## ON THE CONCEPT OF MODES

CHEN Liqun,1)

Department of Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200444, China

Abstract

In order to better understand the concept of modes, the characteristics of modes are analyzed and compared with those of complex modes or nonlinear modes. The connotation of a mode includes the synchronicity of the modal vibrations, the invariance to the initial conditions, the orthogonality among modes, and the superposition of the modal vibrations into the response. A complex mode concerns with the synchronicity but not the invariance, while the orthogonality and the superposition hold only in the state space. A nonlinear mode concerns both with the synchronicity and the invariance, but not the orthogonality nor the superposition.

Keywords： teaching ; mode ; complex mode ; nonlinear mode ; vibration

CHEN Liqun. ON THE CONCEPT OF MODES. MECHANICS IN ENGINEERING[J], 2021, 43(2): 252-255 DOI:10.6052/1000-0879-20-430

17世纪,人们开始对模态有所认识。1733年丹尼尔·伯努利(Daniel Bernoulli)在研究垂直悬挂细线振动时发现存在不同的模态,1742年对振动杆的实验中发现振动为不同模态的叠加[1]。1747年,欧 拉(L. Euler)研究相同弹簧水平连接相同$n$个质点的纵向振动时,不仅精确地求出了$n$个模态,而且证明每个质点的振动是这些模态振动的叠 加[2]。经过拉格朗日(J.L. Lagrange)[3]、瑞利(J.W.S. Rayleigh)[4]、开尔文(T.W. Kelvin)和台特(P.G. Tait)[5]的发展和应用,模态概念成熟完善,成为多自由度振动分析的基础,也是国内外振动教材中[6-10]的重要内容。虽然模态概念是解耦多自由度系统或连续系统的基础,但在教学中往往着重介绍模态分析法,对模态概念没有重点阐述,因此学生缺乏对模态概念的透彻理解。这样,面对模态概念的进一步发展,如复模态和非线性模态,尤其觉得困惑。本文较为详细地解释了模态概念的基本属性,分析了模态概念发展为复模态概念和非线性模态概念时保留或舍弃了模态的哪些属性。这将有助于模态概念的教学,教师可以从更广泛的角度深入理解模态,从而帮助学生全面掌握模态概念。以下讨论主要是针对离散振动系统,连续振动系统基本上也有平行的结论。

## 1 固有模态

$$$\label{eq1} {M\ddot{{x}}+Kx}={\bf0}$$$

## 2 从实模态到复模态

$$$\label{eq2} {M\ddot{{x}}+G\dot{{x}}+Kx}={\bf 0}$$$

$$$\label{eq3} \left( {{\begin{array}{*{20}c} {{\bf 0}} & {{M}} \\ {{M}} & {{G}} \\ \end{array} }} \right)\left\{ {{\begin{array}{*{20}c} {{\ddot{{x}}}} \\ {{\dot{{x}}}} \\ \end{array} }} \right\}+\left( {{\begin{array}{*{20}c} {-{M}} & {{\bf 0}} \\ {{\bf 0}} & {{K}} \\ \end{array} }} \right)\left\{ {{\begin{array}{*{20}c} {{\dot{{x}}}} \\ {{x}} \\ \end{array} }} \right\}={\bf 0}$$$

$$$\label{eq4} {M\ddot{{x}}+C\dot{{x}}+Kx}={\bf 0}$$$

$$$\label{eq5} \left( {{\begin{array}{*{20}c} {{\bf 0}} & {{M}} \\ {{M}} & {{C}} \\ \end{array} }} \right)\left\{ {{\begin{array}{*{20}c} {{\ddot{{x}}}} \\ {{\dot{{x}}}} \\ \end{array} }} \right\}+\left( {{\begin{array}{*{20}c} {-{M}} & {{\bf 0}} \\ {{\bf 0}} & {{K}} \\ \end{array} }} \right)\left\{ {{\begin{array}{*{20}c} {{\dot{{x}}}} \\ {{x}} \\ \end{array} }} \right\}={\bf 0}$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Cannon JT, Dostrovsky S. The Evolution of Dynamics: Vibration Theory from 1687 to 1742. New York: Springer, 1981

Morris K. Mathematical Thought from Ancient to Modern Times (vol. 2). Oxford: Oxford University Press, 1972

Lagrange JL. Analytical Mechanics. Bossonnade A, Vagliente VN, transl. and ed. New York: Springer, 1997

Rayleigh JWS.

The Theory of Sound (Vols. 1 & 2)

New York: Dover, 1945

Kelvin TW, Tait PG. Treatise on Natural Philosophy. Cambridge: Cambridge University Press, 1888

Meirovitch L.

Fundamentals of Vibrations

Boston: McGraw-Hill, 2001

Gomsberg KJ.

Mechanical and Structural Vibrations: Theory and Applications

New York: John Wiley & Sons, 2001

Rao SS.

Mechanical Vibrations

Singapore: Prentice-Hall, 2005

Meirovitch L.

A new method of solution of the eigenvalue problem for gyroscopic systems

AIAA Journal, 1974,12:1337-1342

Foss KA.

Co-ordinates which uncouple the equations of motion of damped linear dynamics systems

ASME Journal of Applied Mechanics, 1958,25(1):361-364

Caughey TK.

Classical normal modes in damped linear dynamic systems

ASME Journal of Applied Mechanics, 1960,27(2):269-271

Rosenberg RM.

On nonlinear vibrations of systems with many degree of freedom

Shaw SW, Pierre C.

Non-linear normal modes and invariant manifolds

Journal of Sound and Vibration, 1991,150:170-173

Kerschen G.

Modal Analysis of Nonlinear Mechanical Systems

New York: Springer, 2014

Vakakis AF, Manevitch LI, Mikhlin YV, et al.

Normal Modes and Localization in Nonlinear Systems

New York: John Wiley and Sons, 1996

Wagg D, Neild S.

Nonlinear Vibration with Control, 2nd edn

New York: Springer, 2015

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