z are used, these terms also break symmetry about point, whereas for odd harmonics 3 z and 5 z are used. Energy balance approach and Lindstedt Poincare perturbation technique is used for dynamic stability of the oscillator. Relation between measured quantity and model parameters are also expressed. For identification of model parameters of 12 pedestrian data at 10 walking speed, least squares identification technique is used. Model results have good agreement with Fourier results. Analysis of measured force in the longitudinal direction reveals that longitudinal walking force has both the odd and even harmonics, no any point of symmetry and presence of nonlinear damping. Rayleigh oscillator is the basis for the longitudinal walking selfsustained oscillator. To produce even harmonics two even order terms 2 x and xx are used, these terms also break symmetry about point. Energy balance approach and Lindstedt Poincare perturbation technique is used for the analysis of the dynamic stability of the oscillator. Relation between measured quantity and model parameters are also expressed. For identification of model parameters of 12 pedestrian data at 10 walking speed, the least squares identification technique is used. The model results have good agreement with the experimentally extracted Fourier results. All the oscillators proposed here are developed for a rigid floor case as experimentally measured data are collected on rigid floor case. For vibration amplitude less than the amplitude of limit cycle, gait pattern does not alter too much; therefore, oscillators developed on rigid floor can be extended to slightly oscillating floor.