英漢雙語(yǔ)彈性力學(xué)11

1、1 3 Summarize1-1 Movement differential equations of elastic objects11-2 Without rotating wave and equal volume wave11-3 Transverse wave and vertical wave11-4 Spherical waveChapter 11 Elastic Wave4 概述概述1-1 1-1 彈性體的運(yùn)動(dòng)微分方程彈性體的運(yùn)動(dòng)微分方程11-2 11-2 無旋波與等容波無旋波與等容波11-3 11-3 橫波與縱波橫波與縱波11-4 11-4 球面波球面波第十一章第十一章 彈性

2、波彈性波5Summarize:When elastic object bears loads in static force equilibrium conditions, not all the parts of object has displacement, distortion and stress. At the beginning of the loads, the parts which are more far from the loads have no impacts . After then ,the displacement , distortion and stres

3、s caused by loads transmit to other places in a finite speed of wave. This wave is called elastic wave. This chapter will first give movement differential equations of elastic objects, then introduce some conceptions of elastic wave and simplify the equations according to different elastic waves, at

4、 last give the speed transmitting formulas of wave in infinite elastic objects. 6概述概述 當(dāng)靜力平衡狀態(tài)下的彈性體受到荷載作用時(shí)，并不是在彈性體的所有各部分都立即引起位移、形變和應(yīng)力。在作用開始時(shí)，距荷載作用處較遠(yuǎn)的部分仍保持不受干擾。在作用開始后，荷載所引起的位移、形變和應(yīng)力，就以波動(dòng)的形式用有限大的速度向別處傳播。這種波動(dòng)就稱為彈性波。 本章將首先給出描述彈性體運(yùn)動(dòng)的基本微分方程，然后介紹彈性波的幾個(gè)概念，針對(duì)不同的彈性波，對(duì)運(yùn)動(dòng)微分方程進(jìn)行簡(jiǎn)化，最后給出波在無限大彈性體中傳播速度公式。711-1 Movem

5、ent differential equations of elastic objectsThe two assumptions are equal to the basic assumptions when we discuss static force questions. So the physic and geometry equations and elastic equations where stress component is expressed by displacement component , still are the same with movement equa

6、tions at any instantaneous time. The only difference is that the equilibrium differential equations of static questions must be substituted by movement differential equations . This chapter we still adopt the assumptions: （1） Elastic objects are ideal elastic objects. （2） The displacement and distor

7、tion are tinny.811-1 11-1 彈性體的運(yùn)動(dòng)微分方程彈性體的運(yùn)動(dòng)微分方程 上述兩條假設(shè)，完全等同于討論靜力問題的基本假設(shè)。因此，在靜力問題中給出的物理方程和幾何方程，以及把應(yīng)力分量用位移分量表示的彈性方程，仍然適用于討論動(dòng)力問題的任一瞬時(shí)，所不同的僅僅在于，靜力問題中的平衡微分方程必須用運(yùn)動(dòng)微分方程來代替。 本章仍然采用如下假設(shè)： （1） 彈性體為理想彈性體。 （2） 假定位移和形變都是微小的。9Toward any tiny object , when we apply dAlembert theory , we must consider stress , body f

8、orce and the inertia force of elastic objects caused by acceleration . In space right-angle coordinate system, the x, y, z directions component of inertia force of every unite volume are:Where is the density of elastic objects.22tu22t 22tw10 對(duì)于任取的微元體，運(yùn)用達(dá)朗伯爾原理，除了考慮應(yīng)力和體力以外，還須考慮彈性體由于具有加速度而產(chǎn)生的慣性力。每單位體積上

9、的慣性力在空間直角坐標(biāo)系的x,y,z方向的分量分別為:其中為彈性體的密度。22tu22t 22tw11Because of the equilibrium relations ,we simplify them and get:The above formulas are called movement differential equations of elastic objects. They and geometry equations and physic equations are the basic equations of movement questions of elasti

10、city mechanics022tuXzyxzxyxx022tYxzyxyzyy022twZyxzyzxzz12 由平衡關(guān)系，并簡(jiǎn)化后得：上式稱為彈性體的運(yùn)動(dòng)微分方程。它同幾何方程和物理方程一起構(gòu)成彈性力學(xué)動(dòng)力問題的基本方程。022tuXzyxzxyxx022tYxzyxyzyy022twZyxzyzxzz13Note 1: geometry equations:xuxyyzwzzywyzxwzuzxyuxxy14 注注1 1：幾何方程：幾何方程xuxyyzwzzywyzxwzuzxyuxxy15Note2: physic equations)(1zyxxE)(1xzyyE)(1yxzzEy

