Moving Force Model Research Articles

Study on Coupled Vibration of Human-plate System under Pedestrian Excitation

In the paper, in order to explore the influence of human-structure interaction (HSI) in the vertical direction on the dynamic response of floors due to human-induced excitation, the formula for the human-plate interaction system is deduced. The pedestrian is in turn modelled as moving force (MF), spring-mass-damper (SMD), and added mass-spring-mass-damper (MSMD). The floor is modelled using a formulation in modal coordinates. Aiming at the formula of human-plate interaction system, the state space method is used to get the instantaneous frequency and instantaneous damping ratio of the floor under walking excitation, and the acceleration responses can be obtained using the Runge-Kutta method. Furthermore, this work studies the influence of pedestrian mass and structural modal mass on structural vibration response. The results show that the floor frequency is observed to decrease and the damping ratio gives the opposite trend when the pedestrian enters the floor. The MF model overestimates the measured response, while the actual results can be well approximated by the results for SMD and MSMD models considering HIS. Accounting for increasing human mass and decreasing modal mass lead to significant increase in the dynamic properties and the structural response.

An equivalent moving force model for consideration of human-structure interaction

• Human-structure interaction (HSI) is incorporated into a moving force model. • A simplified analytical approach is developed for determining the equivalent system properties. • A more comprehensive numerical approach is used to determine the optimum equivalent system properties. • Equivalent damping expressions are proposed to be used in vibration serviceability assessments of structures. • Accurate vibration assessment can now be performed with currently available commercial engineering software. To predict the vibration response of footbridges, many codes of practice use a deterministic moving force (MF) model. This approach may not be well suited for the design of slender, lightweight, low-damping, and low-frequency footbridges because it ignores the pedestrian interaction with the vibrating footbridge. On the other hand, a spring-mass-damper (SMD) model is able to incorporate human mass, stiffness, and damping into the vibration response prediction. However, the SMD model is computationally demanding and not commonly available in engineering practice. To address this shortfall, a framework is proposed to derive a computationally-efficient equivalent MF-structure system to the reference SMD-structure system such that both systems give a similar vibration response metric. Analytical and numerical approaches to the equivalent MF (EMF) system are described in detail and applied to bridges with approximately simply-supported mode shapes. A sensitivity analysis is carried out to show the effects of different pedestrian parameters on the equivalent damping of the EMF system. The effects of pedestrian damping, frequency, and weight are found to be pronounced, while those of dynamic load factors and pedestrian step length are insignificant. Finally, empirical expressions are proposed in a probabilistic framework to determine the equivalent damping for simply-supported low-frequency footbridges as a function of bridge frequency. This work should find use in the serviceability assessment of low-frequency footbridges in engineering practice.

Damping and frequency of human-structure interaction system

Human presence can change the dynamic characteristics of human-structure interaction systems, i.e. particularly their damping and frequency. In many design codes, the pedestrian is regarded as a moving force (MF) while a more complete model, referred to as moving spring-mass-damper (MSMD) has received attention recently. Unlike the MF model, the MSMD model is able to take into account the human damping and stiffness effects. This paper is devoted to determining the instantaneous damping and frequency of the human-structure interaction (HSI) system subjected to a single and multiple pedestrians. Each pedestrian is modelled as a MSMD, and a methodology for determining the system damping and frequency properties is described. A simply supported beam structure is considered and modelled in modal space. The imparted pedestrian vertical force is modelled adopting the first four harmonics of Fourier representation using Young’s dynamic load factors. It is concluded that human presence can significantly increase and decrease damping and frequency of the structure, respectively and it is so important to consider human effect on the system’s damping and frequency in serviceability assessment of footbridges.

AGV-induced floor micro-vibration assessment in LCD factories by using a regressional modified Kanai-Tajimi moving force model

This study explores the floor micro-vibrations induced by the automated guided vehicles (AGVs) in liquid-crystal-display (LCD) factories. The relationships between moving loads and both the vehicle weights and speeds were constructed by a modified Kanai-Tajimi (MKT) power spectral density (PSD) function whose best-fitting parameters were obtained through a regression analysis by using experimental acceleration responses of a small-scale three-span continuous beam model obtained in the laboratory. The AGV induced floor micro-vibrations under various AGV weights and speeds were then assessed by the proposed regressional MKT model. Simulation results indicate that the maximum floor micro-vibrations of the target LCD factory fall within the VC-B and VC-C levels when AGV moves at a lower speed of 1.0 m/s, while they may exceed the acceptable VC-B level when AGV moves at a higher speed of 1.5 m/s. The simulated floor micro-vibration levels are comparable to those of typical LCD factories induced by AGVs moving normally at a speed between 1.0 m/s and 2.0 m/s. Therefore, the numerical algorithm that integrates a simplified sub-structural multi-span continuous beam model and a proposed regressional MKT moving force model can provide a satisfactory prediction of AGV-induced floor micro-vibrations in LCD factories, if proper parameters of the MKT moving force model are adopted.

Dynamic Response of an Elastic-Supported Bridge Beam Subjected to Velocity-Varied Moving Loads

Dynamic response of an elastic-supported bridge under speed-varied moving loads was investigated. A mathematical model of vehicle-bridge coupled oscillation for an elastic-supported bridge was built up by means of 1/4 vehicle model (Mass-Spring-Mass) and Euler-Bernoulli beam theory. And then dynamic equations of vehicle-bridge coupled oscillation in matrix form were established using two former orders general coordinates of an elastic-supported beam and model superposition method. The influences of vehicle-bridge coupled vibration model, elastic-supported stiffness, entrance speeds and acceleration /deceleration of moving loads on the dynamic responses of bridges were studied. Vehicle-bridge coupled vibration model based on 1/4 vehicle model can more accurately describe the dynamic characters of bridges than that based on constant moving force model. Elastic-supported stiffness only has an impact on the fluctuation amplitudes of dynamic responses. The vehicle-induced impact factor is dependent on the entrance speeds, acceleration/deceleration of moving loads and elastic-supported stiffness.

