"This book offers up novel research which uses analytical approaches to explore nonlinear features exhibited by various dynamic processes. Relevant to disciplines across engineering and physics, the asymptotic method combined with the multiple scale method is shown to be an efficient and intuitive way to approach mechanics. Beginning with new material on the development of cutting-edge asymptotic methods and multiple scale methods, the book introduces this method in time domain and provides examples of vibrations of systems. Clearly written throughout, it uses innovative graphics to exemplify complex concepts such as nonlinear stationary and nonstationary processes, various resonances and jump pull-in phenomena. It also demonstrates the simplification ofproblems through using mathematical modelling, by employing the use of limiting phase trajectories to quantify nonlinear phenomena. Particularly relevant to structural mechanics, in rods, cables, beams, plates and shells, as well as mechanical objects commonly found in everyday devices such as mobile phones and cameras, the book shows how each system is modelled, and how it behaves under various conditions. It will be of interest to engineers and professionals in mechanical engineering and structural engineering, alongside those interested in vibrations and dynamics. It will also be useful to those studying engineering maths and physics"--
This book offers up novel research which uses analytical approaches to explore nonlinear features exhibited by various dynamic processes. Relevant to disciplines across engineering and physics, the asymptotic method combined with the multiple scale method is shown to be an efficient and intuitive way to approach mechanics.
| Preface |
|
ix | |
| Authors |
|
xiii | |
|
|
|
1 | (16) |
|
|
|
1 | (10) |
|
1.2 Multiple scale method |
|
|
11 | (6) |
|
Chapter 2 Spring Pendulum |
|
|
17 | (40) |
|
|
|
17 | (2) |
|
|
|
19 | (2) |
|
|
|
21 | (1) |
|
2.4 Non-resonant vibration |
|
|
22 | (8) |
|
2.5 Resonant vibration at simultaneously occurring external resonances |
|
|
30 | (8) |
|
2.6 Steady-state vibration in simultaneously occurring external resonances |
|
|
38 | (5) |
|
|
|
43 | (12) |
|
|
|
55 | (2) |
|
Chapter 3 Kinematically Excited Spring Pendulum |
|
|
57 | (36) |
|
|
|
57 | (1) |
|
3.2 The physical and mathematical model |
|
|
58 | (3) |
|
|
|
61 | (9) |
|
3.4 Resonance oscillations |
|
|
70 | (8) |
|
3.5 Resonance steady-state oscillation |
|
|
78 | (8) |
|
|
|
86 | (5) |
|
|
|
91 | (2) |
|
Chapter 4 Spring Pendulum Revisited |
|
|
93 | (26) |
|
|
|
93 | (1) |
|
4.2 The physical and mathematical model |
|
|
94 | (2) |
|
4.3 Complex representation |
|
|
96 | (3) |
|
|
|
99 | (7) |
|
4.5 Analysis of non-stationary motion |
|
|
106 | (3) |
|
|
|
109 | (3) |
|
|
|
112 | (5) |
|
|
|
117 | (2) |
|
Chapter 5 Physical Spring Pendulum |
|
|
119 | (74) |
|
|
|
119 | (1) |
|
|
|
120 | (4) |
|
|
|
124 | (2) |
|
5.4 Non-resonant vibration |
|
|
126 | (22) |
|
5.5 Resonant vibration in simultaneously occurring external resonances |
|
|
148 | (24) |
|
5.6 Steady-state vibration in simultaneously occurring external resonances |
|
|
172 | (8) |
|
|
|
180 | (11) |
|
|
|
191 | (2) |
|
Chapter 6 Nonlinear Torsional Micromechanical Gyroscope |
|
|
193 | (54) |
|
|
|
193 | (3) |
|
6.2 Operation principle of micromechanical gyroscope |
|
|
196 | (2) |
|
|
|
198 | (3) |
|
6.4 Approximate motion equations for small vibration |
|
|
201 | (2) |
|
6.5 Asymptotic solution for non-resonant vibration |
|
|
203 | (10) |
|
6.6 Resonant vibration in simultaneously occurring external and internal resonances |
|
|
213 | (13) |
|
6.7 Steady-state responses in simultaneously occurring resonances |
|
|
226 | (14) |
|
|
|
240 | (5) |
|
|
|
245 | (2) |
|
Chapter 7 Torsional Oscillations of a Two-Disk Rotating System |
|
|
247 | (32) |
|
|
|
247 | (1) |
|
7.2 Harmonic oscillator with added nonlinear oscillator |
|
|
248 | (1) |
|
7.3 Formulation of the problem |
|
|
249 | (1) |
|
|
|
250 | (5) |
|
7.5 Complex representation |
|
|
255 | (1) |
|
|
|
256 | (1) |
|
|
|
257 | (2) |
|
|
|
259 | (8) |
|
|
|
260 | (7) |
|
7.9 Non-stationary oscillations |
|
|
267 | (9) |
|
7.9.1 Non-damped oscillations |
|
|
268 | (7) |
|
7.9.2 Damped oscillations |
|
|
275 | (1) |
|
|
|
276 | (3) |
|
Chapter 8 Oscillator with a Springs-in-Series |
|
|
279 | (50) |
|
|
|
279 | (1) |
|
|
|
280 | (4) |
|
|
|
284 | (1) |
|
8.4 Non-resonant vibration |
|
|
285 | (12) |
|
|
|
297 | (12) |
|
8.6 Steady-state vibration at main resonance |
|
|
309 | (2) |
|
|
|
311 | (17) |
|
|
|
328 | (1) |
|
Chapter 9 Periodic Vibrations of Nano/Micro Plates |
|
|
329 | (44) |
|
|
|
329 | (3) |
|
9.2 Mathematical formulation |
|
|
332 | (3) |
|
9.3 The Bubnov-Galerkin method with double mode model |
|
|
335 | (2) |
|
9.4 Validation of the proposed approach |
|
|
337 | (1) |
|
9.5 Non-resonant vibration |
|
|
338 | (15) |
|
|
|
353 | (18) |
|
9.6.1 Steady state resonant responses |
|
|
361 | (4) |
|
9.6.2 Ambiguous resonance areas |
|
|
365 | (6) |
|
|
|
371 | (2) |
| References |
|
373 | (16) |
| Index |
|
389 | |
Jan Awrejcewicz is Head of the Department of Automation, Biomechanics and Mechatronics at Lodz University of Technology. His research covers mechanics, material science, biomechanics, applied mathematics, automation, physics and computer sciences, with his main focus being nonlinear processes. He has authored 850 journal papers and is Editor-in-Chief of three international journals. Additionally, Professor Awrejcewicz is recipient of numerous scientific awards including The Alexander von Humboldt Award for research and educational achievements.
Roman Starosta is Professor at the Institute of Applied Mechanics, Poznan University of Technology (PUT), Poland, where he is the head of the Department of Technical Mechanics. His area of research includes dynamics of structures, fluid mechanics, asymptotic methods, and computational systems of algebra. Professor Starosta is a member of the main board of the Polish Society of Theoretical and Applied Mechanics, and chairman of several editions of the conference on Vibrations in Physical Systems.
Grayna Sypniewska-Kamiska is currently Associate Professor at Poznan University of Technology, and has been at the Institute of Applied Mechanics since 1990. Her area of research covers nonlinear dynamics, asymptotic methods, computer methods in the area of applied mechanics of continuous and discrete systems, and inverse problems of heat conduction. She teaches mechanics, analytical mechanics, elasticity theory, mathematical physics as well as algorithmics, programming languages, and computer graphics.
