Gradient-Based Optimization in Nonlinear Structural Dynamics
Total Page:16
File Type:pdf, Size:1020Kb
Downloaded from orbit.dtu.dk on: Dec 21, 2017 Gradient-based optimization in nonlinear structural dynamics Dou, Suguang; Jensen, Jakob Søndergaard Publication date: 2015 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Dou, S., & Jensen, J. S. (2015). Gradient-based optimization in nonlinear structural dynamics. DTU Mechanical Engineering. (DCAMM Special Report; No. S184). General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Gradient-based optimization in nonlinear structural dynamics PhD Thesis Suguang Dou DCAMM Special Report No. S184 April 2015 Gradient-based optimization in nonlinear structural dynamics by Suguang Dou SOLIDDEPT. OF MECHANICS MECHANICAL ENGINEERING Dept.Solid Mechanics of Mechanical Engineering TECHNICALTECHNICAL UNIVERSITYUNIVERSITY OFOF DENMARK DENMARK Title of the thesis: Gradient-based optimization in nonlinear structural dynamics Ph.D. student: Suguang Dou E-mail: [email protected] E-mail: [email protected] Address: Section of Solid Mechanics Department of Mechanical Engineering Technical University of Denmark Nils Koppels Alle,´ Building 404 2800 Kgs. Lyngby Denmark Supervisor: Jakob Søndergaard Jensen E-mail: [email protected] Address: CAMM – Centre for Acoustic-Mechanical Micro Systems Department of Electrical Engineering Technical University of Denmark Nils Koppels Alle,´ Building 352 2800 Kgs. Lyngby Denmark Copyright © 2015 Suguang Dou Preface This thesis is submitted in partial fulfillment of the requirements for obtaining the Ph.D. degree in mechanical engineering at the Technical University of Den- mark (DTU). The Ph.D. project was funded by European Research Council (ERC) through the project “Integrated Analysis & Design in Nonlinear Dynamics (INN- ODYN) ”. The project was carried out in Department of Mechanical Engineer- ing at DTU from May 1st, 2012 to April 30th, 2015. Professor Dr. techn. Jakob Søndergaard Jensen was the supervisor on the project. Part of the work was carried out during a five-month visit to Professor Dr. Steven W. Shaw and his research group at Michigan State University (MSU), United States. First and foremost, I would like to thank my supervisor for his inspiring and valuable support during the project. I am also grateful to Professor Steven Shaw and his Ph.D. student B. Scott Strachan for the productive collaboration. Further thanks are given to Professor Gaetan¨ Kerschen at University of Liege,` Belgium, Professor Cyril Touze´ at ENSTA-ParisTech, France, and Professor Steven Shaw, for the CISM course “Modal Analysis of Nonlinear Mechanical Systems”, held at Udine, Italy, 2012. Additional thanks are given to Associate Professor Silvan Schmid and Ph.D. student Maksymilian Kurek in Department of Micro- and Nan- otechnology at DTU for their active help in fabrication and tests of the optimized micro-mechanical structures. I would like to thank my colleagues in TopOpt Group and in Department of Mechanical Engineering at DTU. They have created a cheerful and inspiring working environment. Special thanks to Gerda Helene Fogt for the great help in many aspects. I also wish to thank my colleagues in Department of Mechanical Engineering at MSU for creating a friendly and welcoming atmosphere. Last but not the least, I wish to thank my family for their support and under- standing, in particular, to my wife, Fangxing Tian, for her love and thoughtful- ness. Lyngby, April 2015 Suguang Dou i Resum´e (in Danish) Ikke-lineariteter i mekaniske strukturer kan give anledning til rige dynamiske effekter. Pa˚ det seneste har ikke-lineær dynamik i mikromekaniske strukturer bidraget til udviklingen af nye Mikro-Elektro-Mekaniske Systemer (MEMS), for eksempel til ”atomic force” mikroskopi, passive frekvensdelere, frekvens stabil- isatorer, og resonator gyroskoper. For at muliggøre design og optimering af disse strukturer, er det nødvendigt at udvide de eksisterende optimeringsmetoder, der typisk er baseret pa˚ lineær dynamik, til at tage de ikke-lineære dynamiske effek- ter med i betragtning. I denne afhandling er der udviklet en metode til at modellere, analysere, karakterisere og optimere strukturer pa˚ basis af deres ikke-lineær dynamiske opførsel. Modelleringen baserer sig pa˚ ikke-lineær finite element analyse og den ikke-lineære frekvensrespons og ikke-lineære svingningsformer er beregnet ved hjælp af en harmonisk balance metode med