Motion is the essence of any mechanical system. Analyzing a system’s dynamical response to distinct motion parameters allows for increased understanding of its performance thresholds and can in turn provide clear data to inform improved system designs.
Modeling of Complex Dynamic Systems: Fundamentals and Applications equips readers with significant insights into nonlinear vibration phenomenology through a combination of advanced mathematical fundamentals and worked-through modeling experiments. To guide them in determining novel stabilization characteristics for complex moving objects, coupled structures, as well as the stochastic stability of mechanical systems, the technical and methodological analysis is accompanied by industry-relevant practical examples, contributing much sought-after applicable knowledge.
The book is intended for use by postgraduate students, academic researchers, and professional engineers alike.
1. 1.1 Introduction to complex numbers - p. 3 2. 1.2 Complex functions and their continuity and differentiability - p. 5 3. 1.3 Integrals of the complex functions - p. 7 4. 1.4 Cauchy integral theorem and formula - p. 9 5. 1.5 Taylor and Laurent series - p. 11 6. 1.6 Residues - p. 12 7. 1.7 The principle of argument - p. 15 8. 1.8 Laplace transform - p. 16 9. 1.9 Fourier transform - p. 18 10. References - p. 21 11. 2. Mathematical methods and applications in the analysis of nonlinear vibrations - p. 23 12. Abstract - p. 23 13. 2.1 Problem formulation - p. 23 14. 2.2 Numerical methods for solving general dynamical systems - p. 24 15. 2.3 Newmark method - p. 26 16. 2.4 Newmark method in the multidimensional case - p. 28 17. 2.5 Improvement of the accuracy by Richardson extrapolation - p. 29 18. 2.6 Harmonic balance method - p. 30 19. References - p. 33 20. 3. Mathematical methods in stochastic stability of mechanical systems - p. 35 21. Abstract - p. 35 22. 3.1 Introduction to probability space and random variables - p. 35 23. 3.2 Random processes - p. 39 24. 3.3 Wiener and white noise process - p. 41 25. 3.4 Markov and diffusion processes - p. 42 26. 3.5 Stochastic integrals and differential equations - p. 43 27. 3.6 Orstein–Uhlenbeck and bounded noise process - p. 48 28. 3.7 Moment Lyapunov exponent and stability - p. 48 29. 3.8 Perturbation method - p. 48 30. 3.9 New generalized transformations for multiple degrees of freedom mechanical systems - p. 50 31. References - p. 51 32. Part II: Stability of vibrations of complex moving objects: Modeling and applications - p. 53 33. 4. Stabilization and critical velocity of a moving mass - p. 55 34. Abstract - p. 55 35. 4.1 Introduction - p. 55 36. 4.2 Model and solution of free waves propagation - p. 56 37. 4.3 Response of the double-supported beam system subjected to a moving constant vertical force P - p. 63 38. 4.4 Stability of vibrations of a complex double-supported beam system - p. 66 39. 4.5 Influence of a compressive axial force on harmonic vibrations of the moving mass subjected to a constant vertical force - p. 73 40. 4.6 Conclusion - p. 77 41. References - p. 78 42. 5. Stability of vibrations of a system of discrete oscillators that are moving at an overcritical velocity - p. 81 43. Abstract - p. 81 44. 5.1 Introduction - p. 81 45. 5.2 Dispersion of complex model and characteristic equation - p. 83 46. 5.3 Determination of stable regions - p. 95 47. 5.4 Conclusion - p. 102 48. References - p. 103 49. 6. Vibrational benefits of the new stabilizer in moving coupled vehicles - p. 107 50. Abstract - p. 107 51. 6.1 Introduction - p. 107 52. 6.2 Mathematical model and critical velocity - p. 108 53. 6.3 Dynamic stability analysis - p. 119 54. 6.4 Influence of the elastic coupling on dynamic stability - p. 122 55. 6.5 Influence of the mass ratio between the connected rail cars on dynamic stability - p. 130 56. 6.6 Influence of the wheelbases on dynamic stability - p. 134 57. 