振動(dòng)學(xué)第二章

1、1.1 Harmonic Motion1.2 Periodic Motion1.3 Vibration TerminologyCHAPTER 1Oscillatory MotionThe study of vibration is concerned with the oscillatory motions of bodies and the forces associated with them. All bodies possessing mass and elasticity are capable of vibration. Thus, most engineering machine

2、s and structures experience vibration to some degree, and their design generally requires consideration of their oscillatory behavior.Oscillatory system can be broadly characterized as linear or nonlinear. For linear system, the principle of superposition holds, and the mathematical techniques avail

3、able for their treatment are well development. In contrast, techniques for the analysis of nonlinear systems are less well known, and difficult to apply. However, some knowledge of nonlinear system is desirable, because all systems tend to become nonlinear with increasing amplitude of oscillation.CH

4、APTER 1Oscillatory MotionThere are two general classes of vibrationsfree and forced. Free vibration takes place when a system oscillates under the action of forces inherent in the system itself, and when external impressed forces are absent. The system under free vibration will vibrate at one or mor

5、e of its natural frequencies, which are properties of the dynamical system established by its mass and stiffness distribution.Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the exci

6、tation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered, and dangerously large oscillations may result. The failure of major structures such as bridges, buildings, or airplane wings is an awesome possibilit

7、y under resonance. Thus, the calculation of the natural frequencies is major importance in the study of vibrations.CHAPTER 1Oscillatory MotionVibrating systems are all subject to damping to some degree because energy is dissipated by friction and other resistances. If the damping is small, it has ve

8、ry little influence on the natural frequencies of the system, and hence the calculations for the natural frequencies are generally made on the basis of no damping. On the other hand, damping is of great importance in limiting the amplitude oscillation at resonance.The number of independent coordinat

9、es requires to describe the motion of a system is called degrees of freedom of the system. Thus, a free particle undergoing general motion in space will have three degrees of freedom, and a rigid body will have six degrees of freedom, i.e., three components of position and three angles defining its

10、orientation. CHAPTER 1Oscillatory MotionFurthermore, a continuous elastic body will require an infinite number of coordinates (three for each point on the body) to describe its motion; hence, its degrees of freedom must be infinite. However, in many cases, parts of such bodies may be assumed to be r

11、igid, and the system may be considered to be dynamically equivalent to one having finite degrees of freedom. In fact, a surprisingly large number of vibration problems can be treated with sufficient accuracy by reducing the system to one having a few degrees of freedom.CHAPTER 1Oscillatory Motion1.1

12、HARMONIC MOTION1.1HARMONIC MOTION1.1HARMONIC MOTION1.1HARMONIC MOTION1.1HARMONIC MOTION1.1HARMONIC MOTION1.1HARMONIC MOTION1.1HARMONIC MOTION1.2PERIODIC MOTION1.2PERIODIC MOTION1.2PERIODIC MOTION三角函數(shù)積化和差公式1.2PERIODIC MOTIONtitenntinsincos1.2PERIODIC MOTION1.2PERIODIC MOTION1.2PERIODIC MOTION1.3VIBRA

13、TION TERMINOLOGY1.3VIBRATION TERMINOLOGY1.3VIBRATION TERMINOLOGY1.3VIBRATION TERMINOLOGY1.3VIBRATION TERMINOLOGYENDFourier transform (傅立葉變換) 傅里葉是一位法國(guó)數(shù)學(xué)家和物理學(xué)家的名字，英語(yǔ)原名是Jean Baptiste Joseph Fourier(1768-1830), Fourier對(duì)熱傳遞很感興趣，于1807年在法國(guó)科學(xué)學(xué)會(huì)上發(fā)表了一篇論文，運(yùn)用正弦曲線來(lái)描述溫度分布，論文里有個(gè)在當(dāng)時(shí)具有爭(zhēng)議性的決斷：任何連續(xù)周期信號(hào)可以由一組適當(dāng)?shù)恼仪€組合而成

14、。當(dāng)時(shí)審查這個(gè)論文的人，其中有兩位是歷史上著名的數(shù)學(xué)家拉格朗日(Joseph Louis Lagrange, 1736-1813)和拉普拉斯(Pierre Simon de Laplace, 1749-1827)，當(dāng)拉普拉斯和其它審查者投票通過(guò)并要發(fā)表這個(gè)論文時(shí)，拉格朗日?qǐng)?jiān)決反對(duì)，在他此后生命的六年中，拉格朗日?qǐng)?jiān)持認(rèn)為傅里葉的方法無(wú)法表示帶有棱角的信號(hào)，如在方波中出現(xiàn)非連續(xù)變化斜率。法國(guó)科學(xué)學(xué)會(huì)屈服于拉格朗日的威望，拒絕了傅里葉的工作，幸運(yùn)的是，傅里葉還有其它事情可忙，他參加了政治運(yùn)動(dòng)，隨拿破侖遠(yuǎn)征埃及，法國(guó)大革命后因會(huì)被推上斷頭臺(tái)而一直在逃避。直到拉格朗日死后15年這個(gè)論文才被發(fā)表出來(lái)。拉格

15、朗日是對(duì)的：正弦曲線無(wú)法組合成一個(gè)帶有棱角的信號(hào)。但是，我們可以用正弦曲線來(lái)非常逼近地表示它，逼近到兩種表示方法不存在能量差別，基于此，傅里葉是對(duì)的。用正弦曲線來(lái)代替原來(lái)的曲線而不用方波或三角波來(lái)表示的原因在于，分解信號(hào)的方法是無(wú)窮的，但分解信號(hào)的目的是為了更加簡(jiǎn)單地處理原來(lái)的信號(hào)。用正余弦來(lái)表示原信號(hào)會(huì)更加簡(jiǎn)單，因?yàn)檎嘞覔碛性盘?hào)所不具有的性質(zhì)：正弦曲線保真度。一個(gè)正弦曲線信號(hào)輸入后，輸出的仍是正弦曲線，只有幅度和相位可能發(fā)生變化，但是頻率和波的形狀仍是一樣的。且只有正弦曲線才擁有這樣的性質(zhì)，正因如此我們才不用方波或三角波來(lái)表示。Fourier transform (傅立葉變換) 傅立葉變換能將滿足一定條件的某個(gè)函數(shù)表示成三角函數(shù)（正弦和/或余弦函數(shù)）或者它們的積分的線性組合。在不同的研究領(lǐng)域，傅立葉變換具有多種不同的變體形式，如連續(xù)傅立葉變換和離散傅立葉變換。最初傅立葉分析是作為熱過(guò)程的解析分析的工具被提出的。傅里葉變換在物理學(xué)、聲學(xué)、光學(xué)、結(jié)構(gòu)動(dòng)力學(xué)、量子力學(xué)、數(shù)論、組合數(shù)學(xué)、概率論、統(tǒng)計(jì)學(xué)、信號(hào)處理、密碼學(xué)、海洋學(xué)、通訊、金融等領(lǐng)域都有著廣泛的應(yīng)用。例如在信號(hào)處理中，傅里葉變換的典型用途是將信號(hào)分解成振幅分量和頻率分量。連續(xù)傅里葉變換離散傅里葉變換Trigonometric functions transfo