11、zyzE)1 (2zxzxE)1 (2xyxyE)1 (216 注注2 2：物理方程：物理方程)(1zyxxE)(1xzyyE)(1yxzzEyzyzE)1 (2zxzxE)1 (2xyxyE)1 (217 Because the displacement component is difficult to be expressed by stress and its derivative, so movement equations of elasticity mechanics are usually solved according to the displacement. Substi

12、tute the elasticity equations where stress components are expressed by displacement component into movement differential equations, and we let:Then we get:zwyxue0)211()1 (2222tuXuxeE0)211()1 (2222tYyeE0)211()1 (2222twZwzeE18 由于位移分量很難用應(yīng)力及其導(dǎo)數(shù)來表示，所以彈性力學(xué)動(dòng)力問題通常要按位移求解。將應(yīng)力分量用位移分量表示的彈性方程代入運(yùn)動(dòng)微分方程，并令：得：zwyxue

13、0)211()1 (2222tuXuxeE0)211()1 (2222tYyeE0)211()1 (2222twZwzeE19These are the basic differential equations of movement equations solved by displacement. They are also called Lame equations. We need boundary conditions to solve Lame equations. Besides these we still need original conditions , because

14、displacement components are the function of time variable . In order to simplify calculation , usually we neglect body force. Now the movement differential equations of elastic objects can be simplified as :)211()1 (2222uxeEtu)211()1 (2222yeEt)211()1 (2222wzeEt20 這就是按位移求解動(dòng)力問題的基本微分方程，也稱為拉密（Lame)方程。 要

15、求解拉密方程，顯然需要邊界條件。除此之外，由于位移分量還是時(shí)間變量的函數(shù)，因此求解動(dòng)力問題還要給出初始條件。 為求解上的簡(jiǎn)便，通常不計(jì)體力，此時(shí)彈性體的運(yùn)動(dòng)微分方程簡(jiǎn)化為：)211()1 (2222uxeEtu)211()1 (2222yeEt)211()1 (2222wzeEt2111-2 Without rotating Wave and equal volume wave 1. Without rotating wavesWithout rotating wave means that in elastic objects , the distortion caused by waves

16、 is not rotating . That means rotating values of three vertical coordinates at any point in the elastic object are zero.xuyzwWhere is potential function of displacement. This displacement is called without rotating displacement, and the elastic wave corresponds to the displacement are called without

17、 rotating wave.),(tzyxSuppose the displacement of elastic objects can be expressed by :u ,v , w2211-2 11-2 無旋波與等容波無旋波與等容波 一、無旋波一、無旋波 所謂無旋波是指在彈性體中，波動(dòng)所產(chǎn)生的變形不存在旋轉(zhuǎn)，即彈性體在任一點(diǎn)對(duì)三個(gè)垂直坐標(biāo)軸的旋轉(zhuǎn)量皆為零。 假定彈性體的位移u,v,w可以表示成為：xuyzw其中 是位移的勢(shì)函數(shù)。這種位移稱為無旋位移。而相應(yīng)于這種位移狀態(tài)的彈性波就稱無旋波。),(tzyx23proving：at any point of elastic object

18、,the rotating value of z axis is :So the rotating values of the three coordinates at any point of the elastic object are zero.yuxz substitute into the formula , we can get:xuy0zSimilarly: 0 x0y24證：在彈性體的任一點(diǎn)處，該點(diǎn)對(duì)z 軸的旋轉(zhuǎn)量即彈性體的任一點(diǎn)對(duì)三個(gè)坐標(biāo)的旋轉(zhuǎn)量都等于零。yuxz 將 代入，可得：xuy0z 同理 0 x0y25In the condition of without rota

19、ting displacement:2zwyxuesouxxxe222similarly: 2yewze2Substitute the above three formulas into movement differential equations without including body force, simplify it we can get wave movement equations of without rotating wave. 26在無旋位移狀態(tài)下2zwyxue從而uxxxe222 同理 2yewze2 將上三式代入不計(jì)體力的運(yùn)動(dòng)微分方程，并簡(jiǎn)化后得無旋波的波動(dòng)方程2