6.7 Conclusion - p. 135 58. Appendix A 6.1 - p. 139 59. References - p. 143 60. 7. Dynamics and stability of a complex rail vehicle system - p. 145 61. Abstract - p. 145 62. 7.1 Introduction - p. 145 63. 7.2 Dispersion relation and critical velocity - p. 146 64. 7.3 Modeling of the complex rail vehicle systems and the characteristic equations - p. 151 65. 7.4 Dynamic stability analysis - p. 160 66. 7.5 Influence of the viscoelastic coupling on dynamic stability of a triple vehicle system - p. 162 67. 7.6 Influence of the secondary suspension on dynamic stability of a vehicle that has four contact points with the rail - p. 166 68. 7.7 Influence of the supercritical velocity and varying parameters of the secondary suspension on boundaries of the stability domain - p. 167 69. 7.8 Conclusion - p. 170 70. References - p. 170 71. 8. Modeling of a three-part viscoelastic foundation and its effect on dynamic stability - p. 173 72. Abstract - p. 173 73. 8.1 Introduction - p. 173 74. 8.2 Dispersion relation and critical velocity - p. 174 75. 8.3 Modeling of railway vehicle system and characteristic equation - p. 178 76. 8.4 Dynamic stability analysis - p. 184 77. 8.5 Parametric study - p. 186 78. 8.6 Influence of the secondary suspension stiffness on the stability domain - p. 190 79. 8.7 Influence of the bogie’s mass on dynamic stability - p. 190 80. 8.8 Influence of the wheelbase size on dynamic stability - p. 192 81. 8.9 Influence of the position of the supports on dynamic stability - p. 194 82. 8.10 Conclusion - p. 195 83. References - p. 196 84. 9. Vibrational instability in a complex moving object: innovative approaches to elastically damped connections between car body components and supports - p. 197 85. Abstract - p. 197 86. 9.1 Introduction - p. 197 87. 9.2 Modeling of the complex coupled models and characteristic equation - p. 199 88. 9.3 Solution method and stability criteria - p. 212 89. 9.4 Results and discussion - p. 214 90. 9.5 Conclusion - p. 229 91. References - p. 230 92. Part III: Linear and nonlinear vibrations: Stabilizing phenomena and applications - p. 235 93. 10. Nonlinear amplitude analysis of shear deformable beams supported by an elastic foundation with variable discontinuity - p. 237 94. Abstract - p. 237 95. 10.1 Introduction - p. 237 96. 10.2 Mathematical model and equations of motion - p. 238 97. 10.3 Natural frequencies and general modes of vibration - p. 244 98. 10.4 Geometric nonlinear vibrations - p. 252 99. 10.5 Conclusions - p. 254 100. References - p. 254 101. 11. Nonlinear vibrational characteristics of damaged beams resting on a Pasternak foundation - p. 257 102. Abstract - p. 257 103. 11.1 Introduction - p. 257 104. 11.2 Mathematical model of intact Reddy–Bickford beam on nonlinear Pasternak foundation - p. 260 105. 11.3 Modeling of damage and the p-FEM formulation - p. 266 106. 11.4 Natural frequencies and the stiffness function - p. 271 107. 11.5 Forced geometrically nonlinear vibrations in the time domain and the Newmark method - p. 275 108. 11.6 Geometrically nonlinear free vibrations in the frequency domain and the Continuation method - p. 281 109. 11.7 Conclusions - p. 285 110. Appendix 1 - p. 285 111. Appendix 2 - p. 294 112. References - p. 297 113. 12. The purpose of an arch in the stability of nonlinear vibrations of coupled structures - p. 301 114. Abstract - p. 301 115. 12.1 Introduction - p. 301 116. 12.2 Problem statement and modeling of a geometrically nonlinear beam–arch coupled system using the p-FEM - p. 303 117. 12.3 Convergence study - p. 310 118. 12.4 Time-domain analysis and improvement of the Newmark method - p. 312 119. 12.5 Harmonic balance and continuation methods in frequency domain analysis - p. 315 120. 12.6 Applications - p. 317 121. 12.7 Conclusion - p. 333 122. Appendix A 12.1 - p. 333 123. References - p. 335 124. 13. Quantitative effect of an axial load on the amplitude stability of rotating nano-beams - p. 337 125. Abstract - p. 337 126. 