大学物理教案机械振动与机械波讲解学习.pdf

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1、资料收集于网络，如有侵权请联系网站删除word 可编辑教学目标1.掌握简谐振动的定义、表达方式、简谐振动的合成方法；了解自由、阻尼、强迫等各类简谐振动的特点和规律。2.掌握振动和波的关系、波的相干条件、叠加原理、驻波的形成条件、驻波的振幅、相位和能量的空间分布，半波损失。3.学会建立波动方程。教学难点多自由体系的小振动第十一章机械振动振动 是指物体或系统在其平衡位置附近的往复运动。(例子：物体位置、电流强度、电压、电场强度、磁场强度等)。物体或系统质点数是无穷的，自由度数也是无穷的，因此存在空间分布和时间分布，需要用偏微分方程描述(如果一个微分方程中出现多元函数的偏导数，或未知函数与几个变量有

2、关，而且未知函数对应几个变量的导数，那么这种微分方程就是偏微分方程。例如弦包含很多的质点，不能用质点力学的定律研究，但是可以将其细分成若干个极小的小段，每小段可以抽象成一个质点，用微分的方法研究质点的位移，其是这点所在的位置和时间变量的函数，根据张力，就可以建立起弦振动的偏微分方程)。一、简谐振动(单自由度体系无阻尼自由小振动)虽然多质点的振动要用偏微分方程描述，但是我们可以简化或只考虑细分成的每一小段，那么就成为单质点单自由度(只需一个坐标变量)的振动。222222222,0cos():0ii tFkkFkx axmmmd xd xax axdtdtxAtAe ei，令特征方程特征根：A(振

3、幅)、(初相位)都是 积分常数，k为倔强系数。在微分方程中所出现的未知函数的导数的最高阶数称为这个方程的阶。形如()()dxP t xQ xdt的方程为线性方程，其特点是它关于未知函数x及其导数dxdt都是一次 的。若()0Q x，则()0dxP t xdt称为 齐次 的线性方程。二阶常系数齐次线性微分方程的解法：12121212121,212cossinttttxcec excc t eixectct由cos()sin()xAtvAt资料收集于网络，如有侵权请联系网站删除word 可编辑按周期定义，cos()cossin()sinAtAtTAtAtT，同时满足以上两方程的T的最小值应为2pw

4、，所以2Tpw=，于是1,2Tnwpn=，w称为 圆频率 或角频率。不像A、，由初始条件决定，w由固有参量k和m决定，与初始条件无关，故称为振子的固有频率。简谐振动的状态的物理量位置和速度随时间变化，但只要t相同，振动的状态就相同，所以t是决定振动状态的物理量，称为 位相。w是位相的变化速率，单位是弧度/秒。由于复数平面上任一点对应一个矢量，还可以用一个旋转矢量 来描述简谐振动。在相空间中，简谐振动由一条椭圆曲线所描述：位移和动量c o s(),si n(xAtpm vmAt满足椭圆方程22221()xpAmA举例：单摆的摆动弹簧振子和单摆都是在弹性力或准弹性力作用下作简谐振动的保守系统，称为

5、 谐振子。由于弹性力是保守力，简谐振动中机械能是守恒的，于是22222222211cos(),sin()221sin(),2212pkpkEkxkAtpm AtpkEmAtmmEEEkA振动的合成与分解同方向、同频率的两简谐振动的合成(矢量法)312123123iiiitxxxxAeA eA eeI.212,0,1,2,kkjjp-=北则12AAA=+，即当两分振动的相位差为p的偶数倍时，合振动的振幅为两分振动振幅之和。II.()2121,0,1,2,kkjjp-=+=北则12AAA=-，即当两分振动的相位差为p的奇文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6

6、Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F

7、4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3

8、M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1

9、U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J

10、7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9

11、T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10

12、N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2资料收集于网络，如有侵权请联系网站删除word 可编辑数倍时，合振动的振幅为两分振动振幅之差。III.21jj-为一般值，则1212AAAAA-+。同方向、不同频率的两简谐振动的合成(三角函数法)参见 拍振动方向垂直的两谐振动的合成(三角法、计算机法)文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码

13、:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2

14、HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 Z

15、T8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档

16、编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z

17、2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4

18、 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2

19、文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2资料收集于网络，如有侵权请联系网站删除word 可编辑()()212121121212212112222222211221221212211coscoscoscossinsincoscoscoscoscossinsincoscoscossinsincoscos2coscossinsinsincoscossixttAyttAxytAAxyxytAAA A

20、xtAjwjjwjjjwjjwjjjjwjjjjjjwjjjwj=-?Ty?=-?t-=-?+-=-=()()()2121212122121122222222112212212122222121221212nsinsinsinsincoscossinsinsinsinsinsincossincoscos2sinsincossin2cos()sintyttAxytAAxyxytAAA AxyxyAAA Ajwjjjwjjwjjjjwjjjjjjwjjjjjj-?Ty?=-?t-=-?+-=-+-=-若频率比为简单整数比，则合成曲线是稳定的封闭的，运动也具有周期性，其轨迹称为李萨如图形。I.若21