20、7uctu2212222122ctwctw22122)21)(1 ()1 (1EcWhere:1c is the transmitting speed of without rotating wave in infinite elastic objects.28uctu2212222122ctwctw22122)21)(1 ()1 (1Ec其中1c 就是無旋波在無限大彈性體中的傳播速度29Equal volume wave means that in the distortion caused by waves in elastic objects, the volume strain is

21、zero. That means the volume of elastic object remains unchanged. 2. Equal volume waveSuppose the displacement u, v, w of elastic objects satisfy the condition that the volume strain is zero:0zwyxueThis displacement is called equal volume displacement and the elastic wave corresponds to this displace

22、ment is called equal volume wave. 30 所謂等容波是指在彈性體內(nèi)，波動(dòng)所產(chǎn)生的變形中體積應(yīng)變?yōu)榱?。即彈性體中任一部分的容積（即體積）保持不變。二、等容波二、等容波 假定彈性體的位移u,v,w滿足體積應(yīng)變?yōu)榱愕臈l件，即：0zwyxue 這種位移稱為等容位移。而相應(yīng)于這種位移狀態(tài)的彈性 波就是等容波。31Because ，so simplify movement differential equations without including body force we can get wave movement equations of equal volum

23、e wave:0euctu2222222222ctwctw22222where)1 ( 22Ec is the transmitting speed of equal volume waves in infinite elastic objects.2c32 由于 ，故不計(jì)體力的運(yùn)動(dòng)微分方程，簡(jiǎn)化后得等容波的波動(dòng)方程： 0euctu2222222222ctwctw22222其中)1 ( 22Ec 就是等容波在無限大彈性體中的傳播速度。2c33 According to without rotating wave and equal volume wave , we give the concl

24、usion without proving that :in elastic objects , stress , strain and speed of particle transmit in the same way and the same speed as displacement.34 對(duì)于無旋波和等容波，我們不加證明地給出如下結(jié)論：在彈性體中，形變、應(yīng)力以及質(zhì)點(diǎn)速度，都將和位移以相同的方式與速度進(jìn)行傳播。351. vertical wavedefinition The particle movement direction of elastic objects is parall

25、el to the transmitting direction of elastic wave. （seen in the Fig.）11-3 Vertical wave and transverse wave36一、縱波定義 彈性體的質(zhì)點(diǎn)運(yùn)動(dòng)方向平行彈性波的傳播方向（圖示）11-3 縱波與橫波縱波與橫波37Let x axis be the transmitting direction of waves, then the displacement components of any point in elastic objects are:),(txuu 00wSo:xueAnd:22x

26、uxe0ye0ze222xuu0202 w38 將x軸取為波的傳播方向，則彈性體內(nèi)任取一點(diǎn)的位移分量都有：),(txuu 00w從而xue而22xuxe0ye0ze222xuu0202 w39Substitute them into movement differential equations without including body force, we can find that the second and third formulas are identical equations. So the first formula can be simplified as:222122x

27、uctuwhere)21)(1 ()1 (1Ec is transmitting speed of vertical wave in elastic objects.1cIt is obvious that transmitting speed of vertical waves is the same as without rotating waves . In fact vertical wave is a kind of without rotating wave. 40代入不計(jì)體力的運(yùn)動(dòng)微分方程，可見其第二、第三式成為恒等式，而第一式簡(jiǎn)化為：222122xuctu其中)21)(1 ()

28、1 (1Ec 為縱波在彈性體中的傳播速度。1c 顯然縱波的傳播速度與無旋波相同。事實(shí)上，縱波就是一種無旋波。41The general solution of wave movement equations of vertical wave is:)()(),(1211tcxftcxftxuThe physic meaning of the general solution is: let us take the first item for example, function at a fixed time is function of x, and can be expressed by

29、curve abc in Fig.(a) (suppose the form is like this ). After , the function becomes:)(11t cxft)(111tctcxfIf let ，then function becomes . Its format is similar to the original function . From the Fig we can see the only difference is transverse coordinate moves in level . So expresses the wave whose

30、speed is along x positive direction .tcxx11)(111tcxf)(11tcxftc 1)(11tcxf1c42 縱波波動(dòng)方程的通解是：)()(),(1211tcxftcxftxu該通解的物理意義：以其第一項(xiàng)為例，函數(shù) 在某一個(gè)固定時(shí)刻將是x的函數(shù)，可以用圖(a)中的曲線abc表示（假設(shè)是這種形狀），在 時(shí)間之后，函數(shù)變?yōu)椋?(11t cxft)(111tctcxf如果令 ，則函數(shù)可寫為 ，其形式同原函數(shù) 完全類同，只是橫坐標(biāo)發(fā)生平移 tcxx11)(111tcxf)(11tcxftc 1見圖。因此 表示以速度 向x軸正向傳播的波。)(11tcxf1c4