13.1 Introduction - p. 337 127. 13.2 Nonlocal elastic constitutive relations - p. 339 128. 13.3 Mathematical formulations of the equations of motion - p. 340 129. 13.4 Solution methodology - p. 341 130. 13.5 Natural frequencies of the rotating nonlocal cantilever nano-beam - p. 343 131. 13.6 Forced vibration of the undamped system - p. 348 132. 13.7 Forced vibration of the damped system - p. 349 133. 13.8 Results and discussion - p. 353 134. 13.9 Numerical discussion of the forced vibration of the undamped system - p. 353 135. 13.10 Numerical discussion of the forced vibration of the damped system - p. 354 136. 13.11 Conclusion - p. 357 137. References - p. 358 138. 14. Coupled multiple plate systems and their stability characteristics - p. 361 139. Abstract - p. 361 140. 14.1 Introduction - p. 361 141. 14.2 Theoretical formulation - p. 362 142. 14.3 Natural frequencies - p. 366 143. 14.4 Stability analysis of the multiplate system using CPT, FSDT, and HSDT - p. 372 144. 14.5 General natural mode shapes of multiplate systems - p. 374 145. 14.6 Numerical analysis - p. 376 146. 14.7 Conclusion - p. 381 147. References - p. 382 148. Part IV: Stochastic stability of structures and mechanical systems: Methodology and examples - p. 385 149. 15. Moment Lyapunov exponents and stochastic stability of the vibrationally isolated laminated plates - p. 387 150. Abstract - p. 387 151. 15.1 Introduction - p. 387 152. 15.2 Problem formulation - p. 389 153. 15.3 Stochastic stability analysis - p. 391 154. 15.4 Averaged equations in the general case of a three-degree-of-freedom-stochastic-system - p. 399 155. 15.5 Moment Lyapunov exponents - p. 401 156. 15.6 Zeroth order perturbation - p. 402 157. 15.7 First-order perturbation - p. 402 158. 15.8 Second-order perturbation - p. 404 159. 15.9 Stochastic stability conditions and discussion - p. 405 160. 15.10 Conclusions - p. 409 161. References - p. 410 162. 16. Higher-order stochastic averaging method in fractional stochastic dynamics - p. 413 163. Abstract - p. 413 164. 16.1 Introduction - p. 413 165. 16.2 Stochastic averaging - p. 414 166. 16.3 Results and discussions - p. 434 167. 16.4 Conclusions - p. 441 168. Appendix A1 Fractional calculus - p. 441 169. Appendix A2 Fractional viscoelasticity - p. 443 170. References - p. 445 171. 17. Parametric stochastic stability of viscoelastic rotating shafts - p. 447 172. Abstract - p. 447 173. 17.1 Introduction - p. 447 174. 17.2 Vibration of a rotating shaft - p. 448 175. 17.3 Stochastic averaging - p. 450 176. 17.4 Moment Lyapunov exponents - p. 456 177. 17.5 Conclusion - p. 472 178. References - p. 473 179. 18. Stochastic stability of circular cylindrical shells - p. 475 180. Abstract - p. 475 181. 18.1 Introduction - p. 475 182. 18.2 Problem formulation and deterministic vibrations - p. 477 183. 18.3 Stochastic vibration analysis - p. 483 184. 18.4 Moment Lyapunov exponents - p. 488 185. 18.5 Stochastic stability conditions and discussion - p. 492 186. 18.6 Conclusions - p. 493 187. Appendix A18.1 - p. 494 188. Appendix A18.2 - p. 496 189. Appendix A18.3 - p. 496 190. References - p. 497 191. 19. Generalized transformations for MDOF stochastic systems - p. 499 192. Abstract - p. 499 193. 19.1 Introduction - p. 499 194. 19.2 Stochastic stability of MDOF systems - p. 502 195. 19.3 Stochastic stability of a coupled rotating shaft system - p. 510 196. 19.4 Conclusions - p. 514 197. Appendix A19 - p. 515 198. References - p. 509 199. Part V: From traditional methods to Artificial Intelligence - p. 519 200. 20. Modeling and applications of markers in machine learning and technical practice - p. 521 201. Index - p. 531
Motion is the essence of any mechanical system. Analyzing a system’s dynamical response to distinct motion parameters allows for increased understanding of its performance thresholds and can in turn provide clear data to inform improved system designs.
