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Asymptotic Multiple Scale Method in Time Domain: Multi-Degree-of-Freedom Stationary and Nonstationary Dynamics

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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 of problems 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.

Author(s): Jan Awrejcewicz, Roman Starosta, Grażyna Sypniewska-Kamińska
Publisher: CRC Press
Year: 2022

Language: English
Pages: 409
City: Boca Raton

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Authors
Chapter 1: Introduction
1.1. Literature review
1.2. Multiple scale method
Chapter 2: Spring Pendulum
2.1. Introduction
2.2. Mathematical model
2.3. Solution method
2.4. Non-resonant vibration
2.5. Resonant vibration at simultaneously occurring external resonances
2.6. Steady-state vibration in simultaneously occurring external resonances
2.7. Stability analysis
2.8. Closing remarks
Chapter 3: Kinematically Excited Spring Pendulum
3.1. Introduction
3.2. The physical and mathematical model
3.3. Asymptotic solution
3.4. Resonance oscillations
3.5. Resonance steady-state oscillation
3.6. Stability
3.7. Closing remarks
Chapter 4: Spring Pendulum Revisited
4.1. Introduction
4.2. The physical and mathematical model
4.3. Complex representation
4.4. Asymptotic solution
4.5. Analysis of non-stationary motion
4.6. Stationary motion
4.7. Stability analysis
4.8. Closing remarks
Chapter 5: Physical Spring Pendulum
5.1. Introduction
5.2. Mathematical model
5.3. Solution method
5.4. Non-resonant vibration
5.5. Resonant vibration in simultaneously occurring external resonances
5.6. Steady-state vibration in simultaneously occurring external resonances
5.7. Stability analysis
5.8. Closing remarks
Chapter 6: Nonlinear Torsional Micromechanical Gyroscope
6.1. Introduction
6.2. Operation principle of micromechanical gyroscope
6.3. Mathematical model
6.4. Approximate motion equations for small vibration
6.5. Asymptotic solution for non-resonant vibration
6.6. Resonant vibration in simultaneously occurring external and internal resonances
6.7. Steady-state responses in simultaneously occurring resonances
6.8. Stability analysis
6.9. Closing remarks
Chapter 7: Torsional Oscillations of a Two-Disk Rotating System
7.1. Introduction
7.2. Harmonic oscillator with added nonlinear oscillator
7.3. Formulation of the problem
7.4. Mathematical model
7.5. Complex representation
7.6. Solution method
7.7. Resonant oscillation
7.8. Steady-state motion
7.8.1. Stability
7.9. Non-stationary oscillations
7.9.1. Non-damped oscillations
7.9.2. Damped oscillations
7.10. Closing remarks
Chapter 8: Oscillator with a Springs-in-Series
8.1. Introduction
8.2. Mathematical model
8.3. Solution method
8.4. Non-resonant vibration
8.5. Resonant vibration
8.6. Steady-state vibration at main resonance
8.7. Stability analysis
8.8. Closing remarks
Chapter 9: Periodic Vibrations of Nano/Micro Plates
9.1. Introduction
9.2. Mathematical formulation
9.3. The Bubnov-Galerkin method with double mode model
9.4. Validation of the proposed approach
9.5. Non-resonant vibration
9.6. Resonant vibration
9.6.1. Steady state resonant responses
9.6.2. Ambiguous resonance areas
9.7. Closing remarks
References
Index