højere ordens harmoniske kompo- nenter medtaget. I karakteriseringen er ikke-lineære modale koblingskoefficien- ter beregnet direkte ud fra den ikke-lineære finite element model. Dette har re- sulteret i udviklingen af en ny klasse af optimeringsproblemer. Den gradient- baserede optimering er udført ved hjælp følsomhedsanalyse baseret pa˚ den ad- jungerede metode, som er specielt velegnet til storskala optimeringsproblemer. Anvendeligheden af den udviklede metode er eksemplificeret ved optimeringen af plane rammekonstruktioner. Arbejdet har vist følgende resultater: amplitude og frekvens af den ikke-lineære resonante respons kan effektivt optimeres; super- harmoniske resonanser kan undertrykkes eller forstærkes; Ikke–lineære karakter- istika (hard/blød˚ ikke linearitet) kan kvalitativt manipuleres; og en størrelsesordens forbedring af væsentlige modale koblingskoefficienter kan opnas.˚ Undersøgelsen har vist lovende mulige applikationer inden for ikke-lineære mikromekaniske resonatorer. Dette baner ogsa˚ vejen for topologioptimering af deres komplekse ikke-lineære dynamik. ii Abstract The intrinsic nonlinearity of mechanical structures can give rise to rich nonlin- ear dynamics. Recently, nonlinear dynamics of micro-mechanical structures have contributed to developing new Micro-Electro-Mechanical Systems (MEMS), for example, atomic force microscope, passive frequency divider, frequency stabi- lization, and disk resonator gyroscope. For advanced design of these structures, it is of considerable value to extend current optimization in linear structural dy- namics into nonlinear structural dynamics. In this thesis, we present a framework for modelling, analysis, characteriza- tion, and optimization of nonlinear structural dynamics. In the modelling, non- linear finite elements are used. In the analysis, nonlinear frequency response and nonlinear normal modes are calculated based on a harmonic balance method with higher-order harmonics. In the characterization, nonlinear modal coupling coefficients are calculated directly from a nonlinear finite element model. Based on the analysis and the characterization, a new class of optimization problems is studied. In the optimization, design sensitivity analysis is performed by using the adjoint method which is suitable for large-scale structural optimization. The op- timization procedure is exemplified by the design of plane frame structures. The work has demonstrated the following results: the amplitude and the frequency of nonlinear resonance peak can be effectively optimized; the super-harmonic reso- nances can be either suppressed or enhanced; the hardening/softening behavior can be qualitatively changed; and an order-of-magnitude improvement of some essential modal coupling coefficients can be achieved. The study has shown promising applications in nonlinear micro-mechanical resonators. It also paves the way for topology optimization of complex nonlinear dynamics. iii Publications The following publications are part of the thesis [P1] S. Dou, J. S. Jensen, Optimization of nonlinear structural resonance using the incremental harmonic balance method, Journal of Sound and Vibration, 334, 239-254, 2015. [P2] S. Dou, B. S. Strachan, S. W. Shaw and J. S. Jensen, Structural optimization for nonlinear dynamic response. (To appear in the “A field guide to nonlin- earity in structural dynamics” issue of Philosophical Transactions A) [P3] S. Dou, J. S. Jensen, Optimization of hardening/softening behavior of plane frame structures using nonlinear normal modes. (Submitted for journal publication) [P4] S. Dou, J. S. Jensen, Optimal design of hardening and softening resonances of plane frame structures. Abstract presented at the 8th European Nonlinear Dynamics Conference, Vienna, Austria, 2014. [P5] S. Dou, J. S. Jensen, Analytical sensitivity analysis and topology optimiza- tion of nonlinear resonant structures with hardening and softening behav- ior. Abstract presented at the 17th U.S. National Congress on Theoretical and Applied Mechanics, East Lansing, Michigan, United States, 2014. iv Contents Preface i Resum´eii Abstract iii Publications iv Contents v 1 Introduction 1 1.1 Motivation and goals ........................... 1 1.2 Current development in nonlinear dynamics ............. 4 1.3 Gradient-based structural optimization ................ 5 1.4 Structure of the thesis ........................... 7 2 Methods for modelling, analysis and characterization