Nonlinear oscillations in physical systems-猫眼课题宝

DOI：

10.1115/1.3172988

发表时间：

1987-03

期刊：

影响因子：

通讯作者：

中科院分区：

其他

文献类型：

作者：

C. Hayashi;D. Mook

根据大众的需求，这本书是1964年那期的再版。非线性振荡围绕着微分方程不能精确求解的系统。然而，在采用近似解方面已经取得了很大的进展。这提供了关于非线性振荡的充分信息。这卷是为了提供这些非线性微分方程的近似解。正如作者所述，“本书中讨论的问题大多与强迫振动有关;此外，在大多数文本中，只处理具有一个自由度的系统。然而，在外力的影响下，这些系统中可能会发生各种各样的振荡。“这本书实现了作者的意图，打开了一个巨大的故事，使有趣的阅读。虽然最初指向电气工程师，但它仍然是机械工程师以及对其他学科感兴趣的研究人员的良好参考。这本书由四个部分组成，共13章，6个附录，和一个优秀的参考书目。第一部分，题为“非线性分析的物理方法”，包括三章。第一章介绍了分析方法。这包括强大的扰动方法（自治和非自治系统）。使用Duffing和货车Der Pol方程提供了示例。我们进展到迭代法（逐次迭代），平均法（自治和非自治），并参照描述函数法的谐波平衡的原则。比较了扰动法与谐波平衡法求解杜芬方程的异同。两者提供了几乎相同的答案。第二章介绍了拓扑方法，它是研究非线性振动的各种现象的有力工具。自治系统的研究依赖于拓扑学方法。本文从静平面上的积分曲线和奇点出发，根据积分曲线的性质，对奇点进行了分类。作者一头扎进了微分方程的标准形式，高阶奇点加上一个有趣的研究极限环。本文展示了状态空间中5种不同类型的奇点。其次重要的方法是（a）等倾线法（图解法），（B）处理自激振荡的Lienard方法，（c）高阶近似法（用本征值和近似值代替初值），（d）双三角形法，（e）图解构造的斜线法，（f）自治和非自治系统的二阶方程。说明性的例子解释这些方法使用Duffing和货车德尔波尔方程。第三章讨论非线性系统的稳定性。Liapunov稳定性打开这一章。这导致了劳斯-赫维茨准则，弗洛凯定理，马蒂厄方程，以及它们各自的功能。在不稳定区域，Hill无穷行列式和Whittaker方法是最突出的。前者深入研究了奇不稳定区域（第一和第三）和偶不稳定区域（第二）的稳定性问题和改进的特征指数近似。这一章包含了大量的例证。第二部分研究稳态强迫振荡。第四章讨论二阶系统周期振荡的稳定性。由于叠加原理不再适用于非线性系统，因此需要其他确定稳定性的方法。二阶系统的稳定性的不同条件是必要的，并且包括各种类型的变分方程。它指出了各种稳定性条件。第5章主要讨论谐波振荡。这里，高次谐波可以忽略，因为基波分量具有与主导外力相同的周期。在导出基本方程之后，这继续与周期性平衡状态及其稳定性加上具有非对称非线性特性的谐波振荡。本章最后以一个具有不对称特性的电路的演示作为结束。第六章讲的是高次谐波振荡。从串联谐振电路中的高次谐波振荡开始，它迁移到并联谐振电路。第七章讨论了次谐波振荡的重要问题。本研究的次谐波振荡的非线性特性与阶数之间的关系，即，二次，三次和五阶强迫函数。它们中的每一个都贡献了1/3的次谐波阶次。这是扩展到耗散和非耗散系统。这遵循由五次函数确定的非线性特性。本章的结论与非线性特征所代表的对称二次函数产生1 /2谐波振荡。实验研究补充了上述主题。第三部分深入研究瞬态中的强迫振荡。第8章开始与谐波振荡，并考虑周期解和他们的稳定性在过渡阶段。利用积分曲线分析了有无耗散系统的谐波振荡。在下一章中，我们将直接讨论次连续统和次谐波

