Vibrational energy distributions in large built-up structures - a wave chaos approach

预测汽车或飞机等大型复杂机械系统对高频振动的响应是一项非常困难的任务。然而，对这类结构中振动能量的分布，包括子部件之间的耦合、阻尼和声辐射形式的能量损失，获得良好的估计对工程师来说是非常重要的。对低振动、低噪音产品的需求不断增加，以满足性能规范和减少噪声污染，这使得预测振动响应特性方面的任何改进都对工业应用具有直接意义。对改进的虚拟原型的需求，而不是使用昂贵和耗时的物理原型，是降低开发成本和时间尺度的另一个应用领域。数值化工具通常基于有限元分析。虽然这些方法在低频区域工作得很好，即处理系统大小数量级的波长，但在中高频区域，它们的计算成本太高。特别是，有限元不能准确地描述所谓的中频问题，在中频问题中，子分量的特征是波长变化很大。尽管有限元适用于处理汽车车身骨架等“刚性”部件，但它不能常规地通过耦合到刚性部件的柔性薄板等“软”部件来捕捉能量传输。预测短波长分量振动贡献的一个常用数值工具是统计能量分析(SEA)；然而，它基于一组限制性假设，到目前为止，这些假设往往很难控制，通常只在高频极限和低阻尼时满足。因此，SEA不能提供使其对工业研发部门中更广泛的最终用户社区具有吸引力所需的可靠性程度。这里提出，波或量子混沌的数学工具可以极大地改善上述情况。Pi Tanner的最新结果表明，通过结合算子论、动力系统理论和小波长渐近论的方法，SEA可以嵌入到更一般的理论中。新方法基于全格林函数关于射线的半经典展开式，并用线性算子来描述非线性射线动力学。由此产生的方法捕捉到了射线动力学中的全部关联，并且与SEA相比，其有效性范围有了很大的改进。这种方法可以彻底改变复杂机械系统中的振动处理方法。它不仅允许(I)给出SEA(海洋用户感兴趣的)适用性的量化界限；它还将(Ii)在SEA不以适度的计算开销应用的情况下提高预测能力；此外，(Iii)它可以很容易地与有限元方法相结合，从而使其成为解决中频问题的理想候选者。通过从半经典方法开始，所作的近似得到了很好的控制，这使得(Iv)系统地将波浪干扰效应(在标准海面处理中没有)纳入该方法中。通过解决上述问题，我们将能够基于先进的数学方法向工程界提供改进的、在概念上全新的解决方法。这项拟议的研究是从EPSRC以跳板奖学金的形式提供的优质资金演变而来的。因此，该项目缺乏跨学科性质，将由Pi Tanner(诺丁汉，数学)和Pi Mace(南安普顿，ISVR，工程)与来自有限元/软件方面的工业合作伙伴(InuTech)和一家提供关于最终用户需求的工程咨询公司(DS2L)共同处理。

Predicting the response of a large, complex mechanical system such as a car or an aeroplane to high frequency vibrations is a remarkably difficult task. Still, obtaining good estimates for the distribution of vibrational energy in such structures, including coupling between sub-components, damping and energy loss in form of acoustic radiation, is of great importance to engineers. An increasing demand for low vibration, low noise products to meet performance specifications and to reduce noise pollution makes any improvement in predicting vibrations response characteristics of immediate interest for industrial applications. Demand for improved virtual prototyping, as opposed to the use of expensive and time-consuming physical prototypes, is another area of application in reducing development costs and time scales. Numerical tools are often based on 'Finite Element Analysis' (FEM). While these methods work well in the low frequency regime, that is, tackling wavelengths of the order of the size of the system, they become too expensive computationally in the mid-to-high frequency regime. In particular, FEM fails to describe accurately so-called mid-frequency problems where sub-components are characterized by a wide variation of wave-lengths. While FEM is suitable for handling 'stiff' elements such as the body frame in a car, it cannot routinely capture energy transport through 'soft' components such as thin, flexible plates coupled to stiff components. A common numerical tool for predicting the vibrational contribution of short wave length components is Statistical Energy Analysis (SEA); it is, however, based on a set of restrictive assumptions which, so far, are often hard to control and generally only fulfilled in the high frequency limit and for low damping. Thus, SEA can not deliver the degree of reliability necessary to make it attractive for a wider end user community in industrial R & D departments. It is suggested here that mathematical tools from wave or quantum chaos can considerably improve the situation sketched above. Recent results by the PI Tanner show that by combining methods ranging from operator theory, dynamical systems theory and small wavelength asymptotics, SEA can be embedded into a more general theory. The new approach is based on semiclassical expansions of the full Green function in terms of rays and describing the nonlinear ray-dynamics in terms of linear operators. The resulting method captures the full correlations in the ray dynamics and has such a much improved range of validity compared to SEA. The method could revolutionise the treatment of vibrations in complex mechanical systems. Not only does it allow (i) to give quantitative bounds for the applicability of SEA (of interest to SEA users); it will also (ii) improve predictive capability in situation where SEA does not apply at a moderate computational overhead; in addition, (iii) it can be easily combined with FEM methods thus making it an ideal candidate for tackling mid-frequency problems. The approximations made are well controlled by starting from a semiclassical approach which makes it possible (iv) to systematically incorporate wave interference effects (absent in standard SEA treatments) into the method.By tackling the issues addressed above we will be able to provide improved and conceptually completely new solution methods to the engineering community based on advanced mathematical methods. The proposed research evolved out of pump-prime EPSRC funding in terms of a Springboard Fellowship. The project is thus by default of interdisciplinary nature and will be tackled jointly by the PI Tanner (Nottingham, Mathematics) and PI Mace (Southampton, ISVR, Engineering) with industrial partners from the FEM/software side (inuTech) and an engineering consulting firm (DS2L) providing input about end-user demands.