Modeling of Complex Dynamic Systems: Fundamentals and Applications equips readers with significant insights into nonlinear vibration phenomenology through a combination of advanced mathematical fundamentals and worked-through modeling experiments. To guide them in determining novel stabilization characteristics for complex moving objects, coupled structures, as well as the stochastic stability of mechanical systems, the technical and methodological analysis is accompanied by industry-relevant practical examples, contributing much sought-after applicable knowledge.
The book is intended for use by postgraduate students, academic researchers, and professional engineers alike.
1. 1.1 Introduction to complex numbers - p. 3 2. 1.2 Complex functions and their continuity and differentiability - p. 5 3. 1.3 Integrals of the complex functions - p. 7 4. 1.4 Cauchy integral theorem and formula - p. 9 5. 1.5 Taylor and Laurent series - p. 11 6. 1.6 Residues - p. 12 7. 1.7 The principle of argument - p. 15 8. 1.8 Laplace transform - p. 16 9. 1.9 Fourier transform - p. 18 10. References - p. 21 11. 2. Mathematical methods and applications in the analysis of nonlinear vibrations - p. 23 12. Abstract - p. 23 13. 2.1 Problem formulation - p. 23 14. 2.2 Numerical methods for solving general dynamical systems - p. 24 15. 2.3 Newmark method - p. 26 16. 2.4 Newmark method in the multidimensional case - p. 28 17. 2.5 Improvement of the accuracy by Richardson extrapolation - p. 29 18. 2.6 Harmonic balance method - p. 30 19. References - p. 33 20. 3. Mathematical methods in stochastic stability of mechanical systems - p. 35 21. Abstract - p. 35 22. 3.1 Introduction to probability space and random variables - p. 35 23. 3.2 Random processes - p. 39 24. 3.3 Wiener and white noise process - p. 41 25. 3.4 Markov and diffusion processes - p. 42 26. 3.5 Stochastic integrals and differential equations - p. 43 27. 3.6 Orstein–Uhlenbeck and bounded noise process - p. 48 28. 3.7 Moment Lyapunov exponent and stability - p. 48 29. 3.8 Perturbation method - p. 48 30. 3.9 New generalized transformations for multiple degrees of freedom mechanical systems - p. 50 31. References - p. 51 32. Part II: Stability of vibrations of complex moving objects: Modeling and applications - p. 53 33. 4. Stabilization and critical velocity of a moving mass - p. 55 34. Abstract - p. 55 35. 4.1 Introduction - p. 55 36. 4.2 Model and solution of free waves propagation - p. 56 37. 4.3 Response of the double-supported beam system subjected to a moving constant vertical force P - p. 63 38. 4.4 Stability of vibrations of a complex double-supported beam system - p. 66 39. 4.5 Influence of a compressive axial force on harmonic vibrations of the moving mass subjected to a constant vertical force - p. 73 40. 4.6 Conclusion - p. 77 41. References - p. 78 42. 5. Stability of vibrations of a system of discrete oscillators that are moving at an overcritical velocity - p. 81 43. Abstract - p. 81 44. 5.1 Introduction - p. 81 45. 5.2 Dispersion of complex model and characteristic equation - p. 83 46. 5.3 Determination of stable regions - p. 95 47. 5.4 Conclusion - p. 102 48. References - p. 103 49. 6. Vibrational benefits of the new stabilizer in moving coupled vehicles - p. 107 50. Abstract - p. 107 51. 6.1 Introduction - p. 107 52. 6.2 Mathematical model and critical velocity - p. 108 53. 6.3 Dynamic stability analysis - p. 119 54. 6.4 Influence of the elastic coupling on dynamic stability - p. 122 55. 6.5 Influence of the mass ratio between the connected rail cars on dynamic stability - p. 130 56. 6.6 Influence of the wheelbases on dynamic stability - p. 134 57. 6.7 Conclusion - p. 135 58. Appendix A 6.1 - p. 139 59. References - p. 143 60. 