21、0jj-=，则21AyxA=II.若2211,AyxAjjp-=-III.若22212212,12xyyAApjj-=+=IV.若222122123,12xyyAApjj-=+=-二、单自由度体系的小振动单自由度 指只需要一个坐标就可以确定系统的位置。1.自由振动势能()V q在平衡位置0qq附近展开得002200021()()()()2qqdVd VV qV qqqqqdqdq1122112212cos(),cos()coscossinsin,coscossinsinxAtyAtxyttttAAwjwjwjwjwjwj=+=+=-=-文档编码:CM2J7F4F9Z2 HF2S9T3M4O4

22、ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文

23、档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9

24、Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O

25、4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z

26、2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4

27、F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M

28、4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2资料收集于网络，如有侵权请联系网站删除word 可编辑第一项为常数，可取为势能的零点。因在稳定平衡位置势能取驻值(导数为0 的点称为函数的驻点，在驻点取得的函数值为驻值，而极值点0 x是指函数在邻域00,xx)内，0fx是函数的最大值或最小值)，第 2 项中的一阶导数为零。记02212qd Vkdq0 xqq得212Vk

29、x考虑到对稳定约束0tr，根据iiiqqtrrr，可得动能2222011221211(),()22iiiiiiiiiiiiiiiTmqmq qmqqtqqqtmtTa q qmxma q其中rrrrrrr于是拉氏函数221122LTVmxkx。代入拉氏方程得200mxkxxx或其中km为振动频率。上述方程有自由振动解：cosxAt。A为振幅，为初相位。附注：拉格朗日方程,22222222211111(1,2,)111()()()222iiiiiiiiiNNNNNiiiim xm yY mzZ iNTm xyzmxyzm xyz(1-1)(1-2)如果讨论是“保守力系”(指力学系统中的力所作之功

30、，仅与起末位置有关，而与具体路径无关。具有此性质的力场，一定可以引入一位置函数(,)V x y z，而此力所作之功为xyzF dxF dyF dzdV，按功与路径无关的性质，dV应为一全微分VVVdVdxdydzxyz，两式比较得,iiiiiiVVVYZxyz，由此得到iiiiiiid mxd xdTmmxdtxdtdt文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2

31、 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4

32、ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文

33、档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9

34、Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O

35、4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z

36、2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4

37、F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2资料收集于网络，如有侵权请联系网站删除word 可编辑(1-6)于是，由(1-1)得0,0,0iiiiiidTVdTVdTVdtxxdtyydtzz引入拉格朗日函数111111(,)NNNNNNLL xyzxyzxyzxyzTV，可将(1-6)式写成(1-7)将方程(1-7)的直角坐标,x y z换成广义坐标，即得描述具有s个自由度系统的拉氏方程。0(1,2,)iidLLisdtqq2.阻尼振动当速度不大时，阻力与速度的一次方成正比，方向相反，即-bR=x运动方程变为m+kbxx=-x，即20+xx+x=(1-8)其中bm，令txAe，

38、代入(1-8)，得220+，解出2i=，其中222=(因为阻尼系数通常很小)。于是2costxAet(1-9)当存在阻尼时，解是随时间减小的。3.受迫振动若系统除存在阻尼外，还有固有性外力(策动力)，()cosF tFt=，则运动方程变为cos+bkFtxx+x=即2c o s+ftxx+x=(1-10)其中Ffm，式(1-10)的通解可写成一个特解与相应的齐次方程的通解(1-9)之和。后者随时间衰减，逐渐趋向于零。其特解试探形式为cosxAt代入(1-10)得2222cossinsincos0Af可解得0(1,2,)iidLLiNdtxx文档编码:CM2J7F4F9Z2 HF2S9T3M4O

39、4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z

40、2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4

41、F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M

42、4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U

43、6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7

44、F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T

45、3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2资料收集于网络，如有侵权请联系网站删除word 可编辑222422222222222222222222222222222222222222222222222-cos-sin-cos-sin-cos-2-sinsin-cossin-sinsin-cossin-ffA2222-1*222242-1-122222-coss

46、in0-1,-1-coscos-sin0sin-tan-ijijijAfAAA is AMtransposeAAffAAAff222arctan-f当时，发生 共振，振幅为fA。举例 1：弦振动方程弦上取一段微元,x xx，在任一时刻t这一段弦所受诸力应当平衡，即张力+惯性力+外力=0。惯性力：2222(,)(,)xxxu x tu x tdxxtt外力：(,)(,)xxxf x t dxfx txx，x均为,x xx中的点。张力：惯性力和外力均垂直于x轴，故张力在x方向的投影的代数和为零。2112212222()cos()cos011cos|11 tan111cos|11tan1xxxT x

47、xT xuxux，1是张力()T x的方向与水平方向的夹角文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M

48、4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U

49、6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7

50、F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T3M4O4 ZT8Y10N1U6Z2文档编码:CM2J7F4F9Z2 HF2S9T