# Identification and Analysis of Nonlinear Systems

In classical mechanical engineering the predominant group of system analysis and identification tools relies on Linear Systems, where research have been carried out for over half a century. Usage of Linear Systems is most widely spread, often due to its simple mathematics and formulation for many engineering problems. Although linearizing is a means for simplifying a problem, it will introduce more or less severe modelling errors. In some cases the errors due to linearizing are too large to be practically acceptable, and therefore nonlinear structures and models are sometimes introduced.This thesis aims in implementing and evaluating some popular methods and algorithms for nonlinear structure analysis and identification, with emphasis on systems having nonlinear terms…

Contents

1 Introduction
2 Theoretic Background
2.1 Linear Systems
2.1.1 Degree of freedom
2.1.2 Basic mechanical system SDOF
2.1.3 Larger mechanical systems MDOF
2.2 Nonlinear Systems
2.2.1 Brief Introduction
2.2.2 Theoretical Representation of Nonlinearities
2.2.3 Presentation of Nonlinear Systems
3 System Identiﬁcation and Analysis
3.1 Introduction
3.2 Linear System Identiﬁcation and Analysis
3.2.1 Non-Parametrical Spectrum Identiﬁcation
3.2.2 SISO Systems
3.2.3 Time Series Models
3.2.4 Special Types of Random Processes
3.2.5 Modal Analysis
3.3 Nonlinear System Identiﬁcation
3.3.1 NARMAX Modelling
3.3.2 Reverse Path Method
3.3.3 Frequency Domain Structure Selection Algorithm
3.3.4 Finding the Nonlinear Nodes
3.4 Nonlinear System Analysis
3.4.1 Harmonic Balance Method
3.4.2 Hilbert Transformation Techniques
3.4.3 Statistical Analysis
4 System Synthesis
4.1 Introduction
4.2 Linear System Synthesis
4.2.1 Time Response Synthesis using Laplace Transformation
4.2.2 Synthesis by Digital Filters
4.3 Nonlinear Systems Synthesis
4.3.1 Analytical Time Responses Synthesis
4.3.2 Synthesis by Ordinary Differential Equation Solvers
4.3.3 Synthesis by Extended Digital Filter Structures
4.4 Synthesis Quality Assessments
ii5 Experimental Evaluation of Black-box Systems
5.1 Background
5.2 The Experiment
5.2.1 The First System
5.2.2 The Second System
5.2.3 The Third System
5.2.4 The Fourth System
5.2.5 The Results
6 Experimental Evaluation of Test-Rig
6.1 Background
6.2 Introduction
6.3 System Model
6.3.1 Linear Part of System Model
6.3.2 Nonlinear Part of System Model
6.3.3 Nonlinear Property Veriﬁcation
6.4 Experimental Setup
6.4.1 Measurement Equipment
6.4.2 Equipment List
6.4.3 Work Materials
6.5 Choice of Performance and Excitation Signals
6.6 Measurement Settings
6.6.1 Measurement Settings, Collection from Signal Calc when using build-in functions
6.6.2 Measurement Settings, Collection from Signal Calc when using excitation signal pro-
duced in matalab
6.6.3 Channels Settings for measurement 1-3
6.6.4 Channels Settings for measurement 4-11
6.7 Results
6.8 Conclusion
7 Summary and Conclusions
7.1 Summary
7.2 Conclusions and Further Research
A Derivations of Impulse-, Step-, and Ramp Invariance
A.1 Derivation of Impulse Invariance
A.2 Derivation of Step Invariance
A.3 Derivation of Ramp Invariance
B Modiﬁed Bootstrap Structure Detection 861 ´ Notation
2 ´ Introduction
3 ´ Theoretical Model
3.1 Simplifications
3.2 Equations of Motion by Newton’s Method
3.3 Equations of Motion by Lagrange’s Method
4 Solution Method
4.1 Marching Procedure
5 Experimental Investigation
5.1 Experimental Set-up
6 Theoretical and Experimental Results
6.1 Results
6.2 Discussion
7 Conclusion
8 References
9 Appendices

Author: Henrik Åkesson, Benny Sällberg

Source: Blekinge Institute of Technology