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234 Smart Technologies for Safety Engineering Figure 6.9 Two-dimensional truss beam, initial configuration 6.3.1 Stiffest Elastic Substructure 6.3.1.1 Optimization Problem The considered structure is to be optimally remodeled to form the stiffest structure with minimum dynamic deflections under the given impact load. The following objective function is thus used: f (Â) = t,M u 2 M (t) (30) where u M(t) denotes the modeled response of the structure with modified element crosssections Â, subject to the following constraints: V = Ailiμi ≤ ˜V Âi > 0 |σi| < ˜σ The index M belongs to the set of the degrees of freedom in which the displacements are being minimized, which in the case considered here are the vertical displacements of the two impacted nodes. The total volume of the material in the initial structural configuration is denoted by ˜V and the maximum admissible stress level by ˜σ . In order to simplify the optimization process, the original optimization problem of Equation (30) with the constraints in Equations (31) has been transformed into a corresponding unconstrained problem. Instead of the original constraints, three quadratic penalty functions [31] have been used to form the following modified objective function: fc(Â) = f (Â) + c1 V − ˜V 2 1V > ˜V + c2 Â i 2 i 10>Âi + c3 [σi(t) + ˜σ ] 2 1− ˜σ>σi (t) + [σi(t) − ˜σ ] 2 1 ˜σ
VDM-Based Remodeling 235 Differentiation of Equation (32) yields the formula for the gradient of the modified objective function: ∂ fc(Â) =−2 ∂ Âl u M(t) t,M ∂u M(t) + 2c1ll(V − ˜V )1V > ˜V + 2c2 ∂ Âl Âl10>Al + 2c3 [σi(t) + ˜σ ] ∂σi(t) 1− ˜σ>σi (t) + [σi(t) − ˜σ ] ∂ Âl ∂σi(t) 1 ˜σ ˜V + 2c2δlk10>Al + 2c3 t,i ∂ Âl ∂ Âk 1 ˜σ
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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 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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106 Smart Technologies for Safety E
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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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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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