Nonlinear vibration

Author: Zhou Jiqing / et al.
Editor-in-Chief: Zheng Lifang
Publisher:
Publish Date: 2001-10-01
Features:
Content Summary This book systematically presents the classical and modern theories and methods of nonlinear vibration. The book is divided into 12 chapters. The first 9 chapters cover the classical aspects of nonlinear vibration, studying the behavior of conservative systems, dissipative systems, self-excited vibration systems, forced vibration systems, and parametric excitation systems from both qualitative and quantitative perspectives. Chapter 3 on motion stability can be studied independently, and Chapter 8 explores some characteristics of multi-degree-of-freedom systems. Chapters 9 to 12 introduce the latest research findings from scholars in recent two decades—point mapping, cell mapping, catastrophe, bifurcation, and chaos phenomena. The appendix includes six computer programs, and each chapter is followed by exercises. The book is concise and to the point, clarifying mathematical derivations while emphasizing the physical essence of the systems. It is rich in content, progressing from simple to complex, and is suitable for teaching. This book can be used as a reference for graduate students or senior undergraduate students in mechanics, mechanical engineering, physics, and related fields, as well as for teachers and technicians in relevant disciplines.
Excerpt:
The mechanism of damping is not yet fully understood. In linear vibration theory, it is often assumed to be linear damping, or even sometimes proportional damping. These assumptions have certain engineering backgrounds and make the differential equations easier to solve, with results of acceptable accuracy. The existence of linear damping causes vibrations in linear systems to decay. In nonlinear systems, nonlinear damping (e.g., negative damping, quadratic damping, hysteresis damping, etc.) may occur. Even without periodic external forces, the system may exhibit periodic solutions. Under the action of a single-frequency periodic external force, the steady-state solutions of forced vibration in nonlinear systems may contain components at the same frequency as the external force, as well as components at different frequencies, such as subharmonics, superharmonics, and ultraharmonics. When the frequency of the external force changes continuously from high to low or vice versa, the amplitude of forced vibration in the system may exhibit jumps, and the positions of these jump points may differ depending on the order of frequency change. For humans, there is only one nature, but people study it from both deterministic and indeterministic perspectives. Linear vibration theory is based on Newtonian classical mechanics and is entirely deterministic. Once the equations are established and initial conditions are given, we can calculate "the future," and if time is given a negative value, we can calculate "the past." Everything is so perfect. In nonlinear systems, some deterministic equations exhibit highly complex behavior under certain parameter conditions. The motion trajectories are chaotic and non-repeating but confined within a certain range. Even with given initial conditions, the state of the system at any future time cannot be determined, and their characteristics require an indeterministic perspective. This discovery broke the deterministic dominance of Newton and Laplace, brought intellectual liberation and innovation, and accelerated the rapid development of the discipline. The above is only a rough introduction to some characteristics of nonlinear vibration. In the following chapters, we will analyze and study them separately.
1.2 Main Content of Nonlinear Vibration Theory
1. Equation Establishment The mechanical problems of nonlinear vibration systems can be described using one or a set of nonlinear differential equations, difference equations, or even algebraic equations. Therefore, equation establishment is the most fundamental issue. Newton's laws, d'Alembert's principle, Lagrange equations, Hamilton's principle, and other principles in mechanics can all be used to establish dynamic equations. The purpose of establishing equations is to solve them, and the process is naturally related to the nature of the problem. However, equation establishment is also related to the accuracy and methods of solution, i.e., the existing means, which are often overlooked by people, especially beginners. To study the most general nonlinear vibration system, one can establish the most general nonlinear equations, theoretically obtain the most general results, and achieve broad applicability and depth. However, this lacks the means to solve them, making it difficult to obtain meaningful results. For simple models, the equations are simple, and solving them is easy, but the conclusions are less general, but they provide a deeper and more specific understanding of the system. Therefore, when establishing equations (i.e., mathematical modeling), one must consider the means of solving the problem and establish equations based on existing capabilities, which has practical significance. In summary, equation establishment depends on the nature of the problem, the current level of science, the requirements of practical problems, research funding, and research time limits, among other factors. It can be said that establishing equations is the most important issue in studying nonlinear vibration.
2. Solution Methods There are many methods for studying nonlinear vibration, which can be broadly divided into experimental and analytical methods, with the latter further divided into qualitative and quantitative analysis methods. Experimental methods involve conducting experiments on real objects or models to draw conclusions. Due to advancements in modern technology, experimental analysis methods have developed rapidly, and the application of electronic computers has expanded the prospects of experimental research.