振动学第二章

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三角函数积化和差公式1.2PERIODIC MOTIONtitenntinsincos1.2PERIODIC MOTION1.2PERIODIC MOTION1.2PERIODIC MOTION1.3VIBRA

13、TION TERMINOLOGY1.3VIBRATION TERMINOLOGY1.3VIBRATION TERMINOLOGY1.3VIBRATION TERMINOLOGY1.3VIBRATION TERMINOLOGYENDFourier transform (傅立叶变换) 傅里叶是一位法国数学家和物理学家的名字，英语原名是Jean Baptiste Joseph Fourier(1768-1830), Fourier对热传递很感兴趣，于1807年在法国科学学会上发表了一篇论文，运用正弦曲线来描述温度分布，论文里有个在当时具有争议性的决断：任何连续周期信号可以由一组适当的正弦曲线组合而成

14、。当时审查这个论文的人，其中有两位是历史上著名的数学家拉格朗日(Joseph Louis Lagrange, 1736-1813)和拉普拉斯(Pierre Simon de Laplace, 1749-1827)，当拉普拉斯和其它审查者投票通过并要发表这个论文时，拉格朗日坚决反对，在他此后生命的六年中，拉格朗日坚持认为傅里叶的方法无法表示带有棱角的信号，如在方波中出现非连续变化斜率。法国科学学会屈服于拉格朗日的威望，拒绝了傅里叶的工作，幸运的是，傅里叶还有其它事情可忙，他参加了政治运动，随拿破仑远征埃及，法国大革命后因会被推上断头台而一直在逃避。直到拉格朗日死后15年这个论文才被发表出来。拉格

15、朗日是对的：正弦曲线无法组合成一个带有棱角的信号。但是，我们可以用正弦曲线来非常逼近地表示它，逼近到两种表示方法不存在能量差别，基于此，傅里叶是对的。用正弦曲线来代替原来的曲线而不用方波或三角波来表示的原因在于，分解信号的方法是无穷的，但分解信号的目的是为了更加简单地处理原来的信号。用正余弦来表示原信号会更加简单，因为正余弦拥有原信号所不具有的性质：正弦曲线保真度。一个正弦曲线信号输入后，输出的仍是正弦曲线，只有幅度和相位可能发生变化，但是频率和波的形状仍是一样的。且只有正弦曲线才拥有这样的性质，正因如此我们才不用方波或三角波来表示。Fourier transform (傅立叶变换) 傅立叶变换能将满足一定条件的某个函数表示成三角函数（正弦和/或余弦函数）或者它们的积分的线性组合。在不同的研究领域，傅立叶变换具有多种不同的变体形式，如连续傅立叶变换和离散傅立叶变换。最初傅立叶分析是作为热过程的解析分析的工具被提出的。傅里叶变换在物理学、声学、光学、结构动力学、量子力学、数论、组合数学、概率论、统计学、信号处理、密码学、海洋学、通讯、金融等领域都有着广泛的应用。例如在信号处理中，傅里叶变换的典型用途是将信号分解成振幅分量和频率分量。连续傅里叶变换离散傅里叶变换Trigonometric functions transfo