Collective Dynamics of Mechanical Systems with Applications to Bridge Modeling

现代的人行桥、悬索桥和其他大型机械结构都是使用行业标准组件设计的，但观察到灾难性的共振振动，需要数百万美元的维修费用。该研究项目的主要目的是对行人集体行为或桥梁承重构件风致相干振动引起的共振振动进行数学建模。这个协同项目旨在开发经过实验验证的非线性理论，该理论将帮助工程师估计(I)行人荷载的动态影响和(Ii)由桥梁悬挂/支撑系统中诱导的集体振荡引起的一系列危险频率。这样的频率无法通过传统的自然频率线性计算来识别，可能会导致错误的、崩溃的设计。这项研究的结果可能会带来更好的安全性和经济效益。该项目专注于一个需要对机械网络进行数学分析和建模的非线性科学领域，包括步行的行人和热闹的桥梁之间的双向相互作用，以及桥梁承重元件在风中引起的振动。这个项目的第一部分寻求开发生物力学启发的行人对桥梁运动的反应模型，以及详细的、但分析上容易处理的人群动力学和相位锁定模型。该项目还试图验证一个假设，即基于脚部放置的横向位置对行人的平衡控制可以在没有人群同步的情况下引发桥梁晃动。该项目的第二部分旨在更好地了解由于风致振动的频率不同于桥梁的固有频率而导致危险的振动和桥梁倒塌的原因。交叉学科的研究运用了应用数学和工程学的方法，包括稳定性和分叉理论、分段光滑和随机动力系统、图论、经典力学和生物力学。

Modern pedestrian and suspension bridges and other large mechanical structures are designed using industry-standard packages, yet disastrous resonant vibrations are observed, necessitating multi-million dollar repairs. The main objective of this research project is to contribute to mathematical modeling of resonant vibrations caused by collective behavior of pedestrians or wind-induced coherent oscillations of load-bearing elements of bridges. This synergistic project aims to develop experimentally-validated nonlinear theory that will help engineers to estimate (i) the dynamical impact of pedestrian loads and (ii) a range of dangerous frequencies due to induced collective oscillations in a bridge's suspension/supporting systems. Such frequencies, which cannot be identified through the conventional linear calculations of natural frequencies, can lead to faulty, collapsing designs. Results from this research may lead to improved safety and economic benefits. This project focuses on an area of nonlinear science entailing mathematical analysis and modeling of mechanical networks, including bidirectional interactions between walking pedestrians and lively bridges and wind-induced oscillations of load-bearing elements of bridges. The first part of this project seeks to develop bio-mechanically inspired models of pedestrians' responses to bridge motion and detailed, yet analytically tractable, models of crowd dynamics and phase-locking. This project also seeks to verify a hypothesis that the balance control of pedestrians based on the lateral position of foot placement can initiate bridge wobbling, without crowd synchronization. The second part of this project aims at better understanding the cause of dangerous vibrations and bridges collapsing as a result of wind-induced oscillations at a frequency different from the natural frequencies of a bridge. The interdisciplinary research utilizes methods from applied mathematics and engineering, including stability and bifurcation theory, piecewise smooth and stochastic dynamical systems, graph theory, classical mechanics, and bio-mechanics.