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236 Smart Technologies for Safety Engineering Table 6.7 Optimum remodeling example: a single step of the optimization algorithm 1. Derivatives of the response ∂u N (t)/∂ Âl and ∂σi(t)/∂ Âl 2. Gradient of the modified objective function ∇ fc(Â) 3. Approximate Hessian of the modified objective function ˜H(Â) 4. Adaptive selection of the step length: repeat Optimization step Â by solving [ ˜H(Â) + λI]Â =−∇fc(Â) Modified objective function at the new point fc(Â + Â) Expected improvement of the modified objective function 1 2 Â λIÂ −∇fc(Â) Actual-to-expected improvement ratio f (Â) − f (Â + Â) actual improvement κ = = 0.5Â λÂ −∇fc(Â) expected improvement If κκmax, then λ := λ/k until κ ≥ 0 Figure 6.10 Optimally remodeled structure (asymmetric impact case) Figure 6.11 Original and optimum vertical displacements of impacted nodes (asymmetric impact case)
VDM-Based Remodeling 237 Figure 6.12 Optimally remodeled 30-element structure (multiple-impact case) 6.3.1.3 Multiple-Impact Loads It is important for further discussion to consider the case of multiple-impact loads. Assume that two impact scenarios are possible: the one just discussed (m1 = 0.1 kg, m2 = 2 kg) and its symmetric counterpart (m1 = 2 kg, m2 = 0.1 kg). The optimization is still based on the algorithm presented in Table 6.7. Since the two considered loads are mutually symmetric, each optimization step should be symmetrized by assuming Âi := Âi + Âī 2 where the īth element is located in the structure symmetrically to the ith element. The structure optimally remodeled for the two considered impacts is presented in Figure 6.12, with the original and optimized displacements shown later in Figure 6.14 (left). In terms of the objective function the result is inferior to the previous one, since the optimum configuration has to account for multiple-impact loads. A general optimization algorithm should eliminate free nodes and, in certain cases, replace two collinear elements with one. Since this is not implemented in its present form, these modifications have been done manually. However, an automatic algorithm is also possible. The total number of structural elements is reduced from 54 in Figure 6.9 to 30 in Figure 6.12. Notice that a similar but significantly simpler solution shown in Figure 6.13 can be obtained by removing two vertical elements between sections 1, 2 and 10, 11, and by replacing each of the two pairs of the remaining upper elements by one with the mean cross-sectional area. The objective function changes only insignificantly as the result (see Figure 6.14, right). This 26-element structure is discussed in the next subsection. 6.3.2 Structural Fuses as Active Elements In the preceding subsection the considered structure has been remodeled to obtain its stiffest elastic substructure (see also Reference [33]). In the next step of the methodology of constructing adaptive structures, outlined at the beginning of this section, controllable elastoplastic Figure 6.13 Near-optimum 26-element structure (multiple-impact case)
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SMART TECHNOLOGIES FOR SAFETY ENGIN
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Copyright C○ 2008 John Wiley & So
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vi Contents 2.7 Versatility of VDM
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viii Contents 6 VDM-Based Remodelin
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Preface The contents of this book c
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xiv About the Authors The team of s
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Organization of the Book The book h
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1 Introduction to Smart Technologie
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Introduction to Smart Technologies
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12 Smart Technologies for Safety En
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3 VDM-Based Health Monitoring of En
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VDM-Based Health Monitoring 39 3.2
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VDM-Based Health Monitoring 41 When
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VDM-Based Health Monitoring 43 appr
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VDM-Based Health Monitoring 45 (7)
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VDM-Based Health Monitoring 47 μ i
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VDM-Based Health Monitoring 49 μ A
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VDM-Based Health Monitoring 51 Figu
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VDM-Based Health Monitoring 53 by a
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VDM-Based Health Monitoring 55 Figu
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VDM-Based Health Monitoring 57 axia
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axial strain 3,2 2,4 1,6 0,8 0 −0
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(a) (b) (c) Figure 3.27 (a) Screw c
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VDM-Based Health Monitoring 63 The
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VDM-Based Health Monitoring 65 μ i
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VDM-Based Health Monitoring 67 1 3
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VDM-Based Health Monitoring 69 3’
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VDM-Based Health Monitoring 73 form
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VDM-Based Health Monitoring 75 Q Q
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VDM-Based Health Monitoring 77 Note
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VDM-Based Health Monitoring 79 wate
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VDM-Based Health Monitoring 81 0 -0
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VDM-Based Health Monitoring 83 −0
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VDM-Based Health Monitoring 85 impl
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VDM-Based Health Monitoring 87 give
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VDM-Based Health Monitoring 89 sour
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VDM-Based Health Monitoring 91 Rela
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VDM-Based Health Monitoring 95 loca
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VDM-Based Health Monitoring 101 8.
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VDM-Based Health Monitoring 103 55.
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5 Adaptive Impact Absorption Piotr
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Adaptive Impact Absorption 155 Fina
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Adaptive Impact Absorption 157 σ 1
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Adaptive Impact Absorption 159 acce
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Adaptive Impact Absorption 161 norm
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Adaptive Impact Absorption 163 Figu
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Adaptive Impact Absorption 165 Zone
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Adaptive Impact Absorption 167 gas
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Adaptive Impact Absorption 169 moun
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Adaptive Impact Absorption 171 of t
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Adaptive Impact Absorption 173 disp
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Adaptive Impact Absorption 175 comp
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Adaptive Impact Absorption 177 of t
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Adaptive Impact Absorption 179 F VD
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Adaptive Impact Absorption 181 forc
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Adaptive Impact Absorption 183 Tabl
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Page 260 and 261: 7 Adaptive Damping of Vibration by
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Acknowledgements Parts of Section
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Acknowledgements 325 Parts of Chap
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328 Index Control strategy (Continu
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330 Index Modification coefficient
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332 Index System analogies, 29, 68,
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