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。

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.