By popular demand, this book is a reprint of the 1964 issue. Nonlinear oscillations revolve about systems whose differential equations cannot be solved exactly. However, great strides have been accomplished in employing approximate solutions. This provides adequate information concerning nonlinear oscillations. This volume is meant to furnish approximate solutions for these nonlinear differential equations. As stated by the author, "That the problems discussed in this book are mostly concerned with forced oscillations; furthermore in most of the text, only systems with one degree of freedom are treated. However, a wide variety of oscillations may occur in these systems under the influence of external forces." This book fulfills the author's intentions and opens up a vast story that makes interesting reading. Although originally pointed towards electrical engineers, it remains a good reference for mechanical engineers plus researchers interested in other disciplines. The book consists of four parts with a total of 13 chapters, 6 appendices, and an excellent bibliography. Part I, entitled "Physical methods of nonlinear analysis," contains 3 chapters. The initial chapter presents analytical methods. This includes the powerful perturbation method (autonomous and nonautonomous systems). Examples are provided using Duffing's and Van Der Pol equations. We progress to the iteration method (successive iteration), averaging method (autonomous and nonautonomous), and the principles of harmonic balance with reference to the describing function method. Comparisons are made in the solution of Duffing's equation between perturbation and harmonic balance. Both furnish almost identical answers. Chapter 2 reports on topological methods which are powerful schemes of investigating various phenomena of nonlinear oscillations. The study of autonomous systems rests upon topological methods. Beginning with integral curves and singular points in the static plane, we continue with the classification of singular points according to the character of the integral curves. The author plunges ahead into canonical forms of the differential equations, singular points of higher order plus an interesting study of limit cycles. The text shows 5 different types of singular points in state space. The next important methods are (a) isocline method (graphical), (b) Lienard's method dealing with self-excited oscillation, (c) higher order approximates (use of eigenvalue and approximations instead of initial values), (d) double delta methods, (e) slope line method of graphical construction, and (f) second order equations of autonomous and nonautonomous systems. Illustrative examples explain these methods using Duffing and Van Der Pol equations. Chapter 3 speaks about stability of nonlinear systems. Liapunov stability opens this chapter. This leads to Routh-Hurwitz criteria, Floquet's theorem, Mathieu's equations, and their respective functions. In the unstable region, Hill's infinite determinant and Whittaker method are the most prominent. The former delves into the stability problem and improved approximation of the characteristic exponent in the odd unstable regions (first and third) and even (second). This chapter contains a large number of illustrative examples. Part II studies forced oscillations in steady state. Chapter 4 treats stability of periodic oscillations in second order systems. Since the principle of superposition is no longer applicable to nonlinear systems, other means of determining stability is required. Different conditions for stability of second order systems are necessary and encompasses various types of variational equations. It points a finger at the various stability conditions. Chapter 5 focuses on harmonic oscillations. Here, higher harmonics can be neglected because the fundamental component possesses a period, the same as that of the predominant external force. After deriving the fundamental equation, this continues with periodic states of equilibrium and its stability plus harmonic oscillations with unsymmetric nonlinear characteristics. The chapter concludes with a demonstration of an electrical circuit having unsymmetrical characteristics. Chaper 6 speaks about higher harmonic oscillations. Beginning with higher harmonic oscillations in series-resonance circuits, it migrates to parallel resonance circuits. Chapter 7 deals with important topics of subharmonic oscillations. Relationships between nonlinear characteristics and order of subharmonic oscillations of this study, i.e., quadratic, cubic and fifth order forcing functions are considered. Each of them contribute to a subharmonic order of 1/3. This is extended to dissipative and nondissipative systems. This follows with nonlinear characteristics determined by quintic function. The chapter concludes with nonlinear characteristics represented by a symmetric quadratic function which produces a 1 /2 harmonic oscillation. Experimental investigations complement the above subjects. Part III delves into forced oscillations in the transient state. Chapter 8 opens up with harmonic oscillation and considers periodic solutions and their stability in the transient stage. Harmonic oscillations are analyzed by means of integral curves with and without dissipative systems. In the next chapter, we jump ahead into subcontinua and subharmonic

Nonlinear oscillations in physical systems

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