Comportamiento dinámico de viaductos cortos considerando la interacción vehículo-vía-estructura-suelo

The dynamic vehicle-track-bridge-soil interaction is studied on high speed lines. The analysis is carried out using a general and fully three dimensional multi-body-finite element-boundary element model, formulated in the time domain to predict vibrations due to the train passage over the bridge. The vehicle is modelled as a multi-body system, the track and the bridge are modelled using finite elements and the soil is considered as a homogeneous half-space by the boundary element method. Usually, moving force model and moving mass model are employed to study the dynamic response of bridges. In this work, the multi-body system allows one to consider the quasi-static and dynamic excitation mechanisms. Soil-structure interaction is taken into account on the dynamic structure behaviour on simply-supported short span bridges. The influence of soil-structure interaction is analysed in both resonant and non-resonant regimes.

A substructure approach tailored to the dynamic analysis of multi-span continuous beams under moving loads

The paper deals with the dynamic analysis of multiply supported continuous beams subjected to moving loads, which in turn can be modelled either as moving forces or moving masses. A dedicated variant of the component mode synthesis (CMS) method is proposed in which the classical primary–secondary substructure approach (SA) is tailored to cope with slender (i.e. Euler–Bernoulli) continuous beams with arbitrary geometry. To do this, the whole structure is ideally decomposed in primary and secondary spans with convenient restraints, whose exact eigenfunctions are used as assumed local modes; the representation of the internal forces is improved with the help of two additional assumed modes for each primary span, while primary–secondary influence functions allow satisfying the kinematical compatibility between adjacent spans; the continuous beam is then re-assembled, and the Lagrange's equations of motion are derived in a compact block-matrix setting for both moving force and moving mass model. Numerical examples demonstrate accuracy and efficiency of the proposed procedure. An application with a platoon of high-speed moving masses confirms that the inertial effects neglected in the moving force model may have a significant impact in the structural response.

Structural analysis of 3D high-speed train–bridge interactions for simple train load models

Three-dimensional models are developed for analysing the dynamic interaction that occurs between high-speed trains and bridges. The reliability and accuracy of developed models are verified by comparing the results from analysing field tests on high-speed trains. A number of train load models are proposed and their performances are compared in order to identify possible models that would reduce the computational and modelling efforts while maintaining suitable accuracy. The results show that at least 16 cars out of a 20-car train should be modelled to achieve results that are comparable to those obtained using the highly detailed 20-car model. Regarding the simplified train load model, more accurate results are obtained employing the 3D moving vehicle model for power cars, the heaviest cars of a high-speed trainset, and a moving force model for other cars, power passenger cars, and passenger cars, compared with highly detailed 20-car model.

On the dynamic response of rectangular plate, with moving mass

In this article, the dynamics of plates under influence of relatively large masses, moving along an arbitrary trajectory on the plate surface is considered. The method consists of transformation of the governing equation into a series of eigenfunctions, which satisfy the boundary conditions of the plate. The method presented in this investigation is general and can be applied to general moving mass and moving force systems as well. Furthermore, the article shows that the response of structures due to moving mass, which has often been neglected in the past, must be properly taken into account because it often differs significantly from the moving force model.

Vibration analysis of bridges under moving vehicles and trains: an overview

AbstractVibration of bridges under moving vehicles and trains is of great theoretical and practical significance in civil engineering. In early studies, a moving‐force model was used where the inertia of the vehicle was small compared with that of the bridge, whereas a moving‐mass model was used where the inertia of the vehicle could not be neglected. During the last three decades, the interaction problem between moving vehicles and bridge structures has attracted much attention. The advent of high‐speed digital computers has made it possible to analyse the interaction problem with more sophisticated bridge and vehicle models. This paper presents an overview of the recent research work on vibration analysis of various types of bridges under the action of moving vehicles and trains. Some remarks are made on the modelling of bridges, vehicles and analysis methods.

Dynamic response of bridges under travelling loads

In spite of a number of analytical and experimental investigations on the dynamic response of bridges to moving vehicle loads, the controlling parameters that govern the response have not been clearly identified. This has, in turn, inhibited the development of rational design procedures. Based on an analytical investigation of the response of a simplified beam model traversed by a moving mass, the present study identifies the governing parameters. The results clearly show why attempts to correlate the response to a single parameter, either the span length or the fundamental frequency, are unsuccessful. Simple design procedures are developed based on relationships between the speed ratio, the weight ratio, and the dynamic amplification factors; and a set of design curves are provided. Key words: dynamic response of bridges, vehicle–bridge interaction, moving force model, moving sprung mass model, dynamic amplification factor.

Numerical Solution for Response of Beams with Moving Mass

An analytical‐numerical method is presented that can be used to determine the dynamic behavior of beams, with different boundary conditions, carrying a moving mass. This article demonstrates the transformation of a familiar governing equation into a new, solvable series of ordinary differential equations. The correctness of the results has been ascertained by a comparison, using finite element models, and very good agreement has been obtained. Furthermore, the article shows that the response of structures due to moving mass, which has often been neglected in the past, must be properly taken into account because it often differs significantly from the moving force model.