SYSTEM ANALYSIS THROUGH BOND GRAPH MODELING by ...

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22 formalism for modeling kinematic joints [Fav98, Zei95, Mas91]. The applications of Lagrangian bond graphs are many [Bro72, McB05a, McB05b, McB05c]. This research provides further insight into the creation of the Lagrangian bond graph <strong>by</strong> noting natural gyrator modulations of system I-element. A similar research path is currently under investigation. This path is concerned with the creation of Lagrange equations given a bond graph structure. By obtaining conservation laws of different energy domains, using bond graphs, Lagrangian- Hamiltonian mechanics can be extended to deal with dissipative elements and nonpotential fields [Muk05, Muk97, Kar77]. 2.3 Object-Oriented Bond Graph Modeling The ability to use models in a plug-and-play fashion gives the modeler a great advantage in the field of design and simulation. Bond graph modeling allows the user to easily model systems that cross engineering domains. The creation of a bond graph library, within an object-oriented framework, allows the designer to create models that cross multiple engineering domains and simultaneously simulate the models [Cel03a, McB03]. The object-oriented nature of the simulation software allows the reuse of models, eliminates equation generation errors, and removes the users from the difficulties of programming the code. The research presented here provides a bond graph library that can represent electrical, mechanical, hydraulic systems, etc. This library does account for
23 dissipated energy in the form of heat. However, complete thermodynamic systems cannot be modeled using this library. Current research in this area involves the creation of a thermodynamic bond graph library. The creation of a bond graph library that deals with thermodynamic systems, in an object-oriented manner, provides the users with the ability to model convective flows [Cel03b, Gre01a, Gre01b]. Similar research in this area is concerned with defining the role of the library designer and the role of the user. The ability of the user to employ the library’s models with confidence that the components operate correctly, without knowing their internal workings, provides the modeler with greater flexibility [Urq03a, Urq03b]. A bond graph methodology, being based on the first law of thermodynamics, helps provide confidence in the correctness of the library components. 2.4 System Efficiency Measurement Through Bond Graph Modeling Power flow information of a bond graph can be used to develop the state equations for a given system. However, once the equations of motion are obtained, often the power flow map and the system’s causal relationships are discarded. As a result, useful information is lost. This research uses the power flow information of a bond graph to develop a method for measuring the efficiency of a system. By monitoring the input power, and the output power, an efficiency measurement can be defined [McB05c].
Page 1 and 2: SYSTEM ANALYSIS THROUGH BOND GRAPH
Page 3 and 4: 3 STATEMENT BY AUTHOR This disserta
Page 5 and 6: 5 DEDICATION To Shannon, for her un
Page 7 and 8: TABLE OF CONTENTS (continued) 4.3.3
Page 9 and 10: 9 LIST OF FIGURES Figure 3.1. Power
Page 11 and 12: LIST OF FIGURES (continued) Figure
Page 13 and 14: LIST OF FIGURES (continued) Figure
Page 15 and 16: 15 LIST OF TABLES Table 3.1. Effort
Page 17 and 18: 17 CHAPTER 1: Introduction 1.1 Prob
Page 19 and 20: 19 Chapter 5 provides a means of me
Page 21: 21 CHAPTER 2: Related Work 2.1 Intr
Page 25 and 26: 25 of a thermo-fluid bond graph can
Page 27 and 28: 27 between the describing parameter
Page 29 and 30: 29 variables associated with it. Na
Page 31 and 32: 31 3.2.2 Bond Graph Junctions Power
Page 33 and 34: 33 power. They are called 1-port el
Page 35 and 36: 35 the conjugate variables. As seen
Page 37 and 38: 37 As seen in figure 3.6 the power
Page 39 and 40: 39 between all of the variables are
Page 41 and 42: 41 Figure 3.11. Possible Causal Ass
Page 43 and 44: 43 Figure 3.14. Possible Causal Ass
Page 45 and 46: 45 3.2.7 Bond Graph Causal Mark Ass
Page 47 and 48: 47 information is obvious from the
Page 49 and 50: 49 3. Use the conjugate variables a
Page 51 and 52: 51 3.2.9 Conversion of Bond Graph V
Page 53 and 54: 53 obtaining dynamic equations, and
Page 55 and 56: 55 3.3.1 Lagrangian to Hamiltonian
Page 57 and 58: 57 dL ∂L ∂L ∂L & (3.22) ∂q
Page 59 and 60: 59 An underlying assumption in the
Page 61 and 62: 61 7. Develop the Hamiltonian for t
Page 63 and 64: 63 The partial of the Lagrangian wi
Page 65 and 66: 65 bond graph shows a single degree
