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VDM-Based Health Monitoring 97 The gradient of the objective function with respect to the modification parameters μk can be obtained from the following relation: ∇g = ∂g ∂μk = 2 t ∂ fi(t) di(t) ∂μk (125) The procedure for calculation of the gradient of response with respect to the modification parameters was presented in the previous section. In every iteration of the optimization procedure, the vector of the modification parameters is updated according to the following formula: μ (p+1) = μ (p) − λ (p) ∇g (126) where λ (p) is a non negative factor. The full procedure of defect identification in dynamics is presented below: Initial computations: Linear responses and the impulse influence matrix for reference points are to be used for updating the modeled responses and calculation of the gradient of the objective function. Linear voltage responses and the voltage impulse influence matrix for all elements included in the set of distortion locations are to be used for updating the distortions and calculation of the gradient of the distortions. In every iteration, for the actual vector of modifications μ (p) , compute: the vector of the distortions using Equation (107); the modeled response in reference points using Equation (112); the distance functions using Equation (123); the gradient of the distortions using Equation (114); the gradient of the reference response using Equation (113); the gradient of the objective function using Equation (125); the updated value of the modification vector μ (p+1) using Equation (126). 3.5.5 Numerical Example The considered circuit of the symmetrical hexagonal topology is shown schematically in Figure 3.54. It consists of 12 uniform resistive elements of resistances 1 k and two ideal capacitors of capacitances C1 = 10 μF and C2 = 100 μF. The circuit is supplied by an ideal current source of intensity 1 mA in the DC case and amplitude 1 mA, zero phase and frequency 50 Hz in the AC case. 3.5.5.1 Modeling Conductance Modifications by Distortions Changes of conductance in resistors R2 and R6 are now introduced, described by the modification parameters μ2 =0.1 and μ6 =0.2. To find the equivalent state of distortions, the linear voltage responses and voltage influence matrices need to be calculated first, using the circuit
98 Smart Technologies for Safety Engineering R 7 3 C 2 R9 C 1 R2 R4 R1 1 R6 2 7 R8 R3 R5 5 4 R10 6 J R11 R 12 Figure 3.54 Numerical example of an electrical circuit equations (98) or (99). In the DC case, the following results are obtained: In the AC case, D u = L u2 = u L 6 L u2 = u L 6 −0.1 ; D 0.4 u = 450 75 75 450 −0.12019 + 0.06525i −0.17981 − 0.06525i 293.3097 − 64.7071i 231.6903 + 64.7071i 231.6903 + 64.7071i 293.3097 − 64.7071i The columns of influence matrices are calculated as voltage responses to unit distortions, imposed consecutively in elements R2 and R6. The values of distortions, modeling the introduced modifications, are calculated from Equation (105): DC : AC : 0 ε2 ε0 −3 0.0956 × 10 = 6 0.4910 × 10−3 0 −3 ε2 (0.2226 + 0.0656i) × 10 = (0.2338 − 0.0531i) × 10−3 ε 0 6 An arbitrary response of the modified circuit can now be quickly calculated as a superposition of the linear and residual responses.
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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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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 171 of t
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Adaptive Impact Absorption 177 of t
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Adaptive Impact Absorption 183 Tabl
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Adaptive Impact Absorption 185 Curr
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Adaptive Impact Absorption 187 (a)
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Adaptive Impact Absorption 189 Vect
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Adaptive Impact Absorption 191 wher
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Adaptive Impact Absorption 193 (a)
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Adaptive Impact Absorption 195 time
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Adaptive Impact Absorption 199 towe
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Adaptive Impact Absorption 201 Tabl
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Adaptive Impact Absorption 205 5.6.
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Adaptive Impact Absorption 207 Unkn
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Adaptive Impact Absorption 209 Figu
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7 Adaptive Damping of Vibration by
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Adaptive Damping of Vibration 253 x
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Adaptive Damping of Vibration 255 t
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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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