31、3Similarly expresses the wave whose speed is along x negative direction . The general solution expresses two waves transmitting in opposite directions (in Fig.b). And the speed is the modulus of wave movement equations.1c)(12tcxf1c1fcabx(a)(b)tc 1tc 1tc 144同理 ，表示以同樣速度 向x軸負(fù)向傳播的波。整個(gè)通解表示朝相反兩個(gè)方向傳播的兩個(gè)波（如

32、圖b），其傳播速度為波動(dòng)方程的系數(shù) 。 1c)(12tcxf1c1fcabx(a)(b)tc 1tc 1tc 1452. Transverse wavedefinition The partial movement direction of elastic object is vertical to the transmitting direction of elastic wave.Transmitting format of transverse wave46二、橫波二、橫波定義 彈性體的質(zhì)點(diǎn)運(yùn)動(dòng)方向垂直于彈性波的傳播方向。橫波的傳播形式47Still we let x axis be t

33、he transmitting direction of wave and y axis be displacement direction of partial. Then the displacement components of any point in the elastic object have:0u),(tx0wso0eand02 u222x02 wSubstitute them into movement differential equations without including body force, we can find that the first and th

34、ird formulas are identical equations. So the second formula can be simplified as:222222xct)1 (22Ec2c is transmitting speed of transverse wave in elastic object. Because volume strain of transverse wave48 仍然將x軸放在波的傳播方向，y軸為質(zhì)點(diǎn)位移方向，則彈性體內(nèi)任取一點(diǎn)的位移分量都有0u),(tx0w從而0e而02 u222x02 w代入不計(jì)體力的運(yùn)動(dòng)微分方程，可見其第一、第三式成為恒等式，第

35、二式簡(jiǎn)化為：222222xct)1 (22Ec2c 為橫波在彈性體中的傳播速度。由于橫波的體積應(yīng)變49General solution to wave movement equations of transverse wave is: ，so transverse wave is equal volume wave0e)()(),(2221tcxftcxftxIt is obvious that the general solution expresses two waves transmitting in opposite directions . Its displacement is a

36、long y direction and the transmitting direction is along x direction . The transmitting speed is a constant .2c50 橫波的波動(dòng)方程的通解為： ，故橫波為等容波。0e)()(),(2221tcxftcxftx顯然，整個(gè)通解表示朝相反兩個(gè)方向傳播的兩個(gè)波，它的位移沿著y方向，而傳播方向是沿著x方向，傳播速度等于常量 。2c51 11-4 Spherical waveIf elastic objects have spherical hole or spherical out surfac

37、e ,when the spherical hole or spherical out surface bears spherical force , the elastic wave transmitting from the hole to the outside or form the surface to the inside is called spherical wave. Spherical wave is symmetrical of sphere. Apply basic differential equations of sphere symmetry, we get:0)

38、22()21)(1 ()1 (2222rrrrkurdrdurdrudENow . If we ignore body force , and use radial inertia force to substitute),(truurr22turrk52 11-411-4 球面波球面波 如果彈性體具有圓球形的孔洞或具有圓球形的外表面，則在圓球形孔洞或圓球形外表面上受到球?qū)ΨQ的動(dòng)力作用時(shí)，由孔洞向外傳播或由外表面向內(nèi)傳播的彈性波，稱為球面波。 球面波是球?qū)ΨQ的。利用球?qū)ΨQ的基本微分方程：0)22()21)(1 ()1 (2222rrrrkurdrdurdrudE 此時(shí)， ，而不計(jì)體力時(shí)，用徑向

39、慣性力),(truurr22tur 代替 ，rk53Then the above formula can be simplified as:We get：0)22()21)(1 ()1 (22222turururruErrrrlet：)21)(1 ()1 (1EcSuppose rurthen is potential function of displacement. Substitute it into formula (a), we get:),(tr0122222122tucrururrurrrrr （a） 54 則上式簡(jiǎn)寫成即得：0)22()21)(1 ()1 (22222turururruErrrr令：)21)(1 ()1 (1Ec 假定rur 則 是位移的勢(shì)函數(shù)。代入（a）式得 ),(tr0122222122tucrururrurrrrr （a） 55So formula (b) can be written asBecaus