Page 67 and 68: 67 of a Lagrangian/Hamiltonian appr
Page 69 and 70: 69 d dt through 3.48 and the simple
Page 71 and 72: 71 since flows sum around a 0-junct
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73 2 2 ( + A′ )& θ − ( A + B
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75 of the bond graph, i.e., they ar
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77 The simplest bond graph equation
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79 2 ( A + B′ )&& φ sin θ + ( A
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81 ω & 3 = θ (3.102) The method d
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83 Also, the bond graph maps the fl
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85 4.2 Dymola The Dymola framework
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87 such that the user can assign a
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89 A possible set of equations for
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91 Vδ1 = V 2 −V1 Vδ 2 = Vi −V
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93 The icon window has been left bl
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95 derive the equations for a bond
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97 ⎡ 1 ⎛ q ⎤ B4 ⎞ 1 ⎛ qB4
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99 f f f 2 3 1 : : : x 2 1 1 1 x 1
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101 4.2.3.1 Structural Singularitie
Page 103 and 104:
103 Figure 4.10. Gear Train Bond Gr
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105 This section shows that bond gr
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107 [Pan88]. Also, note that the co
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109 The iconic representation of th
Page 111 and 112:
111 Figure 4.14. A-Causal Bond As s
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113 The bond models are complete an
Page 115 and 116:
115 Figure 4.19. Three-Port One The
Page 117 and 118:
117 Figure 4.22 shows a bond graph
Page 119 and 120:
119 Figure 4.25. C-Element Model Fi
Page 121 and 122:
121 The transformer model defines t
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123 Figure 4.32. Modulated Effort S
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125 from the bond graph 3-tuple, an
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127 Figure 4.38. Q Sensor Naturally
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129 These models can now be used to
Page 131 and 132:
131 Figure 4.43. Gyroscope Model: E
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133 4.4.2 Inertial Rate Sensor Mode
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135 The gyroscope in figure 4.45 is
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137 4.4.2.3 Roll Gyro Figure 4.50 s
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139 Figure 4.53. Sensor Delays The
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141 As seen in figure 4.55, the pla
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143 The icon labeled Platform_Cntrl
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145 pointed at a fixed inertial poi
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147 in the camera, a pitch command
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149 The icon of the completed model
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151 inertial axes. These inertial c
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153 The yaw achieved response is sh
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155 80 Camera: Pitch Pitch Angle (d
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157 bond graph and drop it into a h
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159 can be used to monitor the syst
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161 5.2.1 Servo Positioning System:
Page 163 and 164:
163 diode is seen in figure 5.3 as
Page 165 and 166:
165 Figure 5.5. Gear Train and Fin
Page 167 and 168:
167 The modulated effort source is
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169 Figure 5.7. Linear Fin Dynamics
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171 Here the electrical dynamics ar
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173 Similarly, the state space equa
Page 175 and 176:
175 Magnitude (dB) 50 0 -50 -100 -1
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177 showing up on terms (3, 4) and
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179 outputs are fin position and se
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181 5.4 Servo Controllers Separate
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183 Figure 5.14. Linear Controller/