7. Dynamics and stability of a complex rail vehicle system - p. 145 61. Abstract - p. 145 62. 7.1 Introduction - p. 145 63. 7.2 Dispersion relation and critical velocity - p. 146 64. 7.3 Modeling of the complex rail vehicle systems and the characteristic equations - p. 151 65. 7.4 Dynamic stability analysis - p. 160 66. 7.5 Influence of the viscoelastic coupling on dynamic stability of a triple vehicle system - p. 162 67. 7.6 Influence of the secondary suspension on dynamic stability of a vehicle that has four contact points with the rail - p. 166 68. 7.7 Influence of the supercritical velocity and varying parameters of the secondary suspension on boundaries of the stability domain - p. 167 69. 7.8 Conclusion - p. 170 70. References - p. 170 71. 8. Modeling of a three-part viscoelastic foundation and its effect on dynamic stability - p. 173 72. Abstract - p. 173 73. 8.1 Introduction - p. 173 74. 8.2 Dispersion relation and critical velocity - p. 174 75. 8.3 Modeling of railway vehicle system and characteristic equation - p. 178 76. 8.4 Dynamic stability analysis - p. 184 77. 8.5 Parametric study - p. 186 78. 8.6 Influence of the secondary suspension stiffness on the stability domain - p. 190 79. 8.7 Influence of the bogie’s mass on dynamic stability - p. 190 80. 8.8 Influence of the wheelbase size on dynamic stability - p. 192 81. 8.9 Influence of the position of the supports on dynamic stability - p. 194 82. 8.10 Conclusion - p. 195 83. References - p. 196 84. 9. Vibrational instability in a complex moving object: innovative approaches to elastically damped connections between car body components and supports - p. 197 85. Abstract - p. 197 86. 9.1 Introduction - p. 197 87. 9.2 Modeling of the complex coupled models and characteristic equation - p. 199 88. 9.3 Solution method and stability criteria - p. 212 89. 9.4 Results and discussion - p. 214 90. 9.5 Conclusion - p. 229 91. References - p. 230 92. Part III: Linear and nonlinear vibrations: Stabilizing phenomena and applications - p. 235 93. 10. Nonlinear amplitude analysis of shear deformable beams supported by an elastic foundation with variable discontinuity - p. 237 94. Abstract - p. 237 95. 10.1 Introduction - p. 237 96. 10.2 Mathematical model and equations of motion - p. 238 97. 10.3 Natural frequencies and general modes of vibration - p. 244 98. 10.4 Geometric nonlinear vibrations - p. 252 99. 10.5 Conclusions - p. 254 100. References - p. 254 101. 11. Nonlinear vibrational characteristics of damaged beams resting on a Pasternak foundation - p. 257 102. Abstract - p. 257 103. 11.1 Introduction - p. 257 104. 11.2 Mathematical model of intact Reddy–Bickford beam on nonlinear Pasternak foundation - p. 260 105. 11.3 Modeling of damage and the p-FEM formulation - p. 266 106. 11.4 Natural frequencies and the stiffness function - p. 271 107. 11.5 Forced geometrically nonlinear vibrations in the time domain and the Newmark method - p. 275 108. 11.6 Geometrically nonlinear free vibrations in the frequency domain and the Continuation method - p. 281 109. 11.7 Conclusions - p. 285 110. Appendix 1 - p. 285 111. Appendix 2 - p. 294 112. References - p. 297 113. 12. The purpose of an arch in the stability of nonlinear vibrations of coupled structures - p. 301 114. Abstract - p. 301 115. 12.1 Introduction - p. 301 116. 12.2 Problem statement and modeling of a geometrically nonlinear beam–arch coupled system using the p-FEM - p. 303 117. 12.3 Convergence study - p. 310 118. 12.4 Time-domain analysis and improvement of the Newmark method - p. 312 119. 12.5 Harmonic balance and continuation methods in frequency domain analysis - p. 315 120. 12.6 Applications - p. 317 121. 12.7 Conclusion - p. 333 122. Appendix A 12.1 - p. 333 123. References - p. 335 124. 