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1、 A TRANSITIONARY NONLINEAR DYNAMICS APPROACH FOR MODELING AND SIMULATING DAMAGE EVOLUTION IN A CANTILEVERED STRUCTURE A Thesis Submitted to the Faculty of Purdue University by Madhura Nataraju In Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical Engineering Au
2、gust 2003 ii ACKNOWLEDGEMENTS My sincere thanks to Dr. Douglas E. Adams for his support, guidance and encouragement throughout this endeavor. It has really been a great experience working with him. Many thanks to Prof. Arvind Raman and Prof. Patricia Davies for showing interest in my work and servin
3、g on my advisory committee. I would like to thank the Army Research Office for their support of this work. I also thank the Herrick laboratories staff for all their help and co-operation. I thank all my colleagues, especially, Harold, Jason, Haroon and Tim for all the help and co-operation over the
4、last couple of years. Thanks to Shankar for his suggestions and timely help during my experiments. Thanks to Ganesh who sat with me for hours just staring at the shaker, waiting for the beams to break. And many thanks to Raja and Santanu for taking care of my meals! Thanks to my family and friends f
5、or their love and blessings. Special thanks to Ajit for his invaluable help and support throughout. Thank you God for giving me the chance to pursue my academic goals and a wonderful and complete life full of amazing people. iii TABLE OF CONTENTS Page LIST OF TABLES.v LIST OF FIGURES. vi ABSTRACT.x
6、CHAPTER 1: INTRODUCTION1 1.1. Prognostics: Progress and Challenges 3 1.2. Fatigue, Fracture and Damage Mechanics8 1.3. Fatigue Damage in Heterogeneous Materials.14 1.4. Further Motivation for the Approach15 1.5. Thesis Statement.21 CHAPTER 2: THEORY AND CONCEPTS.22 2.1. Problem Definition22 2.2. Bas
7、ics of Fracture Mechanics.24 2.2.1. Linear Elastic Fracture Mechanics.25 2.2.2. Fatigue Crack Growth29 2.3. Variable Amplitude Loading35 2.3.1. Damage Summing Methods for Initiation.36 2.3.2. Cycle Counting38 2.3.3. Crack Propagation under Variable Amplitude Loading. .39 2.4. Strengths and Weaknesse
8、s of the LEFM Approach .43 2.5. Continuum Damage Mechanics Approach to Fatigue Damage44 2.5.1. Fatigue Damage Accumulation45 2.6. System Equations48 iv Page 2.7. Nonlinear Dynamics and Damage49 2.8. Summary.54 CHAPTER 3: EXPERIMENTAL METHODOLOGY .56 3.1. Experimental Setup.56 3.2. Results and Discus
9、sion .58 3.2.1. Test I61 3.2.2. Test II.67 3.2.1. Test III70 3.2.1. Test IV .73 3.2.1. Test V.79 3.2.6. Tests on the Shaker80 3.3. Damage Accumulation82 3.4. Summary.83 CHAPTER 4: ANALYTICAL MODEL.85 4.1. Description of Approach.85 4.2. Validation of Trends Observed in Test Cases.90 CHAPTER 5: CONCL
10、USIONS95 5.1. Contributions.95 5.2. Recommendations.97 LIST OF REFERENCES.98 APPENDICES Appendix I: Code to Obtain Transmissibilities .108 Appendix II: Code for Stress Computation .112 Appendix III: Code to Obtain Crack Lengths115 Appendix IV: Analytical Crack Growth Model.119 v LIST OF TABLES Table
11、 Page 3.1 Different Beam Dimensions and Natural Frequencies60 3.2 Test I Excitation and Crack Specifications.61 3.3 Test II Excitation and Crack Specifications67 3.4 Test III Excitation and Crack Specifications71 3.5 Test IV Excitation and Crack Specifications73 3.6 Test V Excitation and Crack Speci
12、fications .76 vi LIST OF FIGURES Figure Page 1.1 Illustration of structural diagnosis and prognosis and the economic/safety benefits of NDE/SHM through condition-based maintenance scheduling as adapted from Cronkhite and Gill (1998).2 1.2 Damage growth versus usage.7 1.3 Development of fatigue damag
13、e15 1.4 (Left) Three-story building model frame with base excitation for simulating seismic input and (Right) building bolted joint instrumented with accelerometers in two orthogonal measurement degrees-of-freedom.16 1.5 (Top) Illustration of structural joint and boundary conditions, (Left-bottom) i
14、llustration of qualitative change in nonlinear nature of response at joint due to gap when loosened, and (Right-bottom) illustration of change in linear nature of response due to loss of preload and sliding friction across joint17 1.6 (Top) Magnitude of transmissibility function between two orthogon
15、al acceleration measurements at the damaged joint, and (Bottom) probability distribution of transmissibility magnitude for damage cases showing trend for increasing damage18 1.7 (Top) Magnitude of nonlinear indicator function magnitude from autoregressive model between two orthogonal acceleration me
16、asurements at the damaged joint, and (Bottom) probability distribution of this function for damage cases showing reversing trend for increasing damage18 1.8 Bell 206L helicopter fuselage with bolted fasteners at hinges with three fully tightened fasteners and one that is gradually loosened .19 vii F
