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。了解振动对于新的设计标准和技术至关重要；它是设计更轻、更环保和更安全的未来一代结构的关键因素。动力学面临的“巨大挑战”是非线性的特性。不幸的是，几乎所有的真实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和计算机能力取得了进展，但他们无法从结构运动方程的精确解中获得洞察力，因为这些方程已经不可能用传统方法求解。同样的问题意味着，通常不可能证明确切的解决方案存在，或者是唯一的。这里的研究计划的第一个目标是通过非传统方法找到准确的结果；使用最先进的机器学习/进化搜索方法，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。由于近似和计算部分消除了对精确解的需要，工程师们首先转向一个更紧迫的问题--寻找运动方程。从基本原理上讲，这通常是不可能的，因为连接过程(例如焊接)的未知物理学掩盖了除了最简单的组合结构之外的所有分析。这个问题是通过发展“系统识别”(SI)方法解决的，其中所需的方程是从测量数据中推断出来的。再一次，线性系统首先被“解决”。虽然线性SI被证明有技术上的困难，但经过50年的发展，它现在已经建立在工作理论和实践中，供工程师们开发。可以说，对于线性系统来说，最强大的技术是“模态分析”；这种方法具有看似神奇的特性，即涉及许多耦合动力系统的问题可以归结为一组不耦合的问题，每个问题都涉及单个质量在其自身的弹簧上振动。不幸的是--就像精确解的情况一样--模态分析不能推广到非线性系统。由于缺乏支撑的通用技术，工程师们被迫开发了一种“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。尽管已经有迹象表明有普遍的方法，但没有一种技术出现，成为“统治所有方法的唯一环”。已经发展了一些版本的非线性模态分析，但没有一个版本表现出线性理论的所有理想特性。该计划的第二个目标将是为非大规模集成电路创建一个完全通用的框架，该框架可以推导出运动方程，并在其预测中获得统计上的置信度。该计划还将考虑解耦非线性系统的新方法--观察非线性模态分析的新方法。这里的研究将为非线性动力学提供非常新的方法。寻找运动方程的新的和通用的方法将被开发出来。给定这些方程，该程序将提供新的求解方法；对以前从未解决过、也不可能使用解析方法求解的问题的准确解决方案。问题包括：非线性微分方程组的精确解；非线性系统到线性系统的精确和近似变换，以及多变量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决方案的研究文化，是一种真正的非线性动力学思维的新方式。在某些方面，新的精确解决方案将与动物学中新物种的发现一样重要；通过剖析它们，一个人可以推进整个学科的知识。

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.