New Ways Forward for Nonlinear Structural Dynamics

结构动力学（或振动理论）是工程学中最重要的领域之一。了解振动对新的设计标准和技术至关重要；它是设计更轻、更环保、更安全的下一代建筑的关键推动者。动力学面临的一个“重大挑战”是非线性的性质。不幸的是，几乎所有的实际结构在某种程度上都是非线性的，并且高度抵制数学分析，因为数学是建立在线性基础上的。尽管工程师们通过使用近似值和计算机能力取得了进展，但他们一直无法从结构运动方程的精确解中获得洞察力，因为这些方程无法用传统方法求解。同样的问题意味着，往往不可能证明精确解的存在，或者是唯一的。这里的研究项目的首要目标是通过非传统方法找到精确的结果；基于PI在非线性动力学和现代机器学习方面30多年的经验，使用最先进的机器学习/进化搜索方法。由于近似和计算在一定程度上消除了对精确解的需要，工程师们转向了一个更紧迫的问题——首先找到运动方程。这通常是不可能从第一性原理，因为未知的物理连接过程（如焊接），模糊了所有的分析，但最简单的建筑结构。通过开发“系统识别”（SI）方法解决了这个问题，从测量数据中推断出所需的方程。同样，线性系统首先被“解决”。虽然线性SI在技术上有一定的困难，但经过五十年的发展，它已经在工作理论和实践中建立起来，可供工程师们利用。可以说线性系统最强大的技术是“模态分析”；这种方法具有一种看似不可思议的性质，即涉及许多耦合动力系统的问题可以简化为一组不耦合的问题，每个问题都涉及在自己的弹簧上振荡的单个质量。不幸的是，在精确解的情况下，模态分析不能推广到非线性系统。由于缺乏基础的通用技术，工程师们被迫开发一种“工具箱”哲学，即不同类型的非线性系统需要不同的非线性SI （NLSI）方法。虽然已经有了通用方法的暗示，但没有一种技术已经成为“统治所有技术的一环”。一些版本的非线性模态分析已经发展，但没有表现出所有的期望性质的线性理论。该计划的第二个目标是为NLSI创建一个完全通用的框架，该框架可以导出运动方程以及其预测的统计置信度。该计划还将考虑解耦非线性系统的新方法-看待非线性模态分析的新方法。本文的研究将为非线性动力学的发展提供新的途径。寻找运动方程的新的和一般的方法将得到发展。给定这些方程，该程序将提供解决它们的新方法；对以前从未解决过的问题的精确解决，也没有使用分析方法解决的前景。问题将包括：非线性微分方程的精确解；非线性系统到线性系统的精确和近似变换，以及多元系统的精确和近似解耦（非线性模态分析）。创造一种期望找到精确解的研究文化，是思考非线性动力学的一种真正的新方式。在某些方面，新的精确解将与发现动物学新物种一样重要；通过解剖它们，我们可以在整个学科中推进知识。

Structural Dynamics (or the Theory of Vibrations), is one of the most important fields of Engineering. Understanding vibrations is vital for new design standards and technologies; it is a key enabler in the design of lighter, greener and safer future-generation structures. A 'Grand Challenge' faced by dynamics is the property of nonlinearity. Unfortunately, almost all real structures are nonlinear to some extent, and highly resistant to mathematical analysis, because mathematics has been built on linear foundations. Although engineers have made progress by using approximations and computer power, they have been denied the insight that comes from exact solutions of structural equations of motion, because those equations have been impossible to solve using traditional methods. The same issue means that it is often impossible to prove that exact solutions even exist, or are unique. The first aim of the programme of research here is to find exact results by non-traditional methods; using state-of-the-art machine learning/evolutionary search methods, based on the PI's 30+ years of experience in nonlinear dynamics and modern machine learning.Because approximation and computation partly removed the need for exact solutions, engineers turned to a more immediately pressing problem - that of finding equations of motion in the first place. This is often impossible from first principles because the unknown physics of joining processes (e.g. welding), obscures the analysis of all but the simplest built-up structures. The problem was solved by developing 'system identification' (SI) methods, where the required equations were inferred from measured data. Again, linear systems were 'solved' first. Although linear SI proved to have technical difficulties, after fifty years of development, it is now established in working theory and practice which engineers can exploit. Arguably the most powerful technology for linear systems is that of 'modal analysis'; this method has the seemingly miraculous property that problems involving many coupled dynamical systems can be reduced to a set of uncoupled problems, each involving a single mass oscillating on its own spring. Unfortunately - as in the case of exact solutions - modal analysis does not generalise to nonlinear systems. Lacking an underpinning general technology, engineers have been forced to develop a 'toolbox' philosophy, whereby different types of nonlinear systems require different nonlinear SI (NLSI) methods. Although there have been hints at general approaches, no one technology has emerged as 'the one ring to rule them all'. Some versions of nonlinear modal analysis have been developed, but none exhibit all the desirable properties of the linear theory. The second aim of this programme will be to create a completely general framework for NLSI, which can derive equations of motion together with statistical confidences in their predictions. The programme will also consider new approaches to decoupling nonlinear systems - new ways of looking at nonlinear modal analysis.The research here will provide very new ways forward in nonlinear dynamics. New and general ways of finding equations of motion will be developed. Given the equations, the programme will provide new ways to solve them; exact solutions to problems which have never been solved before and do not have the prospect of solution using analytical methods. Problems will include: exact solution of nonlinear differential equations; exact and approximate transformation of nonlinear systems into linear ones, and the exact and approximate decoupling of multivariate systems (nonlinear modal analysis). Creating a research culture with an expectation of finding exact solutions is a truly new way of thinking about nonlinear dynamics. In some ways, new exact solutions will be as important as the discovery of new species in zoology; by dissecting them, one can advance knowledge in the whole subject.