Nonsmooth modal analysis of complex mechanical structures

在结构动力学中，持续的振动可能导致疲劳机制和部件失效。涉及非线性项的系统的振动分析是未知量（位移和速度）的光滑函数，最近取得了令人信服的进展。然而，这是远远不正确的系统在其边界上经历单边接触条件。本研究计划的目的是提供更好的建模，理解和预测的复杂机械结构的振动响应容易单侧接触条件。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。这是解释的事实，解决方案预计将采取的形式，这是具有挑战性的数字捕捉的行波。 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解决非光滑机械系统动态响应中日益增长的复杂性。它的目的是解决振动，疲劳和故障问题，系统地确定潜在的振动共振发生在一个统一的框架称为“非光滑模态分析”。该方法搜索保守自治非光滑力学系统的连续族周期解，在该系统附近通常会出现振动共振。为此，拟议的探索补助金申请将集中在两个主要项目上： 1)将考虑空间中的简化一维系统，例如（1）一端与刚性基础接触的简单杆，（2）两个接触杆共享共同的单侧接触界面，以及（3）一个具有大畸变项的杆，该大畸变项结合了公式中的多项式和非光滑非线性。 2)非光滑模态分析将被开发来处理线性弹性假设下的三维机械系统。数值实现将基于时域边界元法和有限差分格式的高维版本，当以行波为目标时，这些格式具有非常吸引人的功能。还将开发一种无颤振和能量保持的有限元技术。 该研究计划的长期目标是提供有效的数值求解方法，能够帮助工程师努力设计在可能的敌对单边和非光滑条件下运行的更有效，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，在现代涡轮机中，叶片的设计不可避免地倾向于与周围壳体接触。这样的叶片应该安全地运行远离非线性振动共振条件。

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.