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振动分析是未知数（位移和速度）的光滑函数，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，情况远非如此。本研究计划的目的是提供改进的建模，理解和预测容易发生单边接触条件的复杂机械结构的振动响应。在连续介质力学的框架下，由于缺乏适当的数值工具，这种分析目前是不可用的或不完整的。这可以通过这样一个事实来解释，即解决方案预计采用传播冲击波的形式，这在数值上是具有挑战性的。******提出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复杂的问题。它的目的是解决振动，疲劳和失效问题，通过系统地识别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机械系统周期解的连续族，其附近通常会出现振动共振。为此，提出的发现基金申请将侧重于两个主要项目：***1)将考虑空间中简化的一维系统，例如(1)一端与刚性基础接触的简单杆，(2)两个接触杆共享一个共同的单边接触界面，以及(3)一个杆具有大畸变项，结合公式中的多项式和非光滑非线性。***2)将发展非光滑模态分析，以在线性弹性假设下处理三维机械系统。数值实现将以时域边界元方法的高维版本和有限体积方案为基础，当目标是渐进波时，它们具有非常吸引人的特征。一种无抖振和节能的有限元技术也将得到发展。******这项研究计划的长期目标是提供有效的数值解决方法，能够帮助工程师在他们的努力中设计更有效和更安全的机械系统，在可能敌对的单边和非光滑条件下运行。航空航天工业系统将成为首选应用。一个非常重要的例子是，在现代涡轮机械中，叶片的设计不可避免地容易与周围的机壳发生接触。这种叶片应该在非线性振动共振条件下安全地工作。

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.