17、igure Page 1.9 Changes in transmissibility-based damage indicator as a function of transmissibility pairing and stage of loosening in bolt20 2.1 Helicopter rotor blade modeled as a cantilever beam22 2.2 Fatigue life initiation and propagation (Bannantine, 1990)25 2.3 Three loading modes27 2.4 Stress
18、 intensity factor of a double edge notch specimen .28 2.5 Yielding near crack tip and monotonic plastic zone size (Bannantine, 1990).28 2.6 Remote stress range.30 2.7 Three regions of crack growth rate curve30 2.8 Rate of crack growth in 2 mm thick 2024-T3 aluminum specimen as a function of the effe
19、ctive applied stress showing threshold from Harris (1991). .31 2.9 Illustration of four potential wells, each of which defines a different possible damage state that depends on the given initial flaw distribution in a healthy structural dynamic system50 2.10 Transcritical bifurcation model of damage
20、 initiation.52 2.11 Link-up and arrest behaviors in aluminum 2024-T3 specimen with multiple rivets forming lap joint of fuselage for studying multi-site damage (MSD) from Orringer (1991).53 2.12 Asymmetric pitchfork bifurcation model of sudden restrained damage growth describing arrested isolated cr
21、ack growth due to plasticity at tip54 2.13 Subcritical pitchfork bifurcation model of sudden unrestrained damage growth in two DOF system describing link-up of crack growth at multiple sites54 3.1 Photographs of one particular beam before and after the test57 3.2 Schematics of the beam and crack con
22、figuration 60 3.3 A sample input time record (Test I).62 3.4 Transmissibility plots between the root and a point just across the crack at various times during the first test. 3.3a (top) - zoomed in to show modes 3-6. 3.3b (bottom) - zoomed in to show mode 463 viii Figure Page 3.5 Time point that eac
23、h curve on the transmissibility plot (Figure 3.4) represents64 3.6 First 6 mode shapes of a cantilever beam64 3.7 Variation of fourth natural frequency with time for Test I65 3.8 Variation of the fourth natural frequency with crack length for Test I66 3.9 Variation of dc/dt with c for Test I66 3.10
24、Variation of dc/dt with time for Test I.67 3.11 A sample input time record (Test II)68 3.12 Transmissibility between the root and a point across the crack for Test II.69 3.13 Time point that each curve on the transmissibility plot (Figure 3.12) represents69 3.14 Variation of fourth natural frequency
25、 with time for Test II.70 3.15 A sample input time record (Test III)71 3.16 Transmissibility between the root and a point across the crack for Test III72 3.17 Time point that each curve on the transmissibility plot (Figure 3.16) represents72 3.18 Variation of fourth natural frequency with time for T
26、est III .73 3.19 A sample input time record (Test IV)74 3.20 Transmissibility between the root and a point across the crack for Test IV75 3.21 Time point that each curve on the transmissibility plot (Figure 3.20) represents75 3.22 Change in the fourth (top, 3.13a) and second (bottom, 3.13b) natural
27、frequencies with time, for Test IV.76 3.23 Variation of the fourth natural frequency with crack length for Test IV.77 3.24 Effect of the variation of E on the changes in the fourth natural frequency versus changes in the crack length curves.77 3.25 Variation of dc/dt with c for Test IV .78 3.26 Vari
28、ation of dc/dt with time for Test IV78 3.27 A sample input time record (Test V).79 3.28 Change in fourth natural frequency with time for Test V80 3.29 Time response of the shaker in x, y, and z directions81 3.30 Transmissibility between the vertical motion of the shaker fixture and the root of the b
29、eam.82 ix Figure Page 3.31 Variation of damage accumulation with time83 4.1 Change in the fourth natural frequency with time, obtained using the simulation.87 4.2 Trend similar to that of Test I. 4.2a (top) damage accumulation versus time. 4.2b (bottom) crack length versus time90 4.3 Trend similar t
30、o that of Test II. 4.5a (top) damage accumulation versus time. 4.3b (bottom) crack length versus time91 4.4 Trend similar to that of Test III. 4.4a (top) damage accumulation versus time. 4.4b (bottom) crack length versus time92 4.5 Trend similar to that of Test IV. 4.5a (top) damage accumulation ver
31、sus time. 4.5b (bottom) crack length versus time93 4.6 Trend similar to that of Test V. 4.6a (top) damage accumulation versus time. 4.6b (bottom) crack length versus time94 x ABSTRACT Nataraju, Madhura, M.S.M.E., Purdue University, August, 2003. A Transitionary Nonlinear Dynamics Approach for Modeli
32、ng and Simulating Damage Evolution in a Cantilevered Structure. Major Professor: Dr. Douglas E. Adams, School of Mechanical Engineering. Structural systems may be composed of homogeneous or heterogeneous materials such as composites, plastics, ceramics, fabrics and metal-alloys. Heterogeneous struct
33、ures have complicated dynamics of their own in addition to numerous types of damage and failure modes (crack growth, delaminations, fiber breakage, matrix cracking, component failures), which interact in complicated ways that vary tremendously for different initial states, levels of damage accumulat
