Nonsmooth modal analysis of complex mechanical structures

在结构动力学中，持续振动可能会导致疲劳机构和构件失效。含有作为未知量(位移和速度)的光滑函数的非线性项的系统的振动分析最近取得了令人信服的进展。然而，对于在其边界上经历单边接触条件的系统来说，情况远非如此。该研究计划的目的是改进对容易发生单面接触的复杂机械结构的振动响应的建模、理解和预测。在连续介质力学的框架内，由于缺少适当的数值工具，这种分析目前是不可用的或不完整的。这是因为解决方案预计将采取行进冲击波的形式，而这种形式很难在数字上捕捉到。 拟议的研究计划将通过开发创新的数值方法来完成，这些方法能够有效地处理非光滑机械系统动态响应中日益增长的复杂性。它旨在通过在一个称为“非光滑模态分析”的统一框架中系统地识别潜在的振动共振发生，来解决振动、疲劳和故障问题。这种方法搜索保守的自治非光滑力学系统的周期解的连续族，在其附近通常产生振动共振。为此，拟议的探索奖助金申请将侧重于两个主要项目： 1)空间中简化的一维系统将被考虑为(1)一端与刚性地基接触的简支杆，(2)两个接触杆共享一个共同的单向接触界面，(3)一个具有大变形项的杆，在公式中结合了多项式和非光滑非线性。 2)在线弹性假设的基础上，发展了三维力学系统的非光滑模态分析。数值实现将基于高维的时域边界元方法和有限体积格式，这些格式在以逐行波为目标时具有非常吸引人的特征。此外，还将开发一种无抖振、节能的有限元技术。 这一研究计划的长期目标是提供有效的数值解方法，能够帮助工程师努力设计在可能有害的单边和非光滑条件下运行的始终更有效和更安全的机械系统。航空航天工业系统将作为首选应用。一个非常重要的例子是现代透平机械中叶片的设计不可避免地容易与周围的机壳接触。这样的叶片应该在远离非线性振动共振的情况下安全运行。

In structural dynamics, sustained vibration occurrences may lead to fatigue mechanisms and component failures. The vibration analysis of systems involving nonlinear terms that are smooth functions of the unknowns (displacement and velocity) has recently made convincing advances. However, this is far from being true for systems undergoing unilateral contact conditions on their boundary. The objective of this research program is to provide improved modeling, understanding, and prediction of the vibratory responses of complex mechanical structures prone to unilateral contact conditions. In the framework of continuum mechanics, such analysis is currently unavailable or incomplete as the appropriate numerical tools are missing. This is explained by the fact that solutions are expected to take the form of travelling shock waves which are challenging to capture numerically. The proposed research program will be accomplished by developing innovative numerical methods capable of efficiently tackling the ever-growing complexities in the dynamic responses of nonsmooth mechanical systems. It is designed to address vibration, fatigue and failure issues by systematically identifying potential vibratory resonance occurrences in a unified framework called "Nonsmooth Modal Analysis". This method searches for continuous families of periodic solutions of conservative autonomous nonsmooth mechanical systems in the vicinity of which vibratory resonances commonly arise. To this end, the proposed Discovery Grant application will focus on two major items: 1) Simplified one-dimensional systems in space will be considered such as (1) a simple bar with one end in contact with a rigid foundation, (2) two contacting bars sharing a common unilateral contact interface, and (3) one bar with large distortion terms combining polynomial and nonsmooth nonlinearities within the formulation. 2) Nonsmooth Modal Analysis will be developed to tackle mechanical systems in three dimensions with the assumption of linear elasticity. The numerical implementation will be grounded on a higher-dimensional version of the Time-Domain Boundary Element Method and Finite Volumes schemes which enjoy highly appealing features when progressive waves are targeted. A chattering-free and energy-preserving Finite Element technique will also be developed. The long-term goal of this research program is to provide efficient numerical solution methods able to help engineers in their endeavour to design always more efficient and safer mechanical systems operating in possibly hostile unilateral and nonsmooth conditions. Aerospace industrial systems will serve as preferred applications. A very important example is the design of blades inevitably prone to contact occurrences with surrounding casings in modern turbomachines. Such blades should safely function away from nonlinear vibratory resonance conditions.