Page 185 and 186:
185 5.4.3 Non-Linear Control Scheme
Page 187 and 188:
187 Figure 5.17. Content of the Y3
Page 189 and 190:
189 The power signal vector is pass
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191 6 5 (deg) Step: Hinge Moment =
Page 193 and 194:
193 1.5 x 106 5 (deg) Step: Hinge M
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195 PID1 delivers less energy to th
Page 197 and 198:
197 Figure 5.28 shows a clear and c
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199 The step responses and efficien
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201 Figures 5.34 and 5.35 correspon
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203 Figures 5.38 and 5.39 correspon
Page 205 and 206:
205 Figure 5.40. Two Non-Linear Act
Page 207 and 208:
207 Figure 5.43 through 5.47 show t
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209 5.3 5 (deg) Step: Hinge Moment
Page 211 and 212:
211 Figures 5.49 through 5.54 show
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213 20.3 20.2 20 (deg) Step: Hinge
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215 CHAPTER 6: Optimal Gain Compari
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217 need to be optimized further, o
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219 S ref C N 2 πd = (6.3) 4 2 1.5
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221 Figure 6.5 shows a Dymola model
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223 The equations used to execute t
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225 Angle of attack and body accele
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227 A Dymola model of the above aut
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229 1 3-Loop AP: -12.92 G step resp
Page 231 and 232:
231 Figure 6.18. Three Loop AP: Clo
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233 6 3-Loop AP: -12.92 G step resp
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235 than the missile response. Prop
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237 Equation 6.15 shows that C N is
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239 M δ Q * S * d * C ref Mδ I yy
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241 5 0 5 Deg. Fin Deflection, at S
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243 0 -0.5 3 Loop AP: -12.92 G step
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245 can be exploited to place the c
Page 247 and 248:
247 Although equations 6.44 through
Page 249 and 250:
249 Figures 6.32 through 6.34, show
Page 251 and 252:
251 Figures 6.35 and 6.36, show how
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253 6 Complete System: -12.92 G ste
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255 view. Gain set 1 rises higher i
Page 257 and 258:
257 Figures 6.43 and 6.44 show the
Page 259 and 260:
259 compare controller efficiencies
Page 261 and 262:
261 1.2 x 10-6 Complete System: -12
Page 263 and 264:
263 It is interesting to note, that
Page 265 and 266:
265 where −1 T K = −R B P (6.52
Page 267 and 268:
267 ∂J − ∂t * = H 1 ⎛ ∂J
Page 269 and 270:
269 u * = KC −1 * [ y − Du ] (6
Page 271 and 272:
271 Where −1 ⎡1⎤ [ − K D] K
Page 273 and 274:
−1 T ( Q − SR S ) 273 Ch = (6.1
Page 275 and 276:
275 real matrix, and an imaginary m
Page 277 and 278:
277 ~ the characteristic polynomial
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279 6.6.6 Nonlinear Autopilot Resul
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281 15 10 Complete System: -12.92 G
Page 283 and 284:
283 0.07 Complete System: 1 G step
Page 285 and 286:
Page 287 and 288:
287 Shifting the cg towards the nos
Page 289 and 290:
Page 291 and 292:
291 much increase in efficiency red
Page 293 and 294:
293 CHAPTER 7: Summary 7.1 Contribu
Page 295 and 296:
295 Various control schemes were pr
Page 297 and 298:
297 Naturally this expansion is not
Page 299 and 300:
299 Real Hric[1, 2]; Real Lric[1, 1
Page 301 and 302:
Text( extent=[-16, -62; 18, -80], s
Page 303 and 304:
Modelica.Blocks.Interfaces.OutPort
Page 305 and 306:
Text(extent=[36, 38; 78, 24], strin
Page 307 and 308:
307 AB2 = A*AB1; AB3 = A*AB2; for j
Page 309 and 310:
309 for j in 1:n loop Reigvec_outpu
Page 311 and 312:
311 eig3[2] = eig_input.signal[7];
Page 313 and 314:
313 APPENDIX A5: Dymola Models, Mis
Page 315 and 316:
315 APPENDIX B1: Symmetry of Hamilt
Page 317 and 318:
317 APPENDIX B2: Vandermonde Repres
Page 319 and 320:
319 APPENDIX C: Glossary of Terms 2
Page 321 and 322:
321 Cur84 Dym Elm94 Fah99 Fah94 Fav
Page 323 and 324:
323 Mas91 Mat McB05a Maschke, B.,
Page 325:
325 Zar02 Zei95 Zho96 Zarchan, P.,
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