34、ion and loading history, making it very difficult to forecast their remaining useful life in operation. Though there have been abundant, relatively successful efforts to model and predict specific types of failure in complex material and structural systems, this work is directed towards the investig
35、ation of a more universal approach to prognosis that can accommodate the diversity of failure modes exhibited by structures. In the approach used here, damage states in structural systems are associated with quasi-static equilibrium points and, subsequently, all significant damage events are modeled
36、 as bifurcations (qualitative changes) in the set of stable equilibria for the structure-damage system using a low-order nonlinear dynamics model. Different damage states are assumed to interact with one another and with the structural response states resulting in transitions from one damage mode to
37、 another and eventually to the end of remaining useful life. By observing these transitions on the manifold of measured equilibria, the approach associates transitions with bifurcations in structural health to model damage growth for a given failure mode in a homogeneous cantilever xi beam. Testing
38、was done with beams, mounted on a shaker at their roots. Cracks were milled into the beams at chosen locations and they were excited with various inputs. The transmissibility functions were computed from the measured data. The transitions seen in the natural frequencies of the system as the damage p
39、rogressed were studied. Fatigue failure and continuum damage mechanics concepts were reviewed and an analytical model based on the observed phenomena, using low order nonlinear equations was developed to simulate various damage evolution scenarios. 1 1 INTRODUCTION Structural health monitoring (SHM)
40、 is of paramount importance for the safe operation and long life of all structural systems and components. All components and systems are subjected to constant wear and tear. It is therefore extremely beneficial to be able to locate and determine the extent of the damage or deterioration that has oc
41、curred in them. Several researchers have directed their studies towards this area and numerous non- destructive evaluation (NDE) techniques with reasonably high rates of success have been developed. Several technical conferences, workshops and journals dedicated to structural integrity, such as the
42、SPIE Symposia on NDE for Health Monitoring and Diagnostics, International Workshop on SHM, Journal of SHM, etc., indicate the tremendous significance and the huge effort and progress made in this field. Butcher (2000) presented a detailed report that described the concept, goals, capabilities and ch
43、allenges of condition-based monitoring (CBM), in the context of Department of Defense interests. Boller and Biemans (1997) described the importance of structural health monitoring in aircraft in terms of safety, reliability and cost effectiveness and pointed out the problems associated with traditio
44、nal time-based maintenance, manual inspection and repair (e.g. time consuming, error prone). They also summarized the state-of-the-art technologies available. The advantages of SHM can be extended to other civil and transportation systems and structures such as bridges, buildings, ships, offshore dr
45、illing platforms, spacecraft, etc. Doebling et al. (1996) provided a comprehensive review of vibration- based damage identification techniques. Figure 1.1 illustrates how diagnostics and prognostics are used together for CBM. 2 Safety risks with no prognostics Additional use gained through prognosti
46、cs Life Consumption Design life of Structure Prognosis Severe usage Scheduled service without SHM/NDT Diagnosis Mild usage Time in Operation Figure 1.1 Illustration of structural diagnosis and prognosis and the economic/safety benefits of NDE/SHM through condition-based maintenance scheduling as ada
47、pted from Cronkhite and Gill (1998). The existing damage in a component and the damage growth pattern are dependent on factors such as the material properties, input or loading history, presence of initial defects and environmental conditions. Damage initiation, growth, saturation and final failure
48、can occur due to different reasons (fatigue, creep, stress-rupture, fracture and overload, to name a few) and at different points in time, even for similar structures, depending on the above mentioned factors. Moreover, structural systems can refer to assemblies of homogeneous components or heteroge
49、neous material systems consisting of a mixture of composites, ceramics, plastics, metal-alloys and fabrics. The wide variety of damage phenomena (e.g., crack growth, delaminations, fiber breakage, debonding, fastener failures, etc.), exhibited by these systems makes it difficult to forecast their reliability. This thesis is aimed at developing a framework for prognosis that can incorporate the above-menti