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Fractional Stochastic Dynamics in Structural Stability Analysis

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Fractional Stochastic Dynamics in Structural Stability Analysis Fractional Stochastic Dynamics in Structural Stability Analysis by Jian Deng A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Civil Engineering Waterloo, Ontario, Canada, 2013 ©Jian Deng 2013 Author’s Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract The objective of this thesis is to develop a novel methodology of fractional stochastic dynamics to study stochastic stability of viscoelastic systems under stochastic loadings. Numerous structures in civil engineering are driven by dynamic forces, such as seismic and wind loads, which can be described satisfactorily only by using probabilistic models, such as white noise processes, real noise processes, or bounded noise processes. Viscoelastic materials exhibit time-dependent stress relaxation and creep; it has been shown that frac- tional calculus provide a unique and powerful mathematical tool to model such a hereditary property. Investigation of stochastic stability of viscoelastic systems with fractional calcu- lus frequently leads to a parametrized family of fractional stochastic differential equations of motion. Parametric excitation may cause parametric resonance or instability, which is more dangerous than ordinary resonance as it is characterized by exponential growth of the response amplitudes even in the presence of damping. The Lyapunov exponents and moment Lyapunov exponents provide not only the infor- mation about stability or instability of stochastic systems, but also how rapidly the response grows or diminishes with time. Lyapunov exponents characterizes sample stability or insta- bility. However, this sample stability cannot assure the moment stability. Hence, to obtain a complete picture of the dynamic stability,it is important to study both the top Lyapunov ex- ponent and the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the accurate values of theses two exponents. One has to resort to numerical and approximate approaches. The main contributions of this thesis are: (1) A new numerical simulation method is proposed to determine moment Lyapunov exponents of fractional stochastic systems, in which three steps are involved: discretization of fractional derivatives, numerical solution of the fractional equation, and an algorithm for calculating Lyapunov exponents from small data sets. (2) Higher-order stochastic averaging method is developed and applied to investigate stochastic stability of fractional viscoelastic single-degree-of-freedom structures under white noise, real noise, or bounded noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and gyroscopic viscoelastic systems under random excitation, the v Stratonovich equations of motion are set up, and then decoupled into four-dimensional Itô stochastic differential equations, by making use of the method of stochastic averaging for the non-viscoelastic terms and the method of Larionov for viscoelastic terms. An elegant scheme for formulating the eigenvalue problems is presented by using Khasminskii and Wedig’s mathematical transformations from the decoupled Itô equations. Moment Lyapunov exponents are approximately determined by solving the eigenvalue problems through Fourier series expansion. Stability boundaries, critical excitations, and stability index are obtained. The effects of various parameters on the stochastic stability of the system are discussed. Parametric resonances are studied in detail. Approximate analytical results are confirmed by numerical simulations. vi Acknowledgments First and foremost, I like to express my sincere appreciation to my supervisors, Professor Wei-Chau Xie and Professor Mahesh D Pandey, for their valuable advice, support and encouragement. I wish to thank them for introducing me to Waterloo, to the famous ‘‘Waterloo School’’, and to the novel, exciting, and challenging field of research. I would like to extend my sincere gratitude to my committee members, Professor Scott Walbridge, Professor Jeffrey West, Professor Xinzhi Liu, and Professor Lei Xu, for their insightful comments and cares for my research. I am grateful to Professor N. Sri Na- machchivaya from the University of Illinois at Urbana-Champaign, for agreeing to serve on my thesis committee as an external examiner and for carefully reading my thesis and making some encouraging comments. I also wish to thank Professors Robert Gracie,Sriram Narasimhan, Bryan Tolson, John Shortreed, and Arnold Yuan for their comments and cares for my research. Thanks also go to my friends in the mechanics and structural engineering group, with whom I shared many unforgettable days. The financial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the University Network of Excellence in Nuclear Engineering (UN- ENE) in the form of a Research Assistantship is greatly appreciated. vii TO My Family ix Contents List of Figures ................................ xv 1Introduction ................................ 1 1.1 Stochastic Stability of Structures 1 1.1.1 Motivations 1 1.1.2 Gyroscopic Systems 2 1.1.3 Stochastic Stability 4 1.1.4 Lyapunov Exponents and Moment Lyapunov Exponents 7 1.2 Noise Models and Stochastic Differential Equations 10 1.2.1 Noise Models 10 1.2.2 Stochastic Differential Equations 13 1.2.3 Approximation of a Physical Process by a Diffusion Process 16 1.2.4 Numerical Methods for Solving SDEs 17 1.2.5 Stochastic Averaging Method for Approximating SDEs 18 1.3 Ordinary Viscoelasticity & Fractional Viscoelasticity 22 1.3.1 Fractional Calculus 22 1.3.2 Viscoelasticity 24 1.4 Scopes of the Research 29 2A Numerical Method for Moment Lyapunov Exponents of Fractional Systems ................................... 31 2.1 Exponential Kernel and Power Kernel 31 2.2 Numerical Determination of Moment Lyapunov Exponents of Fractional Systems 33 xi 2.2.1 Numerical Discretization of Fractional Derivatives 33 2.2.2 Determination of the pth Moment of Response Variables 35 2.2.3 Determination of the pth Moment Lyapunov Exponent 37 2.2.4 Procedure for Numerical Determination of Moment Lyapunov Exponents of Fractional Systems 40 2.3 Summary 42 3Higher-Order Stochastic Averaging Analysis of SDOF Fractional Viscoelastic Systems ........... 44 3.1 Formulation 44 3.2 Stochastic Averaging Method for Fractional Differential Equations 46 3.3 Fractional Viscoelastic Systems Under Wide-band Noise 48 3.3.1 First-Order Averaging Analysis 48 3.3.2 Second-Order Stochastic Averaging 52 3.3.3 Higher-Order Stochastic Averaging 59 3.3.4 Numerical Results and Discussion 67 3.3.5 Ordinary Viscoelastic Systems 75 3.4 Lyapunov Exponents of Fractional Viscoelastic Systems under Bounded Noise Excitation 78 3.4.1 Lyapunov Exponents 78 3.4.2 Numerical Determination of Lyapunov Exponents 84 3.4.3 Results and Discussion 85 3.5 Moment Lyapunov Exponents of Fractional Systems under Bounded Noise Excitation 89 3.5.1 First-Order Stochastic Averaging 89 3.5.2 Second-Order Stochastic Averaging 93 3.5.3 Higher-Order Stochastic Averaging 94 3.5.4 Results and Discussion 94 xii 3.6 Summary 101 4Stochastic Stability of Coupled Non-Gyroscopic Viscoelastic Systems .................. 103 4.1 Coupled Systems Excited by Wide-Band Noises 103 4.1.1 Formulation 103 4.1.2 Moment Lyapunov Exponents of 4-D Systems 108 4.1.3 Stability Boundary 118 4.1.4 Simulation 122 4.1.5 Stability Index 126 4.1.6 Application: Flexural-Torsional Stability of a Rectangular Beam 127 4.1.7 Results and Discussion 130 4.2 Coupled Systems Excited by Bounded Noise 135 4.2.1 Formulation 135 4.2.2 Parametric Resonances 139 4.3 Summary 154 5Stochastic Stability of Gyroscopic Viscoelastic Systems ........ 156 5.1 Vibration of a Rotating Shaft 156 5.2 Gyroscopic Systems Excited by Wide-band Noise 159 5.2.1 Formulation 159 5.2.2 Moment Lyapunov Exponents 165 5.2.3 Stability Boundary 167 5.2.4 Stability Index 169 5.2.5 Simulation 170 5.2.6 Results and Discussion 172 5.3 Gyroscopic Systems Excited by Bounded Noise 178 5.3.1 Stochastic Averaging 178 xiii 5.3.2 Parametric Resonances 184 5.3.3 Effect of Viscoelasticity 194 5.4 Summary 198 6Conclusions and Future Research .................... 210 6.1 Contributions 210 6.2 Future Research 213 Bibliography ................................. 215 xiv List of Figures 1.1 Illustration of Lyapunov stability 5 1.2 Seismic waves and power spectral densities (El-Centro, U.S.A., 1940) 11 1.3 Power spectral density of a real noise process 11 1.4 Power spectral density of a bounded noise process 14 1.5 Fractional calculus of a function 25 1.6 Creep and recovery for a fractional viscoelastic material 28 3.1 Comparison of simulation and approximate results for fractional SDOF viscoelastic system under white noise excitation 69 3.2 Effect of damping (β) on Moment Lyapunov exponents under Gaussian white noise excitation 69 3.3 Effect of elastic modulus (E) on Moment Lyapunov exponents of fractional SDOF viscoelastic system under white noise excitation 70 3.4 Effect of viscosity (η) on moment Lyapunov exponents of fractional SDOF viscoelastic system under white noise excitation 71 3.5 Effect of retardation (τε) on moment Lyapunov exponents of fractional SDOF viscoelastic system under white noise excitation 71 3.6 Effect of fractional order µ on moment Lyapunov exponents under Gaussian white noise excitation 72 3.7 Effect
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