13. Quantitative effect of an axial load on the amplitude stability of rotating nano-beams - p. 337 125. Abstract - p. 337 126. 13.1 Introduction - p. 337 127. 13.2 Nonlocal elastic constitutive relations - p. 339 128. 13.3 Mathematical formulations of the equations of motion - p. 340 129. 13.4 Solution methodology - p. 341 130. 13.5 Natural frequencies of the rotating nonlocal cantilever nano-beam - p. 343 131. 13.6 Forced vibration of the undamped system - p. 348 132. 13.7 Forced vibration of the damped system - p. 349 133. 13.8 Results and discussion - p. 353 134. 13.9 Numerical discussion of the forced vibration of the undamped system - p. 353 135. 13.10 Numerical discussion of the forced vibration of the damped system - p. 354 136. 13.11 Conclusion - p. 357 137. References - p. 358 138. 14. Coupled multiple plate systems and their stability characteristics - p. 361 139. Abstract - p. 361 140. 14.1 Introduction - p. 361 141. 14.2 Theoretical formulation - p. 362 142. 14.3 Natural frequencies - p. 366 143. 14.4 Stability analysis of the multiplate system using CPT, FSDT, and HSDT - p. 372 144. 14.5 General natural mode shapes of multiplate systems - p. 374 145. 14.6 Numerical analysis - p. 376 146. 14.7 Conclusion - p. 381 147. References - p. 382 148. Part IV: Stochastic stability of structures and mechanical systems: Methodology and examples - p. 385 149. 15. Moment Lyapunov exponents and stochastic stability of the vibrationally isolated laminated plates - p. 387 150. Abstract - p. 387 151. 15.1 Introduction - p. 387 152. 15.2 Problem formulation - p. 389 153. 15.3 Stochastic stability analysis - p. 391 154. 15.4 Averaged equations in the general case of a three-degree-of-freedom-stochastic-system - p. 399 155. 15.5 Moment Lyapunov exponents - p. 401 156. 15.6 Zeroth order perturbation - p. 402 157. 15.7 First-order perturbation - p. 402 158. 15.8 Second-order perturbation - p. 404 159. 15.9 Stochastic stability conditions and discussion - p. 405 160. 15.10 Conclusions - p. 409 161. References - p. 410 162. 16. Higher-order stochastic averaging method in fractional stochastic dynamics - p. 413 163. Abstract - p. 413 164. 16.1 Introduction - p. 413 165. 16.2 Stochastic averaging - p. 414 166. 16.3 Results and discussions - p. 434 167. 16.4 Conclusions - p. 441 168. Appendix A1 Fractional calculus - p. 441 169. Appendix A2 Fractional viscoelasticity - p. 443 170. References - p. 445 171. 17. Parametric stochastic stability of viscoelastic rotating shafts - p. 447 172. Abstract - p. 447 173. 17.1 Introduction - p. 447 174. 17.2 Vibration of a rotating shaft - p. 448 175. 17.3 Stochastic averaging - p. 450 176. 17.4 Moment Lyapunov exponents - p. 456 177. 17.5 Conclusion - p. 472 178. References - p. 473 179. 18. Stochastic stability of circular cylindrical shells - p. 475 180. Abstract - p. 475 181. 18.1 Introduction - p. 475 182. 18.2 Problem formulation and deterministic vibrations - p. 477 183. 18.3 Stochastic vibration analysis - p. 483 184. 18.4 Moment Lyapunov exponents - p. 488 185. 18.5 Stochastic stability conditions and discussion - p. 492 186. 18.6 Conclusions - p. 493 187. Appendix A18.1 - p. 494 188. Appendix A18.2 - p. 496 189. Appendix A18.3 - p. 496 190. References - p. 497 191. 19. Generalized transformations for MDOF stochastic systems - p. 499 192. Abstract - p. 499 193. 19.1 Introduction - p. 499 194. 19.2 Stochastic stability of MDOF systems - p. 502 195. 19.3 Stochastic stability of a coupled rotating shaft system - p. 510 196. 19.4 Conclusions - p. 514 197. Appendix A19 - p. 515 198. References - p. 509 199. Part V: From traditional methods to Artificial Intelligence - p. 519 200. 20. Modeling and applications of markers in machine learning and technical practice - p. 521 201. Index - p. 531
